Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements

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1 Spase econstuction by convex elaxation: Fouie and Gaussian measuements Ma Rudelson Depatment of Mathematics Univesity of Missoui, Columbia Columbia, Missoui 65 Roman Veshynin Depatment of Mathematics Univesity of Califonia, Davis Davis, Califonia Abstact This pape poves best nown guaantees fo exact econstuction of a spase signal f fom few non-adaptive univesal linea measuements. We conside Fouie measuements (andom sample of fequencies of f) and andom Gaussian measuements. The method fo econstuction that has ecently gained momentum in the Spase Appoximation Theoy is to elax this highly non-convex poblem to a convex poblem, and then solve it as a linea pogam. What ae best guaantees fo the econstuction poblem to be equivalent to its convex elaxation is an open question. Recent wo shows that the numbe of measuements (, n) needed to exactly econstuct any -spase signal f of length n fom its linea measuements with convex elaxation is usually O( polylog(n)). Howeve, nown guaantees involve huge constants, in spite of vey good pefomance of the algoithms in pactice. In attempt to econcile theoy with pactice, we pove the fist guaantees fo univesal measuements (i.e. which wo fo all spase functions) with easonable constants. Fo Gaussian measuements, (, n).7 ˆ.5 + log(n/), which is optimal up to constants. Fo Fouie measuements, we pove the best nown bound (, n) = O( log(n) log () log( log n)), which is optimal within the log log n and log 3 factos. Ou aguments ae based on the technique of Geometic Functional Analysis and Pobability in Banach spaces. I. INTRODUCTION Duing the last two yeas, the Spase Appoximation Theoy benefited fom a apid development of methods based on the Linea Pogamming. The idea was to elax a spase ecovey poblem to a convex optimization poblem. The convex poblem can be futhe be endeed as a linea pogam, and analyzed with all available methods of Linea Pogamming. Convex elaxation of spase ecovey poblems can be taced bac in its udimentay fom to mid-seventies; efeences to its ealy histoy can be found in [6]. With the development of fast methods of Linea Pogamming in the eighties, the idea of convex elaxation became tuly pomising. It was put fowad most enthusiastically and successfully by Donoho and his collaboatos since the late eighties, stating fom the seminal pape [5] (see Theoem 8, attibuted thee to Logan, and Theoem 9). Thee is extensive wo being caied out, both in theoy and in pactice, based on the convex elaxation [8], [4], [6], [7], [3], [9], [4], [5], [6], [], [9], [], [], [], [], [4], [5], [3], [3], [6], []. To have theoetical guaantees fo the convex elaxation method, one needs to show that the spase appoximation poblem is equivalent to its convex elaxation. Poving this pesents a mathematical challenge. Known theoetical guaantees wo only fo andom measuements (e.g. andom Gaussian and Fouie measuements). Even when thee is a theoetical guaantee, it involves intactable o vey lage constants, fa wose than in the obseved pactical pefomances. In this pape, we substantially impove best nown theoetical guaantees fo andom Gaussian and Fouie (and nonhamonic Fouie) measuements. Fo the fist time, we ae able to pove guaantees with easonable constants (although only fo Gaussian measuements). Ou poofs ae based on methods of Geometic Functional Analysis, Such methods wee ecently successfully used fo elated poblems [3], []. As a esult, ou poofs ae easonably shot (and hopefully, tanspaent). In Section II, we state the spase econstuction poblem and descibe the convex elaxation method. A guaantee of its coectness is a vey geneal esticted isomety condition on the measuement ensemble, due to Candes and Tao ([5], see [3]). Unde this condition, the econstuction poblem with espect to these measuements is equivalent to its convex elaxation. In Sections III and IV, we impove best nown guaantees fo the spase econstuction fom andom