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t =
�
q2(t)v∗ + Σ1/2Zv,
Y ′′ = √ru∗ + Ξ1/2Z′′.
All steps in the proof of Theorem 4.4 carry over.
The above lemma allows us to derive the following auxiliary characterization of Fn.
Lemma D.3.
lim
n→∞ Fn = sup
r≥0
�F(r) − r2
4 .
(D.10)
Proof. Let r: [0, 1] → R≥0 be a differentiable function. For t ∈ [0, 1], consider (Y, Y ′
t , Y ′′
t ) where
Y is given in (4.1) and Y ′
t , Y ′′
t are defined as
Y ′
t =
�
1 − t
n
u∗u∗⊤ + Ξ1/2Z′Ξ1/2,
Y ′′
t =
�
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-0.08940828591585159,
-0.03729235380887985,
0.5926775336265564,
-0.5142711400985718,
-1.0079035758972168,
0.10878518223762512,
-0.6478254199028015,
0.2614690065383911,
-0.14776240289211273,
0.5087035298347473,
0.48823732137680054,
0.9208517074584961,
-0.681429922580719,
0.13681398332118988,
0.0024287051055580378,
-0.12094118446111679,
0.0018146428046748042,
0.04732130840420723,
0.3417930603027344,
0.4011988639831543,
1.183013677597046,
1.0433140993118286,
-0.018317747861146927,
-0.010113422758877277,
0.17964546382427216,
0.06990678608417511,
0.22982709109783173,
-0.2759803533554077,
-0.3520854413509369,
0.31639525294303894,
0.03123016655445099,
-0.16869960725307465,
0.001100040040910244,
-0.21817179024219513,
-0.480092316865921,
0.18010200560092926,
0.02088540606200695,
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0.2287752777338028,
0.053599078208208084,
0.48796799778938293,
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0.2148202359676361,
0.5687176585197449,
0.005333833396434784,
0.04132267087697983,
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0.03062661923468113,
0.26689329743385315,
0.2457287311553955,
0.04187024384737015,
0.3997717499732971,
-0.6740548610687256,
0.12705308198928833,
0.3671764135360718,
-0.6517276763916016,
0.17721278965473175,
0.22041988372802734,
0.5213707685470581,
-0.14991725981235504,
-0.4120182693004608,
-0.052339818328619,
-0.3556496798992157,
0.2097611129283905,
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-0.5309219360351562,
0.49167072772979736,
0.6525834798812866,
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0.2180134803056717,
0.28082799911499023,
0.18610970675945282,
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0.7909078001976013,
1.1085010766983032,
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0.34639471769332886,
0.1065419614315033,
0.5965030193328857,
0.7734667062759399,
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0.3451496362686157,
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0.6855594515800476,
0.4088280498981476,
0.017662638798356056,
0.18175555765628815,
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0.2833731472492218,
0.09205357730388641,
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-0.5896165370941162,
-0.6311934590339661,
-0.8285969495773315,
0.329237163066864,
0.49089935421943665,
0.2781364619731903,
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0.4205809235572815,
0.4064733386039734,
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0.4192292094230652,
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0.794045627117157,
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0.23079559206962585,
0.45083382725715637,
0.3222350776195526,
0.07493844628334045,
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0.20391033589839935,
-0.6009247899055481,
0.5444976091384888,
-0.029547588899731636,
0.452058345079422,
-0.5378547310829163,
-0.676110029220581,
-0.8230810165405273,
0.04203849658370018,
0.24610181152820587,
-0.15925396978855133,
-0.009223666973412037,
-0.2383117526769638,
-0.10163351893424988,
0.15301822125911713,
0.03521481156349182,
-0.36227479577064514,
-0.4240817725658417,
0.45444992184638977
] |
t , Y ′′
t are defined as
Y ′
t =
�
1 − t
n
u∗u∗⊤ + Ξ1/2Z′Ξ1/2,
Y ′′
t =
�
r(t)u∗ + Ξ1/2Z′′,
with Z′′ ∼ N(0n, In) independent of everything else.
Similar to (C.5) and (C.6), denote by
fn(t) := 1
nE
� �
Rd
�
Rn exp
��
1
n �u⊤Z�v + 1
n �u⊤�u∗�v⊤�v∗ − 1
2n∥�u∥2
2∥�v∥2
2
+ 1
2
�
1 − t
n
�u⊤Z′�u + 1 − t
2n (�u⊤�u∗)2 − 1 − t
4n ∥�u∥4
2
+ r(t)�u⊤�u∗ +
�
r(t)�u⊤Z′′ − r(t)
2 ∥�u∥2
2
�
d �P(�u) d �Q(�v)
�
the free energy associated with (Y, Y ′
t , Y ′′
t ) and by ⟪·⟫n,t the conditional expectation with respect to | [
0.2754862606525421,
-0.47504183650016785,
0.20645026862621307,
-0.18690933287143707,
0.007729013916105032,
0.33531466126441956,
0.17535078525543213,
-0.735561192035675,
0.044010963290929794,
-0.14678563177585602,
-0.3300827145576477,
0.5843106508255005,
-0.329280823469162,
0.2897930443286896,
0.4624035358428955,
0.7072950601577759,
0.7380636930465698,
0.1286313682794571,
-0.40083253383636475,
-0.9351929426193237,
0.3482864201068878,
0.09826293587684631,
0.6770810484886169,
-0.014195832423865795,
-0.08100629597902298,
-0.032664936035871506,
0.7189892530441284,
-0.35734155774116516,
0.36060112714767456,
0.40317675471305847,
-0.37843674421310425,
0.5862158536911011,
-0.27886930108070374,
0.25998732447624207,
0.32795941829681396,
0.08621367067098618,
0.3281138241291046,
0.33774706721305847,
0.009669378399848938,
0.03821684420108795,
0.22552113234996796,
-0.30438894033432007,
-0.03278077766299248,
0.43200501799583435,
-0.7660039663314819,
-0.7480197548866272,
-0.9044088125228882,
-0.2900589406490326,
-0.05548232048749924,
-0.6645849347114563,
0.04521888494491577,
0.2276565581560135,
-0.22931772470474243,
0.3986794352531433,
-0.4391239285469055,
0.2271491289138794,
-0.3980170786380768,
-0.5861777067184448,
0.4699614942073822,
-0.4048396050930023,
0.11667371541261673,
0.03166179358959198,
0.32877683639526367,
0.6870207190513611,
0.021725088357925415,
0.8856091499328613,
0.0806736946105957,
-0.10051967203617096,
0.048608459532260895,
-0.37685614824295044,
-0.16910727322101593,
0.2560077905654907,
-0.41593244671821594,
-0.31750187277793884,
-0.2645149528980255,
0.6196624636650085,
-0.31892162561416626,
0.1080726683139801,
0.3229944407939911,
0.18078012764453888,
0.04147729277610779,
-0.14692354202270508,
0.266876757144928,
-0.15912747383117676,
0.20144711434841156,
-0.020556394010782242,
0.018366340547800064,
0.3696008324623108,
-0.6227654218673706,
0.924466073513031,
0.29942160844802856,
-0.09451079368591309,
-0.15271706879138947,
0.2521842420101166,
-0.32960450649261475,
0.9031484723091125,
-0.2202647179365158,
-0.4876376688480377,
0.5192502737045288,
0.1251632571220398,
-0.25530245900154114,
-0.2781580090522766,
-0.346947580575943,
-0.5477551817893982,
-0.32531827688217163,
-0.6608847379684448,
-0.5016391277313232,
-0.7572645545005798,
-0.3645968437194824,
-0.2516522705554962,
-0.996242105960846,
0.6108686923980713,
0.29510384798049927,
-0.5087447166442871,
-0.3154029846191406,
0.03762241080403328,
0.011197282932698727,
-0.17614701390266418,
-0.08342419564723969,
0.4963037669658661,
0.14106881618499756,
0.6947835087776184,
-0.044815849512815475,
0.1849716454744339,
0.44845306873321533,
0.08736559748649597,
0.42922690510749817,
0.6642555594444275,
-0.506489098072052,
-0.2415638566017151,
0.06653672456741333,
0.6005836129188538,
0.0856865644454956,
0.6165962815284729,
-0.5192509293556213,
-0.4953441917896271,
-0.39059877395629883,
-0.8568938970565796,
-0.3184710741043091,
0.09211627393960953,
-0.5802859663963318,
-0.11508242040872574,
-0.17275531589984894,
0.23385395109653473,
0.39560413360595703,
-0.17729595303535461,
-0.2823871076107025,
0.028262855485081673,
0.036248527467250824,
0.3883676826953888,
-0.19639642536640167,
0.00984114222228527,
-0.5411764979362488,
0.29413825273513794,
-0.7540625333786011,
-0.9831617474555969,
0.15337206423282623,
0.46949559450149536,
-0.4111458957195282,
0.4794212579727173,
0.5009733438491821,
0.18604826927185059,
-0.017867516726255417,
-0.2536757290363312,
0.6461291909217834,
-0.6626298427581787,
0.6009791493415833,
-0.0788203552365303,
-0.040247026830911636,
-0.19845469295978546,
-0.4024736285209656,
-0.11992496252059937,
0.06958436965942383,
0.08631706237792969,
-0.8450212478637695,
-0.3092111051082611,
-0.47864726185798645,
-0.01552774477750063,
0.32601162791252136,
-0.021022174507379532,
-0.5801905989646912,
-0.14565730094909668,
0.1257183849811554,
-0.32945340871810913,
0.31330984830856323,
0.3004082441329956,
0.26796144247055054,
-0.003286498598754406,
0.10804512351751328,
0.006438221782445908,
-0.46431127190589905,
0.2004784643650055,
0.20391643047332764,
-1.2431148290634155,
1.118881106376648,
-0.2464991956949234,
-0.18318741023540497,
-0.21706917881965637,
-0.13503921031951904,
0.08548226207494736,
-0.08206744492053986,
-0.4233780801296234,
0.06117196008563042,
0.3293718993663788,
-0.12420948594808578,
0.4814522862434387,
0.16082219779491425,
-0.1447698324918747,
-0.4855667054653168,
0.3714814782142639,
0.35461562871932983,
-0.10906148701906204,
0.3409430682659149,
-0.31417983770370483,
0.3286515772342682,
-0.5312740802764893,
-0.22762644290924072,
0.2926335632801056,
0.19887958467006683,
0.10035447776317596,
0.7943238615989685,
-0.13436686992645264,
0.3592452108860016,
0.7866635322570801,
-0.10215775668621063,
0.7447803020477295,
0.6496521830558777,
0.018534213304519653,
0.40786492824554443,
-0.6770458817481995,
-0.1463458091020584,
-0.3433806300163269,
0.051119495183229446,
0.4733065366744995,
0.7983384132385254,
0.4270738363265991,
-0.041431184858083725,
0.08655034005641937,
0.21063774824142456,
-0.3661002516746521,
0.23852166533470154,
0.8174237608909607,
-0.3173653185367584,
0.4788552224636078,
-0.4811477065086365,
0.10804158449172974,
-0.12221922725439072,
-0.03457842767238617,
-0.31965363025665283,
-0.7081524133682251,
-0.22575578093528748,
-0.09655365347862244,
0.23021893203258514,
-0.37213262915611267,
-0.36372846364974976,
-0.13279429078102112,
0.1922254115343094,
-0.29469403624534607,
0.21262234449386597,
0.2512836456298828,
-0.3780667185783386,
-0.40331825613975525,
0.5062935948371887,
-0.5090835094451904,
-0.1801481395959854,
-0.3923206627368927,
-1.1961694955825806,
-0.21794870495796204,
0.6216551661491394,
0.7050843834877014,
0.6103748083114624,
-0.29924488067626953,
-0.7454133629798889,
-0.17278225719928741,
-0.4827304780483246,
-0.9830873012542725,
-0.04968172311782837,
0.2790461480617523,
-0.843445897102356,
-0.31407248973846436,
0.025631781667470932,
0.7094173431396484,
0.5257304310798645,
-0.07999082654714584,
-0.27738311886787415,
-0.7092898488044739,
-0.16870693862438202,
-0.2291276454925537,
-0.05665598809719086,
-0.5374303460121155,
-0.17365001142024994,
0.3972215950489044,
-0.06266528367996216,
-0.7664171457290649,
-0.009737554006278515,
0.672881543636322,
-0.0444331020116806,
0.17488324642181396,
0.4960140585899353,
0.06279496848583221,
-0.26831039786338806,
-0.40360260009765625,
-0.11112020909786224,
0.08924193680286407,
0.32275593280792236,
-0.593617856502533,
-0.2935342490673065,
0.51899653673172,
-5.674353122711182,
-0.2462422102689743,
-0.29913684725761414,
0.5414373278617859,
0.025314409285783768,
0.19913248717784882,
1.09375,
0.04444597288966179,
-0.2222503274679184,
-0.24452006816864014,
-0.2839502692222595,
-0.38465672731399536,
-0.3904247581958771,
0.45535245537757874,
0.28477922081947327,
-0.009753436781466007,
0.7056193351745605,
0.49591922760009766,
0.6651586890220642,
0.15543943643569946,
0.23503272235393524,
-0.13364535570144653,
-0.3301570415496826,
-0.10980930924415588,
0.5085133910179138,
0.22339588403701782,
-0.5291869044303894,
0.09871988743543625,
-0.3918517529964447,
-0.13173937797546387,
-0.6071358323097229,
-0.588878870010376,
-0.09353770315647125,
-0.5933535695075989,
0.5741245150566101,
0.07450791448354721,
0.2936827540397644,
-0.3106200397014618,
0.036055807024240494,
-0.5473130941390991,
0.4647485017776489,
1.104231357574463,
0.10760010033845901,
-0.34911176562309265,
-0.10671159625053406,
-0.39822572469711304,
0.08398690074682236,
0.49462947249412537,
-0.24178442358970642,
0.41629570722579956,
-0.1978674978017807,
0.14364346861839294,
-0.08460936695337296,
-0.3408626317977905,
-0.2855418026447296,
-0.37729546427726746,
-0.1851491630077362,
0.11215043067932129,
-0.229607954621315,
0.6482124924659729,
0.3916732966899872,
-0.27955663204193115,
0.3476010859012604,
-0.24612188339233398,
-0.4686839282512665,
0.20592853426933289,
-0.4287291467189789,
-0.21431639790534973,
0.7230172753334045,
0.3095303475856781,
-0.6644359230995178,
-0.25011539459228516,
0.21073076128959656,
-0.648349404335022,
0.44943946599960327,
-0.212311789393425,
-0.0540023073554039,
0.17905957996845245,
-0.009854652918875217,
0.09035085886716843,
-0.3442022204399109,
-0.399432897567749,
0.33302032947540283,
0.5990481972694397,
0.345961332321167,
-0.32758963108062744,
-0.36045128107070923,
0.09574238955974579,
-0.5116652250289917,
0.17641742527484894,
0.4255571961402893,
-0.03568593040108681,
0.7719224095344543,
0.22541983425617218,
0.25087133049964905,
0.2138206958770752,
0.25362199544906616,
-0.3559756278991699,
0.3090857267379761,
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0.5728588104248047,
0.30754488706588745,
0.12169329077005386,
-0.5982333421707153,
-0.4622860848903656,
0.4972752630710602,
-0.1690954864025116,
0.5854693055152893,
0.36206167936325073,
-0.1226259395480156,
-0.150276318192482,
-0.3608283996582031,
-0.7061744332313538,
-0.0909012034535408,
0.38850390911102295,
-0.4098716378211975,
-0.8627098798751831,
0.08852783590555191,
0.6017181277275085,
0.06757830828428268,
-0.15300631523132324,
0.0060929241590201855,
-0.6222872734069824,
-0.713169515132904,
-0.37381723523139954,
0.02713450789451599,
0.0868799164891243,
0.03335638344287872,
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-0.1014104038476944,
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0.143435999751091,
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0.308207243680954,
-0.2091018557548523,
0.4397708773612976,
0.1346011608839035,
0.06989934295415878,
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0.5151594877243042,
0.5568903684616089,
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0.06896103173494339,
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0.3709189295768738,
0.11487672477960587,
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] |
t , Y ′′
t ) and by ⟪·⟫n,t the conditional expectation with respect to
the corresponding Gibbs measure. The rest of the proof follows the skeleton of Theorem 4.4 and we
only highlight the differences.
