AN INTRODUCTION TO COMPRESSIVE SENSING

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1 AN INTRODUCTION TO COMPRESSIVE SENSING Rodrigo B. Platte School of Mathematical and Statistical Sciences APM/EEE598 Reverse Engineering of Complex Dynamical Networks

2 OUTLINE 1 INTRODUCTION 2 INCOHERENCE 3 RIP 4 POLYNOMIAL MATRICES 5 DYNAMICAL SYSTEMS AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 2 / 37

3 THE RICE DSP WEBSITE Resources for papers, codes, and more... References: Emmanuel Candès, Compressive sampling. (Proc. International Congress of Mathematics, 3, pp , Madrid, Spain, 26) Richard Baraniuk, A Lecture on Compressive Sensing. (IEEE Signal Processing Magazine, July 27) Emmanuel Candès and Michael Wakin, An introduction to compressive sampling. (IEEE Signal Processing Magazine, 25(2), pp. 21-3, March 28) m-files and some links are available in the course page AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 3 / 37

4 Some well known CS people: VIDEO LECTURES Emmanuel Candès (Stanford University) Sequence of papers with Terence Tao and Justin Romberg in 24. David Donoho (Stanford University) Richard Baraniuk (Rice University) Ronald A. DeVore (Texas A&M) Anna C. Gilbert (Univ. of Michigan) Jared Tanner (University of Edinburgh)... A good way to learn the basics of CS is to watch these IMA video lectures: IMA New Directions short courses Compressive Sampling and Frontiers in Signal Processing (two weeks long) AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 4 / 37

5 UNDERDETERMINED SYSTEMS cafeperss.com $2 AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 5 / 37

6 S N K-sparse y = UNDERDETERMINED SYSTEMS Θ S (b) t process with a random Gaussian ansform (DCT) matrix. The vector of rement process with =. There are oefficients; the measurement vector y is a length N. However, both the RIP and erated it), and the basis struct the length-n signal Solve lently, its sparse coefficient K-sparse signals, since M Ax = b, there are infinitely many s where s A= isy. m This N is because if and (s m < + N. r) = y for any vector space N ( ) of. Therefor reconstruction algorithm a the signal s sparse coefficie the (N M)-dimensional tr space H = N ( ) + s. Minimum l 2 norm rec Define the l p norm of th ( s p ) p = N i=1 s i p. Th approach to inverse prob type is to find the vector AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 6 / 37

7 erated it), and the basis S UNDERDETERMINED SYSTEMS y Θ S struct the length-n signal Solve lently, its sparse coefficient N = K-sparse signals, since M K-sparse Ax = b, there are infinitely many s where s A= isy. m This N is because if and (s m < + N. r) = y for any vector space N ( ) of. Therefor In CS we want to obtain sparse (b) solutions, i.e., x reconstruction j, for several j algorithm s. a t process with a random Gaussian the signal s sparse coefficie ansform (DCT) matrix. The vector of the (N M)-dimensional tr rement process with =. There are space H = N ( ) + s. oefficients; the measurement vector y is a Minimum l 2 norm rec Define the l p norm of th ( s p ) p = N i=1 s i p. Th approach to inverse prob length AN INTRODUCTION N. However, TO COMPRESSIVE both SENSING the R. PLATTE RIP and MATHEMATICS type AND is STATISTICS to find the vector 6 / 37

erated it), and the basis S UNDERDETERMINED SYSTEMS y Θ S struct the length-n signal Solve lently, its sparse coefficient N = K-sparse signals, since M K-sparse Ax = b, there are infinitely many s

a One option: Minimize x t process with a random Gaussian l1 subject to Ax = b. the signal s sparse coefficie ansform (DCT) matrix. The vector of the (N M)-dimensional tr rement process with =.

