Invertibility and Robustness of Phaseless Reconstruction

INVER TIBILITY AND R OBUSTNESS OF PHASELESS RECONSTRU CTIO N RADU BALAN AND Y ANG W ANG Abstra ct. This pap er is concerned with the question o f reconstructing a vec tor in a finite-dimensional real Hilb ert space when only the magnitudes of the coefficients of the vector under a redundant li near map are known. W e analyze v arious Lipsc hitz b ound s of the nonlinear analysis map and w e establish theoretical p erformance bou n ds of an y recon- struction algorithm. W e show th at robust and stable reconstruction requires add itional redundancy than the critical threshold. 1. Introduction This pap er is concerned w ith the question of reconstructing a v ector x in a finite- dimensional r eal Hilb ert sp ace H of dim en sion n when only the magnitud es of the co ef- ficien ts of the v ector un d er a r ed undant linear map are kno wn. Sp ecifically our problem is to reconstruct x ∈ H up to a global phase factor from the magnitudes {|h x, f k i| , 1 ≤ k ≤ m } where F = { f 1 , · · · , f m } is a frame (complete s ystem) for H . A previous pap er [6] describ ed the imp ortance of this problem to signal pro cessing, in particular to the analysis of sp eec h . Of particular int erest is the case when the co efficients are obtained from a Windo w ed F our ier T ransform (also known as S h ort-Time F our ier T rans- form), or an Undecimated W a v elet T ransform (in audio and image signal pro cessing). A similar problem app ears in Quan tum Information (QI) literature (see e.g. [28]). Ho w- ev er some imp ortant differences are notable: first the unknown ob jects to b e reconstructed are qu an tum states (meaning nonnegativ e symmetric op erators of unit trace); secondly , measuremen ts are p erformed by taking Hilb ert-Schmidt inner pro ducts against some (n on- negativ e) symmetric op erators of rank not necessarily one. In Q I language our problem is to reconstruct rank on e nonnegativ e symmetric op erators fr om measurements against a set of rank one nonnegativ e symmetric op erators. While [6] presents some necessary and su fficien t conditions for reconstruction, the general problem of find ing fast/efficien t algo rithms is still op en. In [4] w e describ e one solution in the case of STFT co efficien ts. R. Balan was sup p orted in part by NSF DMS-1109498. Y. W ang wa s supp orted in part by NS F DMS- 1043032 , and by AFOSR F A9550-12-1-0455. 1 2 R. BALAN AND Y. W ANG F or v ectors in real Hilb ert spaces, the reconstruction pr oblem is easily sho wn to b e equiv alen t to a com binatorial problem. In [7 ] this pr oblem is fur ther p ro v ed to b e equiv alent to a (noncon v ex) optimization pr ob lem. A differen t appr oac h (which we called the algebr aic appr o ach ) was p rop osed in [3]. While it applies to b oth real and complex cases, noisless and n oisy cases, the app roac h requires solving a linear system of size exp onen tially in sp ace dimension. Th is algebraic approac h generalizes the app roac h in [8] wh er e reconstruction is p erformed with complexit y O ( n 2 ) (plus computation of the p rincipal eigen v ecto r for a matrix of s ize n ). Ho wev er this m etho d requires m = O ( n 2 ) frame v ectors. Recen tly the authors of [15] d ev elop ed a conv ex optimization algorithm (a SemiDefinite Program called PhaseLift ) and pro v ed its abilit y to p erform exact reconstru ction in the absence of noise, as well as its stabilit y under noise conditions. In a separate pap er [16], the authors fur th er d ev eloped a s im ilar algorithm in the case of wind ow ed DFT transforms. Inspired by the PhaseLift and MaxCut algorithms, but operating in the co efficien ts space, the authors of [44] prop osed a SemiDefinite Pr ogram called PhaseCut . They show th e algorithm yields the exact solution in the absence of noise und er similar conditions as PhaseLift. The pap er [5] presents an iterativ e regularized least-square algorithm for inv ertin g the nonlinear map and compares its p erformance to a Cramer-Rao lo w er b ound f or this pr oblem in th e real case. Th e p ap er also present s some new injectivit y r esults w hic h are incorp orated in to this pap er. A different approac h is prop osed in [1]. There the auth ors use a 4-term p olarization iden tit y together w ith a family of sp ectral expander graphs to design a fr ame of b ound ed redund ancy ( m n ≤ 236) th at yields an exact reconstruction algorithm in the absence of noise. The authors of [23 ] study several robustness b ounds to the p hase reco v ery p roblem in the real case. Ho w ev er their approac h is d ifferen t than ours in several resp ects. First th ey consider a probabilistic setup of this pr oblem, wh ere data x and frame v ectors f j ’s are random v ectors with pr obabilities from a class of subgaussian distributions. Add itionally , their fo cus is on classes of k -sparse signals. Their