Parallel to Lemma C.1, we have that if r(0) ≥ 0 and limn→∞ r(0) = 0, then
fn(0) = Fn + O(r(0)),
fn(1) = �Fn(r(1)).
The analogue of Lemma C.2 gives
f′
n(t) = −1
4E
�
⟪
� �u⊤�u∗
n
− r′(t)
�2
⟫
n,t
�
+ r′(t)2
4
.
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] |
f′
n(t) = −1
4E
�
⟪
� �u⊤�u∗
n
− r′(t)
�2
⟫
n,t
�
+ r′(t)2
4
.
(D.11)
We now show a pair of matching lower and upper bounds on Fn. First comes the lower bound. Using
(D.11) with r(t) = rt for a constant r ≥ 0, we have
Fn = fn(0) = fn(1) −
� 1
0
f′
n(t) dt ≥ �Fn(r) − r2
4 .
30
By Lemma D.2, this implies the lower bound
lim inf
n→∞ Fn ≥ sup
r≥0
�F(r) − r2
4 .
Next we show a matching upper bound. Let ε ∈ [1, 2] and slightly abusing notation, denote by r(t; ε)
the solution to
r′(t) = 1
nE
�
⟪�u⊤�u∗⟫n,t
�
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] |
the solution to
r′(t) = 1
nE
�
⟪�u⊤�u∗⟫n,t
�
,
r(0) = snε,
where sn = n−1/32. The analogue of Lemma C.7 gives
� 2
1
1
n2 E
�
⟪
�
�u⊤�u∗ − E
��
�u⊤�u∗�
n,t
��2
⟫
n,t
�
dε ≤
C
n1/8 ,
(D.12)
for a constant C > 0 independent of n. Using (D.11) with r(t; ε), we have
Fn =
� 2
1
fn(0) dε + o(1) =
� 2
1
�
�Fn(r(1, ε)) −
� 1
0
r′(t, ε)2
4
dt
�
dε + o(1)
≤
� 2
1
� 1
0
�Fn(r′(t, ε)) − r′(t, ε)2
4
dt dε + o(1) ≤ sup
r≥0
�Fn(r) − r2
4 + o(1), | [
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] |
�
dε + o(1)
≤
� 2
1
� 1
0
�Fn(r′(t, ε)) − r′(t, ε)2
4
dt dε + o(1) ≤ sup
r≥0
�Fn(r) − r2
4 + o(1),
where the second equality is by (D.12) and the penultimate inequality holds since �Fn(·) is convex
and non-decreasing. Passing to the limit, we obtain the upper bound
lim sup
n→∞ Fn ≤ sup
r≥0
�F(r) − r2
4 ,
as desired.
To establish Lemma D.1, it remains to verify that the RHSs of (D.5) and (D.10) are equal. We need
the following lemma. | [
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the following lemma.
Lemma D.4. Let f, g: R≥0 → R be non-decreasing, lower semi-continuous, convex functions with
finite f(0), g(0) and monotone conjugates f ∗, g∗ (see Definition G.1). Then
sup
r≥0
sup
q1≥0
inf
q2≥0 f(q1 + r) + g(q2) − q1q2 − r2
2 =
sup
q1,q2≥0
q2
2
2 + q1q2 − f ∗(q2) − g∗(q1).
Proof. Writing fr(x) = f(x + r) and using Proposition G.12, we have
sup
q1≥0
inf
q2≥0 f(q1 + r) + g(q2) − q1q2 =
sup
q1,q2≥0
q1q2 − f ∗
r (q2) − g∗(q1)
= sup
q1≥0
fr(q1) − g∗(q1) =
sup
q1,q2≥0 | [
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] |
sup
q1,q2≥0
q1q2 − f ∗
r (q2) − g∗(q1)
= sup
q1≥0
fr(q1) − g∗(q1) =
sup
q1,q2≥0
q2(q1 + r) − f ∗(q2) − g∗(q1),
where we have used the fact that f ∗∗ = f. Therefore,
sup
r≥0
sup
q1≥0
inf
q2≥0 f(q1 + r) + g(q2) − q1q2 − r2
2 =
sup
q1,q2≥0
�
sup
r≥0
�
q2r − r2
2
�
+ q1q2 − f ∗(q2) − g∗(q1)
�
=
sup
q1,q2≥0
q2
2
2 + q1q2 − f ∗(q2) − g∗(q1),
as claimed.
Applying Lemma D.4 immediately finishes the proof of Lemma D.1.
Proof of Lemma D.1. By Lemma D.4 and the definition (D.9),
sup
r≥0
�F(r) − r2
4 =
sup
qu,qv≥0
γ′ | [
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] |
Proof of Lemma D.1. By Lemma D.4 and the definition (D.9),
sup
r≥0
�F(r) − r2
4 =
sup
qu,qv≥0
γ′
4 q2
u + α
2 γquqv − ψ∗
Ξ
�qu
2
�
− αψ∗
Σ
�qv
2
�
,
where we have used the fact that for a, b ≥ 0, the monotone conjugate of g(x) := bf(ax) is
g∗(x) = bf ∗(x/(ab)).
31
D.4
Proof of Corollary 4.3
Denote M := E
�
u∗v∗⊤ ��� Y
�
. By the Nishimori identity (Proposition G.4),
lim
n→∞
1
ndE
����u∗(�v∗)⊤ − E
�
�u∗(�v∗)⊤ �� Y
���2
F
�
= E
�
Ξ
−1�
E
�
Σ
−1�
− lim
n→∞
1
ndE
����Ξ−1/2MΣ−1/2���
2
F
�
,
(D.13)
lim
n→∞
1
ndE | [
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] |
���2
F
�
= E
�
Ξ
−1�
E
�
Σ
−1�
− lim
n→∞
1
ndE
����Ξ−1/2MΣ−1/2���
2
F
�
,
(D.13)
lim
n→∞
1
ndE
����u∗v∗⊤ − E
�
u∗v∗⊤ ��� Y
����
2
F
�
= 1 − lim
n→∞
1
ndE
�
∥M∥2
F
�
.
(D.14)
Theorem 4.2 and (D.13) imply that
lim
n→∞
1
ndE
����Ξ−1/2MΣ−1/2���
2
F
�
= q∗
uq∗
v,
(D.15)
which, by Proposition 4.1, is positive if and only if (4.6) holds.
We first show (4.10) assuming (4.6). Using assumption (3.3), super-multiplicativity of σmin(·), and
the fact that ∥BC∥F ≥ ∥B∥Fσmin(C), we have
lim
n→∞
1
ndE
�
∥M∥2
F
�
≥ lim
n→∞ | [
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] |
the fact that ∥BC∥F ≥ ∥B∥Fσmin(C), we have
lim
n→∞
1
ndE
�
∥M∥2
F
�
≥ lim
n→∞
1
ndσn(Ξ1/2)2E
����Ξ−1/2MΣ−1/2���
2
F
�
σd(Σ1/2)2
= (inf supp(Ξ))(inf supp(Σ)) lim
n→∞
1
ndE
����Ξ−1/2MΣ−1/2���
2
F
�
,
which is positive by (D.15). This combined with (D.14) implies (4.10).
We then show (4.10) with < replaced with =, assuming that (4.6) is reversed. Using assumption
(3.3), sub-multiplicativity of spectral norm, and the fact that ∥BC∥F ≤ ∥B∥F∥C∥2, we have
lim
n→∞
1
ndE
�
∥M∥2
F
�
≤ lim
n→∞
1
nd
���Ξ1/2���
2
2E | [
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] |
lim
n→∞
1
ndE
�
∥M∥2
F
�
≤ lim
n→∞
1
nd
���Ξ1/2���
2
2E
����Ξ−1/2MΣ−1/2���
2
F
����Σ1/2���
2
2
= (sup supp(Ξ))(sup supp(Σ)) lim
n→∞
1
ndE
����Ξ−1/2MΣ−1/2���
2
F
�
,
which is 0 by (D.15). This combined with (D.14) finishes the proof of the corollary for estimating
u∗v∗⊤. The proofs for estimating u∗u∗⊤, v∗v∗⊤ are similar and omitted.
E
Analysis of the spectral estimator
E.1
Bayes-AMP
We propose an AMP algorithm that operates on Ξ−1AΣ−1 and maintains a pair of iterates ut ∈ | [
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] |
We propose an AMP algorithm that operates on Ξ−1AΣ−1 and maintains a pair of iterates ut ∈
Rn, vt+1 ∈ Rd for every t ≥ 0. Specifically, given for every t ≥ 0 a pair of denoising functions2
gt : Rn → Rn, ft+1 : Rd → Rd, the iterates are initialized at �u−1 = 0n and some �v0 ∈ Rd of user’s
choice, and are updated for every t ≥ 0 according to the following rules:
ut = Ξ−1AΣ−1�vt − btΞ−1�ut−1,
�ut = gt(ut),
ct = 1
n Tr((∇gt(ut))Ξ−1),
vt+1 = Σ−1A⊤Ξ−1�ut − ctΣ−1�vt,
�vt+1 = ft+1(vt+1),
bt+1 = 1 | [
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] |
ct = 1
n Tr((∇gt(ut))Ξ−1),
vt+1 = Σ−1A⊤Ξ−1�ut − ctΣ−1�vt,
�vt+1 = ft+1(vt+1),
bt+1 = 1
n Tr((∇ft+1(vt+1))Σ−1),
(E.1)
where ∇gt(ut) ∈ Rn×n, ∇ft+1(vt+1) ∈ Rd×d denote the Jacobians of gt, ft+1 at ut, vt+1, respec-
tively. For any fixed t ≥ 0, the n, d → ∞ limit of the iterates ut, vt+1 can be described by a
deterministic recursion known as the state evolution. To define the latter, we need a sequence of
preliminary definitions. | [
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0.1782490313053131,
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] |
preliminary definitions.
2Strictly speaking, for every t ≥ 0, we are given two sequences of functions gt, ft+1 indexed by n, d
respectively. See Definition E.1 for a formal treatment of function sequences.
32
First,
define
random
vectors
(U ∗, WU,0, WU,1, · · · , WU,t)
∈
(Rn)t+2
and
(V ∗, WV,1, WV,2, · · · , WV,t+1) ∈ (Rd)t+2 with joint distributions specified below:
U ∗
σ0WU,0
σ1WU,1
...
σtWU,t
∼ P ⊗n ⊗ N(0n(t+1), Φt ⊗ In),
V ∗
τ1WV,1
τ2WV,2
...