8 erated it), and the basis S UNDERDETERMINED SYSTEMS y Θ S struct the length-n signal Solve lently, its sparse coefficient N = K-sparse signals, since M K-sparse Ax = b, there are infinitely many s where s A= isy. m This N is because if and (s m < + N. r) = y for any vector space N ( ) of. Therefor In CS we want to obtain sparse (b) solutions, i.e., x reconstruction j, for several j algorithm s. a One option: Minimize x t process with a random Gaussian l1 subject to Ax = b. the signal s sparse coefficie ansform (DCT) matrix. The vector of the (N M)-dimensional tr rement process with =. There are space H = N ( ) + s. oefficients; the measurement x lp = ( xvector p + x y 2 is p a+ + x N p) 1/p Minimum l 2 norm rec Define the l p norm of th Why p = 1? ( s p ) p = N i=1 s i p. Th Remark: the location of nonzero x j s is not known in advance. approach to inverse prob length AN INTRODUCTION N. However, TO COMPRESSIVE both SENSING the R. PLATTE RIP and MATHEMATICS type AND is STATISTICS to find the vector 6 / 37

9 WHY l 1? Unit ball: l, l 1/2, l 1, l 2, l 4, l x lp = ( x p + + x N p) 1/p or, for p < 1, x lp = ( x p + + x N p) x l = # of nonzero entries in x ideal (?) but leads to a NP-complete problem l p, with p < 1 is not a norm (triangular inequality). Also not practical. l 2 computationally easy but does not lead to sparse solutions. The unique solution of minimum l 2 norm is (pseudo-inverse) x = A (AA ) 1 b AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 7 / 37

10 EXAMPLE SPARSITY AND THE l 1 -NORM (2D CASE) a 1 x 1 + a 2 x 2 = b x2!.5!1!1.5!1.5!1! x 1 AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 8 / 37

11 EXAMPLE l 2 SPARSITY AND THE l 1 -NORM (2D CASE) min x x 1,x x 2 2 subject to a 1 x 1 + a 2 x 2 = b x2 x x 2 2 <.8944!.5!1 x x 2 2 >.8944!1.5!1.5!1! x 1 AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 8 / 37

12 SPARSITY AND THE l 1 -NORM (2D CASE) EXAMPLE l 1 min x 1,x 2 x 1 + x 2 subject to a 1 x 1 + a 2 x 2 = b x2 x 1 + x 2 < 1!.5!1 x 1 + x 2 > 1!1.5!1.5!1! x 1 AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 8 / 37

13 MINIMIZING x l2 Recall Parseval s Formula: f (t) = N k= x kφ k (t), with φ k orthonormal in L 2. N f 2 2 = x k 2. k= Also, l 2 penalizes heavily large values, while small values don t affect the norm significantly. In general will not give a sparse representation! See matlab experiment! (Test-l1-l2.m) AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 9 / 37

14 MINIMIZING x l1 Matlab experimet! (Test-l1-l2.m) Note: solution may not be unique! Solve an optimization problem (in practice O(N 3 ) operations). Several codes are available for CS see: AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 1 / 37

15 A SIMPLE EXAMPLE f (t) = 1 N N k=1 x k sin(πkt) N = 124, number of samples: m = 5 AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 11 / 37

16 A SIMPLE EXAMPLE System of equations: f (t j ) = x k sin(πkt j ), j = k=1 SOLVE: min x l1 subject to Ax = b, where A has 5 rows and 124 columns. A j,k = sin(πkt j ), b j = f (t j ). Matlab code on Blackboard: SineExample.m (uses CVX) AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 11 / 37

17 A SIMPLE EXAMPLE original decoded Recovery of coefficients is accurate to almost machine precision! x x 2 x 2 = AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 11 / 37

18 WHY SPARSITY? Sparsity is often a good regularization criteria because most signals have structure. Take a picture! (this one has pixels) AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 12 / 37

19 WHY SPARSITY? Sparsity is often a good regularization criteria because most signals have structure. Gray scale please! AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 12 / 37

20 WHY SPARSITY? Sparsity is often a good regularization criteria because most signals have structure Find wavelet coefficients. Daubechies(6,2), 3 vanish. moments AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 12 / 37

21 WHY SPARSITY? Sparsity is often a good regularization criteria because most signals have structure. Make 75% of the coefficients zero. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 12 / 37