resu lts sho w that, with high probabil- it y , reco v er y is p ossible fr om a n umber of measuremen ts m that has a similar asymptotic b ehavi or with r esp ect to n and k as in the case of linear measurements (that is w ith the phase). In our pap er we analyze stabilit y b ound s of reconstruction for a fixed fr ame using deterministic an alytic to ols. After that we present asymp totic b eha vior of these b ounds for random frames. Finally , the authors of [9] analyze the ph aseless reconstruction pr oblem f or b oth the r eal and complex case. In the r eal case th e authors obtain the exact upp er Lipschitz constan t for the nonlinear map α F , namely √ B wh ere B is the up p er frame b ound. F or the lo w er Lipsc hitz constant, they giv e an estimate b et w een tw o computable singular eienv alues. Our results ha v e o v erlaps with their results somewhat h ere. Ho w ev er, in this pap er we impr o v e INVER TIBILITY AND ROBUSTNESS OF PHASELESS RECON STRUCTI ON 3 the impro v e the lo we r Lip sc hitz constan t b y giving its exa ct v alue. There a re some significan t differences b etw een th is p ap er and [9]. In addition to studyin g of the Lipschitz prop erty of th e map α F w e f o cus also on tw o r elated b ut d ifferen t settings. First we stud y the robustness of the r econstruction give n a fixed err or allo w ance in measurements. S econd we also consider the Lip sc hitz pr op ert y of th e map α F 2 . The authors of [9] p oin t out that the map α F 2 is n ot bi-Lipschitz. Ho we v er in our pap er w e sho w α F 2 b ecomes b i-Lipsc hitz for a differen t metric on the domain. With this metric (the one indu ced by the n uclear norm on the set of symmetric op erators) the n on lin ear m ap α F 2 is bi-Lipsc hitz with constan ts indicated in Theorem 4.5. F urthermore the same conclusion h olds tru e in the complex case, although this will b e stud ied elsewhere. The organiza tion of the pap er is as follo ws. Section 2 f ormally defines the problem and reviews existing inv ers ion results in the r eal case. Section 3 establishes inf ormation theoretic p erformance b ounds , namely the Cramer-Rao lo we r b ound. Section 4 con tains r obustness measures of any reconstruction algorithm. Section 5 presents a sto c hastic analysis of these b ound s, and is follo w ed by referen ces. 2. Back gr ound Let us denote by H = R n the n-dim en sional real Hilb ert space R n with scalar pr o duct h , i . Let F = { f 1 , · · · , f m } b e a spanning set of m v ecto rs in H . In finite dimension (as it is the case here) su c h a set forms a fr ame . In th e infinite dimen s ional case, the concept of frame inv olve s a stronger p rop erty than completeness (see for instance [17]). W e review additional term in ology and prop erties wh ic h remain still true in the infinite dimensional setting. The set F is a frame if and only if there are tw o p ositiv e constan ts 0 < A ≤ B < ∞ (called frame b oun d s) so that (2.1) A k x k 2 ≤ m X k =1 |h x, f k i| 2 ≤ B k x k 2 . When w e can c ho ose A = B the frame is said tight . F or A = B = 1 the f rame is called Parseval . Th e fr ame matrix corresp ondin g to F is d efined as F = [ f 1 , f 2 , . . . , f m ] with the v ectors f j ∈ F as its column s. W e sh all fr equen tly id entify F with its corresp on d ing frame matrix F . The largest A and sm allest B in (2.1) are called the lower fr ame b ound and upp er fr ame b ound of F , and they are given by (2.2) A = λ max ( F F ∗ ) = σ 2 1 ( F ) , B = λ min ( F F ∗ ) = σ 2 n ( F ) where λ max , λ min denote the largest and sm allest eigen v alues r esp ectiv ely , w h ile σ 1 , σ n de- note the first and n -th singular v alues resp ectiv ely . A set of v ectors F of the n -d im en sional Hilb ert space H is said to b e ful l sp ark if an y su b set of n vec tors is linearly indep endent. F or a ve ctor x ∈ H , the collection of co efficien ts {h x, f j i : 1 ≤ j ≤ m } represen ts the analysis map of vect or x giv en by the fr ame F , and fr om which x can b e completely reconstructed. In the ph aseless reconstruction problem, w e ask the f ollo wing qu estion: C an 4 R. BALAN AND Y. W ANG x b e reconstructed from {|h x, f j i| : 1 ≤ j ≤ m } ? Consider the follo wing equiv alence relation ∼ on H : x ∼ y if and only if y = cx for some unimo du lar constan t c , | c | = 1. S ince we fo cus on the real vecto r s pace H = R n , we ha v e x ∼ y if and only if x = ± y . Clearly the phaseless reconstruction problem cannot distinguish x and y if x ∼ y , so w e will b e lo oking at reconstru ction on ˆ H := H / ∼ = R n / ∼ wh ose elemen ts are giv en by equiv alent classes ˆ x = { x, − x } for x ∈ R n . The analogous analysis map for phaseless r econstru ction is the follo w in g nonlin ear map (2.3) α F : ˆ H → R m + , α F ( ˆ x ) = [ |h x, f 1 i| , |h x, f 2 i| , . . . , |h x, f m i| ] T . Note that α F can also b e viewed as a map from R n to R m + . Throughout