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Jing Lei. Adaptive global testing for functional linear models. Journal of the American Statistical
Association, 109(506):624–634, 2014.
Jing Lei. Network representation using graph root distributions. The Annals of Statistics, 2021.
Lihua Lei. Unified ℓ2→∞ eigenspace perturbation theory for symmetric random matrices. arXiv
preprint arXiv:1909.04798, 2019.
Yicheng Li, Zixiong Yu, Guhan Chen, and Qian Lin. On the eigenvalue decay rates of a class | [
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] |
Yicheng Li, Zixiong Yu, Guhan Chen, and Qian Lin. On the eigenvalue decay rates of a class
of neural-network related kernel functions defined on general domains. Journal of Machine
Learning Research, 25(82):1–47, 2024.
Vince Lyzinski, Daniel L Sussman, Minh Tang, Avanti Athreya, and Carey E Priebe. Perfect
clustering for stochastic blockmodel graphs via adjacency spectral embedding. Electron. J.
Statist., 2014.
Cong Ma, Kaizheng Wang, Yuejie Chi, and Yuxin Chen. Implicit regularization in nonconvex sta- | [
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] |
Cong Ma, Kaizheng Wang, Yuejie Chi, and Yuxin Chen. Implicit regularization in nonconvex sta-
tistical estimation: Gradient descent converges linearly for phase retrieval and matrix completion.
In International Conference on Machine Learning, pages 3345–3354. PMLR, 2018.
Xueyu Mao, Purnamrita Sarkar, and Deepayan Chakrabarti. Estimating mixed memberships
with sharp eigenvector deviations. Journal of the American Statistical Association, 116(536):
1928–1940, 2021. | [
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] |
1928–1940, 2021.
Alexander Meister. Asymptotic equivalence of functional linear regression and a white noise inverse
problem. The Annals of Statistics, pages 1471–1495, 2011.
Shahar Mendelson and Joseph Neeman. Regularization in kernel learning. Annals of statistics,
2010.
James Mercer. Xvi. functions of positive and negative type, and their connection the theory of
integral equations. Philosophical transactions of the royal society of London. Series A, containing | [
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] |
integral equations. Philosophical transactions of the royal society of London. Series A, containing
papers of a mathematical or physical character, 209(441-458):415–446, 1909.
Ha Quang Minh, Partha Niyogi, and Yuan Yao. Mercer’s theorem, feature maps, and smoothing.
In International Conference on Computational Learning Theory, pages 154–168. Springer, 2006.
13
Leon Mirsky. Symmetric gauge functions and unitarily invariant norms. The quarterly journal of
mathematics, 11(1):50–59, 1960. | [
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] |
mathematics, 11(1):50–59, 1960.
Joris M Mooij, Jonas Peters, Dominik Janzing, Jakob Zscheischler, and Bernhard Schölkopf.
Distinguishing cause from effect using observational data: methods and benchmarks. Journal of
Machine Learning Research, 17(32):1–102, 2016.
Krikamol Mu, Bharath Sriperumbudur, Kenji Fukumizu, Arthur Gretton, Bernhard Schölkopf,
et al. Kernel mean shrinkage estimators. Journal of Machine Learning Research, 17(48):1–41,
2016. | [
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] |
et al. Kernel mean shrinkage estimators. Journal of Machine Learning Research, 17(48):1–41,
2016.
Warwick Nash, Tracy Sellers, Simon Talbot, Andrew Cawthorn, and Wes Ford. Abalone. UCI
Machine Learning Repository, 1995. DOI: https://doi.org/10.24432/C55C7W.
Andrew Ng, Michael Jordan, and Yair Weiss. On spectral clustering: Analysis and an algorithm.
Advances in neural information processing systems, 14, 2001.
Serge Nicaise. Jacobi polynomials, weighted sobolev spaces and approximation results of some | [
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] |
Serge Nicaise. Jacobi polynomials, weighted sobolev spaces and approximation results of some
singularities. Mathematische Nachrichten, 213(1):117–140, 2000.
Cheng Soon Ong, Xavier Mary, Stéphane Canu, and Alexander J Smola. Learning with non-positive
kernels. In Proceedings of the twenty-first international conference on Machine learning, page 81,
2004.
Dmitrii M Ostrovskii and Alessandro Rudi. Affine invariant covariance estimation for heavy-tailed | [
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] |
Dmitrii M Ostrovskii and Alessandro Rudi. Affine invariant covariance estimation for heavy-tailed
distributions. In Conference on Learning Theory, pages 2531–2550. PMLR, 2019.
David Pollard. Empirical processes: theory and applications. Ims, 1990.
Ali Rahimi and Benjamin Recht. Random features for large-scale kernel machines. Advances in
neural information processing systems, 20, 2007.
Ali Rahimi and Benjamin Recht. Weighted sums of random kitchen sinks: Replacing minimization | [
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] |
Ali Rahimi and Benjamin Recht. Weighted sums of random kitchen sinks: Replacing minimization
with randomization in learning. Advances in neural information processing systems, 21, 2008.
Lorenzo Rosasco, Mikhail Belkin, and Ernesto De Vito. On learning with integral operators.
Journal of Machine Learning Research, 11(2), 2010.
Sam T Roweis and Lawrence K Saul. Nonlinear dimensionality reduction by locally linear embedding.
science, 290(5500):2323–2326, 2000. | [
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] |
science, 290(5500):2323–2326, 2000.
Patrick Rubin-Delanchy, Joshua Cape, Minh Tang, and Carey E Priebe. A statistical interpretation
of spectral embedding: the generalised random dot product graph. Journal of the Royal Statistical
Society Series B: Statistical Methodology, 84(4):1446–1473, 2022.
Mark Rudelson and Roman Vershynin. Delocalization of eigenvectors of random matrices with
independent entries. Duke Math. J., 2015. | [
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] |
independent entries. Duke Math. J., 2015.
Meyer Scetbon and Zaid Harchaoui. A spectral analysis of dot-product kernels. In International
conference on artificial intelligence and statistics, pages 3394–3402. PMLR, 2021.
Bernhard Schölkopf, Alexander Smola, and Klaus-Robert Müller. Nonlinear component analysis as
a kernel eigenvalue problem. Neural computation, 10(5):1299–1319, 1998.
Tao Shi, Mikhail Belkin, and Bin Yu. Data spectroscopy: Learning mixture models using eigenspaces | [
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] |
Tao Shi, Mikhail Belkin, and Bin Yu. Data spectroscopy: Learning mixture models using eigenspaces
of convolution operators. In Proceedings of the 25th international conference on Machine learning,
pages 936–943, 2008.
Tao Shi, Mikhail Belkin, and Bin Yu. Data spectroscopy: Eigenspaces of convolution operators
and clustering. The Annals of Statistics, pages 3960–3984, 2009.
14
Alex Smola, Zoltán Ovári, and Robert C Williamson. Regularization with dot-product kernels. | [
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] |
14
Alex Smola, Zoltán Ovári, and Robert C Williamson. Regularization with dot-product kernels.
Advances in neural information processing systems, 13, 2000.
Ingo Steinwart and Clint Scovel. Mercer’s theorem on general domains: On the interaction between
measures, kernels, and rkhss. Constructive Approximation, 35:363–417, 2012.
Michel Talagrand. A new look at independence. The Annals of Probability, 24(1):1 – 34, 1996. | [
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] |
Michel Talagrand. A new look at independence. The Annals of Probability, 24(1):1 – 34, 1996.
Minh Tang, Daniel L. Sussman, and Carey E. Priebe. Universally consistent vertex classification
for latent positions graphs. The Annals of Statistics, 41(3):1406 – 1430, 2013. doi: 10.1214/
13-AOS1112. URL https://doi.org/10.1214/13-AOS1112.
Terence Tao and Van Vu. Random matrices: Universality of local eigenvalue statistics. Acta | [
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] |
Terence Tao and Van Vu. Random matrices: Universality of local eigenvalue statistics. Acta
Mathematica, 206(1):127 – 204, 2011. doi: 10.1007/s11511-011-0061-3. URL https://doi.org/
10.1007/s11511-011-0061-3.
Madeleine Udell and Alex Townsend. Why are big data matrices approximately low rank? SIAM
Journal on Mathematics of Data Science, 1(1):144–160, 2019.
Ernesto Araya Valdivia. Relative concentration bounds for the spectrum of kernel matrices. arXiv
preprint arXiv:1812.02108, 2018. | [
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] |
preprint arXiv:1812.02108, 2018.
Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Courna-
peau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der
Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson,
Eric Jones, Robert Kern, Eric Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake
VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, | [
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] |
VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero,
Charles R. Harris, Anne M. Archibald, Antonio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt,
and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in
Python. Nature Methods, 17:261–272, 2020. doi: 10.1038/s41592-019-0686-2.
Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17:395–416, 2007. | [
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] |
Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17:395–416, 2007.
Van Vu and Ke Wang. Random weighted projections, random quadratic forms and random
eigenvectors. Random Structures & Algorithms, 47(4):792–821, 2015.
Daniel E Wagner, Caleb Weinreb, Zach M Collins, James A Briggs, Sean G Megason, and Allon M
Klein. Single-cell mapping of gene expression landscapes and lineage in the zebrafish embryo.
Science, 360(6392):981–987, 2018. | [
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] |
Science, 360(6392):981–987, 2018.
Christopher Williams and Carl Rasmussen. Gaussian processes for regression. Advances in neural
information processing systems, 8, 1995.
Christopher Williams and Matthias Seeger. Using the nyström method to speed up kernel machines.
Advances in neural information processing systems, 13, 2000.
Robert C Williamson, Alexander J Smola, and Bernhard Scholkopf. Generalization performance of | [
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] |
Robert C Williamson, Alexander J Smola, and Bernhard Scholkopf. Generalization performance of
regularization networks and support vector machines via entropy numbers of compact operators.
IEEE transactions on Information Theory, 47(6):2516–2532, 2001.
Jiaming Xu. Rates of convergence of spectral methods for graphon estimation. In International
Conference on Machine Learning, pages 5433–5442. PMLR, 2018.
