Recovery of Third Order Tensors via Convex Optimization

Transcription

1 Recvery f Third Order Tesrs via Cvex Optimizati Hlger Rauhut RWTH Aache Uiversity Lehrstuhl C für Mathematik (Aalysis) Ptdriesch Aache Germay rauhut@mathcrwth-aachede Željka Stjaac RWTH Aache Uiversity Lehrstuhl C für Mathematik (Aalysis) Ptdriesch Aache Germay stjaac@mathcrwth-aachede Abstract We study recvery f lw-rak third rder tesrs frm uderdetermied liear measuremets This atural extesi f lw-rak matrix recvery via uclear rm miimizati is challegig sice the tesr uclear rm is i geeral itractable t cmpute T vercme this bstacle we itrduce hierarchical clsed cvex relaxatis f the tesr uit uclear rm ball based s-called theta bdies a recet ccept frm cmputatial algebraic gemetry Our tesr recvery prcedure csists i miimizati f the resultig ew rms subject t the liear cstraits Numerical results recvery f third rder lw-rak tesrs shw the effectiveess f this ew apprach I INTRODUCTION AND MOTIVATION The recetly itrduced thery f cmpressive sesig eables the recvery f sparse vectrs frm udersampled measuremets via efficiet algrithms such as 1-rm miimizati This ccept exteds t lw-rak matrix recvery where the aim is t recstruct a matrix X R 1 f rak at mst r mi { 1 } frm liear measuremets y = (X) where : R 1! R m with m 1 Hwever the atural apprach f fidig the sluti f the ptimizati prblem mi rak (Z) st (Z) = y (1) ZR 1 is NP-hard Nevertheless it has bee shw that uder suitable cditis the sluti f the cvex ptimizati prblem mi ZR 1 kzk st (Z) =y () recstructs X exactly Here k k detes the uclear rm f a matrix ie kzk = P mi{ 1 } i=1 i where { i } mi{1} i=1 is the set f sigular values f a matrix Z The required umber f Gaussia measuremets scales like m Crmax{ 1 } see [3] [] Here we csider a further extesi f cmpressive sesig t the recvery f lw-rak tesrs X R 1 d frm a small umber f liear measuremets y = (X) where : R 1 d! R m with m 1 d Agai we are led t csider the rak-miimizati prblem mi ZR 1 d rak (Z) st y = (Z) (3) /15/$3100 c 015 IEEE Differet tis f the tesr rak crrespdig t differet decmpsitis are available [1] The CP-rak f a arbitrary tesr X R 1 d is the smallest umber f rak e tesrs that sum up t X where a rak e tesr is f the frm A = u 1 u u d r elemet-wise A i1i i d = u 1 i 1 u i u d i d Expectedly the prblem (3) is NP hard [14] Althugh it is pssible t defie a aalg f the uclear rm k k fr tesrs ad csider the miimizati prblem mi kzk st y = (Z) ZR 1 d the cmputati f k k ad thereby this prblem is NP hard [14] as well fr tesrs f rder d 3 T vercme this difficulty previus appraches t lw-rak tesr recvery ad tesr cmpleti via cvex ptimizati [6] ad [16] fcus the Tucker decmpsiti ad the crrespdig rm defied as sum f uclear rms f the ufldigs (see belw fr the ti f ufldig) Hwever it has bee shw i [18] that i this sceari the ecessary umber f measuremets fr recvery f rak r-tesrs via Gaussia measuremet esembles scales expetially i the dimesi ie m Cr d 1 where r =(r r r) R d Other appraches t ecessarily based cvex ptimizati iclude iterative hard threshldig algrithms [0] [1] recvery by Riemaia ptimizati [13] ad by the ALS methd [10] Hwever all these appraches csider the Tucker the tesr trai [19] r i geeral the hierarchical Tucker decmpsiti [8] A recet paper [4] csiders tesr cmpleti via tesr uclear rm miimizati ad gives theretical aalysis with a sigificat imprvemet the ecessary umber f measuremets fr