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SIAM J. Math. Data Sci., 3(3):987–1012, 2021. 2, 8
[42] Jeffrey T. Leek. Asymptotic conditional singular value decomposition for high-dimensional
genomic data. Biometrics, 67(2):344–352, 2011. 1
[43] Thibault Lesieur, Florent Krzakala, and Lenka Zdeborová. Mmse of probabilistic low-rank
matrix estimation: Universality with respect to the output channel. In 2015 53rd Annual Allerton
Conference on Communication, Control, and Computing (Allerton), pages 680–687, 2015. 2 | [
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] |
Conference on Communication, Control, and Computing (Allerton), pages 680–687, 2015. 2
[44] Yue M. Lu. Householder dice: a matrix-free algorithm for simulating dynamics on Gaussian
and random orthogonal ensembles. IEEE Trans. Inform. Theory, 67(12):8264–8272, 2021. 2
[45] Clément Luneau, Jean Barbier, and Nicolas Macris. Mutual information for low-rank even-order
symmetric tensor estimation. Inf. Inference, 10(4):1167–1207, 2021. 2 | [
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] |
symmetric tensor estimation. Inf. Inference, 10(4):1167–1207, 2021. 2
[46] Pierre Mergny, Justin Ko, and Florent Krzakala. Spectral phase transition and optimal pca in
block-structured spiked models. arXiv preprint arXiv:2403.03695, 2024. 2
[47] Paul Milgrom and Ilya Segal. Envelope theorems for arbitrary choice sets. Econometrica,
70(2):583–601, 2002. 5, 28, 29
[48] Léo Miolane. Fundamental limits of low-rank matrix estimation: the non-symmetric case. arXiv
preprint arXiv:1702.00473, 2017. 1, 2, 5, 15 | [
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] |
preprint arXiv:1702.00473, 2017. 1, 2, 5, 15
[49] Léo Miolane. Fundamental limits of inference: A statistical physics approach. Theses, Ecole
normale supérieure - ENS PARIS ; Inria Paris, June 2019. 2, 15, 50
[50] Marco Mondelli, Christos Thrampoulidis, and Ramji Venkataramanan. Optimal combination
of linear and spectral estimators for generalized linear models. Foundations of Computational
Mathematics, pages 1–54, 2021. 3 | [
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] |
Mathematics, pages 1–54, 2021. 3
[51] Marco Mondelli and Ramji Venkataramanan. Pca initialization for approximate message
passing in rotationally invariant models. In Advances in Neural Information Processing Systems,
volume 34, pages 29616–29629, 2021. 9
[52] Andrea Montanari, Feng Ruan, and Jun Yan. Adapting to unknown noise distribution in matrix
denoising. arXiv preprint arXiv:1810.02954, 2018. 9
[53] Andrea Montanari and Ramji Venkataramanan. Estimation of low-rank matrices via approximate | [
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] |
[53] Andrea Montanari and Ramji Venkataramanan. Estimation of low-rank matrices via approximate
message passing. Ann. Statist., 49(1):321–345, 2021. 1, 3, 9
[54] Andrea Montanari and Yuchen Wu. Fundamental limits of low-rank matrix estimation with
diverging aspect ratios. arXiv preprint arXiv:2211.00488, 2022. 1
[55] Andrea Montanari and Yuchen Wu. Posterior sampling from the spiked models via diffusion
processes. arXiv preprint arXiv:2304.11449, 2023. 3 | [
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] |
processes. arXiv preprint arXiv:2304.11449, 2023. 3
[56] Raj Rao Nadakuditi. OptShrink: an algorithm for improved low-rank signal matrix denoising by
optimal, data-driven singular value shrinkage. IEEE Trans. Inform. Theory, 60(5):3002–3018,
2014. 2, 8
[57] Boaz Nadler. Finite sample approximation results for principal component analysis: a matrix
perturbation approach. Ann. Statist., 36(6):2791–2817, 2008. 2
[58] Aleksandr Pak, Justin Ko, and Florent Krzakala. Optimal algorithms for the inhomogeneous | [
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] |
[58] Aleksandr Pak, Justin Ko, and Florent Krzakala. Optimal algorithms for the inhomogeneous
spiked wigner model. In Advances in Neural Information Processing Systems, volume 36, pages
76409–76424, 2023. 2
[59] Dmitry Panchenko. The Sherrington-Kirkpatrick model. Springer Monographs in Mathematics.
Springer, New York, 2013. 50
[60] Henrik Pedersen, Sebastian Kozerke, Steffen Ringgaard, Kay Nehrke, and Won Yong Kim. | [
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[60] Henrik Pedersen, Sebastian Kozerke, Steffen Ringgaard, Kay Nehrke, and Won Yong Kim.
k-t pca: Temporally constrained k-t blast reconstruction using principal component analysis.
Magnetic Resonance in Medicine, 62(3):706–716, 2009. 1
[61] S. Rangan. Generalized approximate message passing for estimation with random linear mixing.
In IEEE International Symposium on Information Theory (ISIT), 2011. 3
[62] Sundeep Rangan, Philip Schniter, and Alyson K Fletcher. Vector approximate message passing. | [
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] |
[62] Sundeep Rangan, Philip Schniter, and Alyson K Fletcher. Vector approximate message passing.
IEEE Transactions on Information Theory, 65(10):6664–6684, 2019. 3
[63] Galen Reeves. Information-theoretic limits for the matrix tensor product. IEEE Journal on
Selected Areas in Information Theory, 1(3):777–798, 2020. 2
12
[64] R. Tyrrell Rockafellar. Convex analysis. Princeton Landmarks in Mathematics. Princeton
University Press, Princeton, NJ, 1997. 28 | [
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] |
University Press, Princeton, NJ, 1997. 28
[65] Charles M. Stein. Estimation of the mean of a multivariate normal distribution. The Annals of
Statistics, 9(6):1135–1151, 1981. 50
[66] Pei-Chun Su and Hau-Tieng Wu. Data-driven optimal shrinkage of singular values under
high-dimensional noise with separable covariance structure. arXiv preprint arXiv:2207.03466,
2022. 1, 2, 3
[67] O. Tamuz, T. Mazeh, and S. Zucker. Correcting systematic effects in a large set of photometric | [
-0.4024529457092285,
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] |
[67] O. Tamuz, T. Mazeh, and S. Zucker. Correcting systematic effects in a large set of photometric
light curves. Monthly Notices of the Royal Astronomical Society, 356(4):1466–1470, 02 2005. 1
[68] Ramji Venkataramanan, Kevin Kögler, and Marco Mondelli. Estimation in rotationally invariant
generalized linear models via approximate message passing. In International Conference on
Machine Learning (ICML), 2022. 3
[69] Yihong Wu and Harrison H. Zhou. Randomly initialized EM algorithm for two-component | [
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] |
[69] Yihong Wu and Harrison H. Zhou. Randomly initialized EM algorithm for two-component
Gaussian mixture achieves near optimality in O(√n) iterations. Math. Stat. Learn., 4(3-4):143–
220, 2021. 1
[70] Kaylee Y Yang, Timothy LH Wee, and Zhou Fan. Asymptotic mutual information in quadratic
estimation problems over compact groups. arXiv preprint arXiv:2404.10169, 2024. 2
[71] Xiaodong Yang, Buyu Lin, and Subhabrata Sen. Fundamental limits of community detection | [
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] |
[71] Xiaodong Yang, Buyu Lin, and Subhabrata Sen. Fundamental limits of community detection
from multi-view data: multi-layer, dynamic and partially labeled block models. arXiv preprint
arXiv:2401.08167, 2024. 2
[72] Anru R. Zhang, T. Tony Cai, and Yihong Wu. Heteroskedastic PCA: algorithm, optimality, and
applications. Ann. Statist., 50(1):53–80, 2022. 2, 3, 8
[73] Cun-Hui Zhang and Stephanie S. Zhang. Confidence intervals for low dimensional parameters | [
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] |
[73] Cun-Hui Zhang and Stephanie S. Zhang. Confidence intervals for low dimensional parameters
in high dimensional linear models. J. R. Stat. Soc. Ser. B. Stat. Methodol., 76(1):217–242, 2014.
8
[74] Lixin Zhang. Spectral analysis of large dimentional random matrices. PhD thesis, National
University of Singapore, 2007. 37
[75] Yihan Zhang, Hong Chang Ji, Ramji Venkataramanan, and Marco Mondelli. Spectral estimators
for structured generalized linear models via approximate message passing. arXiv preprint | [
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] |
for structured generalized linear models via approximate message passing. arXiv preprint
arXiv:2308.14507, 2023. 3, 6, 37
[76] Yihan Zhang, Marco Mondelli, and Ramji Venkataramanan. Precise asymptotics for spectral
methods in mixed generalized linear models. arXiv preprint arXiv:2211.11368, 2022. 3
[77] Xinyi Zhong, Tianhao Wang, and Zhou Fan. Approximate message passing for orthogonally
invariant ensembles: Multivariate non-linearities and spectral initialization. arXiv preprint
arXiv:2110.02318, 2021. 9 | [
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] |
arXiv:2110.02318, 2021. 9
[78] Yuchen Zhou and Yuxin Chen. Deflated heteropca: Overcoming the curse of ill-conditioning in
heteroskedastic pca. arXiv preprint arXiv:2303.06198, 2023. 3
13
Notation.
All vectors are column vectors. The singular values of a matrix A ∈ Rn×d (where
n ≥ d without loss of generality) are denoted by σ1(A) ≥ · · · ≥ σd(A) ≥ 0 and the corresponding
left/right singular vectors are denoted by u1(A), · · · , ud(A) ∈ Sn−1 and v1(A), · · · , vd(A) ∈ Sd−1. | [
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] |
The (real) eigenvalues of a symmetric matrix B ∈ Rd×d are denoted by λ1(B) ≥ · · · ≥ λd(B)
and the corresponding eigenvectors are denoted by v1(B), · · · , vd(B) ∈ Sd−1 (which will not be
confused with the right singular vectors, whenever they are different, since we will never talk about
both simultaneously for a square asymmetric matrix). We generally put overlines on capital letters
to indicate a scalar random variable, e.g., X ∈ R, whose support is denoted by supp(X). The | [
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] |
to indicate a scalar random variable, e.g., X ∈ R, whose support is denoted by supp(X). The
limit/liminf/limsup in probability are denoted by p-lim, p-liminf, p-limsup. The product distribution
whose i-th (i ∈ [k]) marginal is given by Pi is denoted by P1⊗· · ·⊗Pk, with the shorthand P ⊗k when
all Pi’s are equal to P. The gradient of f : Rn → R, or with abuse of notation, the Jacobian matrix of
F : Rn → Rd are denoted by ∇f ∈ Rn, ∇F ∈ Rd×n. The partial derivative of f(x1, · · · , xn) with | [
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] |
F : Rn → Rd are denoted by ∇f ∈ Rn, ∇F ∈ Rd×n. The partial derivative of f(x1, · · · , xn) with
respect to xi is denoted by either
∂
∂xi f(x1, · · · , xn) or ∂if(x1, · · · , xn). All log and exp are to the
base e. We use the standard notation of sup(S), inf(S) for a subset S ⊂ R. We generally use C > 0
to denote a sufficiently large constant independent of n, d. Its dependence on other parameters will
be specified, though its value may change across passages. We use the standard big O notation.
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] |
be specified, though its value may change across passages. We use the standard big O notation.
A
Proof of Proposition 4.1
We eliminate qu and write a fixed point equation only involving qv:
qv = E
γE
�
αγqvΞ
−2
αγqvΞ
−1+1
�
Σ
−2
γE
�
αγqvΞ
−2
αγqvΞ
−1+1
�
Σ
−1 + 1
.
Denote the RHS by f(qv). Recall that we are only interested in non-negative solutions (qu, qv). So
let us restrict attention on f to the domain R≥0. We have f(0) = 0 and
f ′(qu) = E
γΣ
−2
�
γE
�
αγqvΞ
−2
αγqvΞ
−1+1
�
Σ
−1 + 1
�2
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f ′(qu) = E
γΣ
−2
�
γE
�
αγqvΞ
−2
αγqvΞ
−1+1
�
Σ
−1 + 1
�2
E
αγΞ
−2
�
αγqvΞ
−1 + 1
�2
> 0,
f ′(0) = αγ2E
�
Σ
−2�
E
�
Ξ
−2�
,
f ′′(qv) = −2
E
αγΞ
−2
�
αγqvΞ
−1 + 1
�2
2
E
γ2Σ
−3
�
γE
�
αγqvΞ
−2
αγqvΞ
−1+1
�
Σ
−1 + 1
�3
+E
γΣ
−2
�
γE
�
αγqvΞ
−2
αγqvΞ
−1+1
�
Σ
−1 + 1
�2
E
α2γ2Ξ
−3
�
αγqvΞ
−1 + 1
�3
< 0,
lim
qv→∞ f(qv) = E
γE
�
Ξ
−1�
Σ
−2
γE
�
Ξ
−1�
Σ
−1 + 1
∈ (0, ∞). | [
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] |
−1 + 1
�3
< 0,
lim
qv→∞ f(qv) = E
γE
�
Ξ
−1�
Σ
−2
γE
�
Ξ
−1�
Σ
−1 + 1
∈ (0, ∞).
It then becomes evident that a non-trivial fixed point qv > 0 exists if and only if f ′(0) > 1 and in
this case, the non-trivial fixed point is unique.
Finally, by the first equation in (4.5), there is a non-trivial fixed point qu if and only if there is a
non-trivial fixed point qv, which completes the proof.
B
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] |
non-trivial fixed point qv, which completes the proof.
B
Auxiliary Gaussian channel
We formally introduce here the auxiliary model mentioned in Section 4. Consider a Gaussian channel
with blocklength n, input x∗, output Y , anisotropic Gaussian noise Σ1/2Z and SNR γ:
Y = √γx∗ + Σ1/2Z ∈ Rn,
(B.1)
14
where
(x∗, Z) ∼ P ⊗n ⊗ N(0n, In).
By similar derivations as in Section 4, the posterior distribution of x∗ given Y can be written as
dP(x | Y ) =
1
Zn(γ) exp(Hn(x)) dP ⊗n(x), | [
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] |
dP(x | Y ) =
1
Zn(γ) exp(Hn(x)) dP ⊗n(x),
where the Hamiltonian and the partition function are
Hn(x) := γx∗⊤Σ−1x + √γZ⊤Σ−1/2x − γ
2 x⊤Σ−1x,
Zn(γ) :=
�
Rn exp(Hn(x)) dP ⊗n(x).
Define the free energy as
Fn(γ) := 1
nE[log Zn(γ)].
With P = N(0, 1), Zn(γ) becomes a Gaussian integral that can be computed as below using
Proposition G.1:
Zn(γ) =
1
�
det(γΣ−1 + In)
exp
�1
2
�
γΣ−1x∗ + √γΣ−1/2Z
�⊤�
γΣ−1 + In
�−1�
γΣ−1x∗ + √γΣ−1/2Z
��
.