Yiqiao Zhong and Nicolas Boumal. Near-optimal bounds for phase synchronization. SIAM Journal | [
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] |
Yiqiao Zhong and Nicolas Boumal. Near-optimal bounds for phase synchronization. SIAM Journal
on Optimization, 28(2):989–1016, 2018.
Ding-Xuan Zhou. The covering number in learning theory. Journal of Complexity, 18(3):739–767,
2002.
Huaiyu Zhu, Christopher KI Williams, Richard Rohwer, and Michal Morciniec. Gaussian regression
and optimal finite dimensional linear models. 1997.
15
Appendix
A
Proof of Propositions 1 and 2 | [
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0.41078463196754456,
0.16754268109798431,
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0.26627033948898315,
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0.4197203814983368,
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0.004831977654248476,
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0.08948593586683273,
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0.10591099411249161,
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0.027479559183120728,
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0.4120579659938812,
0.14369304478168488,
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0.2940843105316162,
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0.4182979464530945,
0.1608184427022934,
0.3682391047477722,
0.3973032236099243,
-0.2581078112125397,
-0.15328927338123322,
0.23384371399879456,
-0.34449177980422974,
0.3918793499469757,
0.09430334717035294,
-0.08627533167600632,
0.18499904870986938,
0.42528003454208374,
-0.22494512796401978,
-0.33580055832862854,
0.4178737998008728,
-0.14635832607746124,
0.16091670095920563,
-0.5467308163642883,
0.59944087266922,
0.3966459333896637,
0.7190638184547424,
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0.24118585884571075,
0.4417480230331421,
0.2065327763557434,
0.20029939711093903,
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1.0173203945159912,
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0.1614883989095688,
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0.37265607714653015,
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0.38090404868125916,
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0.09473896026611328,
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0.049465883523225784,
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0.21981146931648254,
0.5738739967346191,
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0.04526422917842865,
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0.4994998872280121,
0.15362094342708588,
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0.16560547053813934,
0.0980234295129776,
-0.12658746540546417,
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0.10133141279220581,
-0.16152386367321014,
-0.3776531219482422,
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0.22224879264831543,
0.28420624136924744,
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0.32658857107162476,
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0.7031554579734802,
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0.08878659456968307,
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] |
and optimal finite dimensional linear models. 1997.
15
Appendix
A
Proof of Propositions 1 and 2
For notational simplicity, we will assume in this section that the eigenvalues and eigenfunctions
are indexed from 0 rather than 1.
A.1
Proof of Proposition 1
We will begin by reducing the problem of verifying our assumptions under ρ to verifying them
under µ, the probability measure associated with the univariate Gaussian distribution N(0, σ2).
Let K : L2
ρ → L2 | [
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] |
Let K : L2
ρ → L2
ρ denote the integral operator associated with the kernel k and the measure ρ, and
let {λi} denote its eigenvalues, arranged in descending order, and {ui} denote their corresponding
eigenfunctions. By the rotation invariance of both k and ρ, the operator K may be written as the
p-fold tensor product
K = K ⊗ · · · ⊗ K
where K : L2
µ → L2
µ denotes the integral operator associated with the kernel k and the univariate | [
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] |
where K : L2
µ → L2
µ denotes the integral operator associated with the kernel k and the univariate
Gaussian measure µ. Let {λi} denote its eigenvalues, arranged in descending order, and {ui}
denote their corresponding eigenfunctions. Then the eigenvalues and eigenfunctions of K and K
are related in the following way (see Shi et al. [2008] or Fasshauer and McCourt [2012]). For every
i, there exists i1, . . . , ip satisfying �p
j=1 ij = i such that
λi =
p
�
j=1
λij
and
ui(x) =
p
�
j=1
uij(xj)
(10) | [
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] |
j=1 ij = i such that
λi =
p
�
j=1
λij
and
ui(x) =
p
�
j=1
uij(xj)
(10)
for all x = (x1, . . . , xp)⊤ ∈ Rp. Now suppose that λi = Θ
�
e−βi�
, then
λi =
p
�
j=1
λij ≍
p
�
j=1
e−βij = e−β �
j ij = e−βi,
and suppose that ∥ui∥∞ = O(esi) for some s < β/2. Then
∥ui∥∞ ≤
p
�
j=1
��uij
��
∞ ≲
p
�
j=1
esij = esi
Therefore to prove that (E) hold under ρ, it suffices to show that it holds under µ.
Shi et al. [2008] provide an explicit formula for the eigenvalues and eigenfunctions of K, which is a | [
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] |
refinement of an earlier result of Zhu et al. [1997].
Let υ := 2σ2/ω2 and let Hi(x) be the ith order Hermite polynomial. Then the eigenvalues and
eigenfunctions of K are given by
λi =
�
2
1 + υ + √1 + 2υ
�
υ
1 + υ + √1 + 2υ
�i
ui(x) = (1 + 2β)1/8
√
2ii!
exp
�
− x2
2σ2
√1 + 2β − 1
2
�
Hi
��1
4 + β
2
�1/4 x
σ
�
.
Therefore we have λi = C1e−βi where
β = log
�1 + υ + √1 + 2υ
υ
�
.
16
We will now show that each ui is uniformly bounded. By a change of variables, we can write ui as
ui(x) =
C2
√
2ii!
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ui(x) =
C2
√
2ii!
e−y2Hi(y)
for some y ∈ R. On the other hand, we have the following inequality due to Indritz [1961]. For all
x ∈ R,
e−x2Hi(x) ≤ 1.09
√
2ii!
Therefore ui(x) ≤ 1.09C2 for all x. We can use the fact that Hi(x) is either odd or even to obtain
an analogous lower bound. Therefore ∥ui∥∞ ≤ 1.09C2 for all i, so s = 0 and (E) holds.
We will now show that Γi = 0 for all i ≥ 1 so that (R) holds with b = +∞ and there is no
requirement on the eigengaps ∆i.
Expanding
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] |
requirement on the eigengaps ∆i.
Expanding
�
ui(x)dµ(x), collecting exponential terms and applying a change-of-variables, one can
calculate that
�
ui(x)dµ(x) = C3
� +∞
−∞
e−y2Hi(y)dy.
It is a standard result that e−y2Hi(y) = 0 as long as i ̸= 0 [Gradshteyn and Ryzhik, 2014], and
therefore
���
ui(x)dµ(x)
�� = 0 for all i ≥ 1, and by (10), we have that Γi = 0 for all i ≥ 1.
A.2
Proof of Proposition 2
For any p ≥ 3, dot-product kernels with respect to the uniform distribution on the sphere exhibit | [
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] |
For any p ≥ 3, dot-product kernels with respect to the uniform distribution on the sphere exhibit
the spectral decomposition
k(x, y) =
∞
�
l=0
λ∗
l
Nl
�
m=1
u∗
l,m(x)u∗
l,m(y)
where the eigenfunctions {u∗
l,m} are the mth spherical harmonic of degree l, Nl = 2l+p−2
l
�l+p−3
p−2
�
=
O(lp−2) is the number of harmonics of each degree, and {λ∗
l } are the distinct eigenvalues [Smola
et al., 2000].
The first spherical harmonic is a constant function, and therefore by the orthogonality of the | [
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] |
The first spherical harmonic is a constant function, and therefore by the orthogonality of the
eigenfunctions in L2
ρ,
�
u∗
l,m(x)dρ(x) = 0 for all l ≥ 1, and therefore Γi = 0 for all i ≥ 1. Therefore
(R) holds with s = +∞, and there are no requirements on the eigengaps.
In addition, Lemma 3 of Minh et al. [2006] shows that the supremum-norm of a spherical harmonic
is upper bounded by
∥u∗
l,m∥∞ ≤
�
Nl
|Sp−1| = O
�
i
p−2
2
�
.
(11) | [
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] |
is upper bounded by
∥u∗
l,m∥∞ ≤
�
Nl
|Sp−1| = O
�
i
p−2
2
�
.
(11)
The eigenvalue decay rates are obtained from Propositions 2.3 and 2.4 of Scetbon and Harchaoui
[2021], and given (11), the condition a > (p2 − 4p + 5)/2 ensures that the conditions for (P) are
met.
B
Proof of Equation (4)
We begin this section with two upper bounds on polynomial and exponential series, which we prove
in Section B.1, and which we will use throughout this proof. | [
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] |
in Section B.1, and which we will use throughout this proof.
Lemma 2. Let α > 1, β > 0 and 0 < γ ≤ 1 for fixed constants. Then the following upper bounds
hold:
∞
�
i=d+1
i−α = O
�
d−α+1�
,
∞
�
i=d+1
e−βiγ = O
�
e−βdγd1−γ�
.
17
Throughout this proof, ε > 0 will denote some constant which may change from line to line, and
even within lines.
To show equation (4), we first note that under (P) with d = Ω(n1/α)
n
�
l=d+1
|λl| ≲
n
�
i=d+1
i−α ≲ d−α+1 = O
�
n
−α−1
α
� | [
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] |
n
�
l=d+1
|λl| ≲
n
�
i=d+1
i−α ≲ d−α+1 = O
�
n
−α−1
α
�
where we have used Lemma 2. In addition, under (E) with d > log1/γ(n1/β), we have that
n
�
l=d+1
|λl| ≲
n
�
l=d+1
e−βlγ ≲ e−βdγd1−γ = n−(1+ε) log(1−γ)/γ(n1/β) = O(n−1)
where we have again used Lemma 2. Now, by the triangle inequality we have that
1
n
n
�
l=d+1
����λl
��� ≤
n
�
l=d+1
|λl| +
n
�
l=d+1
�����
�λl
n − λl
�����
and therefore we are left to show that
n
�
l=d+1
�����
�λl
n − λl
����� =
�
O
�
n−(α−1)/α log(n)
�
under (P) with d = Ω
�
n1/α�
;
O | [
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] |
n
�
l=d+1
�����
�λl
n − λl
����� =
�
O
�
n−(α−1)/α log(n)
�
under (P) with d = Ω
�
n1/α�
;
O
�
n−1�
under (E) with d > log1/γ(n1/β).