recvery f rak r tesrs Hwever slvig this ptimizati prblem as already metied befre is NP-hard As a alterative apprach we build ew tesr rms k - rms which rely s-called theta bdies [17] [7] We treat each etry f a tesr as a plymial variable The idea is t defie a plymial ideal J which vaishes the set R (J) f all rak-e rm-e third rder tesrs This is achieved by takig all mirs f rder tw f every ufldig (t satisfy the rak-e cditi) ad a plymial P ijk x ijk 1 (t satisfy the uit rm cditi) as its basis I this sceari

2 the cvex hull f the set R (J) will be the uit tesr uclear rm ball The uit k -rm balls frm a set f hierarchical relaxatis f the set cv ( R (J)) that is f the tesr uit uclear rm ball We fcus third rder tesrs ad the largest uit rm ball i this set the uit 1 -rm ball We prvide semidefiite prgrams (SDPs) fr cmputig the 1 -rm f a give third rder tesr ad fr recvery f lw-rak third rder tesrs via 1 -rm miimizati The perfrmace f the latter SDP is illustrated with umerical results II NOTATION We dete vectrs with small bld letters matrices ad tesrs with capital bld letters ad sets with capital calligraphic letters The cardiality f a set S will be deted by S ad with [m] we dete the set {1 m} With cv(s) we dete the cvex hull f the set S III TENSORS We are iterested i recvery f lw-rak third rder tesr X R 1 3 frm uderdetermied liear measuremets We defie the Frbeius rm f a tesr X R 1 3 as v ux 1 X X 3 kxk F = t Xi 1i i 3 i 1=1 i =1 i 3=1 A third rder tesr X R 1 3 is a rak e tesr if there exist vectrs u 1 R 1 u R u 3 R 3 such that X = u 1 u u 3 r elemet-wise X i1i i 3 = u 1 i 1 u i u 3 i 3 The CP-rak (r caical rak ad i the fllwig just rak) f a tesr X R 1 3 is the smallest umber f rak e tesrs that sum up t X The the aalg f the matrix uclear rm fr tesrs is ( rx rx kxk =mi c k : X = c k u 1k u k u 3k k=1 k=1 r N u ik =1 fr all i [3] k [r] Hwever i the tesr case cmputig the caical rak f a tesr as well as cmputig the uclear rm f a tesr is i geeral NP-hard see [9] [15] The -th ufldig X {} R Q k[3]\{} k f a tesr X R 1 3 is a matrix defied elemet-wise as X {} i(i 1i 1 i+1 i 3) = X i 1i i 3 We fte use MATLAB tati Specifically fr a third rder tesr X R 1 3 we dete the secd rder subtesr i R 1 btaied by fixig the last idex i 3 t k by X(: :k) Vectrizati f a tesr X R 1 3 cverts a tesr it a clum vectr vec(x) R 13 The rderig f the elemets i vec(x) is t imprtat as lg as it is csistet IV THETA BODIES Sice the tesr uclear rm is i geeral NP-hard t cmpute [14] we prpse a alterative apprach We itrduce ew tesr rms via clsed cvex relaxatis (theta bdies) f the tesr uit uclear rm The cmputati f these rms requires the fllwig defiitis ad tls frm algebraic gemetry Fr a -zer plymial f = P a x i R [x] = R [x 1 x x ] ad a mmial rder we defie a) the multidegree f f by multideg (f) = max Z 0 : a 6= 0 b) the leadig cefficiet f f by LC (f) =a multideg(f) R c) the leadig mmial f f by LM (f) =x multideg(f) d) the leadig term f f by LT (f) = LC (f) LM (f) I this paper we use the graded reverse lexicgraphic rderig (grevlex) rderig see [4] Let J be a plymial ideal i R [x] =R [x 1 x x ] The real algebraic variety f the ideal R (J) is the set f all pits x R where the ideal vaishes ie R (J) ={x R : f(x) =0fr all f J} By Hilbert s basis therem we ca assume that the plymial ideal J is geerated by the fiite set F = {f 1 f f k } f plymials i R [x] We write E J = hf 1 f f k i = D{f i } i[k] r