Therefore, by Proposition G.2,
p-lim
n→∞ Fn(γ) = −1
2E
�
log
�
γΣ
−1 + 1
��
+ 1 | [
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0.49331530928611755,
0.24067902565002441,
-0.18798314034938812,
-0.599256694316864,
0.9078270792961121,
-0.9535936117172241,
0.03982607275247574,
0.29389768838882446,
0.30796578526496887,
-0.15876229107379913,
-0.5460513830184937,
-0.7928079962730408,
-0.22635187208652496,
-0.22733460366725922,
0.9144156575202942,
0.025995930656790733,
0.5562849640846252,
-0.07819629460573196,
-0.34849414229393005,
-0.4987873435020447,
0.3568415939807892,
0.5510513186454773,
-0.12756164371967316,
-0.28707462549209595,
-0.06132804974913597,
-0.20284420251846313,
0.37124162912368774,
-0.5092481374740601,
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0.06314662098884583
] |
��
.
Therefore, by Proposition G.2,
p-lim
n→∞ Fn(γ) = −1
2E
�
log
�
γΣ
−1 + 1
��
+ 1
2γ2E
�
Σ
−2�
γΣ
−1 + 1
�−1�
+ 1
2γE
�
Σ
−1�
γΣ
−1 + 1
�−1�
= 1
2
�
γE
�
Σ
−1�
− E
�
log
�
1 + γΣ
−1���
.
(B.2)
The above functional is nothing but ψΣ(γ) introduced in (4.15) which will play an important role in
characterizing the free energy of the original model (4.1).
C
Proof of Theorem 4.4
Before diving into the proof, we make further notation adjustments for the ease of applying the | [
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] |
Before diving into the proof, we make further notation adjustments for the ease of applying the
interpolation argument. Specifically, we will henceforth assume γ = 1 by incorporating the actual
value of γ into the prior distributions P, Q,
�
R
x2 dP(x) = γ,
�
R
x2 dQ(x) = 1.
This is obviously equivalent to the previous setting. So we can drop the dependence on γ and write
MMSEn, Zn, Fn for MMSEn(γ), Zn(γ), Fn(γ) defined in (4.2), (4.13) and (4.14). | [
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] |
MMSEn, Zn, Fn for MMSEn(γ), Zn(γ), Fn(γ) defined in (4.2), (4.13) and (4.14).
We will also assume that Ξ, Σ are diagonal. This is without loss of generality since the Gaussianity
of P, Q, �
W ensures that both the prior distributions and the noise matrix are rotationally invariant.
Furthermore, we truncate P, Q so that they are supported on [−K, K] for a constant K > 0. The
approximation error in the free energy due to truncation can be made arbitrarily small if K is | [
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] |
approximation error in the free energy due to truncation can be made arbitrarily small if K is
sufficiently large, since the free energy is pseudo-Lipschitz in the prior distribution with respect to
the Wasserstein-2 metric.
The proof follows an interpolation argument [9, 48, 49] with suitable modifications to take care of
the noise heteroscedasticity featured by the covariances Ξ, Σ. To start with, define the interpolating
models:
Yt :=
�
1 − t
n
u∗v∗⊤ + Ξ1/2ZΣ1/2 ∈ Rn×d,
Y u
t :=
� | [
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] |
models:
Yt :=
�
1 − t
n
u∗v∗⊤ + Ξ1/2ZΣ1/2 ∈ Rn×d,
Y u
t :=
�
αq1(t)u∗ + Ξ1/2Zu ∈ Rn,
15
Y v
t :=
�
q2(t)v∗ + Σ1/2Zv ∈ Rd,
where q1(t), q2(t) ≥ 0 are to be determined and
(u∗, v∗, Z, Zu, Zv) ∼ P ⊗n ⊗ Q⊗d ⊗ N(0nd, Ind) ⊗ N(0n, In) ⊗ N(0d, Id).
(C.1)
By definition, Y0 = Y is the model that we would like to understand, and Y u
t , Y v
t are instances of
Gaussian channels in (B.1) whose free energy we have already understood (see (B.2)). The idea is | [
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] |
Gaussian channels in (B.1) whose free energy we have already understood (see (B.2)). The idea is
that Yt serves as a path parametrized by t ∈ [0, 1] from the original model Y to the target models
(Y u
1 , Y v
1 ). The crux of the interpolation argument lies in showing that Yt and (Y u
t , Y v
t ) are equivalent
(at the level of free energy) along the path.
To study the interpolating models (Yt, Y u
t , Y v
t ), define the Hamiltonian
Hn,t(�u, �v; q1, q2) :=
�
1 − t
n
�u⊤Z�v + 1 − t
n
�u⊤�u∗�v⊤�v∗ − 1 − t | [
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t ), define the Hamiltonian
Hn,t(�u, �v; q1, q2) :=
�
1 − t
n
�u⊤Z�v + 1 − t
n
�u⊤�u∗�v⊤�v∗ − 1 − t
2n ∥�u∥2
2∥�v∥2
2
+ αq1�u⊤�u∗ + √αq1�u⊤Zu − αq1
2 ∥�u∥2
2
+ q2�v⊤�v∗ + √q2�v⊤Zv − q2
2 ∥�v∥2
2.
(C.2)
Then the posterior distribution of (u∗, v∗) given (Yt, Y u
t , Y v
t ) is
dP(u, v | Yt, Y u
t , Y v
t ) =
1
Zn,t
exp
�
Hn,t(Ξ−1/2u, Σ−1/2v; q1(t), q2(t))
�
dP ⊗n(u) dQ⊗d(v).
(C.3)
Let the partition function be
Zn,t :=
�
Rd
�
Rn exp
�
Hn,t(Ξ−1/2u, Σ−1/2v; q1(t), q2(t))
�
dP ⊗n(u) dQ⊗d(v)
=
�
Rd
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] |
Zn,t :=
�
Rd
�
Rn exp
�
Hn,t(Ξ−1/2u, Σ−1/2v; q1(t), q2(t))
�
dP ⊗n(u) dQ⊗d(v)
=
�
Rd
�
Rn exp(Hn,t(�u, �v; q1(t), q2(t))) d �P(�u) d �Q(�v).
(C.4)
Define the free energy as
fn(t) := 1
nE[log Zn,t].
(C.5)
The Gibbs bracket ⟨·⟩n,t denotes the expectation with respect to the posterior distribution in (C.3):
⟨g(�u, �v)⟩n,t :=
1
Zn,t
�
Rd
�
Rn g(�u, �v) exp(Hn,t(�u, �v; q1(t), q2(t))) d �P(�u) d �Q(�v),
(C.6)
for any g: Rn × Rd → R such that the expectation exists. That is, | [
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] |
(C.6)
for any g: Rn × Rd → R such that the expectation exists. That is,
⟨g(�u, �v)⟩n,t = E[g(�u∗, �v∗) | Yt, Y u
t , Y v
t ],
where we recall the notation
�u∗ := Ξ−1/2u∗,
�v∗ := Σ−1/2v∗.
(C.7)
We will also use the notation ⟨·⟩n for the Gibbs bracket with respect to the original posterior
dP(u, v | Y ) in (4.11).
Lemma C.1. Consider fn(t) defined in (C.5) with t ∈ {0, 1}. Assume that q1(0), q2(0) satisfy
q1(0) ≥ 0,
q2(0) ≥ 0,
lim
n→∞ q1(0) = lim
n→∞ q2(0) = 0.
Then we have
fn(0) = Fn + O(q1(0) + q2(0)), | [
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] |
q2(0) ≥ 0,
lim
n→∞ q1(0) = lim
n→∞ q2(0) = 0.
Then we have
fn(0) = Fn + O(q1(0) + q2(0)),
(C.8)
lim
n→∞ fn(1) = ψΞ(αq1(1)γ) + αψΣ(q2(1)) + oK,
(C.9)
where limK→∞ oK = 0.
16
Proof. To show the first statement (C.8), let us control fn(0) − Fn. Denoting
H′
n,t(�u, �v; q1, q2) := αq1�u⊤�u∗ + √αq1�u⊤Zu − αq1
2 ∥�u∥2
2 + q2�v⊤�v∗ + √q2�v⊤Zv − q2
2 ∥�v∥2
2
and recalling the Gibbs bracket notation ⟨·⟩n, we have
fn(0) − Fn = 1
nE
�
log Zn,0
Zn
�
= 1
nE
�
log
�
exp
�
H′
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] |
fn(0) − Fn = 1
nE
�
log Zn,0
Zn
�
= 1
nE
�
log
�
exp
�
H′
n,0(�u, �v; q1(0), q2(0))
��
n
�
,
(C.10)
where the outer expectation is over all randomness in u∗, v∗, Zu, Zv. The second equality above
follows since
Hn,0(�u, �v; q1(0), q2(0)) = Hn(�u, �v) + H′
n,0(�u, �v; q1(0), q2(0)).
We will derive double-sided bounds on fn(0) − Fn.
To upper bound it, use Jensen’s inequality E[log(·)] ≤ log E[·] on the partial expectation over Zu, Zv
in (C.10):
fn(0) − Fn ≤ 1
n
E
u∗,v∗
�
log
E
Zu,Zv
��
exp
�
H′ | [
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] |
in (C.10):
fn(0) − Fn ≤ 1
n
E
u∗,v∗
�
log
E
Zu,Zv
��
exp
�
H′
n,0(�u, �v; q1(0), q2(0))
��
n
��
= 1
n
E
u∗,v∗
�
log
�
E
Zu,Zv
�
exp
�
H′
n,0(�u, �v; q1(0), q2(0))
���
n
�
,
where the equality is legit since ⟨·⟩n does not depend on Zu, Zv. By the Gaussian integral formula
(Proposition G.1), the inner expectation equals
E
Zu,Zv
�
exp
�
H′
n,0(�u, �v; q1(0), q2(0))
��
= exp
�
αq1(0)�u⊤�u∗ + q2(0)�v⊤�v∗�
.
Replacing the Gibbs bracket with max, we obtain an upper bound:
fn(0) − Fn ≤ 1
n
E
u∗,v∗
�
log
max | [
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0.4559010863304138,
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] |
Replacing the Gibbs bracket with max, we obtain an upper bound:
fn(0) − Fn ≤ 1
n
E
u∗,v∗
�
log
max
(�u,�v)∈Ξ−1/2[−K,K]n×Σ−1/2[−K,K]d exp
�
αq1(0)�u⊤�u∗ + q2(0)�v⊤�v∗��
≤ αq1(0)∥Ξ∥−1
2 K2 + q2(0)∥Σ∥−1
2 K2 d
n
≤ 2αq1(0)K2
inf supp(Ξ) + 2αq2(0)K2
inf supp(Σ),
(C.11)
where the last inequality holds for all sufficiently large n by (3.3) and n ≍ d.
To lower bound fn − Fn, we use Jensen’s inequality again but this time on the Gibbs bracket in
(C.10):
fn(0) − Fn ≥ 1
nE
��
H′
n,0(�u, �v; q1(0), q2(0))
�
n
�
= 1
n | [
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] |
(C.10):
fn(0) − Fn ≥ 1
nE
��
H′
n,0(�u, �v; q1(0), q2(0))
�
n
�
= 1
n
E
u∗,v∗
��
E
Zu,Zv
�
H′
n,0(�u, �v; q1(0), q2(0))
��
n
�
.
(C.12)
Since (Zu, Zv) ∼ N(0n, In) ⊗ N(0d, Id), the inner expectation equals
E
Zu,Zv
�
H′
n,0(�u, �v; q1(0), q2(0))
�
= αq1(0)�u⊤�u∗ − αq1(0)
2
∥�u∥2
2 + q2(0)�v⊤�v∗ − q2(0)
2
∥�v∥2
2.
So
max
(�u,�v)∈Ξ−1/2[−K,K]n×Σ−1/2[−K,K]d
����
E
Zu,Zv
�
H′
n,0(�u, �v; q1(0), q2(0))
�����
≤ 3
2αq1(0)∥Ξ∥−1
2 nK2 + 3
2q2(0)∥Σ∥−1
2 dK2. | [
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] |
����
E
Zu,Zv
�
H′
n,0(�u, �v; q1(0), q2(0))
�����
≤ 3
2αq1(0)∥Ξ∥−1
2 nK2 + 3
2q2(0)∥Σ∥−1
2 dK2.
Using this, we obtain a lower bound on fn − Fn by replacing the Gibbs bracket on the RHS of (C.12)
with − max|·|:
fn(0) − Fn ≥ − 1
n
E
u∗,v∗
�
max
(�u,�v)∈Ξ−1/2[−K,K]n×Σ−1/2[−K,K]d
����
E
Zu,Zv
�
H′
n,0(�u, �v; q1(0), q2(0))
�����
�
17
≥ −3
2αq1(0)∥Ξ∥−1
2 K2 − 3
2q2(0)∥Σ∥−1
2 K2 d
n
≥ − 3αq1(0)K2
inf supp(Ξ) − 3αq2(0)K2
inf supp(Σ),
(C.13) | [
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] |
2 K2 − 3
2q2(0)∥Σ∥−1
2 K2 d
n
≥ − 3αq1(0)K2
inf supp(Ξ) − 3αq2(0)K2
inf supp(Σ),
(C.13)
where the last inequality holds for all sufficiently large n by (3.3) and n ≍ d. Combining (C.11)
and (C.13) gives the first result (C.8).
We then prove the second statement (C.9). Since fn is pseudo-Lipschitz as a function of the priors,
up to a term oK that vanishes as K → ∞ uniformly over n, it suffices to ignore the truncation at K
and assume P = N(0, γ), Q = N(0, 1). From the definition (C.4) of Zn,t, we have | [
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] |
and assume P = N(0, γ), Q = N(0, 1). From the definition (C.4) of Zn,t, we have
Zn,1 =
�
Rd
�
Rn exp
�
H′
n,1(Ξ−1/2u, Σ−1/2v; q1(1), q2(1))
�
dP ⊗n(u) dQ⊗d(v)
=
�
det(Ξ)
γn det(Ξ/γ + αq1(1)In)
× exp
�1
2
�
αq1(1)�u∗ +
�
αq1(1)Zu�⊤
(Ξ/γ + αq1(1)In)−1�
αq1(1)�u∗ +
�
αq1(1)Zu��
×
�
det(Σ)
det(Σ + q2(1)Id) exp
�1
2
�
q2(1)�v∗ +
�
q2(1)Zv�⊤
(Σ + q2(1)Id)−1�
q2(1)�v∗ +
�
q2(1)Zv��
,
where in the second equality, we use the Gaussian integral formula (Proposition G.1). Therefore
lim
n→∞ fn(1) = lim
n→∞
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] |
lim
n→∞ fn(1) = lim
n→∞
1
nE[log Zn,1]
= 1
2αq1(1)γE
�
Ξ
−1�
− 1
2E
�
log
�
1 + αq1(1)γΞ
−1��
+ α
2 q2(1)E
�
Σ
−1�
− α
2 E
�
log
�
1 + q2(1)Σ
−1��
,
verifying the identity (C.9). In the second equality, we have used Proposition G.2.