(12)
To bound (12), we employ a fine-grained concentration bound due to Valdivia [2018]. We begin
with the polynomial hypothesis. The authors only consider the cases that α, r are natural numbers,
since they draw a comparison between between these values and a Sobolev-type notion of regularity. | [
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] |
since they draw a comparison between between these values and a Sobolev-type notion of regularity.
Inspecting their proofs, they treat the cases r = 0 and r ≥ 1 separately, however their proofs follow
through in exactly the same way when the r ≥ 1 case is replaced with r > 0, in order to cover all
values of α > 2r + 1, r ≥ 0. For the r > 0 case, they derive the following result.
Lemma 3. Suppose that the hypotheses (P) holds with r > 0. Then, with overwhelming probability
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�����
�λi
n − λi
����� ≲ B(i, n) log(n)
where
B(i, n) =
i−α+
α
α−1 (r+ 1
2 )n−1/2
if 1 ≤ i ≤ n
α−1
α
1
2r+1 ;
i−α+1+ α−1
α
(r+ 1
2 )n−1/2
if n
α−1
α
1
2r+1 ≤ i ≤ n
1
2r ;
i−α+r+1n−1/2
if n
1
2r ≤ i ≤ n;
Via some rearrangement we can show that
B(i, n) = O
�
i−α�
,
if 1 ≤ i ≤ n
1
2r ,
and by Lemma 2 we have that
⌊n1/2r⌋
�
l=d+1
�����
�λl
n − λl
����� ≲ log(n)
⌊n1/2r⌋
�
l=d+1
i−α ≲ n− α−1
α log(n)
(13)
In addition, we have that
n
�
l=⌈n1/2r⌉
�����
�λl
n − λl
����� ≲ n−1/2 log(n)
n
�
l=⌈n1/2r⌉ | [
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] |
(13)
In addition, we have that
n
�
l=⌈n1/2r⌉
�����
�λl
n − λl
����� ≲ n−1/2 log(n)
n
�
l=⌈n1/2r⌉
i−α+r+1
≲ n−1/2 log(n) ·
�
n
1
2r
�−α+r+2
≲ n−1 log(n)
≲ n− α−1
α
(14)
18
where we have used that 0 < r < (α − 1)/2. Combining (13) with (14) establishes (12) under (P)
assuming r > 0. The case with r = 0 follows analogous fashion so we omit the details.
We now turn to the hypothesis (E). The authors only explicitly derive a result for the case that | [
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] |
We now turn to the hypothesis (E). The authors only explicitly derive a result for the case that
γ = 1, however following through their proof with Lemma 2 to hand, we obtain the following for
the general case that 0 < γ ≤ 1.
Lemma 4. Suppose that the hypotheses E holds with s > 0. Then, with overwhelming probability
�����
�λi
n − λi
����� ≲ e(−β+δ)iγi1−γn−1/2 log(n)
for all 1 ≤ i ≤ n.
Therefore we have that
∞
�
i=d+1
�����
�λi
n − λi
����� ≲ n−1/2 log(n)
n
�
i=d+1
e(−β+s)ii1−γ
≤ n−1/2 log(n)
n
�
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] |
∞
�
i=d+1
�����
�λi
n − λi
����� ≲ n−1/2 log(n)
n
�
i=d+1
e(−β+s)ii1−γ
≤ n−1/2 log(n)
n
�
i=d+1
e−(β/2+ε)iγi1−γ
≲ n−1/2 log(n)
n
�
i=d+1
e−(βiγ/2+ε)
≲ n−1/2 log(n)n−(1/2+ε)
= O(n−1)
where in the second inequality we have used the assumption that s < β/2 and in the fourth we
have used Lemma 2. The case for s = 0 follows similarly, so we omit the details. Then Equation (4)
is established.
B.1
Proof of Lemma 2
To bound the polynomial series, we upper bound it by as integral as
∞
�
i=d+1
i−α ≤
� ∞
d | [
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] |
To bound the polynomial series, we upper bound it by as integral as
∞
�
i=d+1
i−α ≤
� ∞
d
t−αdt = d−α+1
−α + 1 = O
�
d−α+1�
.
To bound the exponential series, we again employ an integral approximation, and upper bound it
as
∞
�
i=d+1
e−βiγ ≤
� ∞
d
e−βtγdt.
We then apply the substitution u = βtγ to obtain
� ∞
d
e−βtγdt = 1
γ
� ∞
βdγ e−uu(1−γ)/γdu = 1
γ Γ
� 1
γ , βdγ
�
where Γ denotes the incomplete Gamma function. We can then use the fact that Γ(s, x) ≤ e−xxs−1
for s > 0 to obtain the upper bound
1
γ Γ
� 1 | [
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0.19646422564983368
] |
for s > 0 to obtain the upper bound
1
γ Γ
� 1
γ , βdγ
�
≤ 1
γ e−βdγ (βdγ)1/γ−1 = β
γ eβdγd1−γ,
from which we can conclude that
∞
�
i=d+1
e−βiγ = O
�
eβdγd1−γ�
,
as required.
19
C
Proof of Lemma 1
The proof of Lemma 1 will make use of the following theorem due to Ledoux [2001] which is a
corollary of Talagrand’s inequality [Talagrand, 1996].
Theorem 3 (Talagrand’s inequality). Let x = (x1, . . . , xn)⊤ be a vector of independent random | [
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] |
Theorem 3 (Talagrand’s inequality). Let x = (x1, . . . , xn)⊤ be a vector of independent random
variables, and let f : Rn → R be a convex 1-Lipschitz function. Then, for all t ≥ 0,
P (|f(x) − M(f)| ≥ t) ≤ 4 exp
�
−t2/16
�
where M(f) denotes the median of f.
It is easy to verify that the map x → ∥πH(x)∥ is convex and 1-Lipschitz, and so by Talagrand’s
inequality we have that
P (|∥πH(x)∥ − M(∥πH(x)∥)| ≥ t) ≤ 4 exp
�
−t2/16
�
.
(15)
For t ≥ 8, we have that
4 exp
�
−(t − 4)2/16
�
≤ 4 exp
�
−t2/32
�
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�
−t2/16
�
.
(15)
For t ≥ 8, we have that
4 exp
�
−(t − 4)2/16
�
≤ 4 exp
�
−t2/32
�
,
so to conclude the proof, it suffices to show that
|M(∥πH(x)∥) − √q| ≤ 4.
(16)
Let E+ denote the event that ∥πH(x)∥ ≥ √q+4 and let E− denote the event that ∥πH(x)∥ ≤ √q−4.
By the definition of a median, (16) is established if we can show that P(E+) < 1/2 and P(E−) < 1/2.
Let ξ be the mean zero random vector such that x = ¯x + ξ, and let P = (pij)1≤i,j≤n be the | [
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] |
Let ξ be the mean zero random vector such that x = ¯x + ξ, and let P = (pij)1≤i,j≤n be the
orthogonal projection matrix onto H. We have that tr P 2 = tr P = �
i pii = q and |pii| ≤ 1.
Furthermore
∥πH(ξ)∥2 − q =
�
1≤i,j≤n
pijξiξj − q = S1 + S2.
where S1 = �n
1=1 pii(ξ2
i − 1) and S2 = �
1≤i̸=j≤n pijξiξj. We now upper bound the expectations
of S2
1 and S2
2 which will be useful later on for bounding the probabilities of E+ and E−. Before we | [
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0.28956156969070435,
-0.14757737517356873,
0.08967065066099167,
-0.11082889884710312,
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0.2150861769914627,
0.5085575580596924,
-0.7959828972816467,
0.2762598991394043,
0.06488170474767685,
0.38175371289253235,
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0.7152478694915771,
0.7984551787376404,
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0.5507901310920715,
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0.1106361597776413,
0.8672321438789368,
-0.2620161473751068,
0.336204469203949,
-0.5360456705093384,
0.6148753762245178,
-0.6525217294692993,
-0.152362659573555,
0.7109664678573608,
0.274852454662323,
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0.1594264656305313,
-0.48192480206489563,
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0.38376572728157043,
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0.6178128719329834,
0.4617762565612793,
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0.8935926556587219,
0.054139453917741776,
0.26194754242897034,
0.5729662179946899,
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0.1439921259880066,
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0.47365450859069824,
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0.5138057470321655,
0.4438605308532715,
0.3688933253288269,
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0.7233952879905701,
0.4484094977378845,
0.3029356896877289,
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-0.5432380437850952,
-0.775012731552124,
0.7532594799995422,
0.07442836463451385,
0.340565025806427,
-0.24391546845436096,
-0.1659265160560608,
-0.20508809387683868,
0.5750836133956909,
-0.37207964062690735,
-0.3958231508731842,
0.06859827786684036,
-0.08461331576108932
] |
1 and S2
2 which will be useful later on for bounding the probabilities of E+ and E−. Before we
do, notice that since ξi ∈ [0, 1], by Popoviciu’s inequality we have that E(ξ2
i ) ≤ 1/4, and therefore
that E(ξ4
i ) ≤ 1/4. Using this, we have that
E
�
S2
1
�
=
n
�
i=1
p2
iiE
�
ξ2
i − 1
�2 ≤
n
�
i=1
p2
ii
�
Eξ4
i + 1
�
≤ 5q
4 ,
(17)
and that
E
�
S2
2
�
= E
�
i̸=j
pijξiξj
2
=
�
i̸=j
pijE(ξ2
i )E(ξ2
j ) ≤ q
16
(18)
To upper bound the probability of E+, we first observe that by assumption
∥πH(X)∥2 = | [
0.011737826280295849,
-0.21361836791038513,
0.20346251130104065,
-0.3252900540828705,
-0.2109881341457367,
0.2481575757265091,
0.2796589434146881,
-0.6449106335639954,
0.07573196291923523,
-0.7426238656044006,
-0.17924253642559052,
0.32250329852104187,
-0.37939149141311646,
-0.011944727040827274,
0.07311079651117325,
0.5352564454078674,
0.45086291432380676,
0.17883558571338654,
-0.78059321641922,
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0.4452984035015106,
0.046639058738946915,
0.4239819347858429,
0.29648658633232117,
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0.37800857424736023,
0.3150082230567932,
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0.4900512993335724,
0.03348526731133461,
0.07797428965568542,
0.11541987955570221,
0.02719523198902607,
-0.22332659363746643,
0.03200776129961014,
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0.07193975895643234,
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0.3816896080970764,
0.07928741723299026,
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0.08411536365747452,
0.29892849922180176,
0.13963401317596436,
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0.2668032646179199,
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0.18418583273887634,
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0.2420036643743515,
1.1713029146194458,
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0.47666865587234497,
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0.18751980364322662,
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0.5533837676048279,
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0.0904095470905304,
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0.11120776832103729,
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0.5322771668434143,
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1.343977689743042,
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0.009173751808702946,
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0.30201318860054016,
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0.06484077870845795,
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0.6905602812767029,
0.04010600969195366,
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0.2926327586174011,
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0.275039404630661,
0.63666170835495,
0.04085657000541687,
0.33228516578674316,
0.09372355788946152,
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0.17530585825443268,
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0.3302893340587616,
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16
(18)
To upper bound the probability of E+, we first observe that by assumption
∥πH(X)∥2 =
��πH( ¯X)
��2 +
��πH(¯ξ)
��2 ≤ 4√q +
��πH(¯ξ)
��2
and therefore we have that
P(E+) = P (∥πH(X)∥ ≥ √q + 4) ≤ P
�
∥πH(X)∥2 ≥ q + 8√q
�
≤ P
�
∥πH(ξ)∥2 ≥ q + 4√q
�
.