simply J = hfi The its real algebraic variety is the set R (J) ={x R : f i (x) =0 fr all i [k]} Greber bases are crucial fr cmputatis with plymial ideals Defiiti 1 (Greber basis): Fix a mmial rder A basis G = {g 1 g g s } f a plymial ideal J R [x] is a Greber basis (r stadard basis) if fr all f R [x] there exist uique r R [x] ad g J such that f = g + r ad mmial f r is divisible by ay f the leadig mmials i G ie by ay f the LM (g 1 ) LM (g ) LM (g s ) A Greber basis is t uique but the reduced versi (defied belw) is Defiiti : Fix a mmial rder The reduced Greber basis fr a plymial ideal J R [x] is a Greber basis G = {g 1 g g s } fr J such that 1) LC(g i )=1 fr all i [s] ) LM(g i ) des t divide LM(g j ) fr all i 6= j Thrughut the paper R [x] k detes the set f plymials f degree at mst k The fllwig defiiti is cetral fr the defiiti f theta bdies [17] [7] which will be used belw fr defiig ur ew tesr rms Defiiti 3 ( [7]): Let J be a ideal i R [x] A plymial f R [x] is k-ss md J if there exists a fiite set f

3 plymials h 1 h h t R [x] k such that f P t j=1 h j P t md J ie if f j=1 h j J We recall that a degree e plymial is als kw as liear plymial Defiiti 4 (Theta bdy [7]): Let J R [x] be a ideal Fr a psitive iteger k the k-th theta bdy f a ideal J is TH k (J) :={x R : f (x) 0 By defiiti fr every liear f that is k-ss md J} TH 1 (J) TH (J) cv ( R (J)) (4) The theta bdies are clsed cvex sets while cv ( R (J)) may t be clsed We say that a ideal J R [x] is TH k - exact if TH k (J) equals t the clsure f cv ( R (J)) ie t cv ( R (J)) Guaratees cvergece ca be fud i [7] Hwever t ur kwledge e f the existig guaratees apply i ur case Checkig whether a give plymial is k-ss md J usig this defiiti requires kwledge f all liear plymials that are k-ss md J T vercme this difficulty we eed a alterative descripti f TH k (J) As i [1] ad [7] we assume that there are liear plymials i the ideal J ad we csider ly the mmial bases B f R [x] /J The degree f a equivalece class f + J deted by deg (f + J) is the smallest degree f a elemet i the class Each elemet i the basis B = {f i + J} f R [x] /J is represeted by the plymial f i such that deg (f i + J) =deg(f i ) We assume that B = {f i + J} is rdered s that f i+1 grevlex f i We defie the set B k := {f + J B:deg(f + J) k} Defiiti 5 ( -basis [7]): Let J R [x] be a ideal A basis B = {f 0 + J f 1 + J } f R [x] /J is called a -basis 1) if B 1 = {1+J x 1 + J x + J}; ) if deg (f i + J) deg (f j + J) k the f i f j + J is i the R-spa f B k Fr cmputig a -basis f R [x] /J we first eed t cmpute the reduced Greber basis G = {g 1 g s } f the ideal J with respect t a rderig which first cmpares the ttal degree (fr example the grevlex rderig) The a set B = {f 0 + J f 1 + J } will be a -basis f R [x] /J if it ctais all f i + J such that 1) f i is a mmial ) f i is t divisible by ay f the mmials i the set {LT(g i ):i[s]} Fr a -basis B = {f i + J} f R [x] /J we defie [x] Bk t be the clum vectr frmed by all elemets f B k i rder The [x] Bk [x] T B k is a square matrix idexed by B k ad its (i j)-etry is equal t f i f j + J By hypthesis the etries f [x] Bk [x] T B k lie i the R-spa f B k Let { be the uique set f real umbers such that f ìj} P i f j + J = :f+jb k ìj (f + J) Defiiti 6 (k-th cmbiatrial mmet matrix [7]): Let J B ad { ìj} be as abve Let y be a real vectr idexed by B k with y 0 =1 where y 0 is the first etry f y idexed by the