Three Recent Examples of The Effectiveness of Convex Programming in the Information Sciences

Transcription

1 Three Recent Examples of The Effectiveness of Convex Programming in the Information Sciences Emmanuel Candès International Symposium on Information Theory, Istanbul, July 2013

2 Three stories about a theme Often have missing information: (1) Missing phase (phase retrieval) (2) Missing and/or corrupted entries in data matrix (robust PCA) (3) Missing high-frequency spectrum (super-resolution) Makes signal/data recovery difficult This lecture Convex programming usually (but not always) returns the right answer!

3 Story # 1: Phase Retrieval Collaborators: Y. Eldar, X. Li, T. Strohmer, V. Voroninski

4 X-ray crystallography Method for determining atomic structure within a crystal principle typical setup 10 Nobel Prizes in X-ray crystallography, and counting...

5 Importance principle Franklin s photograph

6 Missing phase problem Detectors only record intensities of diffracted rays magnitude measurements only! Fraunhofer diffraction intensity of electrical field 2 ˆx(f 1, f 2 ) 2 = x(t 1, t 2 )e i2π(f1t1+f2t2) dt 1 dt 2

7 Missing phase problem Detectors only record intensities of diffracted rays magnitude measurements only! Fraunhofer diffraction intensity of electrical field 2 ˆx(f 1, f 2 ) 2 = x(t 1, t 2 )e i2π(f1t1+f2t2) dt 1 dt 2 Phase retrieval problem (inversion) How can we recover the phase (or equivalently signal x(t 1, t 2 )) from ˆx(f 1, f 2 )?

8 About the importance of phase...

9 About the importance of phase... DFT DFT

10 About the importance of phase... DFT keep magnitude swap phase DFT

11 About the importance of phase... DFT keep magnitude swap phase DFT

12 X-ray imaging: now and then Röntgen (1895) Dierolf (2010)

13 Ultrashort X-ray pulses Imaging single large protein complexes

14 Discrete mathematical model Phaseless measurements about x 0 C n b k = a k, x 0 2 k {1,..., m} = [m] Phase retrieval is feasibility problem find x subject to a k, x 2 = b k k [m] Solving quadratic equations is NP hard in general

15 Discrete mathematical model Phaseless measurements about x 0 C n b k = a k, x 0 2 k {1,..., m} = [m] Phase retrieval is feasibility problem find x subject to a k, x 2 = b k k [m] Solving quadratic equations is NP hard in general Nobel Prize for Hauptman and Karle ( 85): make use of very specific prior knowledge

16 Discrete mathematical model Phaseless measurements about x 0 C n b k = a k, x 0 2 k {1,..., m} = [m] Phase retrieval is feasibility problem find x subject to a k, x 2 = b k k [m] Solving quadratic equations is NP hard in general Nobel Prize for Hauptman and Karle ( 85): make use of very specific prior knowledge Standard approach: Gerchberg Saxton (or Fienup) iterative algorithm Sometimes works well Sometimes does not

17 PhaseLift a k, x 2 = b k k [m]

18 PhaseLift a k, x 2 = b k k [m] Lifting: X = xx a k, x 2 = Tr(x a k a kx) = Tr(a k a kxx ) := Tr(A k X) a k a k = A k Turns quadratic measurements into linear measurements about xx

19 PhaseLift a k, x 2 = b k k [m] Lifting: X = xx a k, x 2 = Tr(x a k a kx) = Tr(a k a kxx ) := Tr(A k X) a k a k = A k Turns quadratic measurements into linear measurements about xx Phase retrieval: equivalent formulation find X s. t. Tr(A k X) = b k k [m] X 0, rank(x) = 1 Combinatorially hard min rank(x) s. t. Tr(A k X) = b k k [m] X 0

20 PhaseLift a k, x 2 = b k k [m] Lifting: X = xx a k, x 2 = Tr(x a k a kx) = Tr(a k a kxx ) := Tr(A k X) a k a k = A k Turns quadratic measurements into linear measurements about xx PhaseLift: tractable semidefinite relaxation minimize Tr(X) subject to Tr(A k X) = b k k [m] X 0 This is a semidefinite program (SDP) Trace is convex proxy for rank

21 Semidefinite programming (SDP) Special class of convex optimization problems Relatively natural extension of linear programming (LP) Efficient numerical solvers (interior point methods) LP (std. form): x R n SDP (std. form): X R n n minimize c, x subject to a T k x = b k k = 1,... x 0 minimize C, X subject to A k, X = b k k = 1,... X 0 Standard inner product: C, X = Tr(C X)

22 From overdetermined to highly underdetermined Quadratic equations Lift b k = a k, x 2 k [m] b = A(xx ) minimize subject to Tr(X) A(X) = b X 0 Have we made things worse? overdetermined (m > n) highly underdetermined (m n 2 )

23 This is not really new... Relaxation of quadratically constrained QP s Shor (87) [Lower bounds on nonconvex quadratic optimization problems] Goemans and Williamson (95) [MAX-CUT] Ben-Tal and Nemirovskii (01) [Monograph]... Similar approach for array imaging: Chai, Moscoso, Papanicolaou (11)