Lemma C.2 (Free energy variation). For all t ∈ (0, 1),
f ′
n(t) = α
2 q′
1(t)q′
2(t) − 1
2E
��� �u⊤�u∗
n
− q′
2(t)
���v⊤�v∗
n
− αq′
1(t)
��
n,t
�
.
(C.14)
Proof. From the definitions (C.4) and (C.5), we compute
f ′
n(t) = 1
nE
� 1
Zn,t
∂
∂tZn,t
�
= 1
nE
� 1
Zn,t
�
Rd
�
Rn
� ∂ | [
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] |
f ′
n(t) = 1
nE
� 1
Zn,t
∂
∂tZn,t
�
= 1
nE
� 1
Zn,t
�
Rd
�
Rn
� ∂
∂tHn,t(�u, �v; q1(t), q2(t))
�
exp(Hn,t(�u, �v; q1(t), q2(t))) d �P(�u) d �Q(�v)
�
= 1
nE
�� ∂
∂tHn,t(�u, �v; q1(t), q2(t))
�
n,t
�
.
By the definition (C.2), the time derivative of the Hamiltonian is
∂
∂tHn,t(�u, �v; q1(t), q2(t)) = −
1
2
�
(1 − t)n
�u⊤Z�v − 1
n �u⊤�u∗�v⊤�v∗ + 1
2n∥�u∥2
2∥�v∥2
2
+ αq′
1(t)�u⊤�u∗ +
√α
2
�
q1(t)
q′
1(t)�u⊤Zu − α
2 q′
1(t)∥�u∥2
2
+ q′
2(t)�v⊤�v∗ +
1
2
�
q2(t)
q′
2(t)�v⊤Zv − 1
2q′
2(t)∥�v∥2
2.
(C.15)
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] |
2 q′
1(t)∥�u∥2
2
+ q′
2(t)�v⊤�v∗ +
1
2
�
q2(t)
q′
2(t)�v⊤Zv − 1
2q′
2(t)∥�v∥2
2.
(C.15)
18
The expectation of ⟨∂tHn,t⟩n,t can be computed using the Stein’s lemma (Proposition G.7). Indeed,
let us consider the term
E
��
�u⊤Z�v
�
n,t
�
=
n
�
i=1
d
�
j=1
E
�
⟨�ui�vj⟩n,tZi,j
�
=
n
�
i=1
d
�
j=1
E
�
∂⟨�ui�vj⟩n,t
∂Zi,j
�
=
n
�
i=1
d
�
j=1
�
1 − t
n
E
��
�u2
i �v2
j
�
n,t
�
−
n
�
i=1
d
�
j=1
�
1 − t
n
E
�
⟨�ui�vj⟩2
n,t
�
=
�
1 − t
n
E
��
∥�u∥2
2∥�v∥2
2
�
n,t
�
−
�
1 − t
n
E
��
�u⊤�u∗�v⊤�v∗�
n,t
�
,
(C.16) | [
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] |
n,t
�
=
�
1 − t
n
E
��
∥�u∥2
2∥�v∥2
2
�
n,t
�
−
�
1 − t
n
E
��
�u⊤�u∗�v⊤�v∗�
n,t
�
,
(C.16)
where the last step is by the Nishimori identity (Proposition G.4). So the first line of (C.15) upon
taken the Gibbs bracket and the expectation becomes
− 1
2nE
��
�u⊤�u∗�v⊤�v∗�
n,t
�
.
Similar cancellations happen for the second and third lines of (C.15). Putting them together, we
obtain
f ′
n(t) = − 1
2n2 E
��
�u⊤�u∗�v⊤�v∗�
n,t
�
+ α
2nq′
1(t)E
��
�u⊤�u∗�
n,t
�
+ 1
2nq′
2(t)E
��
�v⊤�v∗�
n,t
�
, | [
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] |
2n2 E
��
�u⊤�u∗�v⊤�v∗�
n,t
�
+ α
2nq′
1(t)E
��
�u⊤�u∗�
n,t
�
+ 1
2nq′
2(t)E
��
�v⊤�v∗�
n,t
�
,
which is the same as (C.14) with the parentheses opened up.
In what follows, our strategy is:
1. Show that
�
�u⊤�u∗�
n,t is concentrated around its mean E
��
�u⊤�u∗�
n,t
�
;
2. Choose q2(t) to be the solution to
q′
2(t) = 1
nE
��
�u⊤�u∗�
n,t
�
.
Once Items 1 and 2 are done, we then have
Fn ≈ fn(0) = fn(1) −
� 1
0
f ′
n(t) dt
≈ ψΞ(αq1(1)γ) + αψΣ(q2(1)) −
� 1
0
α
2 q′
1(t)q′
2(t) dt
+
� 1
0
1
2E
��� �u⊤�u∗
n
− q′ | [
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] |
f ′
n(t) dt
≈ ψΞ(αq1(1)γ) + αψΣ(q2(1)) −
� 1
0
α
2 q′
1(t)q′
2(t) dt
+
� 1
0
1
2E
��� �u⊤�u∗
n
− q′
2(t)
���v⊤�v∗
n
− αq′
1(t)
��
n,t
�
dt
≈ ψΞ(αq1(1)γ) + αψΣ(q2(1)) −
� 1
0
α
2 q′
1(t)q′
2(t) dt,
where the first line above uses (C.8) in Lemma C.1; the second line uses (C.9) in Lemma C.1 and
Lemma C.2; the third line uses Items 1 and 2. This will almost lead to the desired characterization of
the free energy Fn in Theorem 4.4:
sup
qv≥0
inf
qu≥0 ψΞ(αγqv) + αψΣ(qu) − α
2 quqv.
Consider the function | [
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0.08941090852022171,
0.6757611036300659,
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0.5166054368019104,
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0.18712033331394196,
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0.6202132105827332,
0.7870417237281799,
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0.1442563384771347,
-0.36562639474868774,
0.38498279452323914,
-0.39089831709861755,
-0.061676423996686935,
0.691188395023346,
0.12025868892669678,
0.12410449981689453,
0.20282812416553497,
0.09800481796264648,
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0.3634710907936096,
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-0.480248361825943,
0.5566325187683105,
0.2511321008205414,
0.08244925737380981,
-0.3198976516723633,
-0.16443000733852386,
0.06601253151893616,
-0.1485082507133484,
0.3482630252838135,
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0.6699257493019104,
-0.8498981595039368,
0.25906166434288025,
0.5607739686965942,
0.16869913041591644,
-0.16210606694221497,
-0.44661179184913635,
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0.12546782195568085,
-0.10653321444988251,
0.6918695569038391,
0.3231884241104126,
0.34211280941963196,
-0.10305794328451157,
-1.053844690322876,
-0.5277038812637329,
0.7018819451332092,
0.41488635540008545,
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0.09112243354320526,
-0.10755143314599991,
-0.4482611119747162,
0.1523401290178299,
0.19308604300022125,
-0.31572797894477844,
-0.1982164978981018,
0.19896745681762695
] |
sup
qv≥0
inf
qu≥0 ψΞ(αγqv) + αψΣ(qu) − α
2 quqv.
Consider the function
ϕt(q1, q2) = 1
n log Zn,t.
Note that ϕt(q1, q2) also depends on n, u∗, v∗, Z, Zu, Zv, and E[ϕt(q1, q2)] = fn(t) where the
expectation is over (u∗, v∗, Z, Zu, Zv) distributed according to (C.1).
19
Lemma C.3 (Free energy concentration). Fix a constant M > 0. There exists a constant C > 0
depending only on K, M, α, Ξ, Σ such that for any t ∈ [0, 1], 0 ≤ q1(t), q2(t) ≤ M and sufficiently
large n,
E[|ϕt(q1, q2) − E[ϕt(q1, q2)]|] ≤ C
√n. | [
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] |
large n,
E[|ϕt(q1, q2) − E[ϕt(q1, q2)]|] ≤ C
√n.
Proof. Fix u∗, v∗. Consider ϕt(q1, q2) as a function of (Z, Zu, Zv). Then
��∇(Z,Zu,Zv)ϕt(q1, q2)
��2
2
= ∥∇Zϕt(q1, q2)∥2
2 + ∥∇Zuϕt(q1, q2)∥2
2 + ∥∇Zvϕt(q1, q2)∥2
2
=
n
�
i=1
d
�
j=1
1
nE
��
1 − t
n
�ui�vj
�
n,t
2
+
n
�
i=1
� 1
nE
�
⟨√αq1�ui⟩n,t
��2
+
d
�
j=1
� 1
nE
�
⟨√q2�vj⟩n,t
��2
≤ 1
n3 E
��
∥�u∥2
2∥�v∥2
2
�
n,t
�
+ αq1
n2 E
��
∥�u∥2
2
�
n,t
�
+ q2
n2 E
��
∥�v∥2
2
�
n,t
�
≤ α
n∥Ξ∥−1
2 ∥Σ∥−1
2 K4 + α
nM∥Ξ∥−1
2 K2 + α
nM∥Σ∥−1
2 K2 ≤ C
n , | [
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] |
�
+ q2
n2 E
��
∥�v∥2
2
�
n,t
�
≤ α
n∥Ξ∥−1
2 ∥Σ∥−1
2 K4 + α
nM∥Ξ∥−1
2 K2 + α
nM∥Σ∥−1
2 K2 ≤ C
n ,
where C > 0 is a constant depending only on α, M, Ξ, Σ, K. The penultimate line is by Cauchy–
Schwarz and the last line holds for all sufficiently large n by (3.3) and n ≍ d. Then by the Gaussian
Poincaré inequality (Proposition G.8),
E
Z,Zu,Zv
�����ϕt(q1, q2) −
E
Z,Zu,Zv[ϕt(q1, q2)]
����
�
≤
�
Var
Z,Zu,Zv[ϕt(q1, q2)] ≤ C
√n.
(C.17) | [
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] |
E
Z,Zu,Zv
�����ϕt(q1, q2) −
E
Z,Zu,Zv[ϕt(q1, q2)]
����
�
≤
�
Var
Z,Zu,Zv[ϕt(q1, q2)] ≤ C
√n.
(C.17)
The above result holds for any fixed u∗, v∗. We then verify that EZ,Zu,Zv[ϕt(q1, q2)] has bounded
difference as a function of u∗, v∗. We do so by bounding the derivatives of EZ,Zu,Zv[ϕt(q1, q2)] with
respect to u∗
i , v∗
j for any i ∈ [n], j ∈ [d]. We have
∂
∂u∗
i
E
Z,Zu,Zv[ϕt(q1, q2)] = 1
n
E
Z,Zu,Zv
�� ∂
∂u∗
i
Hn,t(�u, �v; q1, q2)
�
n,t
�
,
(C.18) | [
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0.7361904382705688,
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0.1595533937215805,
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1.0872626304626465,
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0.5272358655929565,
0.37355560064315796,
-0.18585588037967682,
0.25499552488327026,
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0.02538597770035267,
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0.6063219904899597
] |
∂
∂u∗
i
E
Z,Zu,Zv[ϕt(q1, q2)] = 1
n
E
Z,Zu,Zv
�� ∂
∂u∗
i
Hn,t(�u, �v; q1, q2)
�
n,t
�
,
(C.18)
and a similar expression holds for the derivative with respect to v∗
j . Recall the definition of Hn,t from
(C.2). We have
∂
∂u∗
i
Hn,t(Ξ−1/2u, Σ−1/2v; q1, q2) = 1 − t
n
v⊤Σ−1v∗(Ξ−1)i,iui + αq1(Ξ−1)i,iui,
where we have used the fact that Ξ is diagonal. Therefore,
����
∂
∂u∗
i
Hn,t(Ξ−1/2u, Σ−1/2v; q1, q2)
���� ≤ 1
n · dK2∥Σ∥−1
2
· ∥Ξ∥−1
2 K + αq1 · ∥Ξ∥−1
2 K ≤ C, | [
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] |
∂
∂u∗
i
Hn,t(Ξ−1/2u, Σ−1/2v; q1, q2)
���� ≤ 1
n · dK2∥Σ∥−1
2
· ∥Ξ∥−1
2 K + αq1 · ∥Ξ∥−1
2 K ≤ C,
where C > 0 is a constant depending only on α, M, K, Ξ, Σ. The last inequality holds for all
sufficiently large n. A similar bound holds for
∂
∂v∗
j Hn,t(Ξ−1/2u, Σ−1/2v; q1, q2). This, by (C.18),
implies that EZ,Zu,Zv[ϕt(q1, q2)] as a function of (u∗, v∗) satisfies the bounded difference property
with ci = C/n (see Proposition G.9). So by Proposition G.9,
E
u∗,v∗
�����
E
Z,Zu,Zv[ϕt(q1, q2)] −
E
u∗,v∗
�
E | [
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0.04721640422940254,
-0.005970577243715525,
-0.2770949602127075,
-0.564068615436554,
-0.5175352692604065,
-0.27618345618247986,
0.14266015589237213,
0.2387775331735611,
0.018752584233880043,
-0.3737841844558716,
-0.25078022480010986,
-0.43316900730133057,
0.3208663761615753,
-0.15763770043849945,
0.09312233328819275,
0.7048909068107605,
-0.3339795172214508,
0.3872353434562683,
-0.1323329210281372,
-0.2179126739501953,
0.3703039288520813,
0.5257142186164856,
-0.7609999775886536,
-0.19017812609672546,
0.5009777545928955,
-0.22838419675827026,
0.003758177626878023,
0.5565902590751648,
-0.4725337326526642,
0.16631028056144714,
0.02898568846285343,
0.09654998034238815,
-0.3280576467514038,
0.8506485819816589,
0.726043164730072,
-0.17791493237018585,
0.05797486752271652,
-0.04211607947945595,
1.284642219543457,
-0.5132313370704651,
0.6307124495506287,
-0.2608361840248108,
-0.3243999183177948,
-0.17431461811065674,
-0.22182884812355042,
-0.18674799799919128,
0.0073632244020700455,
-0.4862001836299896,
-0.9672627449035645,
-0.18096035718917847,
0.3417758345603943,
-0.1707240343093872,
0.10386759042739868,
0.04442235454916954,
-0.6405593156814575,
-0.25451862812042236,
-0.7467053532600403,
0.7180739045143127,
-0.34549465775489807,
-0.10378483682870865,
0.23275889456272125,
-0.29417282342910767,
-0.2556174099445343,
-0.48274439573287964,
-1.242908000946045,
0.38965532183647156,
-0.3982086777687073,
-0.31628987193107605,
0.14356403052806854,
0.1178227961063385,
0.28676024079322815,
-0.32802483439445496,
-0.041622698307037354,
-0.756689190864563,
0.03904612362384796,
-0.9855208992958069,
0.0008205410558730364,
-0.010898957960307598,
0.8885828256607056,
-0.05909980088472366,
-0.7327247858047485,
0.02200707606971264,
0.761798083782196,
0.5551705956459045,
-0.04727637395262718,
-0.36388659477233887,
0.22329187393188477,
0.2598639130592346,
-0.11910602450370789,
-0.15146496891975403,
-0.2670263946056366,
0.7653244733810425,
-0.004894094541668892,
0.29646289348602295,
-0.47189536690711975,
0.2500365376472473,
0.16837678849697113,
-0.08421267569065094,
0.14031367003917694,
-0.700867772102356,
-0.08173103630542755,
-0.3003086447715759,
0.380189448595047,
-0.17612041532993317,
-0.11207135021686554,
0.012491787783801556,
-0.02378244511783123,
-0.17255370318889618,
0.14015063643455505,
0.2600997984409332,
0.23486626148223877,
-0.1740955114364624,
0.29316428303718567,
-0.5017129182815552,
0.31350332498550415,
0.46061980724334717,
-0.38622570037841797,
-0.41546496748924255,
-0.8905754089355469,
-0.08047516644001007,
-0.07889731228351593,
-0.19880428910255432,
-0.07885916531085968,
-0.0898316353559494,
0.2940998375415802,
-0.2588067650794983,
-0.08122891187667847,
-0.20968760550022125,
-0.5184935331344604,
-1.0355900526046753,
-0.4824884831905365,
0.07413926720619202,
0.6727379560470581,
-0.3725677728652954,
-0.2790665328502655,
-0.053554411977529526,
-0.27146267890930176,
-1.0844515562057495,
0.18357239663600922,
-0.15056879818439484,
0.0247920174151659,
0.5713089108467102,
0.4078952372074127,
0.7625998258590698,
-0.2074977457523346,
-0.3113517165184021,
-0.392898291349411,
0.8521002531051636,
-0.07249996066093445,
0.21887020766735077,
0.45731350779533386,
-0.2764636278152466,
-0.33579033613204956,
0.5773021578788757,
0.21999914944171906,
-0.12838491797447205,
0.6237310767173767,
0.06822609156370163,
0.4204443693161011,
1.4687775373458862,
0.35022979974746704,
-0.7324723601341248,
0.5774462819099426,
-0.44854313135147095,
0.2215910404920578,
0.31340357661247253,
0.4520794153213501,
0.4944535195827484,
0.2459721863269806,
-0.6777838468551636,
0.1710261106491089,
0.4181331992149353,
-0.15526224672794342,
-0.025974877178668976,
0.09828053414821625,
-0.28699490427970886,
0.4263666868209839,
0.3745460510253906,
0.806117832660675,
-0.14631907641887665,
0.09507512301206589,
-0.10147212445735931,
-0.08189261704683304,
0.012083504348993301,