basis elemet 1+J The k-th cmbiatrial mmet matrix M Bk (y) f J is the real matrix idexed by B k whse (i j)-etry is [M Bk (y)] ij = P:f+JB k ìj y Fially the fllwig therem gives us a alterative descripti f the theta bdies Therem 1 ( [7]): The k-th theta bdy f J TH k (J) is the clsure f Q Bk (J) = R y R B k : M Bk (y) 0y 0 =1 where R detes the prjecti the variables y 1 y V THE TENSOR k -NORM Let us w prvide the auced hierarchical clsed cvex relaxatis f the tesr uit uclear rm ball These lead t tesr rms that have t bee csidered befre - at least up t ur kwledge Remark 1: A similar but smewhat easier apprach t the e explaied i detail fr third rder tesrs belw ca be used t defie clsed cvex relaxatis f the matrix uclear uit rm ball I this sceari all these relaxatis cicide with the rigial matrix uit uclear rm ball [] I the tesr case we cat expect t btai equality f all relaxatis t the tesr rm because cmputig the latter wuld the t be NP-hard i geeral Still the equality i the matrix case suggests that these relaxatis are useful apprximatis t the uclear rm i the tesr case First recall that the set f all mirs f rder tw f a matrix A is {det(a IJ ):I [m] J [] I = J =} where A IJ R detes the submatrix f A btaied by deletig all rws i [m] \I ad all clums j [] \J Fr tatial purpses we defie the fllwig plymials i R [x] =R [x 111 x 11 x 1 3 ] 1 (x) = x ijk xîĵˆk + x iĵˆk x îjk ijkîĵˆk S 1 (x) = x ijk xîĵˆk + x iĵk x îjˆk ijkîĵˆk S 3 (x) = x ijk xîĵˆk + x ijˆk x îĵk ijkîĵˆk S 3 X 1 X X 3 g(x) = x ijk 1 i=1 j=1 k=1 with the crrespdig sets f subscripts S 1 = ijkîĵˆk : i<î j < ĵ k ˆk S = ijkîĵˆk : i î j < ĵ k < ˆk S 3 = ijkîĵˆk : i<î j ĵ k < ˆk where i î [ 1 ] j ĵ [ ] ad k ˆk [ 3 ] These plymials crrespd t the rder tw mirs f the ufldigs f a tesr Lemma 1 ( []): A tesr X R 1 3 is a rak e Frbeius rm e tesr if ad ly if g(x) =0 ad (X) =0 fr all ijkîĵˆk S [3]

4 A third rder tesr X R 1 3 is a rak-e tesr if ad ly if all three ufldigs X {1} R 1 3 X {} R 13 ad X {3} R 3 1 are rak-e matrices Ntice that (X) = 0 fr all ijkîĵˆk S is equivalet t the statemet that the -th ufldig X {} is a rak e matrix ie that all its mirs f rder tw vaish Our aim is t fid a relaxati f the tesr uit uclear rm ball I rder t apply the ccept f theta bdies we eed t cme up with a plymial ideal J 3 R [x] = R [x 111 x 11 x 1 3 ] such that its algebraic variety is f the frm R (J 3 )= x : g (x) =0ad (x) =0 fr all ijkîĵˆk S [3] T this ed we defie the plymial ideal J 3 = hg 3rd i where G 3rd = : ijkîĵˆk S [3] [{g} (5) Therem ( []): The basis G 3rd defied i (5) frms the reduced Greber basis f the ideal J 3 = hg 3rd i with respect t the grevlex rder Based the mmet matrix M B1 (y) the 1 -rm f a tesr X ca be cmputed via the semidefiite prgram where mi t st M(t y X) 0 ty X 1 X X 3 M(t y Z) =tm 0 + Z ijk M ijk + i=1 j=1 k=1 9X M i X i= j=1 ym i j (6) with = P i k=3 M(k 1) + j ad with matrices M 0 M ijk M i j R(13+1) (13+1) as defied i Table II We the prpse t recver a lw-rak tesr X frm uderdetermied liear measuremets b = (X) via k - miimizati ie which is equivalet t arg mi tyz arg mi Z kzk k st (Z) =b st M(t y Z) 0 ad (Z) =b Remark : As already metied befre the abve Greber basis G 3rd ca be btaied by takig all mirs f rder tw f all three ufldigs f the tesr X R 1 3 (t csiderig the same mir twice) Oe might thik that the 1 -rm btaied i this way crrespds t a (weighted) sum f the uclear rms f the ufldigs which