the p ap er we w ill not mak e an explicit distinction u n less suc h a d istinction is necessary . Th us th e phaseless reconstruction problems aims to reconstruct ˆ x ∈ ˆ H from the map α F ( x ). W e sa y a frame F is phase r etrievable if one can reconstruct ˆ x ∈ ˆ H for all ˆ x , or in other words, α F is inj ectiv e on ˆ H . The main ob jectiv e of this pap er is to analyze robustn ess and stabilit y of the inv er s ion map, and to giv e p erformance b oun ds of an y reconstruction algorithm. Before pro ceeding further w e first review existing r esults on injectivit y of the nonlinear map α F . In general a subset Z of a top ological space is said generic if its op en in terior is dense. Ho w ev er in the follo wing statemen ts, the term generic refers to Zarisky top ology: a set Z ⊂ K n × m = K n × · · · × K n is said generic if Z is dense in K n × m and its complement is a finite union of zero sets of p olynomials in nm v ariables with co efficien ts in the fi eld K (here K = R ). Theorem 2.1. L et F b e a fr ame in H = R n with m elements. Then the fol lowing hold true: (1) The fr ame F is phase r etrievable in ˆ H if and only i f for any disjoint p artition of the fr ame set F = F 1 ∪ F 2 , either F 1 sp ans R n or F 2 sp ans R n . (2) If F is phase r etrievable in ˆ H then m ≥ 2 n − 1 . F urthermor e, for a generic F with m ≥ 2 n − 1 the map α F is phase r etrievable in ˆ H . (3) L et m = 2 n − 1 . Then F is phase r etrievable in ˆ H if an only if F is ful l sp ark. (4) L et (2.4) a 0 := min k x k = k y k =1 m X j =1 |h x, f j i| 2 |h y , f j i| 2 ≥ 0 . Then (2.5) m X k =1 |h x, f k i| 2 |h y , f k i| 2 ≥ a 0 k x k 2 k y k 2 . Then F is pha se r etrievable in ˆ H if and only if a 0 > 0 . INVER TIBILITY AND ROBUSTNESS OF PHASELESS RECON STRUCTI ON 5 (5) F or any x ∈ R n define the matrix R ( x ) by (2.6) R ( x ) := m X j =1 |h x, f j i| 2 f j f ∗ j . Then R ( x ) ≥ a 0 k x k I wher e I is the identity matrix and a 0 is given by (2.4). In other wor ds, λ min ( R ( x )) ≥ a 0 k x k 2 . F urthermor e a 0 = m in k x k =1 λ min ( R ( x )) . Pro of. The r esults (1)-(3 ) are in [6], an d (4) -(5) are in [5]. 3. Informa tion Theoretic Performance Bounds In this section w e d eriv e exp ressions for th e Fisher Information Matrix and obtain p er- formance b oun ds for reconstruction algorithms in the noisy case. Consider the follo w ing n oisy measur emen t pro cess: (3.1) y k = |h x, f k i| 2 + ν k , ν k ∼ N (0 , σ 2 ) , 1 ≤ k ≤ m where the noise m o del is A W GN (additiv e w hite Gaussian noise): eac h r andom v ariable ν k is in dep end en t and normally d istributed w ith zero mean and σ 2 v ariance. Consider the noiseless case first (that is ν k = 0). Ob viously one cannot obtain the exact v ector x ∈ H du e to the global s ign am biguit y . Instead the b est outcome is to id en tify (that is, to estimate) th e class ˆ x = { x, − x } from α F ( x ). As s uc h, we fix a d isj oin t partition of the punctured Hilb ert space H , R n \ { 0 } = Ω 1 ∪ Ω 2 , su c h that Ω 2 = − Ω 1 . W e make the choice that the v ector x b elongs to Ω 1 . Hence an y estimator of x is a map ω : R m − → Ω 1 ∪ { 0 } . Denote by ˚ Ω 1 its interior as a subset of R n . A t ypical such decomp osition is Ω 1 = n [ k =1 n x ∈ R n : x k ≥ 0 , x j = 0 x j = 0 for j < k o . Note its in terior is giv en by ˚ Ω 1 = { x ∈ R n , x 1 > 0 } . Under these assumptions we compute the Fisher Information matrix (see [32]). This is giv en b y (3.2) ( I ( x )) k ,j = E  ( ∇ log L ( x ))( ∇ log L ( x )) T  where the lik elihoo d fun ction L ( x ) is giv en by (3.3) L ( x ) = p ( y | x ) = 1 (2 π ) m/ 2 σ m exp  − 1 2 σ 2 k y − α F ( x ) k 2  . After some algebra (see [5]) we obtain (3.4) I ( x ) = 4 σ 2 R ( x ) , R ( x ) = m X j =1 |h x, f j i| 2 f j f T j . Note the matrix R ( x ) is exactly the same as the matrix introd uced in (2.6). Th us we obtain the follo wing results: 6 R. BALAN AND Y. W ANG Theorem 3.1. The fr ame F is phase r etrievable i f and only if the Fisher information matrix I ( x ) is invertible for any x 6 = 0 . F u rthermor e, when F is phase r etrievable ther e i s a p ositive c onstant a 0 > 0 so that (3.5) I ( x ) ≥ 4 a 0 σ 2 k x k 2 I wher e I is the n × n identity. This allo ws to state the follo w ing p erformance b ound result (see [32] for d etails on the Cramer-Rao lo w er b ound). Theorem 3.2. Assume x ∈ ˚ Ω 1 . L et ω : R m → Ω 1 b e any u nbiase d estimator for x . Then its c ovarianc e matrix is b ounde d b elow by the Cr amer-R ao lower b ound: (3.6) Co v[ ω ( y )] ≥ ( I ( x )) − 1 = σ 2 4 ( R ( x )) − 1 . F urthermor e, any efficient estimator (that is, any unb i ase d estimator ω that achieves the Cr amer-R ao L ower Bound (3.6)) has the c ovarianc e matrix b ounde d fr om ab ove by (3.7) Co v[ ω ( y )] ≤ σ 2 4 a 0 k x k 2 I and Me an-Squar e err or b ounde d ab ove by (3.8) MSE( ω ) = E h k ω ( y ) − x k 2 i ≤ nσ 2 4 a 0 k x k 2 . 