Fouie (and nonhamonic Fouie) measuements and Gaussian measuements (Theoem 3. and 4. espectively). II. THE SPARSE RECONSTRUCTION PROBLEM AND ITS CONVEX RELAXATION We want to econstuct an unnown signal f C n fom linea measuements Φf C, whee Φ is some nown n matix, called the measuement matix. In the inteesting case < n, the poblem is undedetemined, and we ae inteested in the spasest solution. We can state this as the optimization poblem minimize f subject to Φf = Φf, () whee f = suppf is the numbe of nonzeo coefficients of f. This poblem is highly non-convex. So we will conside its convex elaxation: minimize f subject to Φf = Φf, ()

2 whee f p denotes the l p nom thoughout this pape, ( n f i p ) /p. Poblem () can be classically efomulated as the linea pogam n minimize t i subject to t f t, Φf = Φf, which can be efficiently solved using geneal o special methods of Linea Pogamming. Then the main question is: Unde what conditions on Φ ae poblems () and () equivalent? In this pape, we will be inteested in the exact econstuction, i.e. we expect that the solutions to () and () ae equal to each othe and to f. Results fo appoximate econstuction can be deived as consequences, see [4]. Fo exact econstuction to be possible at all, one has to assume that the signal f is -spase, that is supp(f), and that the numbe of measuements = (, n) has to be at least twice the spasity. Ou goal will be to find sufficient conditions (guaantees) fo the exact econstuction. The numbe of measuements (, n) should be ept as small as possible. Intuitively, the numbe of measuements should be of the ode of, which is the tue dimension of f, athe than the nominal dimension n. Vaious esults that appeaed ove the last two yeas demonstate that many natual measuement matices Φ yield exact econstuction, with the numbe of measuements (, n) = O( polylog(n)), see [], [4], [5], [3]. In Sections III and IV, we impove best nown estimates on fo Fouie (and, moe geneally, nonhamonic Fouie) and Gaussian matices espectively. A geneal sufficient condition fo exact econstuction is the esticted isomety condition on Φ, due to Candes and Tao ([5], see [3]). It oughly says that the matix Φ acts as an almost isomety on all O()-spase vectos. Pecisely, we define the esticted isomety constant δ to be the smallest positive numbe such that the inequality C( δ ) x Φ T x C( + δ ) x (3) holds fo some numbe C > and fo all x and all subsets T {,..., n} of size T, whee Φ T denotes the T matix that consists of the columns of Φ indexed by T. The following theoem is due to Candes and Tao ([5], see [3]). Theoem. (Resticted Isomety Condition): Let Φ be a measuement matix whose esticted isomety constant satisfies δ 3 + 3δ 4. (4) Let f be an -spase signal. Then the solution to the linea pogam () is unique and is equal to f. This theoem says that unde the esticted isomety condition (4) on the measuement matix Φ, the econstuction poblem () is equivalent to its convex elaxation () fo all -spase functions f. A poblem with the use of Theoem. is that the esticted isomety condition (4) is usually difficult to chec. Indeed, the numbe of sets T involved in this condition is exponential in. As a esult, no explicit constuction of a measuement matix is pesently nown that obeys the esticted isomety condition (4). All nown constuctions of measuement matices ae andomized. III. RECONSTRUCTION FROM FOURIER MEASUREMENTS Ou goal will be to econstuct an -spase signal f C n fom its discete Fouie tansfom evaluated at = (, n) points. These points will be chosen at andom and unifomly in {,..., n }, foming a set Ω. The Discete Fouie tansfom ˆf = Ψf is defined by the DFT matix Ψ with enties Ψ ω,t = exp( iπωt/n), ω, t {,..., n }. n So, ou measuement matix Φ is the submatix of Ψ consisting of andom ows (with indices in Ω). To be able to apply Theoem., it is enough to chec that the esticted isomety condition (4) holds fo the andom matix Φ with high pobability. The poblem is what is the smallest numbe of ows (, n) of Φ fo which this