Using the definitions of S1 and S2, it follows that
P(E+) ≤ P (S1 + S2 ≥ 4√q) ≤ P (S1 ≥ 2√q) + P (S2 ≥ 2√q)
20
By Markov’s inequality, we have that
P (S1 ≥ 2√q) = P
�
S2
1 ≥ 4q
�
≤ E
�
S2
1
�
4q
≤ 5
16
and similarly that
P (S2 ≥ 2√q) = P
�
S2
2 ≥ 4q
�
≤ E | [
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] |
�
S2
1 ≥ 4q
�
≤ E
�
S2
1
�
4q
≤ 5
16
and similarly that
P (S2 ≥ 2√q) = P
�
S2
2 ≥ 4q
�
≤ E
�
S2
2
�
4q
≤ 1
64.
It therefore follows that P(E+) < 1/2.
We upper bound P(E−) in a similar fashion.
Since
∥πH(x)∥ ≥ ∥πH(ξ)∥, we have that
P (E−) = P(∥πH(x)∥ ≤ √q − 4) ≤ P(∥πH(ξ)∥ ≤ √q − 4) = P(∥πH(ξ)∥2 ≤ q − 8√q + 16).
Again, recalling the definitions of S1 and S2 we have that
P (E−) ≤ P (S1 + S2 ≤ −8√q + 16) ≤ P(S1 ≤ 8 − 4√q) + P(S2 ≤ 8 − 4√q).
As before, applying Markov’s inequality we have that | [
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] |
As before, applying Markov’s inequality we have that
P(S1 ≤ 8 − 4√q) ≤ P(S2
1 ≥ 16q − 8√q) ≤ P
�
S2
1 ≥ 14q
�
≤ E
�
S2
1
�
14q
≤ 5
56
where we have twice used the assumption that q ≥ 16. Similarly
P(S2 ≤ 8 − 4√q) ≤ P
�
S2
2 ≥ 14q
�
≤ E
�
S2
2
�
14q
≤
1
224.
It therefore follows that P (E−) < 1/2, thereby establishing (16) and concluding the proof.
D
Proof of the claim that ∥πJ(¯b)∥ ≤ 2|J|1/4
In this section, we prove the claim made in the proof of Theorem 1 that
∥πJ(¯b)∥ ≤ 2|J|1/4,
(19) | [
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] |
In this section, we prove the claim made in the proof of Theorem 1 that
∥πJ(¯b)∥ ≤ 2|J|1/4,
(19)
with overwhelming probability, which we require in order to invoke Lemma 1. For notational
convenience, we shall assume in an n-dimensional space, rather than an (n − 1) dimensional space
as this is immaterial in our big-O bounds.
Recall that ¯b is a constant vector with entries in [0, 1], and let 1 denote the all-ones vector. Let | [
-0.2372683584690094,
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] |
ξ be some value such that 8a < ξ < b/2, which exists by assumption (R), and let d′ denote the
smallest index such that
|λd′+1| = O
�
n−1/ξ�
.
This implies that under (P), d′ = O
�
n1/ξα�
, and under (E), d′ < log1/γ(n1/ξβ). Clearly d′ ≤ d
and so J ⊂ {d′ + 1, . . . , n}, and therefore
∥πJ(¯b)∥ ≤
��π{d′+1,...,n}(1)
�� .
In addition, we observe that
��π{d′+1,...,n}(1)
��2 = n −
��π{1,...,d′}(1)
��2 .
Since |J| = Ω(n), it will suffice to show that
1
n
��π{1,...,d′}(1)
��2 = 1 − Ω
�
n−1/2−Ω(1)�
(20)
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] |
1
n
��π{1,...,d′}(1)
��2 = 1 − Ω
�
n−1/2−Ω(1)�
(20)
21
with overwhelming probability.
Let �Ud′ and Ud′ denote the n × d′ matrices with entries
�Ud′(i, j) = �uj(i),
Ud′(i, j) = uj(xi)
n1/2
respectively. Then we have
1
n
��π{1,...,d′}(1)
��2 = 1
n
����Ud′ �U ⊤
d′1
���
2
= 1
n
����U ⊤
d′1
���
2
= 1
n
���(�Ud′W)⊤1
���
2
,
where W is an orthogonal matrix which we will define later one. By the triangle inequality, we
have that
1
n
���(�Ud′W)⊤1
���
2
≥ 1
n
��U ⊤
d′1
��2 − 1
n
����Ud′W − U ⊤
d′
���
2
∥1∥2 = 1
n | [
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] |
have that
1
n
���(�Ud′W)⊤1
���
2
≥ 1
n
��U ⊤
d′1
��2 − 1
n
����Ud′W − U ⊤
d′
���
2
∥1∥2 = 1
n
��U ⊤
d′1
��2 −
����Ud′W − U ⊤
d′
���
2
and therefore to show (20), we need to show that
n−1 ��U ⊤
d′1
��2 = 1 − O
�
n−1/2−Ω(1)�
(21)
and
����Ud′W − U ⊤
d′
��� = O
�
n−1/4−Ω(1)�
(22)
with overwhelming probability.
D.1
Bounding (21)
To bound (21), we begin by using the the inequality (c1 + c2)2 ≤ 2(c2
1 + c2
2) to write
n−1 ��U ⊤
d′1
��2 = n−1
d′
�
l=1
� n
�
i=1
ul(xi)
n1/2
�2
=
d′
�
l=1
� n
�
i=1
ul(xi)
n
�2
≥
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] |
n−1 ��U ⊤
d′1
��2 = n−1
d′
�
l=1
� n
�
i=1
ul(xi)
n1/2
�2
=
d′
�
l=1
� n
�
i=1
ul(xi)
n
�2
≥
d′
�
l=1
��
ul(x)dρ(x)
�2
− 2
d′
�
l=1
� n
�
i=1
ul(xi)
n
−
�
ul(x)dρ(x)
�2
.
To bound the first term, we observe that since {ui}∞
i=1 forms an orthonormal basis for L2
ρ(X), we
have that
∞
�
l=1
��
ul(x)dρ(x)
�2
= 1
and therefore
d′
�
l=1
��
ul(x)dρ(x)
�2
= 1 −
∞
�
l=d′+1
��
ul(x)dρ(x)
�2
=: 1 − Γd′+1
By assumption,
Γd′+1 = O
�
λb
d′+1
�
= O
�
n−b/ξ�
= O
�
n−1/2−Ω(1)� | [
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��
ul(x)dρ(x)
�2
=: 1 − Γd′+1
By assumption,
Γd′+1 = O
�
λb
d′+1
�
= O
�
n−b/ξ�
= O
�
n−1/2−Ω(1)�
where we have used the assumption that b > ξ/2, and therefore
d′
�
l=1
��
ul(x)dρ(x)
�2
= 1 − O
�
n−b/ξ�
= 1 − O
�
n−1/2−Ω(1)�
,
as required.
22
Now to bound the second term we use Hoeffding’s inequality to obtain that
�����
n
�
i=1
ul(xi)
n
−
�
ul(x)dρ(x)
����� = O
�
∥ul∥∞
log1/2(n)
n1/2
�
.
Under (P), we have that for all l ≤ d′,
∥ul∥∞ ≲ d′r ≲ nr/ξα = O
�
n1/2ξ�
= O(n1/16−Ω(1)) | [
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] |
n1/2
�
.
Under (P), we have that for all l ≤ d′,
∥ul∥∞ ≲ d′r ≲ nr/ξα = O
�
n1/2ξ�
= O(n1/16−Ω(1))
where we have used that r < (α − 1)/2 ≤ α/2, and that ξ > 8, and under (E)
∥ul∥∞ ≲ esd′γ ≲ es log(n)/ξβ = O
�
n1/2ξ�
= O(n1/16−Ω(1))
where we have used that s < β/2. Therefore, since d′ = O
�
n1/8�
, we have that
d′
�
l=1
� n
�
i=1
ul(xi)
n
−
�
ul(x)dρ(x)
�2
≲ d′ �
n1/16−1/2�2
= O
�
n−3/4�
which is O
�
n−1/2−Ω(1)�
as required.
D.2
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] |
�2
≲ d′ �
n1/16−1/2�2
= O
�
n−3/4�
which is O
�
n−1/2−Ω(1)�
as required.