has already bee treated i the papers [6] [11] That is there exist R such that kx {1} k + kx {} k + kx {3} k = kxk 1 The example f a cubic tesr X R preseted i Table I shws that this is t the case Frm the first ad the secd tesr i Table I we btai =0 Similarly the first ad X R kx {1} k kx {} k kx {3} k kxk p p p p p p TABLE I TENSORS X R ARE REPRESENTED IN THE SECOND COLUMN AS X =[X (: : 1) X (: : )] THE THIRD FOURTH AND FIFTH COLUMN REPRESENT THE NUCLEAR NORMS OF THE FIRST SECOND AND THE THIRD UNFOLDING OF A TENSOR X RESPECTIVELY THE LAST COLUMN CONTAINS THE 1 -NORM WHICH WAS COMPUTED NUMERICALLY the third tesr ad the first ad furth tesr give =0 ad =0 respectively Thus the 1 -rm is t a weighted sum f the uclear rms f the ufldigs I additi the last tesr shws that the 1 -rm is t the maximum f the rms f the ufldigs Remark 3 (cmplexity): The psitive semidefiite matrix M used either fr lw-rak tesr recvery r cmputig the 1 - rm f a third rder tesr X R is f dimesi (1+ 3 ) (1 + 3 ) If a := 3 detes the ttal umber f etries f a tesr X the y is a vectr f at mst a(a+1) O(a ) variables Therefre the semidefiite prgram fr cmputig the 1 -rm as well as the semidefiite prgram fr lw-rak tesr recvery has plymial cmplexity We remark that this apprach fr defiig relaxatis f uclear rm ca als be exteded t geeral dth rder tesrs see [] VI NUMERICAL EXPERIMENTS I this secti we prvide recvery results fr third rder tesrs btaied by miimizig the 1 -rm We directly build the matrix M defied i (6) where matrices M 0 M ijk M i j are listed i Table II Due t symmetry ly the -zer etries f the upper triagle part f these matrices are specified The elemets f the -basis are give via their represetative i the secd clum The fucti f : Z 3! R is defied as f (i j k) = (i 1) 3 +(j 1) 3 + k +1 The last clum lists the set T p1p p 3 = {(i j k (p i ) iq ):1i<p j< p 1 k<p 3 3 } where Q := {i : p i 6= i } ad ˆ i = i +1 fr all i [3] We preset the recvery results fr third rder tesrs X R 1 3 i Table III We csider cubic ad cubic tesrs f raks e ad tw Fr fixed dimesis 1 3 umber f measuremets m ad fixed rak we perfrmed 00 simulatis We say that ur algrithm succeeds t recver the rigial tesr X R 1 3 if the elemet-wise differece betwee the rigial tesr X 0 ad the tesr X = arg mi Z: (Z)= (X) kzk 1 is at mst 10 6 With m max we

5 M -basis M pq (p q) Rage f i î j ĵk ˆk M (1 1) 1 ( ) M ijk x ijk 1 (1f(i j k)) Tˆ1 ˆ ˆ 3 M f x ijk 1 ( ) 1 (f(i j k)f(i j k)) Tˆ1 ˆ ˆ 3 \{(1 1 1)} M 3 f 3 x x iĵk ijˆk 1 (f(i j k)f(i ĵ ˆk)) Tˆ1 ĵˆk 1 (f(i j ˆk)f(i ĵk)) Tˆ1 ĵˆk M 4 f 4 x ijk xîĵˆk 1 (f(i j k)f(î ĵ ˆk)) Tîĵˆk 1 (f(i ĵk)f(î j ˆk)) Tîĵˆk 1 (f(i ĵ ˆk)f(î j k)) Tîĵˆk 1 (f(i j ˆk)f(î ĵk)) Tîĵˆk M 5 f 5 x ijk xîjˆk 1 (f(i j k)f(î j ˆk)) Tîˆ ˆk 1 (f(i j ˆk)f(î j k)) Tîˆ ˆk M 6 f 6 x ijk xîĵk 1 (f(i j k)f(î ĵk)) Tîĵˆ 3 1 (f(i ĵk)f(î j k)) Tîĵˆ 3 M 7 f 7 xîjk x ijk 1 (f(i j k)f(î j k)) Tîˆ ˆ 3 M 8 f 8 x iĵk x ijk 1 (f(i j k)f(i ĵk)) Tˆ1 ĵˆ 3 M 9 f 9 x ijˆkx ijk 1 (f(i j k)f(i j ˆk)) Tˆ1 ˆ ˆk TABLE II MATRICES USED IN THE DEFINITION OF M IN (6) dete the maximal umber f measuremets fr which we d t recvery ay f the 00 geerated tesrs ad m mi detes the miimal umber f measuremets fr which we recvered all 00 tesrs (the success f recvery is 00/00) We use liear mappigs : R 1 3! R m defied with tesrs k R 1 3 via [ (X)] (k) =hx ki fr k [m] We chse the k as stchastically idepedet tesrs with iid Gaussia N 0 1 m etries We geerate