22 WHY SPARSITY? Sparsity is often a good regularization criteria because most signals have structure. Restored image from 25% of the coefficients. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 12 / 37

23 WHY SPARSITY? Sparsity is often a good regularization criteria because most signals have structure Relative error 3%. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 12 / 37

24 WHY SPARSITY? Sparsity is often a good regularization criteria because most signals have structure Keep only 2% of the coefficients, set 98% to zero. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 12 / 37

25 WHY SPARSITY? Sparsity is often a good regularization criteria because most signals have structure. Reconstructed image from 2% of the coefficients. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 12 / 37

26 SPARSITY IS NOT SUFFICIENT FOR CS TO WORK! Example: A is a finite difference matrix A maps a sparse vector x into another sparse vector y = AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 13 / 37

27 SPARSITY IS NOT SUFFICIENT FOR CS TO WORK! Example: A is a finite difference matrix A maps a sparse vector x into another sparse vector y = A few samples of y are likely to be all zeros! AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 13 / 37

28 SPARSITY IS NOT SUFFICIENT FOR CS TO WORK! The image below is sparse in physical domain and Haar wavelet coefficients. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 14 / 37

29 A GENERAL APPROACH Sample coefficients in a representation by random vectors. N y = < y, ψ k > ψ k, k=1 ψ k are obtained from orthogonalized Gaussian matrices. Ax = y Ψ Ax = Ψ y Θx = z AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 15 / 37

30 INCOHERENCE + SPARSITY IS NEEDED INCOHERENCE sparse representation sample here AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 16 / 37

31 INCOHERENCE + SPARSITY IS NEEDED INCOHERENCE sparse representation sample here THEOREM (CANDÈS, ROMBERG, TAO) Assume that x is S-sparse and that we are given K Fourier coefficients with frequencies selected uniformly at random. Suppose that the number of observations obeys K C S log N. Then minimizing l 1 reconstructs x exactly with overwhelming probability. In details, if the constant C is of the form 22(δ + 1), then the probability of success exceeds 1 O(N δ ). AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 16 / 37

32 INCOHERENCE + SPARSITY IS NEEDED NUMERICAL EXPERIMENT Signal recovered from Fourier coefficients: original decoded Code FourierSampling.m. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 16 / 37

33 INCOHERENT SAMPLING Let (Φ, Ψ) be orthonormal bases of R n. f (t) = n x i ψ i (t) and y k = f, ϕ k, k = 1,..., m. i=1 Representation matrix: Ψ = [ψ 1 ψ 2 ψ n ] Sensing matrix: Φ = [ϕ 1 ϕ 2 ϕ n ] COHERENCE BETWEEN Φ AND Ψ µ(φ, Ψ) = n max ϕ k, ψ j. 1 j,k n Remark: µ(φ, Ψ) [1, n] Upper bound: Cauchy-Schwarz Lower bound: Ψ T Φ is also orthonormal, hence ϕk, ψ j 2 = 1 max j ϕ k, ψ j 1/ n AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 17 / 37

34 f (t) = A GENERAL RESULT FOR SPARSE RECOVERY n x i ψ i (t) and y k = f, ϕ k, k = 1,..., m. i=1 Consider the optimization problem: min x x R n l 1 subject to y k = Ψx, ϕ k, k = 1,..., m. THEOREM (CANDÈS AND ROMBERG, 27) Fix f R n and suppose that the coefficient sequence x of f in the basis Ψ is s-sparse. Select m measurements in the Φ domain uniformly at random. Then if m C µ 2 (Φ, Ψ) S log(n/δ) for some positive constant C, the solution of the problem above is exact with probability exceeding 1 δ. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 18 / 37

35 EXAMPLES OF INCOHERENT BASES Φ is the identity (ϕ k (t) = δ(t k)) and Ψ is the Fourier basis. The time-frequency pair obeys µ(φ, Ψ) = 1. noiselets and Haar wavelets have coherence 2. random matrices are largely incoherent with any fixed basis Ψ (about 2 log n). Matlab example: measurementsl1.m AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 19 / 37