-0.10527674853801727,
-0.16013844311237335,
-0.6040779948234558,
0.32299941778182983,
-0.2422143965959549,
-0.17321182787418365,
-0.07197623699903488,
-0.6499231457710266,
0.4540814459323883,
0.7393274307250977,
-0.4963395893573761,
-0.1707482635974884,
0.14206074178218842,
-0.09443587809801102,
-0.39798682928085327,
0.10225925594568253,
-0.27257463335990906,
-0.29835832118988037,
-0.2509506046772003,
-0.12480785697698593,
-0.3839404582977295,
0.11013864725828171,
-0.12350235134363174,
0.023785151541233063,
0.5283641815185547,
0.45577389001846313,
0.49631208181381226,
0.5654962062835693,
-0.7917945384979248,
0.11453156918287277,
0.04012995585799217,
-0.32281795144081116,
-0.08807142078876495,
0.04589960724115372,
0.34095168113708496,
0.42792418599128723,
-0.6205406188964844,
0.21231485903263092,
-0.47838595509529114,
0.09160357713699341,
-0.5139362215995789,
-0.08726836740970612,
0.27123069763183594,
0.08131855726242065,
-0.6565176248550415,
-0.664395809173584,
-0.24415040016174316,
0.432003915309906,
0.2649613916873932,
-0.9422929883003235,
0.6895109415054321,
0.7506684064865112,
-0.08609974384307861,
0.26988649368286133,
0.5088828206062317,
0.45651158690452576,
0.5512300133705139,
-0.622859537601471,
0.26344114542007446,
-0.5765237808227539,
0.5836524367332458,
-0.42931631207466125,
-0.3088677227497101,
0.6743812561035156,
0.2949713170528412,
0.15271241962909698,
-0.20725807547569275,
-0.11000300943851471,
-0.2973316013813019,
0.3558213412761688,
-0.08041699975728989,
-0.15865764021873474,
-0.3712039589881897,
-0.7574223875999451,
-0.1687147170305252,
0.41059213876724243,
0.25964757800102234,
0.27057236433029175,
-0.3042728900909424,
0.4299579858779907,
0.2275964915752411,
-0.03560395538806915,
0.4483932852745056,
-0.12840008735656738,
-0.11223083734512329,
-0.2330949455499649,
0.5003461837768555,
-1.0068730115890503,
0.4029691517353058,
-0.020551536232233047,
0.29507413506507874,
-0.10655201971530914,
-0.6326887011528015,
-0.23510275781154633,
-0.13545913994312286,
-0.17991992831230164,
1.1806105375289917,
-0.049107685685157776,
0.13662607967853546,
0.23392678797245026,
-0.7117773294448853,
-1.0068143606185913,
0.762337327003479,
0.3153460621833801,
-0.3464946746826172,
0.3745916485786438,
-0.3748725354671478,
-0.01867470145225525,
0.5817038416862488,
-0.3764069378376007,
-0.3666810095310211,
-0.3432706892490387,
0.6247318387031555
] |
E
u∗,v∗
�����
E
Z,Zu,Zv[ϕt(q1, q2)] −
E
u∗,v∗
�
E
Z,Zu,Zv[ϕt(q1, q2)]
�����
�
≤
�
Var
u∗,v∗
�
E
Z,Zu,Zv[ϕt(q1, q2)]
�
≤ C
√n.
(C.19)
Finally using (C.17) and (C.19) and the triangle inequality,
E[|ϕt(q1, q2) − E[ϕt(q1, q2)]|]
≤
E
u∗,v∗
�
E
Z,Zu,Zv
�����ϕt(q1, q2) −
E
Z,Zu,Zv[ϕt(q1, q2)]
����
�
+
����
E
Z,Zu,Zv[ϕt(q1, q2)] −
E
u∗,v∗
�
E
Z,Zu,Zv[ϕt(q1, q2)]
�����
�
≤ C
√n,
concluding the proof.
20
Suppose a, b ≥ 0 are constants. With q1 = sna2, q2 = snb2, we can write Hn,t in (C.2) as | [
-0.04262879118323326,
0.02475759945809841,
0.6944721341133118,
-0.3268802762031555,
-0.11920758336782455,
0.24332523345947266,
0.6036558151245117,
-1.0476089715957642,
-0.12971915304660797,
-0.39211520552635193,
-0.3659985065460205,
0.28715577721595764,
-0.28945285081863403,
0.2336605191230774,
0.5564512014389038,
0.7913941144943237,
0.6537660956382751,
0.4052078127861023,
-0.6472430229187012,
-0.9139226675033569,
0.24027404189109802,
0.09418971091508865,
0.6738200783729553,
-0.04272084683179855,
-0.010676921345293522,
0.05501197278499603,
0.46447286009788513,
-0.5766479969024658,
0.10828249156475067,
0.4222649335861206,
0.12768328189849854,
0.6424019932746887,
-0.06373049318790436,
0.2405538409948349,
0.3068803548812866,
-0.3403870761394501,
0.9091715812683105,
0.33219605684280396,
0.392415314912796,
-0.22593282163143158,
0.024811528623104095,
-0.40951335430145264,
0.598889946937561,
0.5351827144622803,
-0.23569078743457794,
-0.6176521182060242,
-1.249967336654663,
-0.08545857667922974,
0.17643682658672333,
-0.1857670396566391,
-0.3313046991825104,
-0.12178117036819458,
0.3036092221736908,
0.4388483762741089,
-0.6138805747032166,
-0.1512041687965393,
-0.43198922276496887,
-0.36849138140678406,
0.6700555682182312,
-0.44036003947257996,
0.5395403504371643,
0.152737095952034,
-0.09803341329097748,
0.7078032493591309,
-0.26420870423316956,
0.80069500207901,
-0.10858802497386932,
0.4972643256187439,
-0.6813751459121704,
-0.08116763830184937,
0.09670555591583252,
0.06199251115322113,
-0.31598421931266785,
-0.24123942852020264,
0.07540328800678253,
0.3123883306980133,
-0.4984411299228668,
-0.300446480512619,
0.1951872706413269,
-0.056491028517484665,
-0.1776488870382309,
-0.30728694796562195,
0.4487382769584656,
-0.4234185218811035,
0.1334158033132553,
0.07226931303739548,
0.34755685925483704,
0.17415684461593628,
-0.33166006207466125,
0.934039831161499,
0.27124613523483276,
-0.5041090846061707,
0.05444100871682167,
-0.1124800369143486,
-0.2967405915260315,
0.3339293897151947,
-0.03450364992022514,
-0.41783323884010315,
0.685893177986145,
0.12395820021629333,
-0.22102726995944977,
-0.6238330602645874,
-0.44125744700431824,
-0.4162561297416687,
-0.7999362349510193,
-0.46720215678215027,
-0.672676146030426,
-0.6144318580627441,
-0.0359204076230526,
0.22702430188655853,
-1.0622117519378662,
0.13962069153785706,
-0.13697700202465057,
-0.3741433918476105,
0.07535959035158157,
0.10350139439105988,
-0.02633000910282135,
-0.1538430154323578,
0.1054736077785492,
0.5633179545402527,
0.0923008993268013,
0.6887626051902771,
0.31213024258613586,
0.7735950946807861,
0.44437721371650696,
-0.33054831624031067,
0.5341320037841797,
0.6398093700408936,
-0.6237980127334595,
-0.21425826847553253,
0.10493931919336319,
0.8637795448303223,
0.15933984518051147,
0.3399917185306549,
-0.5231180787086487,
-0.8543705940246582,
-0.4868752956390381,
-0.41708800196647644,
-0.7456059455871582,
-0.3402481973171234,
-0.25913554430007935,
-0.09767711907625198,
-0.29135027527809143,
0.2109399437904358,
0.8738237619400024,
-0.11109279096126556,
-0.28278109431266785,
0.17141291499137878,
0.005687293130904436,
0.15674039721488953,
0.12420176714658737,
0.4809018671512604,
-0.49511635303497314,
0.2816217243671417,
-0.5758808255195618,
-0.14028456807136536,
0.1890234500169754,
0.1391134411096573,
-0.18376527726650238,
0.11238361150026321,
0.3947787284851074,
0.41734039783477783,
-0.07387636601924896,
-0.007453346159309149,
0.6757357716560364,
-0.3105427920818329,
0.22415608167648315,
-0.2138228863477707,
-0.3028731644153595,
0.010121539235115051,
-0.4211744964122772,
-0.22749915719032288,
-0.08548139780759811,
-0.085910364985466,
-0.43852290511131287,
0.03388836979866028,
-0.10611298680305481,
0.25963136553764343,
0.4266505539417267,
0.14108289778232574,
-0.7099191546440125,
-0.07823366671800613,
-0.028606709092855453,
-0.29511284828186035,
0.4409826397895813,
0.3005067706108093,
0.34494549036026,
0.11577348411083221,
0.3891560137271881,
0.04044315963983536,
-0.21856467425823212,
0.049951840192079544,
0.41065114736557007,
-1.1461693048477173,
1.3249481916427612,
0.30983811616897583,
-0.4004587233066559,
-0.010600905865430832,
-0.44809380173683167,
-0.01838001050055027,
-0.09817865490913391,
-0.0005728485994040966,
0.37123528122901917,
0.3428446650505066,
-0.28201642632484436,
0.6647409796714783,
0.09545128792524338,
0.144802525639534,
-0.9432065486907959,
0.3266333043575287,
0.1478610783815384,
0.2668437659740448,
-0.0846886932849884,
-0.494233101606369,
0.39029455184936523,
-0.16904623806476593,
0.14118051528930664,
-0.268318235874176,
-0.010748322121798992,
-0.25504907965660095,
0.9009937644004822,
-0.3699212372303009,
0.20211349427700043,
0.6173879504203796,
-0.22952356934547424,
0.1984744369983673,
0.7624187469482422,
-0.10300025343894958,
0.19064432382583618,
-0.5133851170539856,
-0.14242304861545563,
0.05104420334100723,
0.3115517199039459,
0.39970409870147705,
0.12662814557552338,
0.4645632207393646,
0.20970577001571655,
0.09116271883249283,
0.4893035888671875,
-0.6985337734222412,
0.2997910678386688,
0.4767312705516815,
0.00288402265869081,
0.21965546905994415,
-0.08066819608211517,
0.029398243874311447,
-0.17279238998889923,
-0.13938985764980316,
-0.6803260445594788,
-0.22711266577243805,
-0.08803191781044006,
-0.126087486743927,
0.18133816123008728,
-0.2929316759109497,
-0.04059714823961258,
-0.024410823360085487,
0.3657180666923523,
-0.015839647501707077,
0.14506608247756958,
0.36742672324180603,
-0.20724895596504211,
-0.2589983642101288,
0.18375691771507263,
-0.3576405942440033,
-0.12269099801778793,
0.25612375140190125,
-1.1078757047653198,
-0.48767074942588806,
0.7467105388641357,
0.5536375641822815,
0.15207038819789886,
-0.3331214189529419,
-0.8666556477546692,
-0.041195355355739594,
-0.4341661334037781,
-0.5441171526908875,
0.10235036909580231,
0.23320937156677246,
-0.6399503350257874,
-0.36774134635925293,
-0.12182266265153885,
0.609477162361145,
0.33274611830711365,
-0.4317151606082916,
-0.4238751530647278,
-0.26479610800743103,
-0.1268390715122223,
-0.49395957589149475,
-0.527529776096344,
-0.695077121257782,
0.08165623992681503,
0.888258695602417,
0.11091410368680954,
-0.732995867729187,
0.06825675815343857,
0.2735153138637543,
-0.2559768855571747,
0.3075767159461975,
0.45299017429351807,
0.15138587355613708,
-0.05956263840198517,
-0.0909501239657402,
0.01496881153434515,
-0.03428354486823082,
-0.26740938425064087,
-0.899790346622467,
-0.20659200847148895,
0.573087215423584,
-5.769162178039551,
-0.2575016915798187,
-0.6497533917427063,
0.5210542678833008,
-0.40979090332984924,
-0.13084626197814941,
0.4275128245353699,
-0.22920192778110504,
0.05922709032893181,
-0.15539734065532684,
0.022507615387439728,
0.1959017664194107,
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0.40653133392333984,
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0.9373313188552856,
0.04336678981781006,
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0.3393508195877075,
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0.318968802690506,
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-0.718011200428009,
0.046665437519550323,
0.37412622570991516,
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0.1733669638633728,
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0.5219089984893799,
0.08870125561952591,
0.005122060887515545,
0.5538902878761292,
0.46175503730773926,
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0.5431240797042847,
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0.3573586046695709,
0.6377396583557129,
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0.5253325700759888,
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0.3180374801158905,
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0.09111873060464859,
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0.09511455148458481,
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0.42653247714042664,
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0.5152485370635986,
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0.35404109954833984,
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0.2731107771396637,
0.18501397967338562,
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1.0333738327026367,
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0.16838933527469635,
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0.12905260920524597,
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0.5079572796821594,
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0.45091304183006287,
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0.5414040088653564,
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0.07585606724023819,
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0.045479778200387955,
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0.005604675505310297,
0.39679622650146484,
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1.0429587364196777,
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] |
20
Suppose a, b ≥ 0 are constants. With q1 = sna2, q2 = snb2, we can write Hn,t in (C.2) as
Hn,t(�u, �v; sna2, snb2) = Hn,t(�u, �v) + Hu
n,a(�u) + Hv
n,b(�v),
where
Hn,t(�u, �v) :=
�
1 − t
n
�u⊤Z�v + 1 − t
n
�u⊤�u∗�v⊤�v∗ − 1 − t
2n ∥�u∥2
2∥�
vv∥2
2,
Hu
n,a(�u) := αsna2�u⊤�u∗ + a√αsn�u⊤Zu − αsna2
2
∥�u∥2
2,
Hv
n,b(�v) := snb2�v⊤�v∗ + b√sn�v⊤Zv − snb2
2
∥�v∥2
2.