τt+1WV,t+1
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] |
...
σtWU,t
∼ P ⊗n ⊗ N(0n(t+1), Φt ⊗ In),
V ∗
τ1WV,1
τ2WV,2
...
τt+1WV,t+1
∼ Q⊗d ⊗ N(0d(t+1), Ψt+1 ⊗ Id),
(E.2)
where we recall that for A ∈ Rm×n, B ∈ Rp×q, their Kronecker product is
A ⊗ B =
A1,1B
· · ·
A1,nB
...
...
...
Am,1B
· · ·
Am,nB
∈ Rmp×nq.
The covariance matrices Φt = (Φr,s)0≤r,s≤t, Ψt+1 = (Ψr+1,s+1)0≤r,s≤t ∈ R(t+1)×(t+1) are
given by the (t + 1) × (t + 1) principal minors of two infinite-dimensional symmetric matrices | [
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] |
given by the (t + 1) × (t + 1) principal minors of two infinite-dimensional symmetric matrices
Φ := (Φr,s)r,s≥0, Ψ := (Ψr+1,s+1)r,s≥0 whose elements are in turn given recursively below:
Φ0,0 = p-lim
n→∞
1
n(�v0)⊤Σ−1�v0,
Φ0,s = lim
n→∞
1
nE
�
f0(V ∗)⊤Σ−1fs(Vs)
�
,
s ≥ 1,
Φr,s = lim
n→∞
1
nE
�
fr(Vr)⊤Σ−1fs(Vs)
�
,
r, s ≥ 1,
Ψr+1,s+1 = lim
n→∞
1
nE
�
gr(Ur)⊤Ξ−1gs(Us)
�
,
r, s ≥ 0.
(E.3)
Furthermore, for t ≥ 0, σt, τt+1 > 0 are defined as
σ2
0 := Φ0,0 = p-lim
n→∞
1
n(�v0)⊤Σ−1�v0,
σ2
t := Φt,t = lim
n→∞
1
nE
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] |
σ2
0 := Φ0,0 = p-lim
n→∞
1
n(�v0)⊤Σ−1�v0,
σ2
t := Φt,t = lim
n→∞
1
nE
�
ft(Vt)⊤Σ−1ft(Vt)
�
,
t ≥ 1,
τ 2
t+1 := Ψt+1,t+1 = lim
n→∞
1
nE
�
gt(Ut)⊤Ξ−1gt(Ut)
�
,
t ≥ 0.
(E.4)
With the above definitions,
note that each WU,t, WV,t+1
is marginally distributed as
N(0n, In), N(0d, Id), respectively.
Next, define two sequences of deterministic scalars (µt, νt+1)t≥0:
µ0 = lim
n→∞
λ
nE
��
Σ−1V ∗, f0(V ∗)
��
,
µt = lim
n→∞
λ
nE
��
Σ−1V ∗, ft(Vt)
��
,
t ≥ 1,
νt+1 = lim
n→∞
λ
nE
��
Ξ−1U ∗, gt(Ut)
��
,
t ≥ 0,
(E.5) | [
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] |
n→∞
λ
nE
��
Σ−1V ∗, ft(Vt)
��
,
t ≥ 1,
νt+1 = lim
n→∞
λ
nE
��
Ξ−1U ∗, gt(Ut)
��
,
t ≥ 0,
(E.5)
where f0 is determined by the initializer �v0; see Assumption (A1) below.
With these, for t ≥ 0, define random vectors
Ut = µtΞ−1U ∗ + σtΞ−1/2WU,t,
Vt+1 = νt+1Σ−1V ∗ + τt+1Σ−1/2WV,t+1.
(E.6)
Finally, we need the notion of (uniformly) pseudo-Lipschitz functions.
Definition E.1 (Pseudo-Lipschitz functions). A function ϕ: Rk×m → Rℓ×m is called pseudo-
Lipschitz of order j ≥ 1 if there exists L > 0 such that
1
√
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] |
Lipschitz of order j ≥ 1 if there exists L > 0 such that
1
√
ℓ
∥ϕ(x) − ϕ(y)∥F ≤ L
√
k
∥x − y∥F
�
1 +
� 1
√
k
∥x∥F
�j−1
+
� 1
√
k
∥y∥F
�j−1�
,
(E.7)
for every x, y ∈ Rk×m.
33
We will consider sequences of functions ϕi : Rki×m → Rℓi×m indexed by i → ∞ though the index i
is often suppressed. A sequence of functions (ϕi : Rki×m → Rℓi×m)i≥1 (with increasing dimensions
(ki)i≥1, (ℓi)i≥1) is called uniformly pseudo-Lipschitz of order j if there exists a constant L such that
for every i ≥ 1, (E.7) holds. | [
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] |
for every i ≥ 1, (E.7) holds.
The assumptions below are imposed on the initializer �v0 and denoising function (gt, ft+1)t≥0.
(A1) �v0 is independent of �
W but may depend on v∗.3 Assume that
p-lim
d→∞
1
d
���v0��2
2,
p-lim
d→∞
1
n(�v0)⊤Σ−1�v0
exist and are finite. There exists a uniformly pseudo-Lipschitz function f0 : Rd → Rd of
order 1 such that
lim
d→∞
1
dE[⟨f0(V ∗), f0(V ∗)⟩] ≤ p-lim
d→∞
1
d
���v0��2
2
and for every uniformly pseudo-Lipschitz function ϕ: Rd → Rd of finite order, the following | [
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] |
2
and for every uniformly pseudo-Lipschitz function ϕ: Rd → Rd of finite order, the following
two limits exist, are finite and equal:
p-lim
d→∞
1
d
�
�v0, ϕ(V ∗)
�
= lim
d→∞
1
dE[⟨f0(V ∗), ϕ(V ∗)⟩].
Let �ν ∈ R, �τ ∈ R≥0. For any s ≥ 1,
lim
d→∞
1
dE
�
f0(V ∗)⊤Σ−1fs(�νΣ−1V ∗ + �τΣ−1/2WV )
�
exists and is finite, where WV ∼ N(0d, Id) is independent of V ∗.
(A2) Let �ν ∈ R, and T ∈ R2×2 be positive definite. For any r, s ≥ 1,
lim
n→∞
λ
nE
��
Σ−1V ∗, fr(�νΣ−1V ∗ + Σ−1/2N)
��
,
lim
d→∞
1
dE
� | [
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] |
lim
n→∞
λ
nE
��
Σ−1V ∗, fr(�νΣ−1V ∗ + Σ−1/2N)
��
,
lim
d→∞
1
dE
�
fr(�νΣ−1V ∗ + Σ−1/2N)⊤Σ−1fs(�νΣ−1V ∗ + Σ−1/2N ′)
�
exist and are finite, where (N, N ′) ∼ N(02d, T ⊗ Id) is independent of V ∗.
Let �µ ∈ R, and S ∈ R2×2 be positive definite. For any r, s ≥ 0,
lim
n→∞
λ
nE
��
Ξ−1U ∗, gr(�µΞ−1U ∗ + Ξ−1/2M)
��
,
lim
d→∞
1
dE
�
gr(�µΞ−1U ∗ + Ξ−1/2M)⊤Ξ−1gs(�µΞ−1U ∗ + Ξ−1/2M ′)
�
exist and are finite, where (M, M ′) ∼ N(02n, S ⊗ In) is independent of U ∗. | [
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] |
�
exist and are finite, where (M, M ′) ∼ N(02n, S ⊗ In) is independent of U ∗.
We now give the state evolution result for the AMP in (E.1), which is proved in Appendix F.
Proposition E.1 (State evolution for AMP in (E.1)). For every t ≥ 0, let (gt : Rn → Rn)n≥1 and
(ft+1 : Rd → Rd)d≥1 be uniformly pseudo-Lipschitz of finite order subject to Assumption (A2).
Consider the AMP iteration in (E.1) defined by (gt, ft+1)t≥0 and initialized at �u−1 = 0n and | [
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Consider the AMP iteration in (E.1) defined by (gt, ft+1)t≥0 and initialized at �u−1 = 0n and
some �v0 ∈ Rd subject to Assumption (A1). For any fixed t ≥ 0, let (ϕ: R(t+2)n → R)n≥1 and
(ψ: R(t+2)d → R)d≥1 be uniformly pseudo-Lipschitz of finite order. Then,
p-lim
n→∞ ϕ(u∗, u0, u1, · · · , ut) − E[ϕ(U ∗, U0, U1, · · · , Ut)] = 0,
(E.8a)
p-lim
d→∞
ψ(v∗, v1, v2, · · · , vt+1) − E[ψ(V ∗, V1, V2, · · · , Vt+1)] = 0,
(E.8b)
where (Us, Vs+1)0≤s≤t are defined in (E.6). | [
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] |
(E.8b)
where (Us, Vs+1)0≤s≤t are defined in (E.6).
3Practically one can think of the dependence of �v0 on v∗ being given by some side information. However,
here AMP is used solely as a proof technique, and we can consider initializers with impractical access to v∗.
34
Given Ut, Vt+1, the Bayes-optimal (in terms of mean square error) choice of (gt, ft+1)t≥0 is given
by the conditional expectations. Specifically, for any t ≥ 0 and u ∈ Rn, v ∈ Rd,
g∗
t (u) := E[U ∗ | Ut = u],
f ∗
t+1(v) := E[V ∗ | Vt+1 = v]. | [
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] |
g∗
t (u) := E[U ∗ | Ut = u],
f ∗
t+1(v) := E[V ∗ | Vt+1 = v].
(E.9)
We call (E.1) with gt = g∗
t , ft+1 = f ∗
t+1 the Bayes-AMP.
If P = Q = N(0, 1), by (E.2) and (E.6), (U ∗, Ut) and (V ∗, Vt+1) are jointly Gaussian with mean
zero and covariance:
�
In
µtΞ−1
µtΞ−1
µ2
tΞ−2 + σ2
t Ξ−1
�
∈ R2n×2n,
�
Id
νt+1Σ−1
νt+1Σ−1
ν2
t+1Σ−2 + τ 2
t+1Σ−1
�
∈ R2d×2d,
respectively. Therefore using Proposition G.5, g∗
t , f ∗
t+1 admit the following explicit formulas:
g∗
t (u) = µtΞ−1(µ2
tΞ−2 + σ2
t Ξ−1)−1u = µt(µ2
tΞ−1 + σ2 | [
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] |
g∗
t (u) = µtΞ−1(µ2
tΞ−2 + σ2
t Ξ−1)−1u = µt(µ2
tΞ−1 + σ2
t In)−1u,
f ∗
t+1(v) = τt+1Σ−1(τ 2
t+1Σ−2 + τ 2
t+1Σ−1)−1v = νt+1(ν2
t+1Σ−1 + τ 2
t+1Id)−1v.
(E.10)
Under the above choice, the state evolution recursion for µt, σt, νt+1, τt+1 in (E.4) and (E.5) becomes:
for all t ≥ 1,
µt = lim
n→∞
λ
nE
��
Σ−1V ∗, νt(ν2
t Σ−1 + τ 2
t Id)−1Vt
��
= λ
δ E
�
ν2
t Σ
−2
ν2
t Σ
−1 + τ 2
t
�
,
(E.11a)
νt+1 = lim
n→∞
λ
nE
��
Ξ−1U ∗, µt(µ2
tΞ−1 + σ2
t In)−1Ut
��
= λE
�
µ2
tΞ
−2
µ2
tΞ
−1 + σ2
t
�
,
(E.11b)
σ2
t = lim
n→∞
1
nE | [
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] |
tΞ−1 + σ2
t In)−1Ut
��
= λE
�
µ2
tΞ
−2
µ2
tΞ
−1 + σ2
t
�
,
(E.11b)
σ2
t = lim
n→∞
1
nE
�
V ⊤
t (ν2
t Σ−1 + τ 2
t Id)−1νtΣ−1νt(ν2
t Σ−1 + τ 2
t Id)−1Vt
�
= 1
δ E
�
ν4
t Σ
−3
(ν2
t Σ
−1 + τ 2
t )2
�
+ 1
δ E
�
ν2
t τ 2
t Σ
−2
(ν2
t Σ
−1 + τ 2
t )2
�
= 1
δ E
�
ν2
t Σ
−2
ν2
t Σ
−1 + τ 2
t
�
,
(E.11c)
τ 2
t+1 = lim
n→∞
1
nE
�
U ⊤
t (µ2
tΞ−1 + σ2
t In)−1µtΞ−1µt(µ2
tΞ−1 + σ2
t In)−1Ut
�
= E
�
µ4
tΞ
−3
(µ2
tΞ
−1 + σ2
t )2
�
+ E
�
µ2
tσ2
t Ξ
−2
(µ2
tΞ
−1 + σ2
t )2
�
= E
�
µ2
tΞ
−2
µ2
tΞ
−1 + σ2
t
�
,
(E.11d) | [
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] |
−1 + σ2
t )2
�
+ E
�
µ2
tσ2
t Ξ
−2
(µ2
tΞ
−1 + σ2
t )2
�
= E
�
µ2
tΞ
−2
µ2
tΞ
−1 + σ2
t
�
,
(E.11d)
where we have used the definitions (E.6) of Ut, Vt+1, the joint distribution (E.2) of
(U ∗, WU,t), (V ∗, Vt+1), the convergence of the empirical spectral distributions of Σ, Ξ, and Proposi-
tion G.2.