4. R obustne ss Measures for Reconstruction In this section w e analyze the robu stness of deterministic phaseless reconstruction. Addi- tionally w e connect the constan t a 0 in tro duced earlier in Th eorem 2.1 to quan tities d ir ectly computable from the frame F . A natural approac h is to analyze th e stabilit y in the wo rst case scenario, for whic h w e consider the follo wing measures. Denote d ( x, y ) := min ( k x − y k , k x + y k ). F or an y x ∈ R n define (4.1) Q ε ( x ) = max { y : k α F ( x ) − α F ( y ) k≤ ε } d ( x, y ) ε . The size of Q ε ( x ) measures th e worst case stabilit y of the reconstru ction f or the vect or x , under the assumption that the total noise lev el is con trolled b y ε . W e also study the global stabilit y b y analyzing the measures (4.2) q ε := max k x k =1 Q ε ( x ) , q 0 := lim sup ε → 0 q ε , q ∞ := su p ε> 0 q ε . Here k . k denotes usual Euclidian norm. Note th at Q ε ( x ) has the scaling prop ert y Q ε ( x ) = Q | c | ε ( cx ) for any real c 6 = 0. Thus it is natural to fo cus on unit ve ctors x . INVER TIBILITY AND ROBUSTNESS OF PHASELESS RECON STRUCTI ON 7 W e in tro duce now some quantitie s that pla y k ey roles in the estimation of these robustn ess measures. F or the f r ame F let F = [ f 1 , f 2 , · · · , f m ] b e its frame matrix. Denote by F [ S ] = { f k , k ∈ S } the sub set of F indexed b y a subset S ⊆ { 1 , 2 , · · · , m } , and b y F S the fr ame matrix corresp ond ing to F [ S ] (whic h is the m atrix with vec tors in F [ S ] as its columns). Set (4.3) A [ S ] := σ 2 n ( F S ) = λ min ( F S F ∗ S ) , where as usual σ n and λ min denote the n -th singular v alue and the minimal eigen v alue, resp ectiv ely . Note that A [ S ] is in fact the lo w er frame b ound of F [ S ]. Let S denote the collect ion of subsets S of { 1 , 2 , · · · , m } so that dim (span( F [ S c ])) < n , where S c = { 1 , 2 , · · · , m } \ S is the complement of S . In other words, r ank( F S c ) < n . Denote b y ∆ and ω the follo wing expressions : ∆ = min S p A [ S ] + A [ S c ] (4.4) ω = min S ∈S σ n ( F S ) . (4.5) All of them dep end of course on F . Ho wev er since w e fi x F thr oughout the p ap er, we shall without confusion n ot explicitly reference F in the notation f or s implicit y as there will not b e an y confusion. C learly (4.6) ∆ ≤ ω . Prop osition 4.1. L et ε > 0 . Then the stability me asur ement function Q ε ( x ) is giv e n by (4.7) Q ε ( x ) = 1 ε max w 1 ,w 2 min  k w 1 k , k w 2 k  under the c onstr aints 1 2 ( w 1 + w 2 ) = x and (4.8) m X j =1 min  |h f j , w 1 i| 2 , |h f j , w 2 i| 2  =   F ∗ S w 1   2 +   F ∗ S c w 2   2 ≤ ε 2 , wher e S := S ( w 1 , w 2 ) = { j : |h f j , w 1 i| ≤ |h f j , w 2 i|} . Pro of. F or an y x, y ∈ R n let w 1 = x + y and w 2 = x − y . Then x = 1 2 ( w 1 + w 2 ) and y = 1 2 ( w 1 + w 2 ). It is easy to c hec k that f or S = { j : |h f j , w 1 i| ≤ |h f j , w 2 i|} w e ha v e |h f j , x i| − |h f j , y i| =  ±h f j , w 1 i j ∈ S, ±h f j , w 2 i j ∈ S c . In other w ords, (4.9) |h f j , x i| − |h f j , y i| = min( |h f j , w 1 i| , |h f j , w 2 i| ) . Let F b e the fr ame matrix of F . W e th u s ha v e    α F ( x ) − α F ( y )    2 = X j ∈ S |h f j , w 1 i| 2 + X j ∈ S c |h f j , w 2 i| 2 =   F ∗ S w 1   2 +   F ∗ S c w 2   2 . Note that d ( x, y ) = m in( k w 1 k , k w 2 k ). The prop osition now follo ws. 8 R. BALAN AND Y. W ANG The ab o v e prop osition allo ws us to establish the follo wing stabilit y result for the worst case s cenario. Theorem 4.2. Assume that the fr ame F i s phase r etrievable. L et A > 0 b e the lower fr ame b ound for the fr ame F and let τ := min { σ n ( F S ) : S ⊆ { 1 , . . . , m } , rank( F S ) = n } . (A) F or any ε > 0 we have (4.10) min n 1 ε , 1 ω o ≤ q ε ≤ 1 ∆ . (B) If ε < τ then q ε = 1 ω . Conse que ntly q 0 = 1 ω . (C) F or any nonzer o x ∈ R n and any 0 < ε < δ x we have (4.11) Q ε ( x ) = 1 √ A , wher e δ x := 2 τ max( k f j k ) + τ min n |h f j , x i| : h f j , x i 6 = 0 o . (D) The upp er b ound q ∞ e quals the r e cipr o c al of ∆ : (4.12) q ∞ = 1 ∆ . Pro of. T o pr o v e (A) w e first establish the upp er b oun d in (4.10). Let x ∈ R n . By Prop osition 4.1 we hav e Q ε ( x ) = 1 ε max w 1 ,w 2 min  k w 1 k , k w 2 k  under the constrain ts 1 2 ( w 1 + w 2 ) = x and   F ∗ S w 1   2 +   F ∗ S c w 2   2 ≤ ε 2 for some S . Now assum e with ou t loss of generalit y that k w 1 k ≤ k w 2 k . Th en ε 2 k w 1 k 2 ≥   F ∗ S w 1   2 +   F ∗ S c w 2   2 k w 1 k 2 ≥ σ 2 n ( F S ) + σ 2 n ( F S c ) k w 2 k 2 k w 1 k 2 ≥ ∆ . It f ollo ws that 1 ε min  k w 1 k , k w 2 k  ≤ 1 ∆ . Th us Q ε ( x ) ≤ 1 ∆ . T o establish the lo w er b ound in (4.10) we constru ct for any ε > 0 an x ∈ R n and v ectors w 1 , w 2 satisfying th e imp osed constrain ts. Let S b e a su bset of { 1 , 2 , . . . , m } such that rank( F S c ) < n and σ n ( F S ) = ω . C ho ose v 1 , v 2 ∈ R n with the prop ert y k v 1 k = k v 2 k = 1 and k F ∗ S v 1 k = ω , F ∗ S c v 2 = 0 . INVER TIBILITY AND ROBUSTNESS OF PHASELESS RECON STRUCTI ON 9 Set t = min n ε ω , 1 o , and w 1 = t v 1 . Hence k w 1 k = t ≤ 1. No w w e s elect an s ∈ R s o that k w 1 + sv 2 k = 2. This is alw a ys p ossible s ince s 7→ k w 1 + s v 2 k is con tin uous and k w 1 + 0 v 2 k = t ≤ 1 ≤ 2 ≤ k w 1 + 3 v 2 k . Set w 2 = s v 2 so k w 1 + w 2 k = 2. W e hav e | s | = k sv 2 k ≥ k w 1 + s v 2 k − k w 1 k = 2 − t ≥ 1 . Th us k w 2 k ≥ k w 1 k . No w let x = 1 2 ( w 1 + w 2 ) and y = 1 2 ( w 1 − w 2 ) . W e hav e then k α F ( x ) − α F ( y ) k 2 = m X j =1 min ( |h f j , w 1 i| 2 , |h f j , w 2 i| 2 ) ≤ X j ∈ S |h f j , w 1 i| 2 + X j ∈ S c |h f j , w 2 i| 2 = t 2 ω 2 ≤ ε 2 . F urthermore d ( x, y ) = min( k w 1 k , k w 2 k ) = k w 1 k = t. Hence for this x we ha v e Q ε ( x ) ≥ d ( x, y ) ε = m in n 1 ε , 1 ω o . It f ollo ws that q ε ≥ min { 1 ε , 1 ω } . No w by taking ε > 0 sufficiently small w e hav e q ε ≥ 1 ω . W e now pro v e (B). Assume that ε ≤ min { σ n ( F S ) : rank( F S ) = n } . T hen clearly w e ha v e ε ≤ ω . Thus by (4.10) w e ha v e q ε ≥ 1 ω . Again for eac h x ∈ R n with k x k = 1 w e consider w 1 , w 2 for the estimat ion of q ε ( x ). Th e constrain t k w 1 + w 2 k = 2 implies either k w 1 k ≥ 1 or k w 2 k ≥ 1. Without loss of generalit y we assume th at k w 1 k ≥ 1. F or the constrain t   F ∗ S w 1   2 +   F ∗ S c w 2   2 ≤ ε 2 for some S , assum e that ran k( F S ) = n then we hav e k F ∗ S w 1 k ≥ σ n ( F S ) k w 1 k ≥ min { σ n ( F S ) : rank( F S ) = n } > ε. This is a con tradiction. So r ank( F S ) < n and hence ε 2 ≥   F ∗ S w 1   2 +   F ∗ S c w 2   2 ≥   F ∗ S c w 2   2 ≥ ω 2 k w 2 k 2 . Th us k w 2 k ≤ ε ω . Prop osition 4.1 n o w yields q ε = 1 ω , proving part (B). No w w e prov e (C). W e go b ac k to the form ulation in Prop osition 4.1. Q ε ( x ) = 1 ε max w 1 ,w 2 min  k w 1 k , k w 2 k  under the constrain ts 1 2 ( w 1 + w 2 ) = x and   F ∗ S w 1   2 +   F ∗ S c w 2   2 ≤ ε 2 10 R. BALAN AND Y. W ANG where S := S ( w 1 , w 2 ) = { j : |h f j , w 1 i| ≤ |h f j , w 2 i|} . Since α F is inj ectiv e, either rank( F S ) = n or r ank( F S c ) = n by Theorem 2.1 (1). Without loss of generalit y w e assu me rank( F S ) = n . Th us ε ≥ k F ∗ S w 1   ≥ τ k w 1 k . So k w 1 k ≤ ε/τ . W e sho w that for an y k ∈ S c w e m ust ha v e h f k , x i = 0. Assume otherwise and write w 2 = 2 x − w 1 , L x := min {|h f j , x i| : h f j , x i 6 = 0 } . Then |h f k , w 2 i| ≥ 2 |h f k , x i| − |h f k , w 1 i| ≥ 2 L x − max ( k f j k ) k w 1 k ≥ 2 L x − m ax ( k f j k ) ε τ > ε. This is a con tradiction. Thus f or k ∈ S c w e ha v e h f k , x i = 0 and |h f j , w 2 i| = |h f j , 2 x − w 1 i| = |h f j , w 1 i| . It f ollo ws that   F ∗ S w 1   2 +   F ∗ S c w 2   2 = k F ∗ w 1 k 2 ≤ ε 2 . Th us k w 1 k ≤ ε/ √ A and hence Q ε ( x ) ≤ 1 √ A . No w w e sho w the b ound can b e ac hiev ed. Let w 1 satisfy k F ∗ w 1 k = √ A k w 1 k = ε . Such a w 1 alw a ys exists. Then clearly w 1 and w 2 = 2 x − w 1 satisfy the required constraints, and it is easy to chec k that min ( k w 1 k , k w 2 k ) = k w 1 k = ε/ √ A . Finally we pro v e (D). By the r esult at part (A), q ∞ ≤ 1 ∆ . It is therefore sufficien t to sho o w that Q ε ( x ) ≥ 1 ∆ for some x and ε . Let S 0 b e the s u bset th at ac hiev es th e minimum in (4.4). Let u, v ∈ H b e un it eigen v ectors corresp ond ing to the lo west eigen v alues of F S 0 F ∗ S 0 and F S c 0 F ∗ S c 0 resp ectiv ely . Th us k F ∗ S 0 u k 2 = A [ S 0 ] , k F ∗ S c 0 v k 2 = A [ S c 0 ] Let x = ( u + v ) / 2 and ε = ∆, an d set w 1 = u , w 2 = v . Th en by Prop osition 4.1 Q ε ( x ) ≥ min( k w 1 k , k w 2 k ) ε = 1 ∆ since m X j =1 min( |h f j , w 1 i| 2 , |h f j , w 2 i| 2 ) ≤ k F ∗ S 0 w 1 k 2 + k F ∗ S c 0 w 2 k 2 = ε 2 This concludes the pro of. Remark. I t may seem strange that Q ε ( x ) = 1 √ A for all x 6 = 0 and su fficien tly sm all ε w hile q 0 = 1 ω , w here ω is typical ly muc h smaller than √ A . The reason is that for Q ε ( x ) = 1 √ A to hold, ε dep end s on x . Thus we cann ot exc hange the order of lim sup ε → 0 and max k x k =1 . Related to the study of stabilit y of phaseless reconstruction is the study of Lipsc hitz prop erty of the map α F on ˆ H := R n / ∼ . W e analyze th e bi-Lipsc h itz b ound s of b oth α F and α F 2 , w hic h is simply the map α F with all en tries squared, i.e. α F 2 ( x ) := [ |h f j , x i| , . . . , |h f m , x i| ] T . INVER TIBILITY AND ROBUSTNESS OF PHASELESS RECON STRUCTI ON 11 W e shall consider tw o distance functions on ˆ H = R n / ∼ : the s tand ard distance d ( x, y ) := min( k x − y k , k x + y k ) an d the distance d 1 ( x, y ) := k xx ∗ − y y ∗ k 1 where k X k 1 denotes the nucle ar norm of X , whic h is the sum of all singular v alues of X . Sp ecifically we are in terested in examining the lo cal and global b eha vior of the ratios (4.13) U ( x, y ) := k α F ( x ) − α F ( y ) k d ( x, y ) , V ( x, y ) := k α F 2 ( x ) − α F 2 ( y ) k d 1 ( x, y ) . W e first inv estigate the b oun d s for U ( x, y ). F or this the upp er b ound is relativ ely straigh t- forw ard. Let w 1 = x − y and w 2 = x + y . W e ha v e already sho wn in the pr o of of T h eorem 4.2 u sing (4.9) that k α F ( x ) − α F ( y ) k 2 = m X j =1 min ( |h f j , w 1 i| 2 , |h f j , w 2 i| 2 ) ≤ min n m X j =1 |h f j , w 1 i| 2 , m X j =1 |h f j , w 2 i| 2 o ≤ B min n k w 1 k 2 , k w 2 k 2 o = B d 2 ( x, y ) , where B is the up p er frame b ound of th e fr ame F . Th us U ( x, y ) h as an upp er b ound U ( x, y ) ≤ √ B . F urthermore, the b ound is sharp. T o s ee this, pic k a unit v ecto r x ∈ R n suc h that P m j =1 |h f j , w 1 i| 2 = B and set y = 2 x . Then U ( x, y ) = √ B . T o study the lo w er b ound U ( x, y ) w e now consider the follo wing quan tities: ρ ε ( x ) := inf { y : d ( x,y ) ≤ ε } U ( x, y ) , ρ ( x ) := lim inf { y : d ( x,y ) → 0 } U ( x, y ) = lim inf ε → 0 ρ ε ( x ) , ρ 0 := inf x ρ ( x ) , ρ ∞ := inf d ( x,y ) > 0 U ( x, y ) . W e apply the equalit y U 2 ( x, y ) = P m j =1 min ( |h f j , w 1 i| 2 , |h f j , w 2 i| 2 ) min ( k w 1 k 2 , k w 2 k 2 ) where again w 1 = x − y and w 2 = x + y . Now fix x and let d ( x, y ) < ε . Wi thout loss of generalit y w e ma y assume k y − x k < ε . T h us k w 1 k < ε and k w 2 − 2 x k = k w 1 k < ε . Let S = { j, h f j , x i 6 = 0 } and set (4.14) ε 0 ( x ) := min k ∈ S |h f k , x i| max k ∈ S k f k k . Note for an y w 1 with k w 1 k < ε 0 and k ∈ S we hav e |h f k , w 2 i| = | 2 h f k , x i − h f k , w 1 i| ≥ 2 |h f k , x i| − |h f k , w 1 i| ≥ 2 ε 0 ( x ) k f k k − k w 1 kk f k k ≥ |h f k , w 1 i| , 12 R. BALAN AND Y. W ANG whereas for k ∈ S c w e h a v e |h f k , w 2 i| = |h f k , w 1 i| . Hence min ( |h f j , w 1 i| 2 , |h f j , w 2 i| 2 ) = |h f j , w 1 i| 2 for all j whenev er ε < ε 0 ( x ). It follo ws th at U 2 ( x, y ) = P m j =1 |h f j , w 1 i| 2 k w 1 k 2 = m X j =1   h w 1 k w 1 k , f j i   2 . Th us U 2 ( x, y ) ≥ A where A is the low er frame b ound for the frame F . F urthermore this lo w er b ound is ac h iev ed w henev er w 1 = x − y is an eigen vect or corresp onding to the s mallest eigen v alue of F F ∗ . This imp lies that ρ ε ( x ) = √ A whenev er ε < ε 0 ( x ). Consequently ρ ( x ) = √ A . W e ha ve the f ollo win g theorem: Theorem 4.3. Assume that the fr ame F is phase r etrievable. L et A , B b e the lower and upp er fr ame b ounds for the fr ame F , r esp e ctively and for e ach x ∈ R n , let ε 0 ( x ) b e given in (4.14) . Then (1) U ( x, y ) ≤ √ B for any x, y ∈ R n with d ( x, y ) > 0 . (2) Assume that ε < ε 0 ( x ) . Then ρ ε ( x ) = √ A . Conse quently ρ ( x ) = ρ 0 = √ A . (3) ∆ = ρ ∞ ≤ ω ≤ ρ 0 = ρ ( x ) = √ A . (4) The map α F is bi- Lipschitz with (optimal) upp er Lipschitz b ound √ B and lower Lipschitz b ound ρ ∞ : ρ ∞ d ( x, y ) ≤ k α F ( x ) − α F ( y ) k ≤ √ B d ( x, y ) , ∀ x , y ∈ ˆ H Pro of. W e hav e already pro v ed (1) and (2) of the theorem in the discussion. It remains only to pro v e (3) s in ce (4) is just a restatemen t of (1) and (3). Note that ρ 2 ∞ = in f d ( x,y ) > 0 U 2 ( x, y ) = inf w 1 ,w 2 6 =0 P m j =1 min ( |h f j , w 1 i| 2 , |h f j , w 2 i| 2 ) min ( k w 1 k 2 , k w 2 k 2 ) . F or any w 1 , w 2 , assume withou t loss of generalit y that 0 < k w 1 k ≤ k w 2 k . Let S = { j : |h f j , w 1 i| ≤ |h f j , w 2 i|} . Set v 1 = w 1 / k w 1 k , v 2 = w 2 / k w 2 k and t = k w 2 k / k w 1 k ≥ 1. Then P m j =1 min ( |h f j , w 1 i| 2 , |h f j , w 2 i| 2 ) min ( k w 1 k 2 , k w 2 k 2 ) = X j ∈ S |h f j , v 1 i| 2 + t 2 X j ∈ S c |h f j , v 2 i| 2 ≥ X j ∈ S |h f j , v 1 i| 2 + X j ∈ S c |h f j , v 2 i| 2 ≥ ∆ 2 . Hence ρ ∞ ≥ ∆ . Let S and u, v ∈ H b e normalized (eigen) ve ctors that achiev e the b ound ∆, that is: k u k = k v k = 1 , X k ∈ S |h u, f k i| 2 + X k ∈ S c |h v , f k i| 2 = ∆ . INVER TIBILITY AND ROBUSTNESS OF PHASELESS RECON STRUCTI ON 13 Set x = u + v and y = u − v . Then , follo wing [9] k α F ( x ) − α F ( y ) k 2 = X k ∈ S | |h u + v , f k i| − |h u − v , f k i| | 2 + X k ∈ S c | |h u + v , f k i| − |h u − v , f k i| | 2 ≤ 4 X k ∈ S |h u, f k i| 2 + X k ∈ S c |h v , f k i| 2 ! = 4∆ . On the other hand d ( x, y ) = min ( k x − y k , k x + y k ) = 2 . Th us w e obtain k α F ( x ) − α F ( y ) k d ( x, y ) ≤ ∆ . The theorem is no w pr o v ed. Remark. T he t w o qu an tities, ρ ∞ and q ∞ satisfy ρ ∞ = 1 q ∞ . Ho wev er there are s u btle differences b etw een q ε ( x ) and ρ ε ( x ) so that the simple relationship ρ ε ( x ) = 1 /q ε ( x ) do es not usually hold. Remark. The u pp er Lipsc hitz b ound √ B has b een obtained indep endently in [9]. The lo w er Lipschitz b ound we obtained here stren gh tens th e estimat es giv en in [9 ]. Sp ecifically their estimate for ρ ∞ reads σ ≤ ρ ∞ ≤ √ 2 σ where σ = min S max( σ n ( F S ) , σ n ( F S c )) Clearly σ ≤ ∆ ≤ √ 2 σ . W e conclude th is section b y turning ou r atten tion to the analysis of V ( x, y ). A m