Fix A ≥ 2. Define
ϕu(a) =
1
nsn
log
�
Rn exp
�
Hu
n,a(�u)
�
d �P(�u),
ϕv(b) =
1
nsn
log
�
Rd exp
�
Hv
n,b(�v)
�
d �Q(�v),
ξu
n(sn) :=
sup | [
-0.2679750621318817,
-0.010288991034030914,
0.33889949321746826,
-0.3695093095302582,
0.053584158420562744,
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0.09132891148328781,
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0.5392970442771912,
0.9024933576583862,
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0.24779483675956726,
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0.5111817121505737,
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0.5437756180763245,
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0.561320960521698,
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0.06910096108913422,
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0.10503485798835754,
0.9503083229064941,
0.43292585015296936,
0.7927702069282532,
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0.2019345760345459,
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0.16999757289886475,
0.6716051697731018,
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0.7027555704116821,
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0.04937068000435829,
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0.3311704397201538,
0.159498929977417,
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0.4503619372844696,
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0.04873960092663765,
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0.5804002285003662,
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0.15830154716968536,
0.5041279196739197,
0.09552719444036484,
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] |
Rn exp
�
Hu
n,a(�u)
�
d �P(�u),
ϕv(b) =
1
nsn
log
�
Rd exp
�
Hv
n,b(�v)
�
d �Q(�v),
ξu
n(sn) :=
sup
1/2≤a≤A+1/2
E[|ϕu(a) − E[ϕu(a)]|],
ξv
n(sn) :=
sup
1/2≤b≤A+1/2
E[|ϕv(b) − E[ϕv(b)]|],
(C.20)
Υu(�u) :=
1
nsn
∂
∂aHu
n,a(�u),
Υv(�v) :=
1
nsn
∂
∂bHb
n,b(�v).
Denote by ⟨g(u∗, v∗)⟩n,a,b the conditional expectation of g(u∗, v∗) given
Yt,
Y u
t (a) := a√αsnu∗ + Ξ1/2Zu,
Y v
t (b) := b√snv∗ + Σ1/2Zv,
(C.21)
where the expectation is with respect to the distribution in (C.1).
Corollary C.4. Let sn = n−1/32 and A ≤
� | [
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0.9462641477584839,
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0.49085307121276855,
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] |
Corollary C.4. Let sn = n−1/32 and A ≤
�
M/sn − 1/2 for a constant M > 0 independent of n.
Then there exists a constant C > 0 depending only on K, α, Ξ such that for all sufficiently large n,
ξu
n(sn) ≤
C
sn
√n.
Proof. Note that if b = 0, t = 1, it holds that Hn,1(�u, �v; sna2, 0) = Hu
n,a(�u) and ϕ1(sna2, 0) =
snϕu(a). The conclusion then follows immediately from Lemma C.3 since by the assumption on A,
q1 = sna2 ∈ [0, M] for any 1/2 ≤ a ≤ A + 1/2.
Lemma C.5. Let sn = n−1/32. For all A ≥ 2,
1
A − 1
� A
1
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Lemma C.5. Let sn = n−1/32. For all A ≥ 2,
1
A − 1
� A
1
E
�����Υu(�u) − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
�
da ≤ C
�
1
√nsn
+
�
ξun(sn)
�
,
where C > 0 only depends on α, Ξ, K.
Proof. By the triangle inequality,
E
�����Υu(�u) − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
�
≤ E
�����Υu(�u) − ⟨Υu(�u)⟩n,a,b
���
�
n,a,b
�
+ E
�����⟨Υu(�u)⟩n,a,b − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
�
.
(C.22)
We will bound the two terms on the RHS separately. We first bound
1
A − 1
� A
1
E
�����Υu(�u) − ⟨Υu(�u)⟩n,a,b
���
�
n,a,b
�
da
≤
�
1
A − 1 | [
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] |
1
A − 1
� A
1
E
�����Υu(�u) − ⟨Υu(�u)⟩n,a,b
���
�
n,a,b
�
da
≤
�
1
A − 1
� A
1
E
���
Υu(�u) − ⟨Υu(�u)⟩n,a,b
�2�
n,a,b
�
da
�1/2
.
(C.23)
21
The first two derivatives of ϕu are
(ϕu)′(a) =
1
nsn
�
Rn exp
�
Hu
n,a(�u)
� ∂
∂aHu
n,a(�u) d �P(�u)
�
Rn exp
�
Hun,a(�u)
�
d �P(�u)
= ⟨Υu(�u)⟩n,a,b,
(C.24a)
(ϕu)′′(a) =
1
nsn
�� ∂
∂aHu
n,a(�u)
�2�
n,a,b
−
� ∂
∂aHu
n,a(�u)
�2
n,a,b
+
� ∂2
∂a2 Hu
n,a(�u)
�
n,a,b
= nsn
��
Υu(�u)2�
n,a,b − ⟨Υu(�u)⟩2
n,a,b
�
+ α
n
�
2
�
�u⊤�u∗�
n,a,b −
�
∥�u∥2
2
�
n,a,b
�
.
(C.24b) | [
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] |
��
Υu(�u)2�
n,a,b − ⟨Υu(�u)⟩2
n,a,b
�
+ α
n
�
2
�
�u⊤�u∗�
n,a,b −
�
∥�u∥2
2
�
n,a,b
�
.
(C.24b)
Since there exists C > 0 depending only on α, Ξ, K such that for all sufficiently large n,
����
α
n
�
2
�
�u⊤�u∗�
n,a,b −
�
∥�u∥2
2
�
n,a,b
����� ≤
4αK2
inf supp(Ξ) =: C,
(C.25)
the second result (C.24b) above implies that for all sufficiently large n,
��
Υu(�u) − ⟨Υu(�u)⟩n,a,b
�2�
n,a,b
≤
1
nsn
((ϕu)′′(a) + C).
Consequently,
� A
1
E
���
Υu(�u) − ⟨Υu(�u)⟩n,a,b
�2�
n,a,b
�
da ≤
1
nsn | [
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0.1849730908870697,
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0.03162091597914696,
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0.5429307222366333,
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0.22188161313533783,
0.19635143876075745,
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0.1534358114004135,
0.08938037604093552,
0.19871100783348083,
0.38458478450775146,
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0.20500430464744568,
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0.5301305651664734,
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0.30215129256248474,
0.5221984386444092,
0.31800177693367004,
0.02787640132009983,
-0.6553044319152832,
-0.6627243757247925,
-0.04487411677837372,
-0.30811235308647156,
0.9260385632514954,
0.2539340555667877,
0.24815087020397186,
-0.16915857791900635,
-0.687956690788269,
-0.5625123381614685,
0.4809994697570801,
0.12553322315216064,
0.2704128623008728,
0.1192888468503952,
-0.30352699756622314,
-0.17176096141338348,
0.254281610250473,
0.009242193773388863,
-0.4010957181453705,
0.291640043258667,
0.6039522290229797
] |
≤
1
nsn
((ϕu)′′(a) + C).
Consequently,
� A
1
E
���
Υu(�u) − ⟨Υu(�u)⟩n,a,b
�2�
n,a,b
�
da ≤
1
nsn
(E[(ϕu)′(A)] − E[(ϕu)′(1)] + C(A − 1)).
(C.26)
To proceed, we compute for any a,
E[(ϕu)′(a)] = E
�
⟨Υu(�u)⟩n,a,b
�
,
(C.27)
which is by (C.24a). By definition,
Υu(�u) = 1
n
�
2αa�u⊤�u∗ +
�
α/sn�u⊤Zu − αa∥�u∥2
2
�
,
whose expectation is therefore given by
E
�
⟨Υu(�u)⟩n,a,b
�
= 2αa
n E
��
�u⊤�u∗�
n,a,b
�
+
√α
n√sn
E
��
�u⊤Zu�
n,a,b
�
− αa
n E
��
∥�u∥2
2
�
n,a,b
�
.
(C.28) | [
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] |
�
= 2αa
n E
��
�u⊤�u∗�
n,a,b
�
+
√α
n√sn
E
��
�u⊤Zu�
n,a,b
�
− αa
n E
��
∥�u∥2
2
�
n,a,b
�
.
(C.28)
Using Stein’s lemma (Proposition G.7), the middle term is equal to
√α
n√sn
E
��
�u⊤Zu�
n,a,b
�
=
√α
n√sn
n
�
i=1
E
�
Zu
i ⟨�ui⟩n,a,b
�
=
√α
n√sn
n
�
i=1
E
� ∂
∂Zu
i
⟨�ui⟩n,a,b
�
= aα
n E
��
∥�u∥2
2
�
n,a,b
�
− aα
n E
��
�u⊤�u∗�
n,a,b
�
.
Therefore,
E
�
⟨Υu(�u)⟩n,a,b
�
= αa
n E
��
�u⊤�u∗�
n,a,b
�
,
(C.29)
and
���E
�
⟨Υu(�u)⟩n,a,b
���� ≤ aC,
(C.30)
where C depends only on α, Ξ, K. | [
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] |
n,a,b
�
,
(C.29)
and
���E
�
⟨Υu(�u)⟩n,a,b
���� ≤ aC,
(C.30)
where C depends only on α, Ξ, K.
Using the last inequality, we can further upper bound the RHS of (C.26) by CA
nsn for some C depending
only on α, Ξ, K. Putting this back to (C.23), we get
1
A − 1
� A
1
E
�����Υu(�u) − ⟨Υu(�u)⟩n,a,b
���
�
n,a,b
�
da ≤
�
CA
nsn(A − 1) ≤
�
2C
nsn
,
(C.31)
22
since A ≥ 2.
Now it remains to bound the second term on the RHS of (C.22) which can be written as
E
�����⟨Υu(�u)⟩n,a,b − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
� | [
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] |
E
�����⟨Υu(�u)⟩n,a,b − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
�
= E[|(ϕu)′(a) − E[(ϕu)′(a)]|]
(C.32)
using (C.24a). To further bound the RHS, consider the following two functions
a �→ ϕu(a) +
2αK2a2
inf supp(Ξ),
a �→ E[ϕu(a)] +
2αK2a2
inf supp(Ξ).
They are both differentiable and convex for all sufficiently large n since their second derivatives are
non-negative by (C.25) and (C.24b). Applying Proposition G.10 with the above two functions, taking | [
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] |
non-negative by (C.25) and (C.24b). Applying Proposition G.10 with the above two functions, taking
the expectation and using the triangle inequality, we have that for any 1 ≤ a ≤ A, 0 < a′ ≤ 1/2,
E[|(ϕu)′(a) − E[(ϕu)′(a)]|] ≤ E[(ϕu)′(a + a′)] − E[(ϕu)′(a − a′)] + 3ξu
n(sn)/a′ +
8αK2a′
inf supp(Ξ).