Inspecting the expressions, we realize that
µt = λσ2
t ,
νt+1 = λτ 2
t .
(E.12)
This allows us to only track the recursion of µt, νt+1:
µt = λ
δ E
�
λνtΣ
−2
λνtΣ
−1 + 1
�
,
νt+1 = λE
�
λµtΞ
−2
λµtΞ
−1 + 1
�
. | [
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] |
µt = λ
δ E
�
λνtΣ
−2
λνtΣ
−1 + 1
�
,
νt+1 = λE
�
λµtΞ
−2
λµtΞ
−1 + 1
�
.
Thus, the fixed point (µ∗, ν∗) of the above recursion must satisfy:
µ∗ = λ
δ E
�
λν∗Σ
−2
λν∗Σ
−1 + 1
�
,
ν∗ = λE
�
λµ∗Ξ
−2
λµ∗Ξ
−1 + 1
�
.
(E.13)
Note that upon a change of variable
qu := ν∗
λ ,
qv := δµ∗
λ ,
(E.14)
the fixed point equation (E.13) coincides with that in the characterization of the free energy; see (4.5).
35
E.2
Spectral estimator from Bayes-AMP | [
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] |
35
E.2
Spectral estimator from Bayes-AMP
Under Gaussian priors, the Bayes-AMP algorithm specified by (E.1) and (E.10) naturally suggests a
spectral estimator with respect to a matrix which is a non-trivial transformation of A. In what follows,
we provide a heuristic derivation of this spectral estimator. Its performance guarantee (Theorem 5.1)
will be proved in Appendix E.4.
Suppose, informally, that µt, σt, νt+1, τt+1, ut, vt+1, ct, bt+1 converge (under the sequential limits | [
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] |
n → ∞, t → ∞) to µ∗, σ∗, ν∗, τ ∗, u, v, c∗, b∗, respectively, in the sense that, e.g.,
lim
t→∞ p-lim
n→∞
1
√n
��ut − u
��
2 = 0.
Recall that (µ∗, ν∗) solves the fixed point equation (E.13), and from (E.12), the following relation
holds:
µ∗ = λσ∗2,
ν∗ = λτ ∗2.
(E.15)
So denoting
G := λ(λµ∗Ξ−1 + In)−1 ∈ Rn×n,
F := λ(λν∗Σ−1 + Id)−1 ∈ Rd×d,
(E.16)
by the design of g∗
t , f ∗
t+1 in (E.10), we have that �ut, �vt+1 converge to
�u = µ∗�
µ∗2Ξ−1 + σ∗2In
�−1
u = Gu,
�v = ν∗�
ν∗2Σ−1 + τ ∗2Id
�−1
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] |
�u = µ∗�
µ∗2Ξ−1 + σ∗2In
�−1
u = Gu,
�v = ν∗�
ν∗2Σ−1 + τ ∗2Id
�−1
v = Fv,
respectively, and the limiting Onsager coefficients b∗, c∗ are given by:
b∗ = lim
n→∞
1
n Tr(FΣ−1) = 1
δ E
�
λ
λν∗ + Σ
�
,
c∗ = lim
n→∞
1
n Tr(GΞ−1) = E
�
λ
λµ∗ + Ξ
�
. (E.17)
At the fixed point of (E.1), we have
u = Ξ−1AΣ−1�v − b∗Ξ−1�u = Ξ−1AΣ−1Fv − b∗Ξ−1Gu,
v = Σ−1A⊤Ξ−1�u − c∗Σ−1�v = Σ−1A⊤Ξ−1Gu − c∗Σ−1Fv.
(E.18)
Upon rearrangement, (E.18) is equivalent to
�Gu = Ξ−1AΣ−1Fv,
�Fv = Σ−1A⊤Ξ−1Gu,
(E.19)
where
�G := In + b∗Ξ−1G ∈ Rn×n, | [
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�Gu = Ξ−1AΣ−1Fv,
�Fv = Σ−1A⊤Ξ−1Gu,
(E.19)
where
�G := In + b∗Ξ−1G ∈ Rn×n,
�F := Id + c∗Σ−1F ∈ Rd×d.
(E.20)
We further introduce the notation:
�G := �GG−1 ∈ Rn×n,
�F := �FF −1 ∈ Rd×d,
(E.21)
so that (E.19) can be rewritten as
�GGu = Ξ−1AΣ−1Fv,
�FFv = Σ−1A⊤Ξ−1Gu,
or
�G1/2Gu = �G−1/2Ξ−1AΣ−1 �F −1/2 · �F 1/2Fv,
�F 1/2Fv = �F −1/2Σ−1A⊤Ξ−1 �G−1/2 · �G1/2Gu.
The key observation is that this is a pair of singular vector equations for the matrix
A∗ := �G−1/2Ξ−1AΣ−1 �F −1/2 ∈ Rn×d
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A∗ := �G−1/2Ξ−1AΣ−1 �F −1/2 ∈ Rn×d
(E.22)
with respect to left/right singular vectors (up to rescaling)
�G1/2Gu ∈ Rn,
�F 1/2Fv ∈ Rd
and singular value 1. Using the definitions (E.16), (E.20) and (E.21), we verify that the two expressions
of A∗ in (5.3) and (E.22) are equal.
By the state evolution result (Proposition E.1), u, v behave (in the sense of (E.8)) as
µ∗Ξ−1u∗ + σ∗Ξ−1/2WU,
ν∗Σ−1v∗ + τ ∗Σ−1/2WV ,
for WU ∼ N(0n, Id), WV ∼ N(0d, Id) independent of each other and of u∗, v∗. This suggests that | [
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] |
for WU ∼ N(0n, Id), WV ∼ N(0d, Id) independent of each other and of u∗, v∗. This suggests that
Ξ( �G1/2G)−1u1(A∗),
Σ( �F 1/2F)−1v1(A∗)
(E.23)
are effective estimates of u∗, v∗. Simple algebra reveals that the above vectors, upon suitable rescaling,
are precisely �u, �v in (5.4).
36
E.3
Right edge of the bulk
Before proceeding with the proof of Theorem 5.1, we provide a characterization of σ2(A∗), i.e., the
right edge of the bulk of the spectrum of A∗. This bulk is related to the spectrum of the non-spiked | [
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] |
random matrix
W ∗ := λ(λ(µ∗ + b∗)In + Ξ)−1/2�
W(λ(ν∗ + c∗)Id + Σ)−1/2.
(E.24)
We first present a characterization of σ1(W ∗) and then relate it to σ2(A∗). Define random variables:
Ξ
∗ :=
λ
λ(µ∗ + b∗) + Ξ,
Σ
∗ :=
λ
λ(ν∗ + c∗) + Σ.
(E.25)
Define functions c, s: (sup supp(Ξ
∗), ∞) → (0, ∞) as
c(α) = E
�
Ξ
∗
α − Ξ
∗
�
,
s(α) = sup supp(Σ
∗)c(α).
Define the implicit function β : (sup supp(Ξ
∗), ∞) → (0, ∞) as, for any α ∈ (sup supp(Ξ
∗), ∞),
the unique solution in (s(α), ∞) to
1 = 1
δ E
�
Σ
∗
β − c(α)Σ
∗
�
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] |
∗), ∞),
the unique solution in (s(α), ∞) to
1 = 1
δ E
�
Σ
∗
β − c(α)Σ
∗
�
.
(The existence and uniqueness of the solution is easy to see.) Next, define ψ: (sup supp(Ξ
∗), ∞) →
(0, ∞) as ψ(α) = αβ(α). It is known that ψ is differentiable and the set of its critical points is a
nonempty finite set [75]. Let α◦ ∈ (sup supp(Ξ
∗), ∞) be the largest critical point, i.e.,
1 = 1
δ E
�
Σ
∗2
(β(α) − c(α)Σ
∗)2
�
E
�
Ξ
∗2
(α − Ξ
∗)2
�
Finally, denote
σ∗
2 :=
�
ψ(α◦).
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] |
1 = 1
δ E
�
Σ
∗2
(β(α) − c(α)Σ
∗)2
�
E
�
Ξ
∗2
(α − Ξ
∗)2
�
Finally, denote
σ∗
2 :=
�
ψ(α◦).
(E.26)
The characterization of σ1(W ∗) requires an extra technical assumption on the random variables Ξ, Σ,
which is the same as in [75].
(A3) For any c > 0,
lim
β↓s E
�
Σ
∗
β − cΣ
∗
�
= lim
β↓s E
�
Σ
∗
β − cΣ
∗
�2
= ∞.
where s := c · sup supp(Σ
∗). Furthermore,
lim
α↓sup supp(Ξ
∗)
E
�
Ξ
∗
α − Ξ
∗
�
= ∞.
Lemma E.2. Let Assumption (A3) hold. Consider W ∗ defined in (E.24). Then, we have
p-lim
n→∞ σ1(W ∗) = σ∗ | [
0.11257287114858627,
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0.42129960656166077,
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] |
p-lim
n→∞ σ1(W ∗) = σ∗
2.
Proof. Note that W ∗W ∗⊤ is a separable covariance matrix. Its largest eigenvalue is characterized in
[22]. The explicit formulas we need are due to [75]. To apply their results, one simply observes that
the covariances (as in the context of separable covariance matrices) of W ∗ are
Ξ∗ :=
√
λ(λ(µ∗ + b∗)In + Ξ)−1/2,
Σ∗ :=
√
λ(λ(ν∗ + c∗)Id + Σ)−1/2,
whose limiting spectral distributions are given by the distributions of Ξ
∗, Σ
∗ in (E.25). | [
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] |
whose limiting spectral distributions are given by the distributions of Ξ
∗, Σ
∗ in (E.25).
Lemma E.3. Consider A∗ defined in (5.3). Then
p-lim
n→∞ σ2(A∗) = σ∗
2.
Proof. By Weyl’s inequality, σ3(W ∗) ≤ σ1(A∗) ≤ σ1(W ∗). We have already shown in Lemma E.2
that σ1(W ∗) converges to σ∗
2. The almost sure weak convergence of the empirical spectral distribution
of W ∗ [74, Theorem 1.2.1] implies that σ3(W ∗) (and indeed σk(W ∗) for any constant k relative to
n, d) must also converge to the same limit σ∗
2.
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] |
n, d) must also converge to the same limit σ∗
2.
37
E.4
Proof of Theorem 5.1
We suppose throughout the proof that the condition (5.1) holds. Then, by Proposition 4.1 and
the change of variable (E.14), the fixed point equation (E.13) admits a unique non-trivial solution
(µ∗, ν∗) ∈ R2
>0. Construct matrices F, G as in (E.16) using such µ∗, ν∗. Define also the random
variables
G :=
λ
λµ∗Ξ
−1 + 1
,
F :=
λ
λν∗Σ
−1 + 1
whose distributions are the limiting spectral distributions of G, F, respectively. | [
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λ
λν∗Σ
−1 + 1
whose distributions are the limiting spectral distributions of G, F, respectively.
Now consider the denoising functions:
ft+1(vt+1) = Fvt+1,
gt(ut) = Gut.
With this choice, the AMP iteration (E.1) becomes
ut = Ξ−1AΣ−1Fvt − bΞ−1Gut−1,
vt+1 = Σ−1A⊤Ξ−1Gut − cΣ−1Fvt,
(E.27)
where
b = 1
n Tr(FΣ−1),
c = 1
n Tr(GΞ−1).
Note that as n → ∞, b, c converge to b∗, c∗ in (E.17).
Recall from (E.15) the definition of σ∗, τ ∗. We initialize (E.27) with
�u−1 = 0n,
�v0 = F(ν∗Σ−1v∗ + τ ∗Σ−1/2w), | [
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] |
�u−1 = 0n,
�v0 = F(ν∗Σ−1v∗ + τ ∗Σ−1/2w),
where w ∼ N(0d, Id) is independent of everything else. Accordingly, one can take f0 in Assump-
tion (A1) to be f0(v) = ν∗FΣ−1v. Under the above AMP initializer, the state evolution initializers
in (E.3) and (E.5) specialize to
µ0 = lim
n→∞
λ
nE
�
V ∗⊤Σ−1FΣ−1V ∗�
ν∗ = λ
δ E
�
FΣ
−2�
ν∗ = µ∗,
σ2
0 = p-lim
n→∞
ν∗2
n v∗⊤Σ−1FΣ−1FΣ−1v∗ + τ ∗2
n w⊤Σ−1/2FΣ−1FΣ−1/2w
= 1
δ E
�
F
2Σ
−3�
ν∗2 + 1
δ E
�
F
2Σ
−2�
τ ∗2 = σ∗2, | [
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] |
n w⊤Σ−1/2FΣ−1FΣ−1/2w
= 1
δ E
�
F
2Σ
−3�
ν∗2 + 1
δ E
�
F
2Σ
−2�
τ ∗2 = σ∗2,
where the last equalities for both chains of computation are by (E.11). Since the parameters µt, σt
are initialized at the non-trivial fixed point (µ0, σ0) = (µ∗, σ∗), the state evolution recursion (E.11)
will stay at the fixed point (µt, σt, νt+1, τt+1) = (µ∗, σ∗, ν∗, τ ∗) across all t ≥ 0.