D.2
Bounding (22)
To bound (22), we begin by defining the d′ × d′ diagonal matrices �Λd′ and Λd′ with diagonal entries
�Λd′(i, i) :=
�λi
n ,
Λd′(i, i) := λi,
respectively, and the n × d′ matrices �Φd′ and Φd′ with entries
�Φd′(i, j) = �λ1/2
j
�uj(i),
Φd′(i, j) = λ1/2
j
uj(i),
respectively. Then in matrix notation we have that
�Φd′ = n1/2 �Ud′ �Λ1/2
d′ ,
Φd′ = n1/2Ud′Λ1/2
d′
Now, we decompose �Ud′Wd′ − Ud′ as
�Ud′Wd′ − n−1/2Ud′ =
� | [
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d′ ,
Φd′ = n1/2Ud′Λ1/2
d′
Now, we decompose �Ud′Wd′ − Ud′ as
�Ud′Wd′ − n−1/2Ud′ =
�
n−1/2 �
�Φd′Wd′ − Φd′
�
+ �Ud′
�
Wd′Λ1/2
d′
− �Λ1/2
d′ Wd′
��
Λ−1/2
and so we have that
����Ud′Wd′ − Ud′
��� ≤ λ−1/2
d′
�
n−1/2 ����Φd′Wd′ − Φd′
���
F +
���Wd′Λ1/2
d′
− �Λ1/2
d′ Wd′
���
�
.
By the construction of d′ we have that λd′ = Ω(n−1/ξ) = Ω(n−1/8) and therefore λ−1/2
d′
= O
�
n1/16�
.
Therefore to show (22), it will suffice to show that
����Φd′Wd′ − Φd′
���
F = O
�
n
3
16 −Ω(1)�
(23)
and
���Wd′Λ1/2
d′
− �Λ1/2
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] |
����Φd′Wd′ − Φd′
���
F = O
�
n
3
16 −Ω(1)�
(23)
and
���Wd′Λ1/2
d′
− �Λ1/2
d′ Wd′
��� = O
�
n− 5
16 −Ω(1)�
.
(24)
To obtain the bound (23), we make use of the following result which is Equation 3.8 of Tang et al.
[2013].
23
Lemma 5. The exists an orthogonal matrix Wd (constructed as in (28)) such that
����ΦdWd − Φd
���
F = O
�
log1/2(n)
|λd| − |λd+1|
�
with overwhelming probability.
Now, let
d⋆ := arg max
i≥d′ {|λi| − |λi+1|} ,
(25)
then by the assumption (R), we have that
|λd⋆| − |λd⋆+1| = Ω(λa | [
-0.01765148900449276,
-0.26880526542663574,
0.4455709755420685,
-0.37677639722824097,
-0.09303390979766846,
0.11941949278116226,
-0.07369916886091232,
-0.9701130390167236,
0.15720567107200623,
-0.3157872259616852,
-0.44087883830070496,
0.562985897064209,
-0.2770880162715912,
0.11424598842859268,
0.08289855718612671,
0.6461483836174011,
0.8702695369720459,
0.25797760486602783,
-0.5422546863555908,
-0.9111800789833069,
0.4844319522380829,
-0.09328802675008774,
0.39981573820114136,
0.3456866145133972,
-0.1800108253955841,
0.1967390775680542,
0.7209970951080322,
-0.07392153888940811,
0.0945386216044426,
0.3671100437641144,
-0.20113740861415863,
0.2449580579996109,
0.3069305121898651,
-0.061772629618644714,
0.37217995524406433,
0.20561063289642334,
0.5559836626052856,
0.3151492476463318,
0.28025421500205994,
0.20719774067401886,
0.07698766142129898,
-0.14992432296276093,
0.31403470039367676,
0.5324931740760803,
-0.32336950302124023,
-0.641415536403656,
-1.4729845523834229,
0.1497383415699005,
0.03238627314567566,
-0.5744333267211914,
-0.23126178979873657,
-0.04502367973327637,
0.2893153429031372,
0.5093714594841003,
-0.21968680620193481,
0.21535839140415192,
-0.6587265133857727,
-0.3144935965538025,
0.45733320713043213,
-0.18387430906295776,
0.27977311611175537,
-0.09212905168533325,
0.16916559636592865,
0.44125795364379883,
0.041262801736593246,
0.9884332418441772,
-0.000820659683085978,
0.1499110460281372,
-0.693989634513855,
-0.06210318207740784,
0.1453733891248703,
0.19875024259090424,
-0.2663378417491913,
-0.06630901247262955,
-0.20272865891456604,
-0.11117541790008545,
-0.5693859457969666,
-0.0952240601181984,
0.3251117169857025,
0.02282048389315605,
-0.1237531378865242,
-0.3552977740764618,
0.4151018261909485,
-0.4734267294406891,
0.150868758559227,
0.41531360149383545,
-0.002850367221981287,
0.41874489188194275,
-0.8510766625404358,
0.36793267726898193,
-0.05369870737195015,
-0.19064678251743317,
-0.09703914821147919,
0.02816501446068287,
0.09539421647787094,
0.732171356678009,
-0.11654414981603622,
-0.16054505109786987,
0.621855616569519,
0.4707757234573364,
-0.13948586583137512,
-0.2768162488937378,
0.053654465824365616,
-0.4205230474472046,
-0.3715982139110565,
-0.654326319694519,
-0.5026256442070007,
-0.3753689229488373,
-0.24031613767147064,
0.2325681746006012,
-0.990408718585968,
0.7663403153419495,
-0.015294766053557396,
-0.7828059196472168,
0.15008732676506042,
-0.016712801530957222,
-0.0627933070063591,
-0.03269181773066521,
0.06733354181051254,
0.6156186461448669,
-0.2003234475851059,
0.6741229295730591,
0.045488391071558,
0.4820765256881714,
0.26732271909713745,
-0.11123475432395935,
0.2665727138519287,
0.8846025466918945,
-0.8471990823745728,
-0.07474054396152496,
0.33594876527786255,
0.6422116756439209,
0.06904232501983643,
0.2503545582294464,
-0.465335488319397,
-0.7943674325942993,
-0.4195197522640228,
-0.3181322515010834,
-0.9506751298904419,
-0.3567703068256378,
-0.2588600516319275,
-0.36792856454849243,
-0.5310466885566711,
0.17398616671562195,
0.27135270833969116,
0.135206937789917,
-0.11653749644756317,
-0.12698543071746826,
-0.3757464587688446,
0.08152806758880615,
-0.16167546808719635,
0.41211551427841187,
-0.7285486459732056,
0.09014950692653656,
-0.4991241991519928,
-0.22572660446166992,
0.032336074858903885,
-0.11495935916900635,
-0.4828997850418091,
0.3796757757663727,
0.231218159198761,
-0.09701238572597504,
-0.11453384160995483,
-0.2463298887014389,
0.40821313858032227,
-0.794679582118988,
0.8460689187049866,
-0.29559704661369324,
-0.015062298625707626,
0.08173432946205139,
-0.40230005979537964,
-0.2858816683292389,
-0.33093446493148804,
0.22868886590003967,
-0.08039151132106781,
-0.24703510105609894,
-0.15123550593852997,
-0.35175737738609314,
0.2486579865217209,
0.28221866488456726,
-0.8184047341346741,
-0.23740287125110626,
-0.025594105944037437,
-0.3279055953025818,
0.6005050539970398,
0.4374801516532898,
0.3424288034439087,
0.20557551085948944,
0.49673885107040405,
0.25496432185173035,
-0.26754218339920044,
-0.30594050884246826,
0.19501854479312897,
-1.0016838312149048,
1.3671396970748901,
0.17381370067596436,
-0.43736010789871216,
0.03824063390493393,
-0.14974138140678406,
0.31427592039108276,
0.061670802533626556,
0.07822159677743912,
0.21456503868103027,
0.5041086673736572,
-0.05588000267744064,
0.3320199251174927,
-0.01680317521095276,
0.32727035880088806,
-0.5399532318115234,
0.425989031791687,
0.21791088581085205,
-0.07421709597110748,
0.2915429174900055,
-0.3794592618942261,
0.189668208360672,
-0.17477022111415863,
0.1221611350774765,
-0.06699458509683609,
0.19430047273635864,
0.1475689709186554,
0.9162886142730713,
-0.027049465104937553,
0.08411535620689392,
0.6249490976333618,
0.12280862033367157,
0.8539267182350159,
0.8912107944488525,
0.2444230318069458,
0.43598368763923645,
-0.802978515625,
-0.3357876241207123,
0.05684953182935715,
0.15931762754917145,
0.23093512654304504,
0.2853038012981415,
0.6440011858940125,
0.1457088440656662,
-0.008058219216763973,
0.3847920894622803,
-0.50843346118927,
0.49031633138656616,
0.3902103006839752,
-0.16153579950332642,
0.09379547089338303,
-0.06773843616247177,
-0.19160372018814087,
-0.09245943278074265,
0.03516634926199913,
-0.07376493513584137,
-0.6605386734008789,
0.15532457828521729,
-0.13871031999588013,
0.4421955347061157,
-0.5511826276779175,
-0.046141818165779114,
0.04208565875887871,
-0.21587343513965607,
-0.3491979241371155,
0.489504337310791,
0.2290201634168625,
-0.23166753351688385,
-0.22210818529129028,
0.2645372748374939,
-0.440529465675354,
-0.294814795255661,
0.2668308913707733,
-1.0459636449813843,
0.022802237421274185,
0.8341212868690491,
0.636039674282074,
0.18352389335632324,
-0.39591678977012634,
-1.112605094909668,
-0.27700722217559814,
-0.5107215642929077,
-0.5651851296424866,
-0.1653732806444168,
0.14563395082950592,
-0.9724849462509155,
-0.5220951437950134,
0.20475061237812042,
0.798520028591156,
0.2981169819831848,
-0.1675274819135666,
-0.4578711986541748,
-0.32040318846702576,
-0.11630440503358841,
-0.3171798288822174,
-0.47657185792922974,
-0.5051766633987427,
-0.36264926195144653,
0.4319652020931244,
-0.16215120255947113,
-0.7031859755516052,
0.2260245829820633,
0.6273307800292969,
-0.3486343324184418,
-0.140056774020195,
0.3749675154685974,
0.10873743146657944,
-0.22031545639038086,
-0.1740991473197937,
-0.14980877935886383,
-0.16828963160514832,
-0.11005210876464844,
-0.837592363357544,
-0.7223188877105713,
0.8666317462921143,
-5.773746490478516,
0.05167356878519058,
-0.655898928642273,
0.4367336630821228,
-0.30408039689064026,
0.0319681353867054,
0.21306395530700684,
0.04084153100848198,
0.1916259080171585,
-0.22984804213047028,
-0.25234395265579224,
-0.1707695573568344,
-0.25681617856025696,
0.33959323167800903,
-0.16412097215652466,
0.41754376888275146,
0.4341270923614502,
-0.0062605938874185085,
0.3577847182750702,
0.3020176589488983,
0.20749889314174652,
-0.14180411398410797,
-0.3565155863761902,
-0.07809560000896454,