otiv ation for studying it is that in p ractical problems the noise is often add ed d irectly to α F 2 as in (3.1) rather than to α F . Such noise mod el is used in many studies of phaseless reconstruction, e.g. in the Phaselift algorithm [15], or in the IRLS algorithm in [5]. Let Sym n ( R ) denote th e set of n × n symmetric matrices o v er R . It is a Hilb ert s p ace with the stand ard inner pro d u ct given b y h X , Y i := tr( X Y T ) = tr( X Y ). The n onlinear map α F 2 actually indu ces a linear map on Sym n ( R ). W rite X = xx T for any x ∈ R n . Then the en tries of α F 2 ( x ) are (4.15) ( α F 2 ( x )) j = |h f j , x i| 2 = x T f j f T j x = tr ( F j X ) = h F j , X i , where F j := f j f T j . No w w e denote b y A th e lin ear op er ator A : Sym n ( R ) − → R m with en tries ( A ( X )) j = h F j , X i = tr( F j X ) . Let S p,q n b e the s et of n × n real symmetric matrices that ha v e at most p p ositiv e and q negativ e eigen v alues. Thus S 1 , 0 n denotes the s et of n × n real symmetric n on -n egativ e definite matrices of rank at most one. Note that sp ectral d ecomp osition easily shows th at X ∈ S 1 , 0 n if and only if X = xx T for some x ∈ R n . 14 R. BALAN AND Y. W ANG The follo wing lemma will b e useful in this analysis Lemma 4.4. The fol lowing ar e e qu ivalent. (A) X ∈ S 1 , 1 n . (B) X = xx T − y y T for some x, y ∈ R n . (C) X = 1 2 ( w 1 w T 2 + w 2 w T 1 ) for some w 1 , w 2 ∈ R n . F urthermor e, for X = 1 2 ( w 1 w T 2 + w 2 w T 1 ) its nucle ar norm is k X k 1 = k w 1 kk w 2 k . Pro of. (A) ⇒ (B) is a direct result of sp ectral decomp osition, which yields X = β 1 u 1 u T 1 − β 2 u 2 u T 2 for some u 1 , u 2 ∈ R n and β 1 , β 2 ≥ 0. Thus X = xx T − y y T where x := √ β 1 u 1 and y := √ β 2 u 2 . (B) ⇒ (C) is prov ed directly by setting w 1 = x − y and w 2 = x + y . W e no w pro v e (C) ⇒ (A) by computing the eigen v alues of X = 1 2 ( w 1 w T 2 + w 2 w T 1 ). Ob viously rank( X ) ≤ 2. Let λ 1 , λ 2 b e the t w o (p ossib ly) n on zero eigenv alues of X . Then λ 1 + λ 2 = tr { X } = h w 1 , w 2 i , λ 2 1 + λ 2 2 = t r { X 2 } = ( k w 1 k 2 k w 2 k 2 + |h w 1 , w 2 i| 2 ) / 2 . Solving for eigen v alues we obtain λ 1 = 1 2 ( h w 1 , w 2 i + k w 1 kk w 2 k ) , λ 2 = 1 2 ( h w 1 , w 2 i − k w 1 kk w 2 k ) . Hence, b y Cauc hy-Sc h w artz inequalit y , λ 1 ≥ 0 ≥ λ 2 whic h pro v es X ∈ S 1 , 1 n . F urthermore, it also sho ws that the nuclea r n orm of X is k X k 1 = | λ 1 | + | λ 2 | = k w 1 kk w 2 k . No w we analyze V ( x, y ). Paralle l to th e stud y of U ( x, y ) we consider the follo w ing quan tities: µ ε ( x ) := inf { y : d ( x,y ) ≤ ε } V ( x, y ) , µ ( x ) := lim inf { y : d ( x,y ) → 0 } V ( x, y ) = lim inf ε → 0 µ ε ( x ) , µ 0 := inf x µ ( x ) , µ ∞ := inf d ( x,y ) > 0 V ( x, y ) . as wel l as the up p er b oun d sup d 1 ( x,y ) > 0 V ( x, y ). B y (4.15 ) we ha ve |h f j , x i| 2 − |h f j , y i| 2 = h F j , X i w here F j = f j f T j and X = xx T − y y T . Hence V 2 ( x, y ) = P m j =1 |h F j , X i| 2 k X k 2 1 . Set w 1 = x − y and w 2 = x + y and apply Lemma 4.4 w e obtain (4.16) V 2 ( x, y ) = P m j =1 |h f j , w 1 i| 2 |h f j , w 2 i| 2 k w 1 k 2 k w 2 k 2 . INVER TIBILITY AND ROBUSTNESS OF PHASELESS RECON STRUCTI ON 15 W e can immediately obtain th e u pp er b oun d: V ( x, y ) ≤  sup k e 1 k =1 , k e 2 k =1 m X j =1 |h f j , e 1 i| 2 |h f j , e 2 i| 2  1 / 2 =  max k e k =1 m X j =1 |h f j , e i| 4  1 / 2 =: Λ F 2 where Λ F denotes the op erator norm of the linear analysis op erator T : H → R m , T ( x ) = ( h x, f k i ) m k =1 defined b et w een the Euclidian sp ace H = R n and the Banac h sp ace R m endo w ed with the l 4 -norm: (4.17) Λ F =  max k x k =1 m X k =1 |h x, f k i| 4  1 / 4 Note also that Λ F 2 = max k x k =1 λ max ( R ( x )) where R ( x ) wa s d efined in (2.6). An immediate b ound is Λ F ≤ √ B max k f k k with B the upp er frame b ound of F . Fix x 6 = 0 and let d ( x, y ) → 0. T h en either y → x or y → − x . Without loss of generalit y w e assume that x → y . Thus w 1 = x − y → 0 and w 2 = x + y → 2 x . Ho w ev er w 1 / k w 1 k can b e an y unit v ector. Th us µ 2 ( x ) = 1 k x k 2 inf k u k =1 m X j =1 |h f j , x i| 2 |h f j , u i| 2 = 1 k x k 2 inf k u k =1 h R ( x ) u, u i = 1 k x k 2 λ min ( R ( x )) where R ( x ) was int ro du ced in (2.6 ). Thus we obtain µ 2 ( x ) = 1 k x k 2 λ min ( R ( x )) , µ 2 0 = min k u k =1 λ min ( R ( u )) . On the other hand note inf d ( x,y ) > 0 V 2 ( x, y ) = inf w 1 ,w 2 6 =0 P m j =1 |h f j , w 1 i| 2 |h f j , w 2 i| 2 k w 1 k 2 k w 2 k 2 = min k u k =1 λ min ( R ( u )) = a 2 0 . where a 0 w as introd uced in (2.4). Thus we pr o v ed : Theorem 4.5. Assume the fr ame F is phase r etrievable. Then µ ( x ) = 1 k x k p λ min ( R ( x )) , (4.18) µ ∞ = µ 0 = min u : k u k =1 p λ min ( R ( u )) = √ a 0 . (4.19) F urthermor e α F 2 is bi-Lipschitz with upp er Lipschitz b ound Λ F 2 and lower Lipschitz b ound µ 0 : µ 0 d 1 ( x, y ) ≤ k α F 2 ( x ) − α F 2 ( y ) k ≤ Λ F 2 d 1 ( x, y ) Remark. Note that the distance d ( ., . ) is not equiv alent to d 1 ( ., . ). Th eorem 4.5 no w also implies that α F 2 is not b i-Lipsc hitz with resp ect to the distance d ( ., . ) on ˆ H . This fact w as p oint ed out in [9]. 