Lemma E.4. Let
�ut := �G1/2Gut,
�vt+1 := �F 1/2Fvt+1,
(E.28)
where �F, �G are defined in (E.21). Suppose the condition (5.1) holds. Then
lim | [
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] |
(E.28)
where �F, �G are defined in (E.21). Suppose the condition (5.1) holds. Then
lim
t→∞ p-lim
n→∞
|⟨�ut, u1(A∗)⟩|
∥�ut∥2
= lim
t→∞ p-lim
n→∞
���
�vt+1, v1(A∗)
���
∥�vt+1∥2
= 1
(E.29)
and
p-lim
n→∞ σ1(A∗) = 1,
(E.30)
where A∗ is defined in (E.22).
Proof. Denoting
et
1 := ut − ut−1,
et+1
2
:= vt+1 − vt,
(E.31)
for any t ≥ 1 and using the notation �F, �G in (E.20), we have from (E.27) that
�Gut = Ξ−1AΣ−1Fvt + b∗Ξ−1Get
1 + (b∗ − b)Ξ−1Gut−1,
38
�Fvt+1 = Σ−1A⊤Ξ−1Gut + c∗Σ−1Fet+1
2
+ (c∗ − c)Σ−1Fvt. | [
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] |
1 + (b∗ − b)Ξ−1Gut−1,
38
�Fvt+1 = Σ−1A⊤Ξ−1Gut + c∗Σ−1Fet+1
2
+ (c∗ − c)Σ−1Fvt.
Recalling the notation �F, �G from (E.21) and multiplying �G−1/2 (resp. �F −1/2) on both sides of the
first (resp. second) equation above, we arrive at
�G1/2Gut = A∗ · �F 1/2Fvt + b∗ �G−1/2Ξ−1Get
1 + (b∗ − b) �G−1/2Ξ−1Gut−1,
�F 1/2Fvt+1 = A∗⊤ · �G1/2Gut + c∗ �F −1/2Σ−1Fet+1
2
+ (c∗ − c) �F −1/2Σ−1Fvt.
Using the definition of �ut, �vt+1 in (E.28), we rewrite the above as
�ut = A∗�vt + et
u,
�vt+1 = A∗⊤�ut + et+1
v
,
(E.32)
where | [
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] |
�ut = A∗�vt + et
u,
�vt+1 = A∗⊤�ut + et+1
v
,
(E.32)
where
et
u := b �G−1/2Ξ−1Get
1 + (b∗ − b) �G−1/2Ξ−1Gut−1,
et+1
v
:= c �F −1/2Σ−1Fet+1
2
+ (c∗ − c) �F −1/2Σ−1Fvt.
(E.33)
Let us focus on �ut and only prove the first equality in (E.29). The proof of the second one is similar
and will be omitted. Eliminating �vt from (E.32) gives
�ut = A∗A∗⊤�ut−1 + A∗et
v + et
u.
Unrolling this recursion for s steps, we get:
�ut+s =
�
A∗A∗⊤�s
�ut + �et,s
u ,
(E.34)
where
�et,s
u :=
s
�
r=1
�
A∗A∗⊤�s−r
(A∗et+r
v
+ et+r
u | [
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] |
�ut+s =
�
A∗A∗⊤�s
�ut + �et,s
u ,
(E.34)
where
�et,s
u :=
s
�
r=1
�
A∗A∗⊤�s−r
(A∗et+r
v
+ et+r
u
).
(E.35)
Taking 1
n∥·∥2
2 on both sides of (E.34) and take the sequential limits of n → ∞, t → ∞, s → ∞, we
have the left hand side:
lim
s→∞ lim
t→∞ p-lim
n→∞
1
n
���ut+s��2
2 = lim
t→∞ p-lim
n→∞
1
n
���ut��2
2
= lim
t→∞ p-lim
n→∞
1
n
��� �G1/2Gut���
2
2
= lim
t→∞ p-lim
n→∞
1
n
��� �G1/2G1/2ut���
2
2
= lim
t→∞ µ2
tE
�
Ξ
−2(1 + b∗Ξ
−1G)G
�
+ σ2
t E
�
Ξ
−1(1 + b∗Ξ
−1G)G
�
= µ∗2E
�
Ξ
−2(1 + b∗Ξ
−1G)G
�
+ µ∗
λ E
� | [
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] |
tE
�
Ξ
−2(1 + b∗Ξ
−1G)G
�
+ σ2
t E
�
Ξ
−1(1 + b∗Ξ
−1G)G
�
= µ∗2E
�
Ξ
−2(1 + b∗Ξ
−1G)G
�
+ µ∗
λ E
�
Ξ
−1(1 + b∗Ξ
−1G)G
�
= λµ∗2E
�
λ(µ∗ + b∗) + Ξ
Ξ
�
λµ∗ + Ξ
�2
�
+ µ∗E
�
λ(µ∗ + b∗) + Ξ
�
λµ∗ + Ξ
�2
�
∈ (0, ∞).
(E.36)
Next, we have that
lim
t→∞ p-lim
n→∞
1
n
��et
1
��2
2 = lim
t→∞ p-lim
d→∞
1
d
��et+1
2
��2
2 = 0.
(E.37)
To prove the first statement on et
1, the strategy is to write the LHS of (E.37) in terms of the state
evolution parameters and prove that the latter quantities converge. We start with | [
0.6985498666763306,
-0.36561864614486694,
0.2832608222961426,
0.08974155783653259,
-0.3691268563270569,
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0.29982006549835205,
-1.0230780839920044,
0.3626430034637451,
-0.4857478737831116,
-0.07660849392414093,
0.7883055806159973,
-0.10904122143983841,
0.21440917253494263,
-0.10919618606567383,
0.591949999332428,
0.5366008281707764,
-0.026093969121575356,
-0.9649773240089417,
-0.7506731748580933,
0.24328266084194183,
0.10336696356534958,
0.6542491316795349,
0.25430017709732056,
-0.4593591094017029,
0.7511823177337646,
0.36691322922706604,
-0.07887113839387894,
0.4855268895626068,
0.2836422622203827,
-0.06480779498815536,
0.40946176648139954,
-0.5731617212295532,
-0.17799009382724762,
-0.3265400230884552,
-0.1288984715938568,
0.6345043778419495,
0.17337611317634583,
0.3414444327354431,
-0.7115958333015442,
0.19973935186862946,
-0.32633447647094727,
0.17125940322875977,
0.3411319851875305,
-0.34230655431747437,
-0.39226382970809937,
-0.9377893209457397,
-0.24817055463790894,
-0.4226384460926056,
-1.0929319858551025,
-0.2637775242328644,
-0.06939529627561569,
-0.12595708668231964,
0.6822925806045532,
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0.31388935446739197,
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0.6923519372940063,
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] |
evolution parameters and prove that the latter quantities converge. We start with
p-lim
n→∞
1
n
��ut − ut−1��2
2 =
�
µ2
tE
�
Σ
−2�
+ σ2
t E
�
Σ
−1��
+
�
µ2
t−1E
�
Σ
−2�
+ σ2
t−1E
�
Σ
−1��
− 2
�
µtµt−1E
�
Σ
−2�
+ Φt,t−1E
�
Σ
−1��
39
= 2σ∗2E
�
Σ
−1�
− 2Φt,t−1E
�
Σ
−1�
,
where the first equality is by Proposition G.3 and the joint distribution of WU,t, WU,t−1 in (E.2), and
the second one holds since the state evolution parameters stay at the fixed point upon initialization. | [
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] |
So it remains to verify that Φt,t−1 → σ∗2 as t → ∞. To see this, note that according to the state
evolution recursion (E.3),
Φt,t−1 = lim
n→∞
1
nE
�
V ⊤
t F ⊤Σ−1FVt−1
�
= ν∗2
δ E
�
F
2Σ
−3�
+ Ψt,t−1
δ
E
�
F
2Σ
−2�
,
Ψt+1,t = lim
n→∞
1
nE
�
U ⊤
t G⊤Ξ−1GUt−1
�
= µ∗2E
�
G
2Ξ
−3�
+ Φt,t−1E
�
G
2Ξ
−2�
.
Eliminating Ψt,t−1 from the first equation, we arrive at
Φt,t−1 = ν∗2
δ E
�
F
2Σ
−3�
+ 1
δ E
�
F
2Σ
−2��
µ∗2E
�
G
2Ξ
−3�
+ Φt−1,t−2E
�
G
2Ξ
−2��
= 1
δ E
�
λ2Σ
−3
(λν∗Σ
−1 + 1)2
�
ν∗2 + 1
δ E
�
λ2Σ
−2
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] |
�
G
2Ξ
−3�
+ Φt−1,t−2E
�
G
2Ξ
−2��
= 1
δ E
�
λ2Σ
−3
(λν∗Σ
−1 + 1)2
�
ν∗2 + 1
δ E
�
λ2Σ
−2
(λν∗Σ
−1 + 1)2
�
×
�
E
�
λ2Ξ
−3
(λµ∗Ξ
−1 + 1)2
�
µ∗2 + E
�
λ2Ξ
−2
(λµ∗Ξ
−1 + 1)2
�
Φt−1,t−2
�
,
whose only fixed point is σ∗2 by the relations in (E.11). This concludes the proof of the first equality
in (E.37). The proof of the second equality is analogous and, hence, omitted.
By using (E.37) and the fact that b → b∗, c → c∗ as n → ∞ (see (E.17)), we obtain
lim
t→∞ p-lim
n→∞
1
n
��et
u
��2
2 = lim
t→∞ p-lim
d→∞
1
d | [
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] |
lim
t→∞ p-lim
n→∞
1
n
��et
u
��2
2 = lim
t→∞ p-lim
d→∞
1
d
��et+1
v
��2
2 = 0.
(E.38)
Note that the operator norm of A∗ is almost surely bounded uniformly in n by Weyl’s inequality, sub-
multiplicativity of matrix norms and the Bai–Yin law [3]. This together with the triangle inequality
of the ℓ2-norm and (E.38) implies that
lim
s→∞ lim
t→∞ p-lim
n→∞
1
n
���et,s
u
��2
2 = 0,
(E.39)
From this, it follows that the right hand side of (E.34) (upon taken the rescaled squared norm and the | [
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] |
sequential limits) equals
lim
s→∞ lim
t→∞ p-lim
n→∞
1
n
���
�
A∗A∗⊤�s
�ut���
2
2.
We then compute the above term by taking the SVD of A∗. Define two spectral projectors that are
orthogonal to each other:
Π := u1(A∗)u1(A∗)⊤,
Π⊥ := In − Π.
We have
1
n
���
�
A∗A∗⊤�s
�ut���
2
2 = 1
n
���
�
A∗A∗⊤�s
�ut���
2
2 + 1
n
���
�
A∗A∗⊤�s
Π⊥�ut���
2
2.
(E.40)
Using the spectral decomposition
�
A∗A∗⊤�s
=
n
�
i=1
σi(A∗)2sui(A∗)ui(A∗)⊤,
we can write the first term in (E.40) as
1
n
���
�
A∗A∗⊤�s
�ut���
2
2 = 1
n
�����
n
� | [
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] |
we can write the first term in (E.40) as
1
n
���
�
A∗A∗⊤�s
�ut���
2
2 = 1
n
�����
n
�
i=1
σi(A∗)2sui(A∗)ui(A∗)⊤u1(A∗)u1(A∗)⊤�ut
�����
2
2
= σ1(A∗)4s ⟨u1(A∗), �ut⟩2
n
.
(E.41)
40
For the second term in (E.40), we have
1
n
���
�
A∗A∗⊤�s
Π⊥�ut���
2
2 = 1
n
���
�
A∗A∗⊤Π⊥�s
�ut���
2
2
≤ ∥�ut∥2
2
n
max
u∈Sn−1
���
�
A∗A∗⊤Π⊥�s
u
���
2
2
= ∥�ut∥2
2
n
σ1
��
A∗A∗⊤Π⊥�s�2
= ∥�ut∥2
2
n
σ1
�
A∗A∗⊤Π⊥�2s
= ∥�ut∥2
2
n
σ2
�
A∗A∗⊤�2s
= ∥�ut∥2
2
n
σ2(A∗)4s,
where penultimate line follows since
A∗A∗⊤Π⊥ =
n
�
i=2 | [
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] |
2
n
σ2
�
A∗A∗⊤�2s
= ∥�ut∥2
2
n
σ2(A∗)4s,
where penultimate line follows since
A∗A∗⊤Π⊥ =
n
�
i=2
σi(A∗)2ui(A∗)ui(A∗)⊤.
From Lemma E.3 and the assumption that σ∗
2 < 1, we know
p-lim
n→∞ σ2(A∗) = σ∗
2 < 1.
This implies:
lim
s→∞ lim
t→∞ p-limsup
n→∞
1
n
���
�
A∗A∗⊤�s
Π⊥�ut���
2
2 ≤ lim
s→∞ lim
t→∞ p-limsup
n→∞
∥�ut∥2
2
n
σ2(A∗)4s
≤
�
lim
t→∞ p-limsup
n→∞
∥�ut∥2
2
n
��
lim
s→∞ p-limsup
n→∞
σ2(A∗)4s
�
= 0,
(E.42)
where the last equality holds since the limit in the first parentheses is finite by (E.36). | [
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] |
(E.42)
where the last equality holds since the limit in the first parentheses is finite by (E.36).