(C.33)
Then
� A
1
E[(ϕu)′(a + a′)] − E[(ϕu)′(a − a′)] da
= (E[ϕu(A + a′)] − E[ϕu(1 + a′)]) − (E[ϕu(A − a′)] − E[ϕu(1 − a′)])
= (E[ϕu(A + a′)] − E[ϕu(A − a′)]) − (E[ϕu(1 + a′)] − E[ϕu(1 − a′)])
=
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0.20188550651073456,
-0.7140984535217285,
0.10791504383087158,
0.1228320375084877,
0.03688580170273781,
0.057171836495399475,
-0.6258532404899597,
0.11418729275465012,
0.4059634506702423,
0.749090313911438,
0.48667603731155396,
-0.269461989402771,
-0.1418890357017517,
-0.08026270568370819,
0.17864596843719482,
0.30288150906562805,
-0.6260188817977905,
-0.37512683868408203,
0.3211841583251953,
0.38109299540519714,
-0.10378985852003098,
-0.5663365125656128,
0.04567047208547592,
-0.752229630947113,
-0.02434096671640873,
0.10770204663276672,
-0.1980389654636383,
0.19473852217197418,
0.14254269003868103,
0.06579957902431488,
0.3656514883041382,
-0.08402217924594879,
-0.3126363456249237,
-0.19648216664791107,
0.14293037354946136,
1.1061431169509888,
0.07701995223760605,
0.0048028286546468735,
-0.7279316186904907,
-0.22221659123897552,
0.36809611320495605,
0.19888804852962494,
0.4753757119178772,
0.6227655410766602,
-0.36332234740257263,
0.08628034591674805,
0.4492553472518921,
-0.34752094745635986,
-0.02939404733479023,
0.1899808943271637,
0.20329318940639496,
-0.3427139222621918,
-0.46639272570610046,
-0.421559602022171,
-0.4505990743637085,
0.21884985268115997,
-0.5097219347953796,
-0.22205699980258942,
0.3765738606452942,
0.5091532468795776,
-0.35695794224739075,
-0.4212225675582886,
0.21705642342567444,
0.3545292913913727,
-0.07175260037183762,
-0.5161445736885071,
0.9145116209983826,
0.6587009429931641,
-0.15770721435546875,
0.5056189894676208,
0.06949014216661453,
0.5083059668540955,
0.3858270049095154,
-0.15804672241210938,
0.083096943795681,
-0.40523427724838257,
0.27224257588386536,
-0.6483186483383179,
-0.24960710108280182,
0.5797762274742126,
-0.030915234237909317,
0.07420535385608673,
-0.019988732412457466,
-0.16653452813625336,
0.07779871672391891,
0.6070381999015808,
-0.06991058588027954,
-0.2191789746284485,
-0.3501341938972473,
-0.3966665267944336,
-0.046248190104961395,
-0.16925376653671265,
0.16556154191493988,
-0.07488758862018585,
-0.19035102427005768,
0.10400819033384323,
-0.20244088768959045,
0.44188836216926575,
0.5339767336845398,
-0.47704094648361206,
0.17505116760730743,
0.08910984545946121,
0.2935219705104828,
-0.7496191263198853,
-0.13095228374004364,
0.5351686477661133,
0.4492355287075043,
-0.014456729404628277,
-0.6376975774765015,
-0.5392977595329285,
-0.1302819848060608,
-0.3248833417892456,
0.9686925411224365,
0.39854341745376587,
0.31385552883148193,
-0.5963409543037415,
-0.7285887598991394,
-0.2132771760225296,
0.09348171204328537,
0.23323537409305573,
0.4036558270454407,
-0.08466313779354095,
0.42176929116249084,
-0.16017299890518188,
-0.06670790910720825,
-0.008353086188435555,
-0.5056940317153931,
0.1166892796754837,
0.11015370488166809
] |
= (E[ϕu(A + a′)] − E[ϕu(A − a′)]) − (E[ϕu(1 + a′)] − E[ϕu(1 − a′)])
=
� a′
−a′ E[(ϕu)′(A + a)] da −
� a′
−a′ E[(ϕu)′(1 + a)] da
≤ 4a′(A + a′)C ≤ 8a′AC,
where the last step is by (C.27) and (C.30), and C depends only on α, Ξ, K. Using this in (C.33), we
obtain
� A
1
E[|(ϕu)′(a) − E[(ϕu)′(a)]|] da ≤ CA(a′ + ξu
n(sn)/a′),
and the RHS is minimized by a′ =
�
ξun(sn) which lies in the interval (0, 1/2] for all sufficiently
large n due to Corollary C.4. Using this result in (C.32) and integrating over a, we have | [
-0.34947365522384644,
-0.0909389778971672,
0.1531466245651245,
-0.3935600817203522,
-0.06519361585378647,
0.23092667758464813,
0.2783941924571991,
-0.97060227394104,
-0.03781891241669655,
-0.2727733850479126,
-0.36833882331848145,
0.3707435131072998,
-0.2029884308576584,
0.10663803666830063,
0.4907284677028656,
0.6319258213043213,
0.6091118454933167,
0.30291348695755005,
-0.5108227133750916,
-0.39173510670661926,
0.2426089197397232,
-0.028051871806383133,
0.23236383497714996,
0.1943017989397049,
0.008881451562047005,
0.01318212691694498,
0.4582507610321045,
0.08227088302373886,
0.40285420417785645,
0.1079372763633728,
-0.3416734039783478,
0.22631923854351044,
0.3960421085357666,
-0.1766568273305893,
0.3046559989452362,
-0.10881160944700241,
0.6445983052253723,
0.14621497690677643,
0.20175416767597198,
-0.2292885035276413,
-0.3857964277267456,
-0.4278854429721832,
0.3310259282588959,
0.40099242329597473,
-0.5496811270713806,
-0.4646933376789093,
-1.19032883644104,
0.26785406470298767,
0.22474046051502228,
-0.4752059578895569,
-0.2328902930021286,
0.03348434343934059,
-0.09074182063341141,
0.2942168116569519,
-0.3696916997432709,
0.13786493241786957,
-0.5094995498657227,
-0.1483309119939804,
0.44339409470558167,
-0.1697126179933548,
0.4086843729019165,
-0.2984844446182251,
0.4483567774295807,
0.3571120500564575,
-0.16773948073387146,
0.9801761507987976,
-0.3400931656360626,
0.14372210204601288,
-0.28565463423728943,
0.1792232096195221,
0.1233927309513092,
-0.32925543189048767,
-0.5798916816711426,
0.021101035177707672,
-0.34690389037132263,
0.35167205333709717,
-0.40204596519470215,
-0.4257943332195282,
0.3099808990955353,
-0.16232652962207794,
-0.19317489862442017,
-0.35596567392349243,
0.6636056900024414,
-0.7370610237121582,
0.17111380398273468,
0.2930750250816345,
-0.06750714778900146,
0.2846131920814514,
-0.5755268335342407,
0.5319561958312988,
-0.14575234055519104,
-0.20033639669418335,
0.27815085649490356,
0.13955728709697723,
-0.0007188324234448373,
0.3065376281738281,
0.3285992443561554,
-0.3840394914150238,
0.3641451895236969,
0.061749063432216644,
-0.6193580031394958,
-0.18330678343772888,
-0.32057616114616394,
-0.49421024322509766,
-0.1873658001422882,
-0.6422169804573059,
-0.61845463514328,
-0.403405100107193,
-0.5797699093818665,
-0.06165824458003044,
-0.6953288316726685,
0.5538293123245239,
0.040972523391246796,
-0.6807408928871155,
-0.1621287763118744,
-0.2904464900493622,
-0.01849469169974327,
0.2443578541278839,
0.5878201723098755,
0.34346556663513184,
0.21366021037101746,
0.9000190496444702,
0.13415278494358063,
0.35710224509239197,
0.06867463886737823,
0.109285369515419,
0.35801640152931213,
0.6252718567848206,
-0.43848705291748047,
0.06328660994768143,
0.1920778602361679,
0.26625844836235046,
0.5632367134094238,
0.43070197105407715,
-0.31760403513908386,
-0.8891461491584778,
-0.49469760060310364,
-0.47460854053497314,
-0.6581728458404541,
-0.37422293424606323,
-0.2305254191160202,
-0.11786352843046188,
0.08754387497901917,
-0.052179865539073944,
0.3574773073196411,
0.04687035083770752,
-0.15762370824813843,
-0.11780375987291336,
0.012837812304496765,
0.04279584437608719,
0.04274206608533859,
0.2681128978729248,
-0.775364100933075,
0.04350590705871582,
-0.4836856424808502,
-0.08140071481466293,
0.06715787947177887,
-0.008137003518640995,
-0.6290005445480347,
0.21095748245716095,
0.8603079319000244,
0.1441137045621872,
0.2839924395084381,
-0.3326597511768341,
0.9001402854919434,
-0.6961790919303894,
0.2013314962387085,
0.09635823220014572,
-0.22505773603916168,
0.21106001734733582,
-0.26643410325050354,
-0.3467148542404175,
-0.2986912131309509,
0.09655480831861496,
-0.1842459738254547,
-0.20624646544456482,
-0.12431594729423523,
-0.008383572101593018,
0.3907511830329895,
0.2951592803001404,
-0.7339316606521606,
0.05616477504372597,
-0.21836121380329132,
-0.10446631163358688,
0.24375781416893005,
0.15436434745788574,
0.32807043194770813,
-0.21863143146038055,
0.6848758459091187,
0.5741907358169556,
-0.6200881004333496,
-0.08997935801744461,
0.019093215465545654,
-0.842456042766571,
0.8919508457183838,
-0.12438169121742249,
-0.8329179286956787,
-0.08214264363050461,
-0.12125536054372787,
0.08085831254720688,
-0.19165877997875214,
0.21891680359840393,
0.2659398913383484,
0.14307434856891632,
-0.21616971492767334,
0.5556586384773254,
-0.2853206694126129,
0.25438371300697327,
-0.791766881942749,
0.7433527112007141,
0.01634886860847473,
-0.0024768533185124397,
0.31437554955482483,
-0.4969205856323242,
0.5831794738769531,
-0.047834839671850204,
-0.28172197937965393,
0.06469310075044632,
0.2642653286457062,
0.18936091661453247,
0.6068592667579651,
-0.08018634468317032,
-0.030689546838402748,
0.6392297744750977,
0.07768338918685913,
0.6521090865135193,
1.0320786237716675,
0.1607600748538971,
0.7390428185462952,
-0.9844317436218262,
0.20196308195590973,
-0.0918467789888382,
0.2998513877391815,
-0.07174086570739746,
0.44703561067581177,
0.694868803024292,
-0.17021557688713074,
-0.21522770822048187,
0.5336596369743347,
-0.24362137913703918,
0.10561280697584152,
0.181574746966362,
0.051748622208833694,
-0.08593588322401047,
0.3810771405696869,
-0.1138230487704277,
-0.3280602991580963,
-0.1711585819721222,
-0.2425706684589386,
-0.1857684701681137,
-0.16577447950839996,
0.0375232920050621,
0.22542113065719604,
-0.3243842124938965,
-0.1674066036939621,
-0.11141661554574966,
0.21665559709072113,
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0.1291886568069458,
0.06516814231872559,
-0.49387887120246887,
-0.2966820299625397,
0.10758703202009201,
-0.2788735330104828,
0.17495201528072357,
-0.01626521535217762,
-1.359571099281311,
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0.6965665221214294,
0.2990204989910126,
0.03633822128176689,
-0.6591458916664124,
-0.5202438831329346,
-0.3416472375392914,
-0.4888506531715393,
-1.0080803632736206,
-0.1129532903432846,
0.3680316209793091,
-0.8014901876449585,
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0.10001261532306671,
0.5266785621643066,
0.14710621535778046,
0.060594622045755386,
-0.7930461168289185,
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0.2776230573654175,
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0.06423280388116837,
0.2957206666469574,
0.05004725605249405,
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0.6248116493225098,
0.018353566527366638,
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0.08944191038608551,
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0.061460550874471664,
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0.4369490444660187,
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0.4517141878604889,
0.24364474415779114,
0.39231860637664795,
0.5183893442153931,
0.3060458302497864,
0.2680850327014923,
0.69181889295578,
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0.19536824524402618,
0.6453099250793457,
0.05172969400882721,
-0.3997744023799896,
0.4569135904312134,
-0.21623413264751434,
-0.31266409158706665,
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0.3769270181655884,
0.7313505411148071,
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0.12820957601070404,
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0.5544127225875854,
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0.7621616125106812,
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0.2810709774494171,
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0.03503761813044548,
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1.0958479642868042,
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] |
large n due to Corollary C.4. Using this result in (C.32) and integrating over a, we have
1
A − 1
� A
1
E
�����⟨Υu(�u)⟩n,a,b − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
�
da ≤
CA
A − 1 · 2
�
ξun(sn) ≤ 4C
�
ξun(sn),
(C.34)
since A ≥ 2.
Finally, combining (C.22), (C.31) and (C.34) proves the lemma.
Lemma C.6. There exists C > 0 depending only on α, K, Ξ such that for any A ≥ 2,
1
A − 1
� A
1
1
n2 E
��
(�u(1))⊤�u(2) − E
��
(�u(1))⊤�u(2)�
n,a,b
��2�
n,a,b
da ≤ C
�
1
√nsn
+
�
ξun(sn)
�
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] |
��
(�u(1))⊤�u(2) − E
��
(�u(1))⊤�u(2)�
n,a,b
��2�
n,a,b
da ≤ C
�
1
√nsn
+
�
ξun(sn)
�
,
where �u(1) = Ξ−1/2u(1), �u(2) = Ξ−1/2u(2) with u(1), u(2) being two i.i.d. copies from the condi-
tional law of u∗, v∗ given (C.21).
Proof. Since supp(P), supp(Q), ∥Ξ∥−1
2
are all bounded, by the triangle inequality,
1
n
����E
��
Υu(�u(1))(�u(1))⊤�u(2)�
n,a,b
�
− E
�
⟨Υu(�u)⟩n,a,b
�
E
��
(�u(1))⊤�u(2)�
n,a,b
�����
(C.35)
≤ ∥Ξ∥−1
2 K2E
�����Υu(�u) − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
�
.
(C.36)
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] |
n,a,b
�����
(C.35)
≤ ∥Ξ∥−1
2 K2E
�����Υu(�u) − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
�
.
(C.36)
23
On the other hand, let us compute (C.35) on the LHS of the above inequality. Recall from (C.29) in
the proof of Lemma C.5 that
E
��
Υu(�u(1))
�
n,a,b
�
= αa
n E
��
(�u(1))⊤�u(2)�
n,a,b
�
.
(C.37)
Similar to (C.28), we have
E
��
Υu(�u(1))(�u(1))⊤�u(2)�
n,a,b
�
= 2αa
n E
��
(�u(1))⊤�u∗(�u(1))⊤�u(2)�
n,a,b
�
+
√α
n√sn
E
��
(�u(1))⊤Zu(�u(1))⊤�u(2)�
n,a,b
�
− αa
n E
������u(1)���
2
2(�u(1))⊤�u(2)
�
n,a,b
�
. | [
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1.018263339996338,
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0.32614511251449585,
0.5401518940925598,
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0.2794950008392334,
0.2175363004207611,
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0.5296119451522827,
0.07567593455314636,
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0.5531472563743591,
0.13362576067447662,
0.6806871891021729,
0.5011467933654785,
0.2732775807380676,
0.12823118269443512,
0.05649334564805031,
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0.4128198027610779,
0.39676666259765625,
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0.19156064093112946,
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1.007951259613037,
0.2560810148715973,
0.39049896597862244,
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0.327980101108551,
0.30022093653678894,
0.25243815779685974,
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0.035754285752773285,
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0.005471991840749979,
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0.5968977808952332,
0.3516446053981781
] |
+
√α
n√sn
E
��
(�u(1))⊤Zu(�u(1))⊤�u(2)�
n,a,b
�
− αa
n E
������u(1)���
2
2(�u(1))⊤�u(2)
�
n,a,b
�
.
Using Stein’s lemma and following similar derivations leading to (C.29), the second term on the RHS
equals
√α
n√sn
E
��
(�u(1))⊤Zu(�u(1))⊤�u(2)�
n,a,b
�
= aα
n E
������u(1)���
2
2(�u(1))⊤�u(2)
�
n,a,b
�
− 2aα
n E
��
(�u(1))⊤�u(3)(�u(1))⊤�u(2)�
n,a,b
�
+ aα
n E
���
(�u(1))⊤�u(2)�2�
n,a,b
�
.
Therefore,
E
��
Υu(�u(1))(�u(1))⊤�u(2)�
n,a,b
�
= αa
n E
���
(�u(1))⊤�u(2)�2�
n,a,b
�
.
(C.38) | [
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] |
�
.
Therefore,
E
��
Υu(�u(1))(�u(1))⊤�u(2)�
n,a,b
�
= αa
n E
���
(�u(1))⊤�u(2)�2�
n,a,b
�
.
(C.38)
Putting (C.37) and (C.38) together and using Nishimori identity (Proposition G.4), we have that
(C.35) equals
αa
n2
�����E
���
(�u(1))⊤�u(2)�2�
n,a,b
�
− E
��
(�u(1))⊤�u(2)�
n,a,b
�2�����
= αa
n2
������
E
��
(�u(1))⊤�u(2) − E
��
(�u(1))⊤�u(2)�
n,a,b
��2�
n,a,b
������
.