tesrs X R 1 3 f rak r =1via their decmpsiti If X = u v w is its CP-decmpsiti each etry f the vectrs u v ad w is take idepedetly frm the rmal distributi N (0 1) Rak tw tesrs are btaied by summig tw rak e tesrs The last clum i Table III represets the umber f idepedet measuremets which are always eugh fr the recvery f a tesr f a arbitrary rak 1 3 rak m max m mi TABLE III NUMERICAL RESULTS FOR LOW-RANK TENSOR RECOVERY IN R 1 3 We used MATLAB (R008b) fr these umerical experimets icludig SeDuMi 13 fr slvig the semidefiite prgrams ACKNOWLEDGEMENTS We wuld like t thak Berd Sturmfels Daiel Plauma ad Shmuel Friedlad fr helpful discussis ad useful iputs t this paper We wuld als like t thak James Sauders ad Hamza Fawzi fr deeper isights it the matrix case sceari We ackwledge fudig by the Eurpea Research Cucil thrugh the Startig Grat StG 5896 REFERENCES [1] G Blekherma P A Parril R R Thmas Semidefiite Optimizati ad Cvex Algebraic Gemetry SIAM 013 [] E J Cadès Y Pla Tight racle buds fr lw-rak matrix recvery frm a miimal umber f radm measuremets IEEE Trasactis Ifrmati Thery 57(4): [3] V Chadrasekara B Recht P A Parril A S Willsky The Cvex Gemetry f Liear Iverse Prblems Fudatis f Cmputatial Mathematics 1(6): [4] D A Cx J Little D O Shea Ideals Varieties ad Algrithms: A Itrducti t Cmputatial Algebraic Gemetry ad Cmmutative Algebra Spriger 1991 [5] D A Cx J Little D O Shea Usig Algebraic gemetry Spriger Graduate texts i mathematics [6] S Gady B Recht I Yamada Tesr cmpleti ad lw--rak tesr recvery via cvex ptimizati Iverse Prblems 7(): 19pp 011 [7] J Guveia P A Parril R R Thmas Theta Bdies fr Plymial Ideals SIAM Jural Optimizati 0(4): [8] W Hackbusch S Kuh A ew scheme fr the tesr represetati J Furier Aal Appl 15(5): [9] J Håstad Tesr rak is NP-cmplete Jural f Algrithms 11(4): [10] S Hltz T Rhwedder R Scheider The alteratig liear scheme fr tesr ptimizati i the tesr trai frmat SIAM J Sci Cmput 34(): A683 A [11] B Huag C Mu D Gldfarb J Wright Prvable Lw-Rak Tesr Recvery T appear i Pacific Jural f Optimizati 015 [1] T Klda B W Bader Tesr Decmpsitis ad Applicatis SIAM Rev 51(3): [13] D Kresser M Steilecher B Vadereycke Lw-rak tesr cmpleti by Riemaia ptimizati BIT Numerical Mathematics 54(): [14] L-H Lim C J Hillar Mst Tesr Prblems are NP-Hard Jural f the ACM (JACM) 60(6): 45pp 013 [15] L-H Lim S Friedlad Cmputatial cmplexity f tesr uclear rm arxiv [16] J Liu P Musiaski P Wka J Ye Tesr Cmpleti fr Estimatig Missig Values i Visual Data IEEE Trasactis Patter Aalysis ad Machie Itelligece 35(1): [17] L Lvász O the Sha capacity f a graph IEEE Tras Ifrm Thery 5(1): [18] C Mu B Huag J Wright D Gldfarb Square deal: lwer buds ad imprved relaxatis fr tesr recvery I: Prceedigs f the Iteratial cferece machie learig 014 [19] I V Oseledets Tesr-trai decmpsiti SIAM J Sci Cmput 33(5): [0] H Rauhut R Scheider Ž Stjaac Lw-rak tesr recvery via Iterative hard threshldig I: Prceedigs f the Iteratial Cferece Samplig Thery ad Applicatis 013 [1] H Rauhut R Scheider Ž Stjaac Tesr cmpleti i hierarchical tesr represetatis T be published i Cmpressed Sesig ad Its Applicatis (edited by H Bche R Calderbak G Kutyik J Vybiral) 015 [] H Rauhut Ž Stjaac Lw-rak tesr recvery via cvex ptimizati i preparati 015 [3] B Recht M Fazel P A Parril Guarateed miimum-rak sluti f liear matrix equatis via uclear rm miimizati SIAM Rev 5(3): [4] M Yua C-H Zhag O tesr cmpleti via uclear rm miimizati arxiv