Now (E.40) to (E.42) jointly show
lim
s→∞ lim
t→∞ p-lim
n→∞
1
n
���
�
A∗A∗⊤�s
�ut���
2
2 = lim
s→∞ lim
t→∞ p-lim
n→∞ σ1(A∗)4s ⟨u1(A∗), �ut⟩2
n
=
�
lim
s→∞ p-lim
n→∞ σ1(A∗)4s
��
lim
t→∞ p-lim
n→∞
⟨u1(A∗), �ut⟩2
n
�
.
Combining this with (E.36) brings us to the following identity:
1 =
�
lim
s→∞ p-lim
n→∞ σ1(A∗)4s
��
lim
t→∞ p-lim
n→∞
⟨u1(A∗), �ut⟩2
∥�ut∥2
2
�
,
which necessarily implies
p-lim
n→∞ σ1(A∗) = 1, | [
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] |
��
lim
t→∞ p-lim
n→∞
⟨u1(A∗), �ut⟩2
∥�ut∥2
2
�
,
which necessarily implies
p-lim
n→∞ σ1(A∗) = 1,
lim
t→∞ p-lim
n→∞
⟨u1(A∗), �ut⟩2
∥�ut∥2
2
= 1,
as desired.
With Lemma E.4, we can complete the proof of Theorem 5.1.
Proof of Theorem 5.1. The characterization (5.7) of the top two singular values have been obtained
in Lemmas E.3 and E.4. It remains to compute the overlaps which can be done using Lemma E.4 and
41 | [
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] |
in Lemmas E.3 and E.4. It remains to compute the overlaps which can be done using Lemma E.4 and
41
the state evolution (Proposition E.1). Recall the estimators �u, �v in (5.4) and their heuristic derivation
in (E.23). Then
p-lim
n→∞
⟨�u, u∗⟩2
∥�u∥2
2∥u∗∥2
2
= p-lim
t→∞ p-lim
n→∞
⟨Ξut, u∗⟩2
∥Ξut∥2
2∥u∗∥2
2
=
λµ∗
λµ∗ + 1 = η2
u,
establishing the first equality in (5.8). The second equality in (5.8) and other quantities in (5.9)
and (5.10) can be similarly obtained. The proof is completed.
F | [
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] |
and (5.10) can be similarly obtained. The proof is completed.
F
Proof of Proposition E.1
Recall �u∗, �v∗ from (C.7) and let
�A := Ξ−1/2AΣ−1/2 = λ
n �u∗(�v∗)⊤ + �
W.
(F.1)
F.1
Auxiliary AMP and its state evolution
For (˘gt : Rn → Rn, ˘ft+1 : Rd → Rd)t≥0, the iterates of the auxiliary AMP, initialized at ˚u−1 = 0d
and some ˚v0 ∈ Rd, are updated according to the following rules for every t ≥ 0:
˘ut = �A˚vt − ˘bt˚ut−1,
˚ut = ˘gt(˘ut),
˘ct = 1
n
n
�
i=1
∂i˘gt(˘ut)i,
˘vt+1 = �A⊤˚ut − ˘ct˚vt, | [
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] |
˘ut = �A˚vt − ˘bt˚ut−1,
˚ut = ˘gt(˘ut),
˘ct = 1
n
n
�
i=1
∂i˘gt(˘ut)i,
˘vt+1 = �A⊤˚ut − ˘ct˚vt,
˚vt+1 = ˘ft+1(˘vt+1),
˘bt+1 = 1
n
d
�
i=1
∂i ˘ft+1(˘vt+1)i.
(F.2)
The state evolution result associated with the above auxiliary AMP iteration asserts that the distri-
butions of (˘u0, ˘u1, · · · , ˘ut), (˘v1, ˘v2, · · · , ˘vt+1) converge (in the sense of (F.16)) respectively to the
laws of the random vectors ( ˘U0, ˘U1, · · · , ˘Ut), ( ˘V1, ˘V2, · · · ˘Vt+1) defined below:
˘Ut = ˘µt �U ∗ + ˘σt ˘WU,t ∈ Rn, | [
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] |
˘Ut = ˘µt �U ∗ + ˘σt ˘WU,t ∈ Rn,
˘Vt+1 = ˘νt+1 �V ∗ + ˘τt+1 ˘WV,t+1 ∈ Rd,
(F.3)
where
U ∗
˘σ0 ˘WU,0
˘σ1 ˘WU,1
...
˘σt ˘WU,t
∼ P ⊗n ⊗ N(0n(t+1), ˘Φt ⊗ In),
V ∗
˘τ1 ˘WV,1
˘τ2 ˘WV,2
...
˘τt+1 ˘WV,t+1
∼ Q⊗d ⊗ N(0d(t+1), ˘Ψt+1 ⊗ Id),
(F.4)
�U ∗ := Ξ−1/2U ∗ ∈ Rn,
�V ∗ := Σ−1/2V ∗ ∈ Rd.
(F.5)
The parameters ˘µt, ˘νt+1 ∈ R, ˘Φt = (˘Φr,s)0≤r,s≤t, ˘Ψt+1 = (˘Ψr+1,s+1)0≤r,s≤t ∈ R(t+1)×(t+1) are
defined recursively through the following state evolution equations:
˘µ0 = λ lim
d→∞
1
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] |
defined recursively through the following state evolution equations:
˘µ0 = λ lim
d→∞
1
nE
��
�V ∗, ˘f0(�V ∗)
��
,
(F.6)
˘µt = λ lim
d→∞
1
nE
��
�V ∗, ˘ft( ˘Vt)
��
,
t ≥ 1,
(F.7)
˘νt+1 = λ lim
n→∞
1
nE
��
�U ∗, ˘gt( ˘Ut)
��
,
t ≥ 0,
(F.8)
˘Φ0,0 = p-lim
n→∞
1
n
��˚v0��2
2,
(F.9)
˘Φ0,s = lim
n→∞
1
nE
��
˘f0(�V ∗), ˘fs( ˘Vs)
��
,
1 ≤ s ≤ t,
(F.10)
˘Φr,s = lim
n→∞
1
nE
��
˘fr( ˘Vr), ˘fs( ˘Vs)
��
,
1 ≤ r, s ≤ t,
(F.11)
42
˘Ψr+1,s+1 = lim
n→∞
1
nE
��
˘gr( ˘Ur), ˘gs( ˘Us)
��
,
0 ≤ r, s ≤ t,
(F.12) | [
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-0.21615369617938995,
0.08638675510883331,
0.2519144117832184,
0.31872519850730896,
0.614690363407135,
-0.6475928425788879,
0.31098949909210205,
0.22877104580402374,
0.03141487017273903,
-0.38178834319114685,
-0.2084820568561554,
0.24769939482212067,
0.3565351366996765,
1.0189032554626465,
1.1634893417358398,
-0.2825535535812378,
-0.20514774322509766,
0.19687031209468842,
0.21375899016857147,
0.04643688350915909,
-0.22584590315818787,
-0.16290903091430664,
0.20592431724071503,
0.3256708085536957,
-0.6009683609008789,
-0.2814480662345886,
-0.33465662598609924,
-0.4561384320259094,
0.293691486120224,
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0.4548681676387787,
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0.1435713768005371,
0.1801280528306961,
0.12155750393867493,
-0.35328614711761475,
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0.06924612075090408,
0.1418052464723587,
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0.020151451230049133,
0.05507945269346237,
0.13631024956703186,
0.051301777362823486,
0.35685354471206665,
-0.36218732595443726,
0.18080024421215057,
0.5213546752929688,
-0.8461365699768066,
-0.2615440785884857,
0.09141726046800613,
0.20898616313934326,
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-0.7503808736801147,
-0.08898259699344635,
0.05252707377076149,
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0.08025931566953659,
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0.3746178150177002,
0.34316274523735046,
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0.0758679136633873,
0.39036810398101807,
0.19078169763088226,
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0.631824791431427,
1.2184507846832275,
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0.4181668162345886,
0.007215074263513088,
0.495556116104126,
0.5129885077476501,
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0.025904612615704536,
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0.4709002375602722,
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0.8685175776481628,
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0.19895730912685394,
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0.8073753118515015,
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0.44570356607437134,
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0.053833212703466415,
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0.7528352737426758,
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0.4433457553386688,
0.566059947013855,
0.41332685947418213,
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0.6125019788742065,
-0.1437229961156845,
0.320674329996109,
0.0363987572491169,
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-0.48557165265083313,
0.371349036693573,
0.017298098653554916,
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-0.17005036771297455,
0.35299447178840637,
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0.15218262374401093,
0.152877077460289,
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0.32412391901016235
] |
1 ≤ r, s ≤ t,
(F.11)
42
˘Ψr+1,s+1 = lim
n→∞
1
nE
��
˘gr( ˘Ur), ˘gs( ˘Us)
��
,
0 ≤ r, s ≤ t,
(F.12)
where ˘f0 is determined by ˚v0 through Assumption (A4) below. In particular,
˘σ2
0 = p-lim
n→∞
1
n
��˚v0��2
2,
(F.13)
˘σ2
t = lim
n→∞
1
nE
��
˘ft( ˘Vt), ˘ft( ˘Vt)
��
,
t ≥ 1,
(F.14)
˘τ 2
t+1 = lim
n→∞
1
nE
��
˘gt( ˘Ut), ˘gt( ˘Ut)
��
,
t ≥ 0.
(F.15)
We require the following assumptions to guarantee the existence and finiteness of the state evolution
parameters defined above.
(A4) ˚v0 is independent of � | [
-0.03258591890335083,
0.06708939373493195,
0.3872234523296356,
-0.35913026332855225,
-0.09855422377586365,
0.5781044960021973,
0.38991400599479675,
-0.784065842628479,
-0.017554588615894318,
-0.2553858757019043,
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0.5385215878486633,
-0.253231406211853,
0.2597924470901489,
0.22078748047351837,
0.9003707766532898,
1.0257362127304077,
0.35200271010398865,
-0.3872390687465668,
-0.5293915271759033,
0.26416391134262085,
-0.11655975133180618,
0.7778772115707397,
-0.06478044390678406,
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-0.00007273838855326176,
0.351847380399704,
-0.11972509324550629,
0.608284592628479,
0.2731150686740875,
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0.5547032952308655,
0.17455536127090454,
0.1159299686551094,
0.1221461370587349,
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0.3563551902770996,
0.32900771498680115,
0.21983006596565247,
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1.050838828086853,
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] |
parameters defined above.
(A4) ˚v0 is independent of �
W but may depend on �v∗. Assume that
p-lim
d→∞
1
d
��˚v0��2
2
exists and is finite. There exists a uniformly pseudo-Lipschitz function ˘f0 : Rd → Rd of
order 1 such that
lim
d→∞
1
dE
��
˘f0(�V ∗), ˘f0(�V ∗)
��
≤ p-lim
d→∞
1
d
��˚v0��2
2
and for every uniformly pseudo-Lipschitz function ϕ: Rd → Rd of finite order, the following
two limits exist, are finite and equal:
p-lim
d→∞
1
d
�
˚v0, ϕ(�v∗)
�
= lim
d→∞
1
dE
��
˘f0(�V ∗), ϕ(�V ∗)
��
. | [
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] |
p-lim
d→∞
1
d
�
˚v0, ϕ(�v∗)
�
= lim
d→∞
1
dE
��
˘f0(�V ∗), ϕ(�V ∗)
��
.
Let �ν ∈ R, �τ ∈ R≥0. For any s ≥ 1,
lim
d→∞
1
dE
�
˘f0(�V ∗)⊤ ˘fs(�ν �V ∗ + �τWV )
�
exists and is finite, where WV ∼ N(0d, Id) is independent of �V ∗.
(A5) Let �ν ∈ R, and T ∈ R2×2 be positive definite. For any r, s ≥ 1,
lim
n→∞
λ
nE
��
�V ∗, ˘fr(�ν �V ∗ + N)
��
,
lim
d→∞
1
dE
�
˘fr(�ν �V ∗ + N)⊤ ˘fs(�ν �V ∗ + N ′)
�
exist and are finite, where (N, N ′) ∼ N(02d, T ⊗ Id) is independent of �v∗. | [
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] |
�
exist and are finite, where (N, N ′) ∼ N(02d, T ⊗ Id) is independent of �v∗.
Let �µ ∈ R, and S ∈ R2×2 be positive definite. For any r, s ≥ 0,
lim
n→∞
λ
nE
��
�U ∗, ˘gr(�µ�U ∗ + M)
��
,
lim
d→∞
1
dE
�
˘gr(�µ�U ∗ + M)⊤˘gs(�µ�U ∗ + M ′)
�
exist and are finite, where (M, M ′) ∼ N(02n, S ⊗ In) is independent of �U ∗.
Proposition F.1 (State evolution for auxiliary AMP (F.2)). For every t ≥ 0, let (˘gt : Rn → Rn)n≥1 | [
0.1576577126979828,
0.11165450513362885,
0.45237377285957336,
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0.41423866152763367,
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0.5558211207389832,
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0.6356557607650757,
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1.2400329113006592,
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0.7690474987030029,
0.16594386100769043,
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0.2992246150970459,
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0.47995856404304504
] |
Proposition F.1 (State evolution for auxiliary AMP (F.2)). For every t ≥ 0, let (˘gt : Rn → Rn)n≥1
and ( ˘ft+1 : Rd → Rd)d≥1 be uniformly pseudo-Lipschitz of finite order subject to Assumption (A5).