So by the inequality (C.36), for any a ≥ 1,
1
n2 E
��
(�u(1))⊤�u(2) − E
��
(�u(1))⊤�u(2)�
n,a,b
��2�
n,a,b
≤ ∥Ξ∥−1
2 K2
α
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] |
1
n2 E
��
(�u(1))⊤�u(2) − E
��
(�u(1))⊤�u(2)�
n,a,b
��2�
n,a,b
≤ ∥Ξ∥−1
2 K2
α
E
�����Υu(�u) − E
�
⟨Υu(�u)⟩n,a,b
����
�
n,a,b
�
.
Integrating over a ∈ [1, A] and invoking Lemma C.5 concludes the proof.
Lemma C.7 (Overlap concentration). Let R1, R2 : [0, 1] × R2
>0 → R≥0 be two continuous bounded
functions such that their partial derivatives with respect to the second and third arguments are
continuous and non-negative. Let sn = n−1/32. For (ε1, ε2) ∈ [1, 2]2, slightly abusing notation, let | [
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] |
q1(·, ε1, ε2), q2(·, ε1, ε2) be the unique solution to
�q1(0) = snε1
q2(0) = snε2
,
�q′
1(t) = R1(t, q1(t), q2(t))
q′
2(t) = R2(t, q1(t), q2(t))
.
(C.39)
Then there exists a constant C > 0 depending only on K, α, ∥R1∥∞, ∥R2∥∞, Ξ such that for every
t ∈ [0, 1],
� 2
1
� 2
1
1
n2 E
���
�u⊤�u∗ − E
��
�u⊤�u∗�
n,t
��2�
n,t
�
dε1 dε2 ≤ Cn−1/8.
24
Proof. The existence and uniqueness of the solution to the Cauchy problem (C.39) is a direct | [
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] |
24
Proof. The existence and uniqueness of the solution to the Cauchy problem (C.39) is a direct
consequence of the Cauchy–Lipschitz theorem [33, Theorem 3.1, Chapter V]. For any t ∈ [0, 1], the
function Qt(ε1, ε2) = (q1(t, ε1, ε2), q2(t, ε1, ε2)) is a C1-diffeomorphism. Its Jacobian determinant
is given by the Liouville formula [33, Corollary 3.1, Chapter V] and can be lower bounded as
J(ε1, ε2) := det(∇Qt(ε1, ε2))
= s2
n exp
�� t
0
∂2R1(s, Qs(ε1, ε2)) ds +
� t
0
∂3R2(s, Qs(ε1, ε2)) ds
�
≥ s2
n,
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] |
= s2
n exp
�� t
0
∂2R1(s, Qs(ε1, ε2)) ds +
� t
0
∂3R2(s, Qs(ε1, ε2)) ds
�
≥ s2
n,
(C.40)
since the partial derivatives are non-negative by assumptions.
We then view the RHS below as a function of q1, q2 and denote it by
V (q1, q2) = E
���
�u⊤�u∗ − E
��
�u⊤�u∗�
n,t
��2�
n,t
�
.
(C.41)
Denote Ωt = Qt([1, 2]2)/sn and M := max{∥R1∥∞, ∥R2∥∞} + 2. Since R1, R2 ≥ 0 by as-
sumptions, q1, q2 are non-decreasing in t by (C.39). So for i ∈ {1, 2}, and any t ∈ [0, 1] and
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] |
(ε1, ε2) ∈ [1, 2]2,
qi(t, ε1, ε2)/sn ≥ qi(0, ε1, ε2)/sn = εi ≥ 1,
qi(t, ε1, ε2)/sn ≤ qi(1, ε1, ε2)/sn = s−1
n
� 1
0
q′
i(t, ε1, ε2) dt + s−1
n qi(0, ε1, ε2) ≤ s−1
n (∥Ri∥∞ + 2).
We obtain the relation Ωt ⊂ [1, M/sn]2 for any t ∈ [0, 1].
Next, using the change of variable (r1, r2) = Qt(ε1, ε2)/sn, we have
� 2
1
� 2
1
1
n2 E
���
�u⊤�u∗ − E
��
�u⊤�u∗�
n,t
��2�
n,t
�
dε1 dε2
= 1
n2
� 2
1
� 2
1
V (q1(t, ε1, ε2), q2(t, ε1, ε2)) dε1 dε2
= 1
n2
�
Ωt
V (snr1, snr2)s2
n
J(Q−1
t (snr1, snr2)) d(r1, r2)
≤ 1
n2
� M/sn | [
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] |
= 1
n2
�
Ωt
V (snr1, snr2)s2
n
J(Q−1
t (snr1, snr2)) d(r1, r2)
≤ 1
n2
� M/sn
1
� M/sn
1
V (snr1, snr2) dr1 dr2,
(C.42)
where the last step is by (C.40). Further applying the change of variable r1 = a2, we have that for all
r2 ≥ 1,
1
n2
� M/sn
1
V (snr1, snr2) dr1 = 1
n2
� √
M/sn
1
V (sna2, snr2)2a da ≤ 2
�
M/sn
n2
� √
M/sn
1
V (sna2, snr2) da.
(C.43)
Recalling V from (C.41), we recognize that
V (sna2, snr2) = E
���
�u⊤�u∗ − E
��
�u⊤�u∗�
n,a,√r2
��2�
n,a,√r2
�
= E
��
(�u(1))⊤�u(2) − E
��
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] |
���
�u⊤�u∗ − E
��
�u⊤�u∗�
n,a,√r2
��2�
n,a,√r2
�
= E
��
(�u(1))⊤�u(2) − E
��
(�u(1))⊤�u(2)�
n,a,√r2
��2�
n,a,√r2
,
where ⟨·⟩n,a,b is defined in (C.21) and the second equality is by Nishimori identity (Proposition G.4).
Since
�
M/sn ≥ 2 for all sufficiently large n, applying Lemma C.6, we get
1
�
M/sn − 1
� √
M/sn
1
1
n2 V (sna2, snr2) da ≤ C
�
1
√nsn
+
�
ξun(sn)
�
,
25
where C > 0 depends only on α, K, Ξ and ξu
n(sn) is given in (C.20) with A =
�
M/sn. Using this | [
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0.14604495465755463,
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0.0355408601462841,
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0.07120133936405182,
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] |
25
where C > 0 depends only on α, K, Ξ and ξu
n(sn) is given in (C.20) with A =
�
M/sn. Using this
back in (C.43) and then in (C.42), we obtain
� 2
1
� 2
1
1
n2 E
���
�u⊤�u∗ − E
��
�u⊤�u∗�
n,t
��2�
n,t
�
dε1 dε2 ≤ 2C(M/sn)2(1/√nsn +
�
ξun(sn)).
Corollary C.4 guarantees that ξu
n(sn) ≤
C
sn
√n for some C > 0 depending only on α, K, Ξ. By the
choice of sn, we can finally upper bound (up to a positive constant depending only on α, K, Ξ, M)
the RHS above by
1
s2n
�
1
√nsn
+
1
�
sn
√n
�
≤
1
s2.5
n
·
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] |
the RHS above by
1
s2n
�
1
√nsn
+
1
�
sn
√n
�
≤
1
s2.5
n
·
2
n1/4 = 2 · n−11/64 ≤ 2 · n−1/8,
for all sufficiently large n, which completes the proof.
Recalling ⟨·⟩n,t defined in (C.6), let us identify E
��
�u⊤�u∗�
n,t
�
as a function of (t, q1, q2):
1
nE
��
�u⊤�u∗�
n,t
�
= ∆(t, q1, q2).
(C.44)
Note that ∆ is continuous, non-negative (by Nishimori identity) on [0, 1] × R2
≥0 and bounded by K2.
Its partial derivatives with respect to the second and third arguments are continuous and non-negative, | [
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] |
since the correlation between �u∗ and ⟨�u⟩n,t is a non-decreasing function of the SNRs q1, q2.
Lemma C.8 (Fundamental sum rule). In the setting of Lemma C.7, for (ε1, ε2) ∈ [1, 2]2, let
q1(t, ε1, ε2), q2(t, ε1, ε2) be the solution to (C.39) with R2 = ∆ defined in (C.44). Then we have
Fn =
� 2
1
� 2
1
� 1
0
ψΞ(αγq1(1, ε1, ε2)) + αψΣ(q2(1, ε1, ε2)) − α
2 q′
1(t, ε1, ε2)q′
2(t, ε1, ε2) dt dε1 dε2 + o(1).
Proof. Fix (ε1, ε2) ∈ [1, 2]2. By the choice R2 = ∆,
q′ | [
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] |
2(t, ε1, ε2) dt dε1 dε2 + o(1).
Proof. Fix (ε1, ε2) ∈ [1, 2]2. By the choice R2 = ∆,
q′
2(t, ε1, ε2) = ∆(t, q1(t, ε1, ε2), q2(t, ε1, ε2)) = 1
nE
��
�u⊤�u∗�
n,t
�
.
Plugging this into (C.14), integrating over (ε1, ε2) ∈ [1, 2]2 and applying Lemma C.7, we have
f ′
n(t) = α
2 q′
1(t)q′
2(t) + o(1),
where o(1) → 0 as n → ∞, uniformly over t. By Lemma C.1, we conclude
Fn =
� 2
1
� 2
1
fn(0) dε1 dε2 + o(1) =
� 2
1
� 2
1
�
fn(1) −
� 1
0
f ′
n(t) dt
�
dε1 dε2 + o(1)
=
� 2
1
� 2
1
� 1
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] |
fn(0) dε1 dε2 + o(1) =
� 2
1
� 2
1
�
fn(1) −
� 1
0
f ′
n(t) dt
�
dε1 dε2 + o(1)
=
� 2
1
� 2
1
� 1
0
ψΞ(αγq1(1, ε1, ε2)) + αψΣ(q2(1, ε1, ε2)) − α
2 q′
1(t, ε1, ε2)q′
2(t, ε1, ε2) dt dε1 dε2 + o(1),
as desired. Here the first and last equalities are by (C.8) and (C.9), respectively.
Finally, we prove a pair of matching upper and lower bounds, completing the proof of Theorem 4.4.
Lemma C.9 (Lower bound). Let sn = n−1/32 and R2 = ∆. Then
lim inf
n→∞ Fn ≥ sup
qv≥0
inf
qu≥0 F(qu, qv). | [
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] |
lim inf
n→∞ Fn ≥ sup
qv≥0
inf
qu≥0 F(qu, qv).
Proof. Fix an arbitrary qv ≥ 0. Let R1 = qv. Then q1(t, ε1, ε2) = snε1 + tqv and Lemma C.8 gives
Fn =
� 2
1
� 2
1
� 1
0
ψΞ(αγ(snε1 + qv)) + αψΣ(q2(1, ε1, ε2)) − α
2 qvq′
2(t, ε1, ε2) dt dε1 dε2 + o(1)
=
� 2
1
� 2
1
ψΞ(αγqv) + αψΣ(q2(1, ε1, ε2)) − α
2 qvq2(1, ε1, ε2) dε1 dε2 + o(1)
≥ inf
q2≥0 ψΞ(αγqv) + αψΣ(q2) − α
2 qvq2 + o(1) = inf
qu≥0 F(qu, qv) + o(1),
where the second line holds since ψΞ is Lipschitz and q2(0, ε1, ε2) = snε2 = o(1). This completes | [
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] |
where the second line holds since ψΞ is Lipschitz and q2(0, ε1, ε2) = snε2 = o(1). This completes
the proof since the above lower bound holds for all qv ≥ 0.
26
Lemma C.10 (Upper bound). Let sn = n−1/32 and R2 = ∆. Then
lim sup
n→∞ Fn ≤ sup
qv≥0
inf
qu≥0 F(qu, qv).
Proof. We apply Lemma C.8 with
R1(t, q1, q2) = 2αψ′
Σ(∆(t, q1, q2)).
(C.45)
Since ψΞ is Lipschitz and convex,
ψΞ(αγq1(1, ε1, ε2)) = ψΞ(αγ(q1(1, ε1, ε2) − q1(0, ε1, ε2))) + o(1) = ψΞ
�
αγ
� 1
0
q′
1(t, ε1, ε2) dt
�
+ o(1)
≤
� 1
0
ψΞ(αγq′ | [
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] |
�
αγ
� 1
0
q′
1(t, ε1, ε2) dt
�
+ o(1)
≤
� 1
0
ψΞ(αγq′
1(t, ε1, ε2)) dt + o(1),
and similarly
ψΣ(q2(1, ε1, ε2)) ≤
� 1
0
ψΣ(q′
2(t, ε1, ε2)) dt + o(1).
Now Lemma C.8 implies
Fn ≤
� 2
1
� 2
1
� 1
0
ψΞ(αγq′
1(t, ε1, ε2)) + αψΣ(q′
2(t, ε1, ε2)) − α
2 q′
1(t, ε1, ε2)q′
2(t, ε1, ε2) dt dε1 dε2 + o(1)
=
� 2
1
� 2
1
� 1
0
G(q′
2(t, ε1, ε2), q′
1(t, ε1, ε2)) dt dε1 dε2 + o(1),
where
G(qu, qv) := ψΞ(αγqv) + αψΣ(qu) − α
2 quqv.
With the choice of R1 in (C.45) and R2 = ∆, the ODE in Lemma C.7 gives
q′ | [
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] |
2 quqv.
With the choice of R1 in (C.45) and R2 = ∆, the ODE in Lemma C.7 gives
q′
1(t, ε1, ε2) = 2αψ′
Σ(q′
2(t, ε1, ε2)),
which corresponds to the criticality condition of G with respect to qu:
∂1G(q′
2(t, ε1, ε2), q′
1(t, ε1, ε2)) = 0.
Since ψΣ is convex and − α
2 quqv is linear in q2, we have that G is convex in qu. Therefore
G(q′
2(t, ε1, ε2), q′
1(t, ε1, ε2)) = inf
qu≥0 G(qu, q′
1(t, ε1, ε2)) = inf
qu≥0 F(qu, q′
1(t, ε1, ε2)) ≤ sup
qv≥0
inf
qu≥0 F(qu, qv),
which completes the proof.
D | [
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0.011638377793133259,
-0.2125926911830902,
-0.14623433351516724,
-0.9027814269065857,
-0.5751430988311768,
0.4542285203933716,
0.34025710821151733,
-0.02767152152955532,
-0.4261467158794403,
0.32606542110443115,
0.0964641273021698,
0.2661784887313843,
0.42377933859825134,
0.10002677142620087,
0.058322131633758545,
-0.04259200766682625,
0.5715224742889404,
-0.7633285522460938,
-0.05620061978697777,
0.24800889194011688,
0.34063994884490967,
-0.07959575206041336,
-0.5047749876976013,
-0.829657256603241,
-0.28402578830718994,
-0.37475520372390747,
1.0212664604187012,
0.22103121876716614,
0.4814278185367584,
-0.3354185223579407,
-0.8822492957115173,
-0.6311887502670288,
0.37399423122406006,
0.4235575795173645,
0.21524693071842194,
-0.1647835671901703,
-0.30551373958587646,
-0.34310561418533325,
0.4437996745109558,
-0.06969620287418365,
-0.4186190366744995,
0.022067267447710037,
0.3706693947315216
] |
qu≥0 F(qu, q′
1(t, ε1, ε2)) ≤ sup
qv≥0
inf
qu≥0 F(qu, qv),
which completes the proof.