holds? With that numbe, Theoem. immediately implies the following econstuction theoem fo Fouie measuements: Theoem 3. (Reconstuction fom Fouie measuements): A andom set Ω {,..., n } of size (, n) satisfies the following with high pobability. Let f be an -spase signal in C n. Then f can be exactly econstucted fom the values of its Fouie tansfom on Ω as a solution to the linea pogam minimize f subject to ˆ f (ω) = ˆf(ω), ω Ω. The cental emaining poblem, what is the smallest value of (, n), is still open. The best nown estimate is due to Candes and Tao [4]: (, n) = O( log 6 n). (5) The conjectued optimal estimate would be O( log n), which is nown to hold fo nonuniveal measuemets, i.e. fo one spase signal f and fo a andom set Ω []. In this pape, we impove on the best nown bound (5): Theoem 3. (Sample size): Theoem 3. holds with (, n) = O( log(n) log () log( log n)). The dependence on n is thus optimal within the log log n facto and the dependence on is optimal within the log 3 facto. So, ou estimate is especially good fo small, but ou estimate always yields (, n) = O( log 4 n). Rema 3.3: Ou esults hold fo tansfoms moe geneal than the discete Fouie tansfom. One can eplace the DFT matix Ψ by any othogonal matix with enties of magnitude O(/ n). Theoems 3. and 3. hold fo any such matix. In the emainde of this section, we pove Theoem 3.. Let Ω be a andom subset of {,..., n} of size. Recall that the

3 measuement matix Φ that consists of the ows of Ψ whose indices ae in Ω). In view of Theoem 3, it suffices to pove that the esticted isomety constant δ of Φ satisfies Eδ ε (6) wheneve ( log n ) ( log n ) C ε log ε log, (7) whee ε > is abitay, and C is some absolute constant. Let y,..., y denote the ows of the matix Ψ. Dualizing (3) we see that (6) is equivalent to the following inequality: E sup id C T C yi T yi T ε with C = / C. Hee and theeafte, fo vectos x, y C n the tenso x y is the an-one linea opeato given by (x y)(z) = x, y z, whee is the canonical inne poduct on C n. The notation x T stands fo the estiction of a vecto x on its coodinates in the set T. The opeato id C T in (8) is the identity on C T, and the nom is the opeato nom fo opeatos on l T. The othogonality of Ψ can be expessed as id C n = n i= y i y i. We shall e-nomalize the vectos y i, letting x i = n y i. Now we have x i = O() fo all i. The poof has now educed to the following pobabilistic statement, which we intepet as a law of lage numbes fo andom opeatos. Theoem 3.4 (Unifom Opeato Law of Lage Numbes): Let x,..., x n be vectos in C n with unifomly bounded enties: x i K fo all i. Assume that id C n = n n x i x i. Let Ω be a andom subset of {,..., n} of size. Then E sup id C T ε (8) povided satisfies (7) (with constant C that may depend on K). Theoem 3.4 is poved by the techniques developed in Pobability in Banach spaces. The geneal oadmap is simila to [], []. We fist obseve that E = n = id C n, n so the andom opeato whose nom we estimate in (8) has mean zeo. Then the standad symmetization (see [7] Lemma 6.3) implies that the left-hand side of (8) does not exceed E sup ε i whee (ε i ) ae independent symmetic {, }-valued andom vaiables; also (jointly) independent of Ω. Then the conclusion of Theoem 3.4 will be easily deduced fom the following lemma. Lemma 3.5: Let x,..., x, n, be vectos in C n with unifomly bounded enties, x i K fo all i. Then E sup ε i sup whee C (K) log() log n log. Let us show how Lemma 3.5 implies Theoem 3.4. We fist condition on a choice of Ω and apply Lemma 3.5 fo x i, i Ω. Then we tae the expectation with espect to Ω. We then use the a consequence of Hölde inequality, E( X ) (E X ) and the tiangle inequality. Let us denote the left hand side of (8) by E. We obtain: E E sup T n (E + ). It follows that E C, povided that Theoem 3.4 now follows fom ou choice of = (, n). (9) = O(). Hence it is only left to pove Lemma 3.5. Thoughout the poof, B n p and BT p denote the unit ball of the nom p on C n. To this end, we fist eplace Benoulli.v. s ε i by standad