0.647882878780365,
-0.0851450264453888,
-0.45012563467025757,
0.436347097158432,
-0.367845743894577,
-0.564686119556427,
-0.10060463100671768,
-0.34350258111953735,
0.05307365953922272,
-0.35453593730926514,
0.3247244954109192,
0.3450838625431061,
0.4134908616542816,
-0.42230111360549927,
-0.09302502870559692,
-0.6131764650344849,
0.3347261846065521,
0.6031472682952881,
-0.3935543894767761,
-0.42008790373802185,
0.18637213110923767,
-0.08600437641143799,
-0.12486684322357178,
0.4010131359100342,
-0.18384547531604767,
0.46526411175727844,
-0.06710809469223022,
-0.091685950756073,
0.08584579080343246,
0.02836931124329567,
-0.18269416689872742,
-0.33289453387260437,
0.061695415526628494,
0.1718910187482834,
-0.1758134812116623,
0.38506922125816345,
0.23443110287189484,
-0.41498345136642456,
0.17472362518310547,
-0.046671412885189056,
-0.3398646116256714,
0.2935628294944763,
-0.4035341441631317,
-0.20664070546627045,
0.3100951910018921,
-0.042041849344968796,
-0.629993736743927,
0.12934881448745728,
0.3423873782157898,
-0.48790696263313293,
0.4450414478778839,
-0.23282940685749054,
-0.04766671732068062,
-0.3839344382286072,
-0.3814065158367157,
-0.08822261542081833,
0.026240050792694092,
-0.6946989893913269,
-0.13048873841762543,
0.54759281873703,
-0.04997444525361061,
-0.19211351871490479,
-0.008301378227770329,
0.18966759741306305,
-0.10992802679538727,
0.39342963695526123,
0.47111776471138,
-0.12028741836547852,
0.47377246618270874,
0.43455061316490173,
0.10804282873868942,
0.30260396003723145,
0.4281970262527466,
-0.3464658260345459,
0.629837691783905,
-0.14872507750988007,
0.4088820517063141,
0.02264448069036007,
-0.14292125403881073,
-0.44993457198143005,
0.20098987221717834,
0.20323194563388824,
-0.25271594524383545,
0.3970780074596405,
0.6487343907356262,
0.03114686906337738,
-0.03742821887135506,
-0.05360293760895729,
-0.7003986835479736,
-0.3702285885810852,
0.9990360736846924,
-0.30114036798477173,
-0.39078453183174133,
-0.10415719449520111,
0.8805975317955017,
-0.0030353963375091553,
-0.2612650990486145,
0.452669233083725,
-0.242840975522995,
-0.314175546169281,
-0.3267384171485901,
-0.11504906415939331,
0.32896676659584045,
0.5248528718948364,
-0.5469497442245483,
-0.11009799689054489,
0.2565672695636749,
0.40950703620910645,
-0.26479852199554443,
-0.002062947256490588,
0.24251513183116913,
-0.4357791543006897,
0.18633969128131866,
0.23216840624809265,
0.3288167417049408,
-0.2690824866294861,
0.5652547478675842,
0.4320509731769562,
-0.23895440995693207,
-0.5772172212600708,
-0.19540007412433624,
0.4560340344905853,
0.08539766073226929,
0.07948099821805954,
-0.5551567673683167,
0.34557926654815674,
0.09130921214818954,
-0.704081654548645,
0.8841338753700256,
-0.22594670951366425,
-0.028292836621403694,
0.3059656620025635,
0.22859498858451843,
0.2926126718521118,
0.40841802954673767,
-0.7042163014411926,
0.3706572353839874,
0.18986274302005768,
-0.38255342841148376,
0.021318668499588966,
0.16567882895469666,
0.325111448764801,
-0.5340381264686584,
-0.3519916236400604,
0.11672283709049225,
0.5842516422271729,
-0.06172394007444382,
-0.12109741568565369,
-0.2506295144557953,
-0.1338529884815216,
0.22827062010765076,
-0.12664610147476196,
-0.01995488815009594,
0.183074951171875,
0.1934623122215271,
-0.1628817319869995,
0.13490666449069977,
-0.348568856716156,
0.07039876282215118,
0.9196414351463318,
0.9818129539489746,
-0.12834443151950836,
-0.28093501925468445,
-0.2692953050136566,
-0.8143482804298401,
-0.16148804128170013,
0.14101412892341614,
-0.05745325982570648,
0.4862770140171051,
-0.12747728824615479,
0.23183032870292664,
-0.17799779772758484,
0.6379714012145996,
0.3468707501888275,
-0.33104240894317627,
-0.2762630581855774,
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0.7432054281234741,
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0.07586672157049179,
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0.18910926580429077,
0.4522402286529541,
-0.3405468463897705,
-0.3289656341075897,
-0.5997981429100037,
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-0.990003764629364,
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-0.48180481791496277,
-0.5083348155021667,
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-0.30776265263557434,
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-0.36485207080841064,
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-0.7609787583351135,
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-0.779657244682312,
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-0.05207545682787895,
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-0.49268245697021484,
0.041059430688619614,
0.0999252125620842,
0.3870525062084198,
-0.12068986147642136,
-0.661552369594574,
0.015739139169454575,
0.12034773081541061,
0.04454505071043968,
0.28651830554008484,
-0.24497941136360168,
0.706614077091217,
0.20449961721897125,
0.21977724134922028,
-0.32998421788215637,
0.09665151685476303,
0.4880863428115845,
-0.08122554421424866,
0.23331525921821594,
-0.4283916652202606,
-0.2776283919811249,
-0.06763342022895813,
0.8720734119415283,
-0.125967875123024,
-0.40901872515678406,
0.4546425938606262,
0.010565051808953285,
-0.29832082986831665,
-0.018925495445728302,
0.08645008504390717,
0.9450515508651733,
0.148112490773201,
0.39303645491600037,
-0.30603745579719543,
0.3515310287475586,
0.09919778257608414,
0.018885213881731033,
-0.34259432554244995,
-0.6231537461280823,
-0.2065909206867218,
-0.45186394453048706,
0.10163191705942154,
0.5075426697731018,
0.1044667437672615,
0.04616670310497284,
-0.01661202497780323,
0.2341534048318863,
-0.07126842439174652,
-0.2340897023677826,
-0.9357906579971313,
-0.6186503767967224,
-0.08606771379709244,
0.6401377320289612,
0.33796393871307373,
-0.01719476841390133,
-0.2048613280057907,
-0.5020685791969299,
-0.24381905794143677,
-0.32775041460990906,
0.03723260387778282,
0.15069818496704102,
0.09907655417919159,
0.05548176169395447,
0.6351752281188965,
0.03341233730316162,
-0.5645747184753418,
-0.20769065618515015,
0.613894522190094,
-0.2862733006477356,
-0.052992578595876694,
0.402201384305954,
-0.46497827768325806,
-0.15002688765525818,
0.5606531500816345,
-0.03192836046218872,
0.1033029705286026,
0.38880494236946106,
-0.020595071837306023,
0.005368147511035204,
0.5563549399375916,
-0.33949020504951477,
-0.6441971063613892,
0.056671302765607834,
-0.2699289917945862,
0.24461716413497925,
0.19626235961914062,
0.11452848464250565,
0.21007880568504333,
0.5058330297470093,
-0.6390810608863831,
0.5113160014152527,
0.10690274089574814,
0.16511264443397522,
-0.04898005723953247,
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0.5013443231582642,
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0.3717709183692932,
0.43291008472442627,
0.031906235963106155,
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] |
i≥d′ {|λi| − |λi+1|} ,
(25)
then by the assumption (R), we have that
|λd⋆| − |λd⋆+1| = Ω(λa
d′) = Ω(n−a/ξ) = Ω
�
n−1/8+Ω(1)�
.
(26)
Using Lemma 5, we obtain that
����Φd′Wd′ − Φd′
���
F ≤
����Φd⋆Wd⋆ − Φd⋆
���
F = O
�
log1/2(n)
|λd⋆| − |λd⋆+1|
�
= O
�
n1/8−Ω(1)�
,
which establishes (23). Obtaining the bound (24) requires some new concepts, so we dedicate it its
own section.
D.3
Bounding (24)
We now turn our attention to the bound (24). We begin by defining H, the reproducing kernel | [
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] |
We now turn our attention to the bound (24). We begin by defining H, the reproducing kernel
Hilbert space associated with the kernel k, and define the operators KH, �KH : H → H given by
KHf =
�
⟨f, kx⟩H kxdρ(x),
�KH = 1
n
n
�
i=1
⟨f, kxi⟩H kxi
(27)
where kx(·) = k(·, x) and ⟨·, ·⟩H is the inner product in H. These operators are known as the
“extension operators” of K and 1
nK, respectively, and it may be shown that each has the same | [
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] |
“extension operators” of K and 1
nK, respectively, and it may be shown that each has the same
eigenvalues, possibly up to zeros, as its corresponding operator (see e.g. Propositions 8 and 9 of
Rosasco et al. [2010]). We will make use of the following concentration inequality for KH − �KH
which is due to De Vito et al. [2005] and appears as Theorem 7 of Rosasco et al. [2010].
Lemma 6 (Theorem 7 of Rosasco et al. [2010]). The operators KH and �KH are Hilbert-Schmidt,
and
��� �KH − KH
���
HS = O
�
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] |
and
��� �KH − KH
���
HS = O
�
log1/2(n)
n1/2
�
with overwhelming probability.