Consider the auxiliary AMP iteration in (F.2) defined by (˘gt, ˘ft+1)t≥0 and initialized at ˚u−1 = 0n
and some ˚v0 ∈ Rd subject to Assumption (A4). For any fixed t ≥ 0, let (ϕ: R(t+2)n → R)n≥1 and
(ψ: R(t+2)d → R)n≥1 be uniformly pseudo-Lipschitz functions of finite order. Then,
p-lim | [
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] |
(ψ: R(t+2)d → R)n≥1 be uniformly pseudo-Lipschitz functions of finite order. Then,
p-lim
n→∞ ϕ(�u∗, ˘u0, ˘u1, · · · , ˘ut) − E
�
ϕ(�U ∗, ˘U0, ˘U1, · · · , ˘Ut)
�
= 0,
(F.16a)
p-lim
d→∞
ψ(�v∗, ˘v1, ˘v2, · · · , ˘vt+1) − E
�
ψ(�V ∗, ˘V1, ˘V2, · · · , ˘Vt+1)
�
= 0,
(F.16b)
where ( ˘Us, ˘Vs+1)0≤s≤t are defined in (F.3).
43
Proof of Proposition F.1. By definitions of the auxiliary AMP (F.2) and the matrix �A in (F.1), we
have that for every t ≥ 0,
˘ut = �A˚vt − ˘bt˚ut−1 = λ
n
�
�v∗,˚vt�
�u∗ + �
W˚vt − ˘bt˚ut−1, | [
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0.19841530919075012
] |
have that for every t ≥ 0,
˘ut = �A˚vt − ˘bt˚ut−1 = λ
n
�
�v∗,˚vt�
�u∗ + �
W˚vt − ˘bt˚ut−1,
˘vt+1 = �A⊤˚ut − ˘ct˚vt = λ
n
�
�u∗,˚ut�
�v∗ + �
W ⊤˚ut − ˘ct˚vt.
For every t ≥ 0, let us consider a pair of related iterates pt, qt+1 with initialization
�p−1 = 0n,
�q0 = ˚v0
(F.17)
and update rules:
pt = �
W �qt − ℓt�pt−1,
�pt = ˘gt(pt + ˘µt�u∗),
mt = 1
n
n
�
i=1
∂i˘gt(pt + ˘µt�u∗)i,
qt+1 = �
W ⊤�pt − mt�qt,
�qt+1 = ˘ft+1(qt+1 + ˘νt+1�v∗),
ℓt+1 = 1
n
d
�
i=1
∂i ˘ft+1(qt+1 + ˘νt+1�v∗)i,
(F.18) | [
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] |
�qt+1 = ˘ft+1(qt+1 + ˘νt+1�v∗),
ℓt+1 = 1
n
d
�
i=1
∂i ˘ft+1(qt+1 + ˘νt+1�v∗)i,
(F.18)
where ˘µt, ˘νt+1 are as in (F.6) to (F.8).
Informally, the above iterates are related to ˘ut, ˘vt+1 via
pt ‘=’ ˘ut − ˘µt�u∗,
qt+1 ‘=’ ˘vt+1 − ˘νt+1�v∗,
(F.19)
where the ‘equalities’ hold only in the large n limit. These relations will be made formal in the rest
of the proof.
The algorithm (F.18) takes the form of a standard AMP iteration with non-separable denoising | [
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] |
The algorithm (F.18) takes the form of a standard AMP iteration with non-separable denoising
functions as in [12, 30] for which the following state evolution result applies. For any t ≥ 0 and
uniformly pseudo-Lipschitz functions ϕ, ψ of finite order, it holds that
p-lim
n→∞ ϕ(�u∗, p0, · · · , pt) − E
�
ϕ(�U ∗, ˘σ0 ˘WU,0, · · · , ˘σt ˘WU,t)
�
= 0,
p-lim
n→∞ ψ(�v∗, q1, · · · , qt+1) − E
�
ψ(�V ∗, ˘τ1 ˘WV,1, · · · , ˘τt+1 ˘WV,t+1)
�
= 0,
(F.20) | [
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] |
p-lim
n→∞ ψ(�v∗, q1, · · · , qt+1) − E
�
ψ(�V ∗, ˘τ1 ˘WV,1, · · · , ˘τt+1 ˘WV,t+1)
�
= 0,
(F.20)
where (�U ∗, ˘σ0 ˘WU,0, · · · , ˘σt ˘WU,t) and (�V ∗, ˘τ1 ˘WV,1, · · · , ˘τt+1 ˘WV,t+1) are defined in (F.4) and (F.5).
Note that [30] allows additional randomness independent of �
W that goes into the denoising functions.
So the asymptotic guarantee in (F.20) holds for the joint tuple involving �U ∗, �V ∗ as well.
(F.20) immediately implies
p-lim
n→∞ ϕ(u∗, p0 + ˘µ0�u∗, · · · , pt + ˘µt�u∗) − E
� | [
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] |
(F.20) immediately implies
p-lim
n→∞ ϕ(u∗, p0 + ˘µ0�u∗, · · · , pt + ˘µt�u∗) − E
�
ϕ(U ∗, ˘U0, · · · , ˘Ut)
�
= 0,
p-lim
n→∞ ψ(v∗, q1 + ˘ν1�v∗, · · · , qt+1 + ˘νt+1�v∗) − E
�
ψ(V ∗, ˘V1, · · · , ˘Vt+1)
�
= 0,
(F.21)
where we recall the definition of ˘Ut, ˘Vt+1 in (F.3). We will show that
p-lim
n→∞ ϕ(u∗, p0 + ˘µ0�u∗, · · · , pt + ˘µt�u∗) − ϕ(u∗, ˘u0, · · · , ˘ut) = 0,
p-lim
n→∞ ψ(v∗, q1 + ˘ν1�v∗, · · · , qt+1 + ˘νt+1�v∗) − ψ(v∗, ˘v1, · · · , ˘vt+1) = 0,
(F.22) | [
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] |
p-lim
n→∞ ψ(v∗, q1 + ˘ν1�v∗, · · · , qt+1 + ˘νt+1�v∗) − ψ(v∗, ˘v1, · · · , ˘vt+1) = 0,
(F.22)
which, when combined with (F.21), concludes the proof of Proposition F.1.
To show (F.22), suppose that ϕ, ψ are uniformly L-pseudo-Lipschitz of order k. Then by the triangle
inequality,
��ϕ(u∗, p0 + ˘µ0�u∗, · · · , pt + ˘µt�u∗) − ϕ(u∗, ˘u0, · · · , ˘ut)
��
≤ L
�
t
�
s=0
1
√n∥ps + ˘µs�u∗ − ˘us∥2
�
44
×
1 +
�
1
√n∥u∗∥2 +
t
�
s=0
1
√n∥ps + ˘µs�u∗∥2
�k−1
+
�
1
√n∥u∗∥2 +
t
�
s=0
1
√n∥˘us∥2
�k−1
≤ L′
�
t
�
s=0
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] |
t
�
s=0
1
√n∥ps + ˘µs�u∗∥2
�k−1
+
�
1
√n∥u∗∥2 +
t
�
s=0
1
√n∥˘us∥2
�k−1
≤ L′
�
t
�
s=0
1
√n∥ps + ˘µs�u∗ − ˘us∥2
�
×
�
1 +
� 1
√n∥u∗∥2
�k−1
+
t
�
s=0
� 1
√n∥ps + ˘µs�u∗∥2
�k−1
+
t
�
s=0
� 1
√n∥˘us∥2
�k−1�
,
for some L′ depending only on t, k, L. Similar manipulation gives
��ψ(v∗, q1 + ˘ν1�v∗, · · · , qt+1 + ˘νt+1�v∗) − ψ(v∗, ˘v1, · · · , ˘vt+1)
��
≤ L′
�t+1
�
s=1
1
√
d
∥qs + ˘νs�v∗ − ˘vs∥2
�
×
�
1 +
� 1
√
d
∥v∗∥2
�k−1
+
t+1
�
s=1
� 1
√
d
∥qs + ˘νs�v∗∥2
�k−1
+
t+1
�
s=1
� 1
√
d
∥˘vs∥2
�k−1�
. | [
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] |
�
1 +
� 1
√
d
∥v∗∥2
�k−1
+
t+1
�
s=1
� 1
√
d
∥qs + ˘νs�v∗∥2
�k−1
+
t+1
�
s=1
� 1
√
d
∥˘vs∥2
�k−1�
.
Clearly, (F.22) holds if for every t ≥ 0,
p-lim
n→∞
1
√n
��pt + ˘µt�u∗��
2 < ∞,
(F.23)
p-lim
n→∞
1
√n
��˘ut��
2 < ∞,
(F.24)
p-lim
n→∞
1
n
��˘ut − (pt + ˘µt�u∗)
��2
2 = 0,
(F.25)
p-lim
d→∞
1
√
d
��qt+1 + ˘νt+1�v∗��
2 < ∞,
(F.26)
p-lim
d→∞
1
√
d
��˘vt+1��
2 < ∞,
(F.27)
p-lim
d→∞
1
d
��˘vt+1 − (qt+1 + ˘νt+1�v∗)
��2
2 = 0,
(F.28)
which, together with the following statements
˘µt < ∞,
˘σt < ∞,
(F.29)
˘νt+1 < ∞, | [
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] |
��2
2 = 0,
(F.28)
which, together with the following statements
˘µt < ∞,
˘σt < ∞,
(F.29)
˘νt+1 < ∞,
˘τt+1 < ∞,
(F.30)
will be shown in the sequel by induction on t ≥ 0.
Base case.
Consider t = 0. From (F.21),
p-lim
n→∞
1
n
��p0 + ˘µ0�u∗��2
2 = p-lim
n→∞
1
nE
���� ˘U0
���
2
2
�
= ˘σ2
0 + ˘µ2
0E
�
Ξ
−1�
,
(F.31)
where the last equality is by (F.3). Due to (F.13) and Assumption (A4), both ˘µ0 and ˘σ0 are finite, so
(F.29) holds for t = 0. Consequently, (F.23) also holds for t = 0.
Since by (F.17), | [
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] |
(F.29) holds for t = 0. Consequently, (F.23) also holds for t = 0.
Since by (F.17),
(p0 + ˘µ0�u∗) − ˘u0 = ˘µ0�u∗ − λ
n
�
�v∗,˚v0�
�u∗,
therefore (F.25) for t = 0 follows from (F.6) and Assumption (A4). This in turn implies, when
combined with the finiteness of (F.31), that
p-lim
n→∞
1
√n
��˘u0��
2 < ∞,
(F.32)
45
verifying (F.24) for t = 0. Since ˘g0 is uniformly pseudo-Lipschitz of finite order, so is the function
1
n
�n
i=1 ∂i˘g0(·)i. (F.23) to (F.25) (for t = 0) together imply
p-lim
n→∞|m0| < ∞,
p-lim | [
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] |
1
n
�n
i=1 ∂i˘g0(·)i. (F.23) to (F.25) (for t = 0) together imply
p-lim
n→∞|m0| < ∞,
p-lim
n→∞|m0 − ˘c0| = 0.
(F.33)
Using the the pseudo-Lipschitzness of ˘g0 again,
p-lim
n→∞
1
√n
���p0 − ˚u0��
2 = p-lim
n→∞
1
√n
��˘g0(p0 + ˘µ0�u∗) − ˘g0(˘u0)
��
2
≤ p-lim
n→∞ L
��(p0 + ˘µ0�u∗) − ˘u0��
2
√n
1 +
���p0 + ˘µ0�u∗��
2
√n
�k−1
+
���˘u0��
2
√n
�k−1
= 0.
(F.34)
The last equality holds because of (F.25) (for t = 0) and the finiteness of (F.31) and (F.32). | [
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] |
The last equality holds because of (F.25) (for t = 0) and the finiteness of (F.31) and (F.32).
To show (F.28) for t = 0, we use (F.2), (F.17) and (F.18) to write
(q1 + ˘ν1�v∗) − ˘v1 = �
W ⊤(�p0 − ˚u0)
�
��
�
T1
+
�
˘ν1 − λ
n
�
˚u0, �u∗��
�
��
�
T2
�v∗ − (m0 − ˘c0)
�
��
�
T3
˚v0.
By (F.34) and the Bai–Yin law [3],
p-lim
d→∞
1
d∥T1∥2
2 = 0.
(F.35)
Using (F.34) again,
p-lim
n→∞
λ
n
�
˚u0, �u∗�
= p-lim
n→∞
λ
n
�
�p0, �u∗�
= p-lim
n→∞
λ
n
�
˘g0(p0 + ˘µ0�u∗), �u∗�
= lim
n→∞
λ
nE
��
˘g0( ˘U0), �u∗��
= ˘ν1,
(F.36) | [
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] |
�p0, �u∗�
= p-lim
n→∞
λ
n
�
˘g0(p0 + ˘µ0�u∗), �u∗�
= lim
n→∞
λ
nE
��
˘g0( ˘U0), �u∗��
= ˘ν1,
(F.36)
where in the second line, the first equality is by (F.22) and the pseudo-Lipschitzness of ˘g0, and the
second equality is by the definition (F.8). We further show the finiteness of ˘ν1. Note that
˘ν1 ≤ lim
n→∞ λE
� 1
n
���˘g0( ˘U0)
���
2
2
�1/2
E
� 1
n
����U ∗���
2
2
�1/2
.