independent nomal andom vaiables g i, using a compaison pinciple (inequality (4.8) in [7]). Then ou poblem becomes to bound the Gaussian pocess, indexed by the union of the unit Euclidean balls B T in CT fo all subsets I of {,..., n} of size at most. We apply Dudley s inequality (Theoem.7 in [7]), which is a geneal uppe bound on Gaussian pocesses. Let us denote the left hand side of (8) by E. We obtain: E C 3 E sup g i = C 3 E sup x B T g i x i, x C 4 log / N ( B T, δ, u ) du, whee N(Z, δ, u) denotes the minimal numbe of balls of adius u in metic δ centeed in points of Z, needed to cove the set Z. The metic δ in Dudley s inequality is defined by the Gaussian pocess, and in ou case it is [ M ( δ(x, y) = xi, x x i, y ) ] [ ( xi, x + x i, y ) ] [ max x i, z ] z B T = R max x i, x y, i max i x i, x y max i x i, x y

4 whee Hence E C 5 R Hee D,n p R := sup = x T i. log / N ( D,n, X, u ) du. () B T p, We will use containments x X = max i x i, x. D,n D,n KB X, D,n B n, () whee B X denotes the unit ball of the nom X. The second containment follows fom the unifom boundedness of (x i ). We can thus eplace D,n in () by D,n. Compaing () to the ight hand side of (9) we see that, in ode to complete the poof of Lemma 3.5, it suffices to show that K log / N ( D,n, X, u ) du C 6 log() log n log, () with C 6 = C 6 (K). To this end, we will estimate the coveing numbes in this integal in two diffeent ways. Fo big u, we will just use the second containment in (), which allows us to eplace D,n by B n. Lemma 3.6: Let x,..., x, n, be vectos as in Lemma 3.5. Then fo all u > we have whee m = C 7 K log()/u. N(B n, X, u) (n) m, Poof: We use the empiical method of Mauey. Fix a vecto y B n. Define a andom vecto Z Rn that taes values (,...,, sign(y(i)),,..., ) with pobability y(i) each, i =,..., n (all enties of that vecto ae zeo except i-th). Hee sign(z) = z/ z, wheneve z, and othewise. Note that EZ = y. Let Z,..., Z m be independent copies of Z. Using symmetization as befoe, we see that E 3 := Ey m X Z j m m m E X ε j Z j. Now we condition on a choice of (Z j ) and tae the expectation with espect to andom signs (ε j ). Using compaison to Gaussian vaiables as befoe, we obtain m X m X E 4 := E ε j Z j C 7 E g j Z j m = C 7 E max g j Z j, x i. i Fo each i, γ i := m g j Z j, x i is a Gaussian andom vaiable with zeo mean and with vaiance σ i = ( m Z j, x i ) / K m, since Z j, x i x i K. Using a simple bound on the maximum of Gaussian andom vaiables (see (3.3) in [7]), we obtain E 4 C 7 E max i γ i C 8 log max i σ i C 8 log K m. Taing the expectation with espect to (Z j ) we obtain E 3 m E(E 4) C 8K log m. With the choice of m made in the statement of the lemma, we conclude that E 3 u. We have shown that fo evey y B n, thee exists a z C n of the fom z = m m Z j such that y z X u. Each Z j taes n values, so z taes (n) m values. Hence B n can be coveed by at (n) m balls of nom X of adius u. A standad agument shows that we can assume that these balls ae centeed in points of B n. This completes the poof of Lemma 3.6. Fo small u, we will use a simple volumetic estimate. The diamete of B consideed as a set in C n is at most K with espect to the nom X (this was stated as the last containment in ()). It follows that N(B,, u) ( + K/u) fo all >, see (5.7) in [Pi]. The set D,n consists of d(, n) = ) balls of fom B T, thus ( n i N ( D,n, X, u ) d(n, )( + K/u). (3) Now we combine the estimate of the coveing numbe N(u) = log / N ( D,n, X, u ) of Lemma 3.6, and the volumetic estimate (3), to bound the integal in (). Using Stiling s appoximation, we see that d(, n) (C 9 n/). Thus [ ] N(u) C log(n/) + log( + /u) =: N (u), N(u) C log log n =: N (u), u whee C = C (K). Then we bound the integal in () as K A K N(u) du N (u) du + N (u) du A C A [ log(n/) + log( + /A) ] + C log(/a) log log n, whee C = C (K). Choosing A = /, we conclude that the integal in () is at most log(n/) + log + log() log log n. This poves (), which completes the poof of Lemma 3.5 and thus of Theoems 3.4 and 3..