Let {uH,i}i≤d denote the eigenfunctions of KH corresponding to the eigenvalues {λi}i≤d, and let
{�uH,i}i≤d denote the eigenfunctions of �K corresponding to the eigenvalues {�λi/n}i≤d. We define
the “matrices” UH,d and �UH,d whose columns contain the eigenfunctions {uH,i}i≤d and {�uH,i}i≤d,
respectively. These “matrices” are well-defined with matrix multiplication is defined as usual with | [
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0.04820101335644722
] |
respectively. These “matrices” are well-defined with matrix multiplication is defined as usual with
inner products taken in H. We will refer to UH,d, �UH,d and the subspaces spanned by their columns
interchangably.
Let Hd = U ⊤
H,d �UH,d with entries Hd(i, j) = ⟨uH,i, �uH,j⟩H and denote its singular values by ξ1, . . . , ξd.
Then the principal angles between the subspaces UH,d and �UH,d, which we will denote by θ1, . . . , θd,
are define as via ξi = cos(θi). Define the matrix
sin Θ
�
�UH,d, UH,d
� | [
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] |
are define as via ξi = cos(θi). Define the matrix
sin Θ
�
�UH,d, UH,d
�
= diag (sin(θ1), . . . , sin(θd)) .
24
Let d⋆ be as in (25) so that |λd⋆| − |λd⋆+1| = Ω
�
n−1/8+Ω(1)�
. Since d⋆ ≥ d′, by the Davis-Kahan
theorem, we have that
���sin Θ
�
�UH,d′, UH,d′
���� ≤
���sin Θ
�
�UH,d⋆, UH,d⋆
���� ≤
√
2
��� �KH − KH
���
λd⋆ − λd⋆+1
≤
√
2
��� �KH − KH
���
HS
λd⋆ − λd⋆+1
= O
�
n−3/8−Ω(1)�
.
where the second inequality comes from the relationship ∥·∥ ≤ ∥·∥HS, and the final inequality
follows from Lemma 6 and (26). | [
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] |
follows from Lemma 6 and (26).
We now come to constructing the matrix Wd. We denote the singular value decomposition of H as
H = Wd,1ΞW ⊤
d,2 and define the matrix Wd by
W = Wd,1W ⊤
d,2,
(28)
which is known as the “matrix sign” of H. Let �Pd and Pd to be the projections onto the subspaces
�UH,d and UH,d, respectively. Then we have the following decomposition.
Wd′Λ1/2
d′
− �Λ1/2
d′ Wd′ = (Wd′ − Hd′)�Λd′ + Λd′(Hd′ − Wd′)
+ U ⊤
H,d′( �KH − KH)( �Pd′ − Pd′)�UH,d′
+ U ⊤
H,d′( �KH − KH)UH,d′Hd′
(29) | [
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] |
+ U ⊤
H,d′( �KH − KH)( �Pd′ − Pd′)�UH,d′
+ U ⊤
H,d′( �KH − KH)UH,d′Hd′
(29)
We first observe that ∥Hd′∥ ≤ ∥�UH,d′∥∥UH,d′∥ = 1, and by (4),
����Λd′
��� , ∥Λd′∥ = O(1). In addition,
following the same steps as in the proof Lemma 6.7 of Cape et al. [2019] (who prove similar
(in)equalities for finite-dimensional matrices) we have that
��� �Pd′ − Pd′
��� =
���sin Θ
�
�UH,d′, UH,d′
���� = O
�
n−3/8−Ω(1)�
∥Hd′ − Wd′∥ ≤
���sin Θ
�
�UH,d′, UH,d′
����
2
= O
�
n−3/4−Ω(1)�
.
We now turn to bounding ∥U ⊤ | [
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∥Hd′ − Wd′∥ ≤
���sin Θ
�
�UH,d′, UH,d′
����
2
= O
�
n−3/4−Ω(1)�
.
We now turn to bounding ∥U ⊤
H,d( �KH − KH)UH,d∥. To condense notation, let Q = U ⊤
H,d( �KH −
KH)UH,d. We will bound ∥Q∥ using a classical ε-net argument. Let Sd−1
ε
be an ε-net of the
(d − 1)-dimensional unit sphere Sd−1 := {v : ∥v∥ = 1}, that is, a subset of Sd−1 such that for any
v ∈ Sd−1, there exists some wv ∈ Sd−1
ε
such that ∥v − wv∥ < ε. Then we have that
∥Q∥ =
max
v:∥v∥≤1
��v⊤Qv
��
=
max
v:∥v∥≤1
���(v − wv + wv)⊤ Q (v − wv + wv) | [
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] |
∥Q∥ =
max
v:∥v∥≤1
��v⊤Qv
��
=
max
v:∥v∥≤1
���(v − wv + wv)⊤ Q (v − wv + wv)
���
≤
�
ε2 + 2ε
�
∥Q∥ + max
w∈Sd−1
ε
��w⊤Qw
�� .
With ε = 1/3, we have
∥Q∥ ≤ 9
2 max
w∈Sd−1
ε
��w⊤Qw
�� .
Using the definitions (27), its (l, m)th entry can be calculated as
Q(l, m) = 1
n2
n
�
i,j=1
k(xi, xj)ul(xi)um(xj) −
��
k(x, y)ul(x)um(y)dρ(x)dρ(y),
(30)
25
and so for a given w ∈ Sd−1
ε
, we have that
��w⊤Qw
�� =
������
d
�
l,m=1
Q(l, m)w(l)w(m)
������
=
������
d
�
l,m=1
w(l)w(m)
1
n2
n
�
i,j=1
k(xi, xj)ul(xi)um(xj) −
�� | [
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] |
Q(l, m)w(l)w(m)
������
=
������
d
�
l,m=1
w(l)w(m)
1
n2
n
�
i,j=1
k(xi, xj)ul(xi)um(xj) −
��
k(x, y)ul(x)um(y)dρ(x)dρ(y)
������
≤ max
1≤l≤d ∥ul∥∞
������
n
�
i=1
d
�
l,m=1
w(l)w(m)
�um(xi)
n
−
�
um(x)dρ(x)
�
������
where we have used that k(x, y) ≤ 1 for all x, y ∈ X. This is a sum of independent random
variables, so by Hoeffding’s inequality, we that that
P
���w⊤Qw
�� ≥ t
�
≤ 2 exp
�
−
2nt2
max1≤l≤d ∥ul∥4
∞
�
where we have used that �d
l,m=1 w(l)w(m) = 1. The set Sd−1 | [
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] |
�
≤ 2 exp
�
−
2nt2
max1≤l≤d ∥ul∥4
∞
�
where we have used that �d
l,m=1 w(l)w(m) = 1. The set Sd−1
1/3 can be selected so that its cardinality
is no greater than 18d (see, for example, Pollard [1990]), so using a union bound we have that
P (∥Q∥ ≥ t) ≤ P
�
max
w∈Sd−1
1/3
��w⊤Qw
�� ≥ 2
9t
�
≤
�
w ∈ Sd−1
1/3 P
���w⊤Qw
�� ≥ 2
9t
�
≤ 2 · 18d′ exp
�
−
8nt2
81 · max1≤l≤d′ ∥ul∥4
∞
�
≤ 2 exp
�
d′ log(18) −
8nt2
81 · max1≤l≤d′ ∥ul∥4
∞
�
≤ 2 exp
�
C
�
n1/8−Ω(1) − n3/4t2�� | [
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∞
�
≤ 2 exp
�
d′ log(18) −
8nt2
81 · max1≤l≤d′ ∥ul∥4
∞
�
≤ 2 exp
�
C
�
n1/8−Ω(1) − n3/4t2��
where we have used that d′ = O(n1/8−Ω(1)) and max1≤l≤d′ ∥ul∥∞ = O(n1/16). Choosing t =
2n−5/16−Ω(1) we have that
P
�
∥Q∥ ≥ 2n−5/16−Ω(1)�
≤ 2 exp
�
−n1/8�
≤ n−c
for any c > 0 for large enough n. Therefore
∥Q∥ = O
�
n−5/16−Ω(1)�
with overwhelming probability.
Combining the above bounds with (29) we obtain that
���Wd′Λ1/2
d′
− �Λ1/2
d′ Wd′
��� = O
�
n−5/16−Ω(1)�
which establishes (24) and therefore completes the proof. | [
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] |
d′
− �Λ1/2
d′ Wd′
��� = O
�
n−5/16−Ω(1)�
which establishes (24) and therefore completes the proof.
E
Additional details about the experiments
In this section, we provide some additional details and plots relating to the experiments in Section 5.
26
0
200
400
600
800
1000
0
25
50
75
100
Frobenius
norm error
GMM
0
1000
2000
3000
4000
0
200
400
600
800
1000
Abalone
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4000
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0
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rank
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Frobenius
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] |
0
2000 4000 6000 8000 10000
rank
0
200
400
Frobenius
norm error
MNIST
0
2500
5000
7500
10000
rank
0
20
40
60
80
20 Newsgroups
0
2000
4000
6000
rank
0
20
40
60
80
Zebrafish
= 1/2
= 3/2
= 5/2
=
Figure 3: The Frobenius-norm error against rank for low-rank approximations of kernel matrices
constructed from a collection of datasets. The kernel matrices are constructed using Matérn kernels
with a range of smoothness parameters, each of which is represented by a line in each plot. Details | [
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] |
with a range of smoothness parameters, each of which is represented by a line in each plot. Details
of the experiment are provided in Section 5.
E.1
Details about the datasets
GMM is a synthetic dataset of 1,000 simulated data points from a 10-component Gaussian mixture
model with unit isotropic covariances and means of size ten on the axes. Abalone [Nash et al.,
1995] and Wine Quality [Cortez et al., 2009] are popular benchmark datasets which we standarised | [
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] |
1995] and Wine Quality [Cortez et al., 2009] are popular benchmark datasets which we standarised
in the usual way by centering and rescaling each feature to have unit variance. We drop the binary
Sex feature from the Abalone dataset to retain only the continuous features. MNIST [Deng, 2012]
is a dataset of handwritten digits, represented as 28x28 gray-scale pixels which we concatenate
into 784 dimensional vectors. These four datasets might be described as “low-dimensional”, and | [
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] |
into 784 dimensional vectors. These four datasets might be described as “low-dimensional”, and
are thus representative of the theory we present in Section 3.