The first term can be bounded as
p-lim
n→∞
1
nE
����˘g0( ˘U0)
���
2
2
�
≤ p-lim
n→∞ L′E
�
1 +
� 1
√n
��� ˘U0
���
2
�k�2
≤ p-lim | [
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] |
n→∞
1
nE
����˘g0( ˘U0)
���
2
2
�
≤ p-lim
n→∞ L′E
�
1 +
� 1
√n
��� ˘U0
���
2
�k�2
≤ p-lim
n→∞ 2L′
�
1 + E
�� 1
n
��� ˘U0
���
2
2
�k��
,
(F.37)
where the last step is the elementary inequality (a + b)2 ≤ 2(a2 + b2). The RHS above is finite since
1
n
��� ˘U0
���
2
2 = ˘µ2
0
n
����U ∗���
2
2 + ˘σ2
0
n
��� ˘WV,0
���
2
2
(F.38)
whose all moments are finite by finiteness of ˘µ0, ˘σ0. This shows the first bound in (F.30) for t = 0.
Recalling (F.36), we then have
p-lim
n→∞|T2| = 0.
(F.39)
By (F.33),
p-lim | [
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] |
Recalling (F.36), we then have
p-lim
n→∞|T2| = 0.
(F.39)
By (F.33),
p-lim
n→∞|T3| = 0.
(F.40)
Therefore, (F.35), (F.39) and (F.40) altogether verify (F.28) for t = 0.
46
We then show (F.26) for t = 0. Since ˘ν1 < ∞, it suffices to only consider q1. According to (F.17)
and (F.18),
q1 = �
W ⊤�p0 − m0˚v0.
Pseudo-Lipschitzness of ˘g0, finiteness (F.23) (for t = 0) and finiteness (F.33) jointly imply
p-lim
n→∞
1
√n
��q1��
2 < ∞, | [
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] |
p-lim
n→∞
1
√n
��q1��
2 < ∞,
from which (F.26) follows. This combined with (F.28) (for t = 0) also implies (F.27) (for t = 0).
Finally, we are left with the second inequality in (F.30). From the definition (F.15),
˘τ 2
1 = lim
n→∞
1
nE
����˘g0( ˘U0)
���
2
2
�
which has already been shown to be finite; see (F.37). So the base case is finished.
Induction step.
Assume that (F.23) to (F.28) all hold up to the t-th step (for an arbitrary t ≥ 1). We | [
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0.2699371874332428,
0.39747387170791626,
0.23022104799747467,
0.29421594738960266,
0.06986819207668304,
-0.23679794371128082,
0.6395597457885742,
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0.35556766390800476,
0.5773077011108398,
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0.5251235365867615,
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0.7756234407424927,
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-1.1462007761001587,
0.04741405323147774,
0.15259695053100586,
0.13453368842601776,
0.2006818801164627,
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-0.26089945435523987,
0.03356412425637245,
0.14462856948375702
] |
Assume that (F.23) to (F.28) all hold up to the t-th step (for an arbitrary t ≥ 1). We
now show that they hold for t + 1. The idea is similar to the base case. We briefly lay down the key
steps for (F.23) to (F.25) and (F.29), and omit the verification of (F.26) to (F.28) and (F.30).
Using (F.3) and (F.21),
p-lim
n→∞
1
n
��pt+1 + ˘µt+1�u∗��2
2 = ˘σ2
t+1 + ˘µ2
t+1E
�
Ξ
−1�
.
(F.41)
Using the definition (F.14) and the pseudo-Lipschitzness of ˘ft+1,
˘σ2
t+1 = lim
n→∞
1
nE
���� ˘ft+1( ˘Vt+1)
���
2
2
�
≤ lim | [
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] |
˘σ2
t+1 = lim
n→∞
1
nE
���� ˘ft+1( ˘Vt+1)
���
2
2
�
≤ lim
n→∞ 2L′
�
1 + E
�� 1
n
��� ˘Vt+1
���
2
2
�k��
,
for some L′ depending only on k, L. The inequality is obtained in a similar way to (F.37). By
induction hypothesis (F.30) and the compactness of supp(Σ), all moments of
1
n
��� ˘Vt+1
���
2
2 = ˘ν2
t+1
n
����V ∗���
2
2 + ˘τ 2
t+1
n
��� ˘WV,t+1
���
2
2
are finite. Therefore ˘σ2
t+1 < ∞, giving the second bound in (F.29). Similarly, using the definition
(F.7) and Cauchy–Schwarz,
˘µt+1 = lim
d→∞
λ
nE
�� | [
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] |
(F.7) and Cauchy–Schwarz,
˘µt+1 = lim
d→∞
λ
nE
��
�V ∗, ˘ft+1( ˘Vt+1)
��
≤ lim
d→∞
L′λ
√n E
�����V ∗���
2
�
1 +
� 1
√n
��� ˘Vt+1
���
2
�k��
≤ lim
d→∞ L′λE
� 1
n
����V ∗���
2
2
�1/2�
2E
�
1 +
� 1
n
��� ˘Vt+1
���
2
2
�k��1/2
,
which is again finite for the same reason as ˘σt+1, giving the first bound in (F.29). Therefore (F.41) is
also finite, verifying (F.23) for t + 1.
We then show (F.25) for t + 1. Using the recursions (F.2) and (F.18),
pt+1 + ˘µt+1�u∗ − ˘ut+1 = �
W(�qt+1 −˚vt+1)
�
��
�
T ′
1
+
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] |
pt+1 + ˘µt+1�u∗ − ˘ut+1 = �
W(�qt+1 −˚vt+1)
�
��
�
T ′
1
+
�
˘µt+1 − λ
n
�
�v∗,˚vt+1��
�
��
�
T ′
2
�u∗ − (ℓt+1�pt − ˘bt+1˚ut)
�
��
�
T ′
3
.
Consider T ′
1. Since (F.26) to (F.28) are assumed to hold, by pseudo-Lipschitzness of ˘ft+1,
p-lim
n→∞
1
√
d
���qt+1 −˚vt+1��
2 = 0.
(F.42)
This with the Bai-Yin law [3] gives
p-lim
n→∞
1
n∥T ′
1∥2
2 = 0,
(F.43)
47
Consider T ′
2. Recall that ˘µt+1 < ∞. Using (F.7), (F.21) and (F.42) and following the argument
leading to (F.36), we have
p-lim
n→∞|T ′
2| = 0.
(F.44) | [
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] |
leading to (F.36), we have
p-lim
n→∞|T ′
2| = 0.
(F.44)
Consider T ′
3. By the triangle inequality,
���ℓt+1�pt − ˘bt+1˚ut���
2 ≤
���ℓt+1 − ˘bt+1
���
���pt��
2 +
���˘bt+1
���
���pt − ˚ut��
2.
Since (F.26) to (F.28) are assumed to hold, by pseudo-Lipschitzness of 1
n
�d
i=1 ∂i ˘ft+1(·)i,
p-lim
n→∞
���˘bt+1
��� < ∞,
p-lim
n→∞
���ℓt+1 − ˘bt+1
��� = 0.
(F.45)
Similarly, pseudo-Lipschitzness of ˘gt and the hypothesis (F.23) to (F.25) ensures
p-lim
n→∞
1
√n
���pt��
2 < ∞,
p-lim
n→∞
1
√n
���pt − ˚ut��
2 = 0, | [
-0.2640340328216553,
-0.29994556307792664,
0.533701479434967,
-0.1860208660364151,
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] |
p-lim
n→∞
1
√n
���pt��
2 < ∞,
p-lim
n→∞
1
√n
���pt − ˚ut��
2 = 0,
(F.46)
So combining (F.45) and (F.46), we have
p-lim
n→∞
1
n∥T ′
3∥2
2 = 0.
(F.47)
(F.43), (F.44) and (F.47) altogether verify (F.25) for t + 1, and therefore also (F.24) by (F.23).
The verification of (F.26) to (F.28) and (F.30) for t + 1 is completely analogous and we do not repeat
similar arguments. The proof is finally completed.
F.2
Proof of Proposition E.1 | [
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] |
similar arguments. The proof is finally completed.
F.2
Proof of Proposition E.1
We will prove Proposition E.1 by reducing the AMP iteration (E.1) (and its associated state evolution
(E.2) to (E.6)) to the auxiliary AMP (F.2) (and its associated state evolution (F.3) to (F.12), (F.14)
and (F.15)).
Under the following change of variables
ut := Ξ−1/2˘ut,
vt+1 := Σ−1/2˘vt+1,
(F.48)
ft+1(vt+1) := Σ1/2 ˘ft+1(Σ1/2vt+1),
gt(ut) := Ξ1/2˘gt(Ξ1/2ut),
(F.49)
(F.2) becomes
ut = Ξ−1AΣ−1�vt − btΞ−1�ut,
�ut = gt(ut), | [
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] |
gt(ut) := Ξ1/2˘gt(Ξ1/2ut),
(F.49)
(F.2) becomes
ut = Ξ−1AΣ−1�vt − btΞ−1�ut,
�ut = gt(ut),
vt+1 = Σ−1A⊤Ξ−1�ut − cbΣ−1�vt,
�vt+1 = ft+1(vt+1),
where bt+1, ct are equal to ˘bt+1, ˘ct, respectively, but are expressed using the derivatives of ft+1, gt.
Specifically,
ct = 1
n
n
�
i=1
∂
∂˘ut
i
˘gt(˘ut)i = 1
n
n
�
i=1
∂
∂˘ut
i
�
Ξ−1/2gt(Ξ−1/2˘ut)
�
i
= 1
n
n
�
i=1
n
�
j=1
n
�
k=1
(Ξ−1/2)i,j(Ξ−1/2)k,i
∂gt(ut)j
∂ut
k
= 1
n Tr((∇gt)Ξ−1).
The second equality follows since by (F.49),
˘gt(˘ut) = Ξ−1/2gt(Ξ−1/2˘ut). | [
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] |
k
= 1
n Tr((∇gt)Ξ−1).
The second equality follows since by (F.49),
˘gt(˘ut) = Ξ−1/2gt(Ξ−1/2˘ut).
The third equality is by the chain rule for derivatives (Proposition G.6). A similar computation gives
bt+1 = 1
n Tr((∇ft+1)Σ−1).
We now see that under the change of variables (F.48) and (F.49), the AMP iteration (E.1) can be cast
as (F.2). Therefore, applying the same change of variables to the state evolution of (F.2) will produce
the state evolution of (E.1). We describe the required modifications below.
48 | [
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] |
the state evolution of (E.1). We describe the required modifications below.
48
The state evolution result in Proposition F.1 for the AMP in (F.2) says that the iterates
v∗, ˘v1, ˘v2, · · · , ˘vt+1 ∈ Rd and u∗, ˘u0, ˘u1, · · · , ˘ut ∈ Rn converge (in the sense of (F.16a) and (F.16b))
respectively to V ∗, ˘V1, ˘V2, · · · , ˘Vt+1 ∈ Rd and U ∗, ˘U1, ˘U2, · · · , ˘Ut ∈ Rn. Recall that AMPs (E.1)
and (F.2) operate on the following matrices respectively:
�A = λ
n(Ξ−1/2u∗)(Σ−1/2v∗)⊤ + �
W,
Ξ−1AΣ−1 = λ | [
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�A = λ
n(Ξ−1/2u∗)(Σ−1/2v∗)⊤ + �
W,
Ξ−1AΣ−1 = λ
n(Ξ−1u∗)(Σ−1v∗)⊤ + Ξ−1/2�
WΣ−1/2.
In view of (F.48), to obtain the analogous state evolution result for the AMP in (E.1), the definition
(F.3) of ˘Ut, ˘Vt+1 should be multiplied by Ξ−1/2, Σ−1/2 respectively. This gives the new definition of
Ut, Vt+1 in (E.6). By the relation (F.49), the parameters ˘µt, ˘νt+1 in ˘Ut, ˘Vt+1 should be modified as fol-
lows: replace ˘ft( ˘Vt), ˘gt( ˘Ut) in the recursive equations (F.6) to (F.8) with Σ−1/2ft(Vt), Ξ−1/2gt(Ut). | [
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] |
This gives the new definition of µt, νt in (E.5). Similar operations map equations (F.9) to (F.12),
(F.14) and (F.15) to equations (E.3) and (E.4). Finally, under the new definition of Ut, Vt+1, the
convergence result (F.16a) and (F.16b) translates to (E.8a) and (E.8b), which completes the proof.
G
Auxiliary lemmas
Proposition G.1 (Gaussian integral). Let A ∈ Rd×d be a positive-definite matrix and b ∈ Rd. Then
�
Rd exp
�
−1
2x⊤Ax + b⊤x
�
dx =
�
(2π)d
det(A) exp
�1
2b⊤A−1b
�
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] |
�
Rd exp
�
−1
2x⊤Ax + b⊤x
�
dx =
�
(2π)d
det(A) exp
�1
2b⊤A−1b
�
.
Proposition G.2. Let V ∼ Q⊗d where Q is a fixed distribution on R with mean 0. Let B ∈ Rd×d
denote a sequence (indexed by d) of deterministic matrices such that the empirical spectral distribution
of 1
dB converges to the law of a random variable B. Then
lim
d→∞
1
dE
�
V ⊤BV
�
= E
�
V
2�
E
�
B
�
where V ∼ Q.
Proposition G.3. Let
(W1, W2) ∼ N
��
0d
0d
�
,
�
σ2
1
ρ
ρ
σ2
2
�
⊗ Id
�
. | [
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