5 IV. RECONSTRUCTION FROM GAUSSIAN MEASUREMENTS Ou goal will be to econstuct an -spase signal f R n fom = (, n) Gaussian measuements. These ae given by Φf R, whee Φ is a n andom matix ( Gaussian matix in the sequel), whose enties ae independent N(, ) andom vaiables. The econstuction will be achieved by solving the linea pogam (). The poblem again is to find the smallest numbe of measuements (, n) fo which, with high pobability, we have an exact econstuciton of evey -spase signal f fom its measuements Φf? It has ecently been shown in [5], [3], [3] that (, n) = O( log(n/)), (4) and was extended in [] to sub-gaussian measuements. This is asymptotically optimal. Howeve, the constant facto implicit in (4) has not been nown; pevious poofs of (4) yield uneasonably wea constants (of ode, and highe). In fact, thee has not been nown any theoetical guaantees with easonable constants fo Linea Pogamming based econstuctions. So, thee is pesently a gap between theoetical guaantees and good pactical pefomance of econstuction () (see e.g. [3]). Hee we shall pove a fist pactically easonable guaantee of the fom (4): (, n) c [ c + log(n/) ] ( + o()), (5) c = , c =.5. Theoem 4. (Reconstuction fom Gaussian measuements): A n Gaussian matix Φ with > (, n) satisfies the following with pobability ( 3.5 exp ( (, n) ) ) /8. Let f be an -spase signal in R n. Then f can be exactly econstucted fom the measuements Φf as a unique solution to the linea pogam (). Ou poof of Theoem 4. is diect, we will not use the Resticted Isomety Theoem.. The fist pat of this agument follows a geneal method of []. One intepets the exact econstuction as the fact that the (andom) enel of Φ misses the cone geneated by the (shifted) ball of l. Then one embeds the cone in a univesal set D, which is easie to handle, and poves that the andom subspace does not intesect D. Howeve, to obtain good constants as in (5), we will need to (a) impove the constant of embedding into D fom [], and (b) use Godon s Escape Though the Mesh Theoem [8], which is tight in tems of constants. In Godon s theoem, one measues the size of a set S in R n by its Gaussian width w(d) = E sup g, x, x S whee g is a andom vecto in R n whose components ae independent N(, ) andom vaiables (Gaussian vecto). The following is Godon s theoem [8]. Theoem 4. (Escape Though the Mesh (Godon)): Let S be a subset of the unit Euclidean sphee S n in R n. Let Y be a andom (n )-dimensional subspace of R n, distibuted unifomly in the Gassmanian with espect to the Haa measue. Assume that w(s) >. Then Y S = with pobability at least ( 3.5 exp ( / + w(s) ) ) /8. We will now pove Theoem 4.. Fist note that the function f is the unique solution of () if and only if is the unique solution of the poblem minimize f g subject to Φg Ke(Φ) =: Y. (6) Y is a (n )-dimensional subspace of R n. Due to the otation invaiance of the Gaussian andom vectos, Y is distibuted unifomly in the Gassmanian G n,n of (n )-dimensional subspaces of R n, with espect to the Haa measue. Now, is the unique solution to (6) if and only if is the