36 MULTIPLE SOLUTIONS OF MIN l 1 -NORM f (t) = a N 2 + a k cos(πkt) + k=1 Data: f ( 1) = 1, f () = 1, f (1) = 1 N b k sin(πkt), t [ 1, 1] k=1 AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 2 / 37

37 MULTIPLE SOLUTIONS OF MIN l 1 -NORM f (t) = a N 2 + a k cos(πkt) + k=1 Data: f ( 1) = 1, f () = 1, f (1) = 1 N b k sin(πkt), t [ 1, 1] k=1 even function: b k = Solutions of min l 1 : {a 2 = 1, a k = (k 2)}, {a 4 = 1, a k = (k 4)},.. ȦN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 2 / 37

38 THE RESTRICTED ISOMETRY PROPERTY (RIP) How about signals that are not exactly sparse? ISOMETRY CONSTANTS For each s = 1, 2,..., define δ s of a matrix A as the smallest number such that (1 δ s ) x 2 l 2 Ax 2 l 2 (1 + δ s ) x 2 l 2 holds for all s-sparse vectors x. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 21 / 37

39 THE RESTRICTED ISOMETRY PROPERTY (RIP) THEOREM (CANDÈS, 27?) Assume δ 2s < 2 1. Then x := argmin x R n x l1 subject to y = Ax x x l2 C x x s l1 s. where x s is the vector x with all but the largest s components set to. If x is s-sparse (exactly), then the recovery is exact. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 22 / 37

40 RIP - BASIC IDEA want A to preserve norm of s-sparse vectors. Ax 1 Ax should not be small for s-sparse vectors x. want < c x 1 x A(x 1 x 2 ) 2 2 for all s-sparse x. If δ 2 s = 1, then Az 2 = for a 2s-sparse z. z = x 1 x 2 with x 1 and x 2 both s-sparse. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 23 / 37

41 RIP - REMARKS the theorem above is deterministic how does one show that column vectors taken from arbitrarily subsets are nearly orthogonal? isometry constants are shown for random matrices (randomness is back) for Fourier basis m C s log 4 n RIP is too conservative (Donoho, Tanner 21) AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 24 / 37

42 POLYNOMIAL MATRICES Back to Dr. Lai s dynamical system problem: dx dt = F(x(t)), with [F(x(t))] j = k 1 k 2 (a j ) k1 k 2 k m x k1 1 k m (t)... x km m (t) This does not fit in classical CS-results. monomial basis becomes ill-conditioned even for small powers we know condition numbers of Vadermonde depend on where x is evaluated. Some CS results are available for orthogonal polynomials. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 25 / 37

43 ORTHOGONAL POLYNOMIALS For Chebyshev polynomials expansions we have that f (x) N λ k cos(k arccos(x)) k= If we let y = arccos(x) or x = cos(y), f (cos(y)) N λ k cos(ky) A Chebyshev expansion is equivalent to a cosine expansion on the variable y. Results carry over from Fourier expansions but with samples chosen independently according to the chebyshev measure k= dν(x) = π 1 (1 x 2 ) 1/2 dx AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 26 / 37

44 SPARSE LEGENDRE EXPANSIONS Rauhut and Ward (21) proved that the same type sampling applies for Legendre exapasions. How about polynomial expansions as power series? AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 27 / 37

45 Discovering Sparse Polynomials INTRODUCTION INCOHERENCE RIP POLYNOMIAL MATRICES DYNAMICAL SYSTEMS ROBERT THOMPSON S EXPRIMENTS How about if we choose just a few function values? Φ m : m randomly chosen rows of identity matrix. And assume that x is K-sparse. t d according to some distribution in ( 1, 1). f(t d1 ) f(t d2 ) y m =.. f(t dm ) m t t 1... t N x 1 t 1 t t N 1 x 2 = Φ m t N t1 N... tn N x N AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 28 / 37