16 R. BALAN AND Y. W ANG 5. R obustne ss and Size of Red undancy Previous sections establish resu lts on the robustness of phaseless reconstruction for the w orst case scenario. A natur al question is to ask: can “reasonable” rob u stness b e ac hiev ed for a giv en frame, and in particular with small num b er of samples? W e s h all examine how q ∞ scales as the dimension n incr eases. Consider the case where m = 2 n − 1. T his is the minimal redu ndancy r equired for phaseless reconstruction. In this case any frame F would hav e ∆ = ω . Hence w e ha v e min { 1 /ω , 1 /ε } ≤ q ε = 1 /ω . The stabilit y of the reconstruction is th us mostly con trolled by the size of 1 /ω . The question is: ho w big is ω , esp ecially as n in creases? Assume that the frame element s of F are all b ou n ded by L , k f j k ≤ L for all f j ∈ F . Consider the n + 1 elemen ts { f j : j = 1 , . . . , n + 1 } . They are linearly d ep endent so w e can fin d c j ∈ R suc h that P n +1 j =1 c j f j = 0. Without loss of generalit y we ma y assum e | c n +1 | = min {| c j |} . Set v = [ c 1 , c 2 , . . . , c n ] T . L et G = [ f 1 , . . . , f n ]. Then Gv = P n j =1 c j f j = − c n +1 f n +1 . No w all | c j | ≥ | c n +1 | so k v k ≥ √ n | c n +1 | . Thus k Gv k = | c n +1 |k f n +1 k ≤ L √ n k v k . It f ollo ws that σ n ( G ) ≤ L √ n , and hence (5.1) ω ≤ L √ n . Note that here we h a v e considered only the fi rst n + 1 v ectors of the fr ame F . The actual v alue of ω will lik ely deca y muc h faster as n increases. In a p reliminary work w e are able to establish the b ound ω ≤ C L/ √ n 3 where C is indep end en t of n [46 ]. But ev en th is estimate is likely far from optimal. Conjecture 5.1. L et m = 2 n − 1 and k f j k ≤ L for al l f j ∈ F . Then ther e exi st c onstants C > 0 and 0 < β < 1 indep endent of n such that ω ≤ C Lβ − n . A related problem is as follo ws: Consider an n × ( n + k ) matrix F = [ g 1 , g 2 , . . . , g n + k ]. Let τ = min { σ n ( F S ) : S ⊂ { 1 , . . . , n + k } , | S | = n } . Assume that all k g j k ≤ 1. Ho w large can τ b e? F or k = 1 we ha ve already seen that it is b ounded from ab o v e by C / √ n . The preliminary wo rk [46] shows that for k = 1 it is b ounded fr om ab o v e b y C /n 3 2 . Conjecture 5.2. Ther e exists a c onstant C = C ( k ) such that τ ≤ C n k − 1 2 . Th us in the m inimal setting with m = 2 n − 1 it is imp ossible to ac hiev e scale in dep end en t stabilit y for phaseless reconstruction. The same argument s can b e used to show that ev en INVER TIBILITY AND ROBUSTNESS OF PHASELESS RECON STRUCTI ON 17 when m = 2 n + k 0 for s ome fixed k 0 scale ind ep endent stabilit y is n ot p ossible. A natural question is whether scale indep endent stabilit y is p ossible when we increase the r edundancy of the frame. As it turn s out th is is p ossib le via a recen t w ork by W ang [45]. More p r ecisely , the follo wing result follo ws from the m ain r esults in [45]: Theorem 5.3. L et r 0 > 2 and let F = 1 √ n G wher e G is an n × m r andom matrix whose elements ar e i.i.d. normal N (0 , 1) r ando m variables such that m/n = r 0 . Then ther e exist c onstants 0 < ∆ 0 ≤ ω 0 dep endent only on r 0 and not on n such that with high pr ob ability we have ∆ ≥ ∆ 0 , ω ≥ ω 0 . Pro of. P art of the main theorem of [45] prov es the follo w ing result: Let λ > δ > 1 b e fixed . Assume th at A = 1 √ n B where B is an n × N random Gaussian matrix with i.i.d N (0 , 1) en tries such th at N/n = λ . T hen th er e exists a constant c > 0 dep end ing con tin uously and only on λ and δ su c h th at min S ⊆{ 1 ,. ..,N } , | S |≥ δ n σ n ( A S ) ≥ c. The theorem no w readily follo ws. Observe th at b ecause r 0 > 2, in th e expression for ∆ w e ma y c ho ose λ = r 0 δ = r 0 2 > 1 and clearly we hav e ∆ ≥ min S ⊆{ 1 ,. ..,N } , | S |≥ δ n σ n ( F S ) ≥ ∆ 0 . for some ∆ 0 > 0 indep en d en t of n . F or ω w e ma y c ho ose λ = r 0 and δ = r 0 − 1 > 1. Again the theorem of [45] implies that ω ≥ min S ⊆{ 1 ,. ..,N } , | S |≥ δ n σ n ( F S ) ≥ ω 0 . In the theorem the v alues ∆ 0 and ω 0 can b e estimated explicitly . Here with high p roba- bilit y is in the standard sense that the probabilit y is at least 1 − c 0 e − β n for some c 0 , β > 0. Th us scale ind ep endent stable phaseless reconstruction is p ossible when ev er the redun dancy is greater than 2 + δ , δ > 0, at least for ran d om Gaussian m atrices. Ac kno w ledgemen t. T he authors would lik e to thank Matt Fic kus, Dustin Mixon and Jeffrey Sc henk er for ve ry h elpful d iscussions. Referen ces [1] B. A lexeev, A. S. Bandeira, M. Fickus, D. G. 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