unique metic pojection of f onto the subspace Y in the nom. This in tun is equivalent to the fact that is the unique contact point between the subspace Y and the ball of the nom centeed at f: (f + f B n ) Y = {}. (7) (Recall that B n p is the unit ball of the nom p.) Let C f be the cone in R n geneated by the set f + f B n (the cone of a set A R n is defined as {ta a A, t R + }). Then the statement that (7) holds fo all -spase functions f is clealy equivalent to C f Y = {} fo all -spase functions f. (8) We can epesent the cone C f as follows. Let T + = {i f(i) > }, T = {j f(i) < }, T = T + T. Then C f = { t R n } t(i) t(i) + t(i). i T i T + i T c We will now bound the cone C f by a univesal set, which does not depend on f. Lemma 4.3: Conside the spheical pat of the cone, K f = C f S n. Then K f ( + )D, whee D = conv{x S n supp(x) }. Poof: Fix a point x C S n. We have x(i) I, x(i) x(i). i T i T c i T The nom D on R n whose unit ball is D can be computed as L ( ) /, x D = l= i I l (x(i) ) whee L = n/, I l = {(l ) +,..., l}, fo l < L, I L = {(L ) +,..., n}, and (x(i) ) is a non-deceasing eaangement of the sequence ( x(i) ).

6 Set F = F (x) = {i x(i) / }. Since x S n, we have F. Hence, fo any x K thee exists a set E = E(x) {,..., m}, which consists of elements and such that E F I. Theefoe, x can be epesented as x = x +x so that supp(x ) E, x, supp(x ) E c, x /. Set ( B V E = B E E c ) B Ec. Then the above agument shows that K f E = V E =: W. The maximum of x D ove x W is attained at the exteme points of the sets V E, which have the fom x = x + x, whee x S E, and x has coodinates and ±/ with non-zeo coodinates. Notice that since supp(x ), x D x. Thus, fo any exteme point x of V E, x D x D + x D x + x +. The second inequality follows fom supp(x ) and supp(x ) =. This completes the poof of the lemma. To use Godon s escape though the mesh theoem, we have to estimate the Gaussian width of D. Lemma 4.4: w(d) log(e 3/ n/)( + o()). Poof: By definition, ( w(d) = sup g(i) ) /. J = Let p > be a numbe to be chosen late. By Hölde s inequality, we have ( ( w(d) E g(i) ) p/) /p J = By the Stiling s fomula, p/ Γ(p/ + /) Γ(/) i J i J ( ) /p( n ( E g(i) ) p/) /p ( en ) /p ( p/ Γ(p/ + /) ) /p. Γ(/) = ( + p ) + ( ) p/ p + ( + o()). Theefoe, w(d) ( ) en /p ( p+ ) / e ( + o()). Now set p = log( en ). Then w(d) (p + ) / ( + o()) = e log e3/ n ( + o()). To deduce (8) we define S = f K f, whee the union is ove all -spase functions f. Then (8) is equivalent to S Y =. (9) Lemma 4.3 implies that S ( +)D. Then by Lemma 4.4, w(s) ( + )w(d) = ( o()) (, n). Then (9) follows Godon s Theoem 4.. This completes the poof of Theoem 4.. Acnowledgement. Afte this pape was announced, A.Pajo pointed out that Lemma 3.6 was poved by B.Cal in [7], see Pop.3 and below. We also than Emmanuel Candes fo impotant emas. REFERENCES [] E. Candes, J. Rombeg, Quantitative Robust Uncetainty Pinciples and Optimally Spase Decompositions, Foundations of Computational Mathematics, to appea [] E. Candes, J. Rombeg, T. 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