In addition, we consider two “high-dimensional” datasets. 20 Newsgroups [Lang, 1995] is a popular
natural language dataset of messages collected from twenty different “newnews” newsgroups. We
remove stop-word and words which appear in fewer than 5 documents or more than 80% of them, | [
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] |
remove stop-word and words which appear in fewer than 5 documents or more than 80% of them,
and convert each document into a vector using term frequency-inverse document frequency (tf-idf)
features for each word. Zebrafish [Wagner et al., 2018] is a dataset of single-cell gene expression in
a zebrafish embryo taken during their first day of development. We subsample 10% of the cells,
and process the data following the steps in Wagner et al. [2018].
E.2
Frobenius norm errors | [
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] |
and process the data following the steps in Wagner et al. [2018].
E.2
Frobenius norm errors
Figure 3 shows the Frobenius norm error of the low-rank approximations. For the low-dimensional
datasets GMM, Abalone, Wine Quality and MNIST, the Frobenius norm error decays very quickly.
However for the high-dimensional datasets 20 Newsgroups and Zebrafish, the Frobenius norm error
decays much more slowly. As pointed out in the main text, the Frobenius norm error of the 20 | [
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] |
decays much more slowly. As pointed out in the main text, the Frobenius norm error of the 20
Newsgroups dataset does not exhibit the sharp drop between d = 2500 and d = 3000 that the
maximum entrywise error exhibits.
27
E.3
Implementation details
The experiments were performed on the HPC cluster at Imperial College London with 8 cores
and 16GB of RAM. The GMM experiment took less than 1 minute; the Abalone, Wine Quality, | [
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] |
and 16GB of RAM. The GMM experiment took less than 1 minute; the Abalone, Wine Quality,
MNIST and Zebrafish experiments took less than 8 hours; and the 20 Newsgroups experiment took
less than 24 hours.
28 | [
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There is HOPE to Avoid HiPPOs for
Long-memory State Space Models
Annan Yu,1∗
Michael W. Mahoney,2,3,4
N. Benjamin Erichson2,3
1 Center for Applied Mathematics, Cornell University
2 Lawrence Berkeley National Laboratory
3 International Computer Science Institute
4 Department of Statistics, University of California at Berkeley
Abstract
State-space models (SSMs) that utilize linear, time-invariant (LTI) systems are known
for their effectiveness in learning long sequences.
However, these models typically face | [
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for their effectiveness in learning long sequences.
However, these models typically face
several challenges: (i) they require specifically designed initializations of the system matrices
to achieve state-of-the-art performance, (ii) they require training of state matrices on a
logarithmic scale with very small learning rates to prevent instabilities, and (iii) they require
the model to have exponentially decaying memory in order to ensure an asymptotically stable | [
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the model to have exponentially decaying memory in order to ensure an asymptotically stable
LTI system. To address these issues, we view SSMs through the lens of Hankel operator
theory, which provides us with a unified theory for the initialization and training of SSMs.
Building on this theory, we develop a new parameterization scheme, called HOPE, for LTI
systems that utilizes Markov parameters within Hankel operators. This approach allows for | [
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systems that utilizes Markov parameters within Hankel operators. This approach allows for
random initializations of the LTI systems and helps to improve training stability, while also
provides the SSMs with non-decaying memory capabilities. Our model efficiently implements
these innovations by nonuniformly sampling the transfer functions of LTI systems, and
it requires fewer parameters compared to canonical SSMs.
When benchmarked against | [
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it requires fewer parameters compared to canonical SSMs.
When benchmarked against
HiPPO-initialized models such as S4 and S4D, an SSM parameterized by Hankel operators
demonstrates improved performance on Long-Range Arena (LRA) tasks. Moreover, we use
a sequential CIFAR-10 task with padded noise to empirically corroborate our SSM’s long
memory capacity.
1
Introduction
State-space models (SSMs) [14] have gained popularity and success in sequence modeling. | [
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] |
State-space models (SSMs) [14] have gained popularity and success in sequence modeling.
Known for its excellent efficiency and capability of handling long sequences, an SSM lever-
ages the continuous-time linear, time-invariant (LTI) systems. These systems are often defined
by four matrices Γ = (A, B, C, D) as
x′(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),
(1)
and they can be used to model the mappings from input time-series u(·) to the output times- | [
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(1)
and they can be used to model the mappings from input time-series u(·) to the output times-
series y(·), where u(t) ∈ Rm and y(t) ∈ Rp for every t. The (hidden) states, which capture
the latent dynamics, are denoted as x = x(t) ∈ Rn. The system matrices are of dimensions
A ∈ Cn×n, B ∈ Cn×m, C ∈ Cp×n, and D ∈ Cp×m. Often, the size n of the state vector x
is much larger than m and p, which allows us to memorize information about the past inputs | [
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] |
is much larger than m and p, which allows us to memorize information about the past inputs
u|(−∞,t] in the state vector x(t) and retrieve it later to compute y via C.
∗Corresponding author: [email protected].
1
arXiv:2405.13975v1 [cs.LG] 22 May 2024
The so-called S4 [14] and S4D [13] models both set m = p = 1, and they differ in the structural
requirement of A. This framework was later generalized to the case where m, p > 1 by the S5 | [
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requirement of A. This framework was later generalized to the case where m, p > 1 by the S5
model [27] via the parallel scans. Another line of research involves making the state transition
rule A depend on the input u, along which the two most notable models are Liquid-S4 [17] and
Mamba [11], where the latter model achieves the state-of-the-art performance on large-scale
real-world datasets.
However, SSMs typically need to be initialized and trained (very) carefully. A randomly ini- | [
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] |
However, SSMs typically need to be initialized and trained (very) carefully. A randomly ini-
tialized SSM has suboptimal performance, but the so-called high-order polynomial projection
operators (HiPPO) [29, 12, 15] can be used to empirically improve it. Subsequent work has con-
sidered stably transforming HiPPO matrices into simplified structures [13, 32]. The empirical
success of the HiPPO framework is traditionally ascribed to interpreting the state vector x(t) | [
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] |
success of the HiPPO framework is traditionally ascribed to interpreting the state vector x(t)
as the projection of the inputs u|(−∞,t] onto some orthogonal polynomial basis. However, such
a claim is not completely satisfiable — indeed, the most popular HiPPO-LegS framework is not
the Legendre-basis projection operator, as it misses the scaling factor 1/t (see [15, Thm. 5.]).
On the other hand, even a properly initialized SSM needs to be trained with care. One of- | [
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] |
On the other hand, even a properly initialized SSM needs to be trained with care. One of-
ten needs to set a smaller learning rate for the matrix A [14], and the LTI systems require
reparameterization to be trained stably [30].
To address these issues, we analyze SSMs through the lens of Henkel Operator theory. Specifi-
cally, we use the Hankel singular value decomposition (HSVD) to analyze an operator defined
by Γ. The decay of the Hankel singular values tells how “complicated” the LTI system is. If | [
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by Γ. The decay of the Hankel singular values tells how “complicated” the LTI system is. If
the Hankel singular values of Γ decay fast, then it informally means that our Γ cannot capture
the complex patterns in the input space {u(·)}; in fact, the theory of reduced-order model-
ing (ROM) says that Γ can be well-approximated by a reduced system with a much smaller
state-space dimension n [9].
We find that the decay of the Hankel singular values can be used for predicting the performance | [
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] |
We find that the decay of the Hankel singular values can be used for predicting the performance
of an SSM, and that every previous effort in proposing a good initialization and training scheme
can be viewed as an effort to avoid fast-decaying Hankel singular values. This is reminiscent of
the works by [22] and [23] that connect the singular values of the weight matrices to a deep neural
network’s performance. In particular, we prove that a random initialization leads to a low-rank | [
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network’s performance. In particular, we prove that a random initialization leads to a low-rank
LTI system (i.e., its Hankel singular values decay rapidly), whereas the HiPPO initialization is
high-rank (see Theorem 1). This serves as an alternative theory with practical consequences and
explains why the HiPPO initialization is preferred over a random one. Moreover, the stability
of the LTI systems depends heavily on the parameters themselves (see Theorem 2). Even a | [
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] |
of the LTI systems depends heavily on the parameters themselves (see Theorem 2). Even a
high-degree LTI system may become a low-degree one by a small amount of perturbation. This
explains why one needs to train A on a logarithmic scale with a reduced learning rate in practice.
Based on these insights, we propose a completely different parametrization of the LTI systems.
Instead of parametrizing the continuous-time systems by matrices A, B and C, we parameterize | [
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] |
Instead of parametrizing the continuous-time systems by matrices A, B and C, we parameterize
our LTI systems by the Markov parameters of the so-called discrete Hankel operator (HOPE).
A discrete Hankel operator is defined by a doubly infinite Hankel matrix, and is naturally
associated with a discrete LTI system, and with a continuous-time system via the bilinear
transform with ∆t = 1 [9].
While this continuous-time LTI system acts on our sequential | [
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] |
transform with ∆t = 1 [9].
While this continuous-time LTI system acts on our sequential
data, the optimization algorithms are applied to the Markov parameters of the Hankel matrix.
(See Figure 1.) In contrast to Theorem 1 and Theorem 2 for a canonical SSM, we prove that
a random Hankel matrix has slowly decaying singular values (see Theorem 3) and a global
stability to perturbation (see Theorem 4). Hence, unlike a canonical SSM, our HOPE-SSM | [
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] |
stability to perturbation (see Theorem 4). Hence, unlike a canonical SSM, our HOPE-SSM
does not need to be carefully initialized by predesigned matrices, and the Markov parameters
of the LTI systems need no reparameterization and enjoy the same learning rate as other
model parameters; hence, less hyperparameter tuning is needed. We show that our HOPE-SSM
can be implemented by nonuniformly sampling the transfer function, which shares the same
2
Continuous LTI System
Discrete LTI System
Bilinear
Transform | [
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