46 Sparse Polynomial Discovery INTRODUCTION INCOHERENCE RIP POLYNOMIAL MATRICES DYNAMICAL SYSTEMS How Well Does It Work? ROBERT THOMPSON S EXPERIMENTS 1d polynomial recovery for N =36,uniformsampling 1d polynomial recovery for N =36,Chebyshevsampling k/m.5.5 k/m m/n m/n Each pixel, 5 experiments: choose random polynomial with k non-zero Gaussian i.i.d coefficients, measure m samples, attempt to recover polynomial coefficients. Sampling at Chebyshev points give (very) slightly better results than uniform points. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 29 / 37

47 Comparison With Chebyshev-Sparse Polynomials ROBERT THOMPSON S EXPERIMENTS Consider linear combinations of Chebyshev polynomials: y = N i=1 T i(t), T i (t) = cos(i arccos(t)) Φ m : m randomly chosen rows of identity matrix. And assume that x is K-sparse. t d according to some distribution in ( 1, 1). f(t d1 ) f(t d2 ) y m =.. f(t dm ) m T (t ) T 1 (t )... T N (t ) x 1 T (t 1 ) T 1 (t 1 )... T N (t 1 ) x 2 = Φ m T (t N ) T 1 (t N )... T N (t N ) x N AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 3 / 37

48 ow Well Does It Work? INTRODUCTION INCOHERENCE RIP POLYNOMIAL MATRICES DYNAMICAL SYSTEMS ROBERT THOMPSON S EXPERIMENTS Vandermonde 1d polynomial recovery for N =36,Chebyshevsampling Chebyshev Sparse 1d Chebyshev polynomial recovery, N = k/m k/m m/n m/n.9 Using Chebyshev basis functions, we realize improvement as m increases. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 31 / 37

49 Increasing m helps ROBERT THOMPSON S EXPERIMENTS Columns of C are orthogonal. All vectors will be distinguishable if we use full C. If we use less than full C, orthogonality is lost, some vectors start to become indistinguishable. V C AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 32 / 37

50 Discovering 2-DROBERT Sparse THOMPSON S Polynomials EXPERIMENTS What about 2-D polynomials? In natural basis: f(t, u) = x ij t i u j i+j=..q (t d,u d ) according to some distribution in ( 1, 1) ( 1, 1). f(t d1,u d1 ) f(t d2,u d2 ) y m =.. f(t dm,u dm ) m 1 t u t u t 2 u 2... x 1 t 1 u 1 t 1 u 1 t 2 1 u x 1 = Φ m.. x 1 x 11.. x 2 1 t N u N t N u N t 2 N u2 N.... AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 33 / 37

51 How Well ROBERT Does THOMPSON S It Work? EXPERIMENTS 2d polynomial recovery, N = k/m m/n.9 Similar to 1-d results. Again increasing m doesn t change much. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 34 / 37

52 Example - Logistic Map ROBERT THOMPSON S EXPERIMENTS (BACK TO DYNAMICAL SYSTEMS) x n+1 = f(x n )=rx n (1 x n ) Coefficient vector: (, r, r,,...) We can recover the system equation in chaotic regime taking about 1 sample pairs or more. 1 Sampling the logistic map, m =1 1 6 Recovery error for logistic map, r = xn c c n m AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 35 / 37

53 How Well Does It Work? INTRODUCTION INCOHERENCE RIP POLYNOMIAL MATRICES DYNAMICAL SYSTEMS ROBERT THOMPSON S EXPERIMENTS (BACK TO DYNAMICAL SYSTEMS) Sensitive to the dynamics determined by r. (Bifurcation diagram: Wikipedia). AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 36 / 37

54 FINAL REMARKS As previously pointed by Dr. Lai recovery seems impractical with monomial basis of large degree. Change of basis to orthogonal polynomials result in full coefficients. Considering small degree expansions in high dimensions what is the optimal sampling strategy? How about a system of PDEs? For example, u t = u(1 u) uv + u v t = v(1 v) + uv + v Thanks! In particular to Robert Thompson and Wen Xu. AN INTRODUCTION TO COMPRESSIVE SENSING R. PLATTE MATHEMATICS AND STATISTICS 37 / 37

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