D
Proofs of Theorem 4.2 and Corollary 4.3
D.1
Proof of (4.7)
We compute the derivative of Fn(γ):
F′
n(γ) = 1
nE
�Z′
n(γ)
Zn(γ)
�
= 1
nE
��
1
2√γn �u⊤Z�v + 1
n �u⊤�u∗�v⊤�v∗ − 1
2n �u⊤�u�v⊤�v
�
n
�
=
1
2n2 E
��
�u⊤�u∗�v⊤�v∗�
n
�
,
(D.1)
where the last step follows similar calculations in (C.16). Since Fn(γ) → supqv infqu F(qu, qv) as
n → ∞, we have F′
n(γ) →
∂
∂γ supqv infqu F(qu, qv). To compute the RHS, note that
sup
qv≥0 | [
-0.49739575386047363,
-0.2992528975009918,
0.720349133014679,
-0.38973915576934814,
-0.45750683546066284,
0.45162567496299744,
0.4908013343811035,
-0.7110274434089661,
0.054476846009492874,
-0.5744495987892151,
-0.15960538387298584,
0.3897711932659149,
-0.3471386432647705,
-0.08114385604858398,
0.2745150029659271,
0.6697939038276672,
0.7257147431373596,
-0.1163325309753418,
-0.523739218711853,
-1.0364357233047485,
0.2838820219039917,
-0.18837828934192657,
0.3859347403049469,
-0.2924763858318329,
0.41465815901756287,
-0.29985174536705017,
0.3003079295158386,
-0.22138911485671997,
0.3761664628982544,
0.3477226495742798,
-0.050411272794008255,
0.46973568201065063,
-0.09480378031730652,
-0.3133672773838043,
-0.15153899788856506,
0.16635368764400482,
0.6768056750297546,
0.25594040751457214,
0.6315069794654846,
-0.22250114381313324,
0.38706687092781067,
-0.359623521566391,
0.01574496366083622,
0.47891470789909363,
-0.2908360958099365,
-0.5265169143676758,
-1.112764596939087,
-0.20625010132789612,
0.15366187691688538,
-0.7018333077430725,
-0.3530457019805908,
0.4496771991252899,
0.1650320589542389,
0.14252503216266632,
-0.2820660173892975,
-0.06300584226846695,
-0.373090922832489,
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0.5863773822784424,
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0.3457099199295044,
0.038361962884664536,
0.5232399106025696,
-0.857628345489502,
0.9335582852363586,
-0.02139635756611824,
0.3848258852958679,
-0.3758822977542877,
-0.08049850910902023,
0.3941974639892578,
0.03476300463080406,
-0.2188703566789627,
-0.23504571616649628,
0.06234792619943619,
0.22626277804374695,
-0.4040969908237457,
-0.08526480197906494,
0.2384999841451645,
0.22299566864967346,
-0.0371418260037899,
-0.3521849513053894,
0.19256411492824554,
-0.637603759765625,
0.29410648345947266,
0.1933867633342743,
0.47812536358833313,
0.8152890205383301,
-0.6292361617088318,
0.6118170619010925,
0.33083608746528625,
-0.6945868730545044,
0.1430414766073227,
-0.02567259408533573,
0.492788702249527,
0.6384977102279663,
0.07472480833530426,
-0.07225199788808823,
0.4360864758491516,
0.3368290066719055,
0.10822075605392456,
-0.1985529512166977,
-0.29045602679252625,
-0.6807872653007507,
-0.41730186343193054,
-0.6140559911727905,
-0.5850606560707092,
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0.24468408524990082,
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0.16796092689037323,
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0.09048029035329819,
0.19399797916412354,
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0.34007832407951355,
0.6803304553031921,
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0.5024898052215576,
0.4969969689846039,
-0.33234864473342896,
0.5307948589324951,
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0.6507352590560913,
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0.37264129519462585,
0.023515427485108376,
-0.7771125435829163,
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-0.8414004445075989,
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0.050406575202941895,
0.2835484743118286,
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0.4521373510360718,
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-0.34272897243499756,
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0.28830620646476746,
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0.19704920053482056,
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0.768231987953186,
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0.3430580198764801,
-0.42039743065834045,
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0.08752866089344025,
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0.2765744924545288,
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1.4012227058410645,
0.00068520667264238,
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0.4131242632865906,
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1.079843282699585,
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] |
n → ∞, we have F′
n(γ) →
∂
∂γ supqv infqu F(qu, qv). To compute the RHS, note that
sup
qv≥0
inf
qu≥0 F(qu, qv) = sup
qv≥0
�
ψΞ(αγqv) + inf
qu≥0
�
αψΣ(γqu) − α
2 γquqv
��
,
27
and the value of the infimum does not depend on γ. Therefore, we have
∂
∂γ sup
qv≥0
inf
qu≥0 F(qu, qv) = ψ′
Ξ(αγq∗
v)αq∗
v = αq∗
uq∗
v
2
,
(D.2)
where first equality is by the envelope theorem from [47, Corollary 4] and the last equality follows
since the extremizers q∗
u, q∗
v solve a pair of equations in (4.16). | [
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] |
since the extremizers q∗
u, q∗
v solve a pair of equations in (4.16).
On the other hand, we relate F′
n(γ) to the MMSE (4.2) as follows:
MMSEn(γ) = 1
ndE
����u∗(�v∗)⊤ −
�
�u�v⊤�
n
��2
F
�
= 1
ndE
�
∥�u∗∥2
2∥�v∗∥2
2 +
���
�u�v⊤�
n
��2
F − 2(�u∗)⊤�
�u�v⊤�
n�v∗�
= Tr(Ξ−1)
n
Tr(Σ−1)
d
− 1
ndE
��
�u⊤�u∗�v⊤�v∗�
n
�
,
(D.3)
where the last step is by Nishimori identity (Proposition G.4). Combining the above with (D.1)
and (D.2), we conclude
MMSEn(γ) → E
�
Ξ
−1�
E
�
Σ
−1�
− q∗
uq∗
v,
as claimed.
D.2
Proof of (4.8) | [
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] |
and (D.2), we conclude
MMSEn(γ) → E
�
Ξ
−1�
E
�
Σ
−1�
− q∗
uq∗
v,
as claimed.
D.2
Proof of (4.8)
Recall Y from (4.1) and define for some γ′ ≥ 0,
Y ′ :=
�
γ′
n u∗u∗⊤ + Ξ1/2Z′Ξ1/2,
(D.4)
where Z′ ∈ Rn×n is a symmetric random matrix independent of u∗, v∗ with Z′
i,i
i.i.d.
∼ N(0, 2) and
Z′
i,j
i.i.d.
∼ N(0, 1) for all 1 ≤ i < j ≤ n. By similar derivations as before, the free energy associated
with (Y, Y ′) is given by
Fn(γ, γ′) = 1
nE
�
log
�
Rd
�
Rn exp
�
Hn(Ξ−1/2u, Σ−1/2v)
+ 1
2
�
γ′
n u⊤Ξ−1Y ′Ξ−1u − γ′ | [
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] |
Fn(γ, γ′) = 1
nE
�
log
�
Rd
�
Rn exp
�
Hn(Ξ−1/2u, Σ−1/2v)
+ 1
2
�
γ′
n u⊤Ξ−1Y ′Ξ−1u − γ′
4n(u⊤Ξ−1u)2
�
dP ⊗n(u) dQ⊗n(v)
�
,
where Hn is given in (4.12). Denote by ⟪·⟫n the Gibbs bracket with respect to the corresponding
Hamiltonian. Let
F(γ, γ′) :=
sup
qu,qv≥0
γ′
4 q2
u + αγ
2 quqv − ψ∗
Ξ
�qu
2
�
− αψ∗
Σ
�qv
2
�
,
(D.5)
where f ∗ denotes the monotone conjugate of a convex non-decreasing function f : R≥0 → R; see
Definition G.1. Basic properties of monotone conjugate can be found in [64, §12]. | [
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] |
Definition G.1. Basic properties of monotone conjugate can be found in [64, §12].
The following lemma, proved in Appendix D.3, characterizes the high-dimensional limit of Fn(γ, γ′).
Lemma D.1. For all γ, γ′ ≥ 0,
lim
n→∞ Fn(γ, γ′) = F(γ, γ′),
Let us show how (4.8) can be derived from Lemma D.1.
Proof of (4.8). Let
MMSEu
n(γ, γ′) := 1
n2 E
����u∗(�u∗)⊤ − E
�
�u∗(�u∗)⊤ �� Y, Y ′���2
F
�
.
Following similar derivations as in Appendix D.1, one can verify the following two identities:
∂2Fn(γ, γ′) =
1
4n2 E
�
⟪
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] |
∂2Fn(γ, γ′) =
1
4n2 E
�
⟪
�
�u⊤�u∗�2⟫
n
�
,
MMSEu
n(γ, γ′) = Tr(Ξ−1)2
n2
− 1
n2 E
�
⟪
�
�u⊤�u∗�2⟫
n
�
.
(D.6)
28
Therefore, the MMSE in (4.3) can be written as
MMSEu
n(γ) = lim
γ′↓0 MMSEu
n(γ, γ′) = Tr(Ξ−1)2
n2
− 4 lim
γ′↓0 ∂2Fn(γ, γ′).
By Lemma D.1 and Proposition G.11,
lim sup
n→∞ lim
γ′↓0 ∂2Fn(γ, γ′) ≤ lim
γ′↓0 ∂2F(γ, γ′).
The envelope theorem from [47, Corollary 4] allows us to compute the RHS:
lim
γ′↓0 ∂2F(γ, γ′) = q∗
u
2
4 ,
where (q∗
u, q∗
v) are the maximizer of
sup
qu,qv≥0
α
2 γquqv − ψ∗
Ξ
�qu
2
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] |
u
2
4 ,
where (q∗
u, q∗
v) are the maximizer of
sup
qu,qv≥0
α
2 γquqv − ψ∗
Ξ
�qu
2
�
− αψ∗
Σ
�qv
2
�
=
sup
(qu,qv)∈C(γ,α)
ψΞ(αγqv) + αψΣ(γqu) − αγ
2 quqv,
where the equality is by Proposition G.12. Putting the above together, we have
lim sup
n→∞
1
n2 E
���
�u⊤�u∗�2�
n
�
= lim sup
n→∞ lim
γ′↓0
1
n2 E
�
⟪
�
�u⊤�u∗�2⟫
n
�
≤ q∗
u
2,
lim inf
n→∞ MMSEu
n(γ) ≥ E
�
Ξ
−1�2
− q∗
u
2.
(D.7)
By a symmetric argument, we also have
lim sup
d→∞
lim
γ′↓0
1
d2 E
���
�v⊤�v∗�2�
n
�
≤ q∗
v
2.
(D.8) | [
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] |
By a symmetric argument, we also have
lim sup
d→∞
lim
γ′↓0
1
d2 E
���
�v⊤�v∗�2�
n
�
≤ q∗
v
2.
(D.8)
To find an upper bound on MMSEu
n(γ), note that by (4.7) and (D.3):
(q∗
uq∗
v)2 = lim
n→∞
1
(nd)2 E
��
�u⊤�u∗�v⊤�v∗�
n
�2 ≤ lim inf
n→∞
� 1
n2 E
���
�u⊤�u∗�2�
n
��� 1
d2 E
���
�v⊤�v∗�2�
n
��
.
This combined with (D.7) and (D.8) implies
lim
n→∞
1
n2 E
���
�u⊤�u∗�2�
n
�
= q∗
u
2,
lim
d→∞
1
d2 E
���
�v⊤�v∗�2�
n
�
= q∗
v
2,
which concludes the proof in view of the relation
MMSEu
n(γ) = Tr(Ξ−1)2
n2
− 1
n2 E
��� | [
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0.17536376416683197,
0.4282960891723633,
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0.050133541226387024,
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0.1334766000509262,
0.37091606855392456
] |
�
= q∗
v
2,
which concludes the proof in view of the relation
MMSEu
n(γ) = Tr(Ξ−1)2
n2
− 1
n2 E
���
�u⊤�u∗�2�
n
�
.
D.3
Proof of Lemma D.1
We assume γ = γ′ = 1 by formally absorbing them into P, Q:
�
R
x2 dP(x) =
�
γ′,
�
R
x2 dQ(x) =
γ
√γ′ ,
so that we can drop the dependence on γ, γ′ in notation such as Fn, F. We then truncate P, Q at a
sufficiently large constant K > 0 so that they have bounded supports.
Recall Y from (4.1) and define for r ≥ 0, Y ′′ := √ru∗ + Ξ1/2Z′′ where Z′′ ∼ N(0n, In) is | [
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] |
Recall Y from (4.1) and define for r ≥ 0, Y ′′ := √ru∗ + Ξ1/2Z′′ where Z′′ ∼ N(0n, In) is
independent of everything else. The free energy �Fn associated with (Y, Y ′′) is
�Fn = 1
nE
�
log
�
Rd
�
Rn exp
�
Hn(Ξ−1/2u, Σ−1/2v) + ru⊤Ξ−1u∗ + √ru⊤Ξ−1/2Z′′ − r
2u⊤Ξ−1u
�
dP ⊗n(u) dQ⊗d(v)
�
,
where Hn is given in (4.12). A straightforward adaptation of the proof of Theorem 4.4 yields a
characterization of the limit of �Fn. Let
�F(r) := sup
qv≥0
inf
qu≥0 ψΞ
��
γ′(αqv + r)
�
+ αψΣ
� γ
√γ′ qu
�
− α
2 quqv.
(D.9)
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�F(r) := sup
qv≥0
inf
qu≥0 ψΞ
��
γ′(αqv + r)
�
+ αψΣ
� γ
√γ′ qu
�
− α
2 quqv.
(D.9)
29
Lemma D.2. For all r ≥ 0,
lim
n→∞
�Fn(r) = �F(r).
Proof. To obtain the result, we execute the interpolation argument as in the proof of Theorem 4.4 on
the Hamiltonian of the following interpolating models:
Yt =
�
1 − t
n
u∗v∗⊤ + Ξ1/2ZΣ1/2,
Y u
t =
�
αq1(t)u∗ + Ξ1/2Zu,
Y v
t =
�
q2(t)v∗ + Σ1/2Zv,
Y ′′ = √ru∗ + Ξ1/2Z′′.
All steps in the proof of Theorem 4.4 carry over. | [
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