24 Quadratic equations: geometric view I

25 Quadratic equations: geometric view I

26 Quadratic equations: geometric view I

27 Quadratic equations: geometric view I

28 Quadratic equations: geometric view I

29 Quadratic equations: geometric view I

30 Quadratic equations: geometric view II

31 Exact phase retrieval via SDP Quadratic equations b k = a k, x 2 k [m] b = A(xx ) Simplest model: a k independently and uniformly sampled on unit sphere of C n if x C n (complex-valued problem) of R n if x R n (real-valued problem)

32 Exact phase retrieval via SDP Quadratic equations b k = a k, x 2 k [m] b = A(xx ) Simplest model: a k independently and uniformly sampled on unit sphere of C n if x C n (complex-valued problem) of R n if x R n (real-valued problem) Theorem (C. and Li ( 12); C., Strohmer and Voroninski ( 11)) Assume m n. With prob. 1 O(e γm ), for all x C n, only point in feasible set {X : A(X) = b and X 0} is xx

33 Exact phase retrieval via SDP Quadratic equations b k = a k, x 2 k [m] b = A(xx ) Simplest model: a k independently and uniformly sampled on unit sphere of C n if x C n (complex-valued problem) of R n if x R n (real-valued problem) Theorem (C. and Li ( 12); C., Strohmer and Voroninski ( 11)) Assume m n. With prob. 1 O(e γm ), for all x C n, only point in feasible set {X : A(X) = b and X 0} is xx Injectivity if m 4n 2 (Balan, Bodmann, Casazza, Edidin 09)

34 Numerical results: noiseless recovery (a) Smooth signal (real part) (b) Random signal (real part) Figure: Recovery (with reweighting) of n-dimensional complex signal (2n unknowns) from 4n quadratic measurements (random binary masks)

35 How is this possible? How can feasible set {X 0} {A(X) = b} have a unique point? [ ] x y Intersection of 0 with affine space y z

36 Correct representation Rank-1 matrices are on the boundary (extreme rays) of PSD cone

37 My mental representation

38 My mental representation

39 My mental representation

40 My mental representation

41 My mental representation

42 With noise b k x, a k 2 k [m] Noise aware recovery (SDP) minimize A(X) b 1 = k Tr(a ka k X) b k subject to X 0 Signal ˆx obtained by extracting first eigenvector (PC) of solution matrix

43 With noise b k x, a k 2 k [m] Noise aware recovery (SDP) minimize A(X) b 1 = k Tr(a ka k X) b k subject to X 0 Signal ˆx obtained by extracting first eigenvector (PC) of solution matrix In same setup as before and for realistic noise models, no method whatsoever can possibly yield a fundamentally smaller recovery error [C. and Li (2012)]

44 Numerical results: noisy recovery Figure: SNR versus relative MSE on a db-scale for different numbers of illuminations with binary masks

45 Numerical results: noiseless 2D images original image Gaussian masks Courtesy S. Marchesini (LBL) binary masks error with 8 binary masks

46 Story #2: Robust Principal Component Analysis Collaborators: X. Li, Y. Ma, J. Wright

47 The separation problem (Chandrasekahran et al.) M = L + S M: data matrix (observed) L: low-rank (unobserved) S: sparse (unobserved)

48 The separation problem (Chandrasekahran et al.) M = L + S M: data matrix (observed) L: low-rank (unobserved) S: sparse (unobserved) Problem: can we recover L and S accurately? Again, missing information

49 Motivation: robust principal component analysis (RPCA) PCA sensitive to outliers: breaks down with one (badly) corrupted data point x 11 x x 1n x 21 x x 2n x d1 x d2... x dn

50 Motivation: robust principal component analysis (RPCA) PCA sensitive to outliers: breaks down with one (badly) corrupted data point x 11 x x 1n x 11 x x 1n x 21 x x 2n x 21 x x 2n.... = x d1 x d2... x dn x d1 A... x dn

51 Robust PCA Data increasingly high dimensional Gross errors frequently occur in many applications Image processing Web data analysis Bioinformatics... Occlusions Malicious tampering Sensor failures... Important to make PCA robust

52 Gross errors Observe corrupted entries Users Movies A A A Y ij = L ij + S ij (i, j) Ω obs L low-rank matrix S entries that have been tampered with (impulsive noise) Problem Recover L from missing and corrupted samples

53 The L+S model (Partial) information y = A(M) about M }{{} object = L }{{} low rank + S }{{} sparse A????? A?????????? A? A????? A?? RPCA Dynamic MR data = low-dimensional structure + corruption video seq. = static background + sparse innovation Graphical modeling with hidden variables: Chandrasekaran, Sanghavi, Parrilo, Willsky ( 09, 11) marginal inverse covariance of observed variables = low-rank + sparse

54 When does separation make sense? Low-rank component cannot be sparse: L =

55 When does separation make sense? Low-rank component cannot be sparse: L = Sparse component cannot be low rank: S =

56 Low-rank component cannot be sparse L = Incoherent condition [C. and Recht ( 08)]: column and row spaces not aligned with coordinate axes (singular vectors are not sparse)

57 Low-rank component cannot be sparse A A M = Incoherent condition [C. and Recht ( 08)]: column and row spaces not aligned with coordinate axes (singular vectors are not sparse)

58 Sparse component cannot be low-rank x 1 x 2 x n 1 x n x 2 x n 1 x n x 1 x 2 x n 1 x n A x 2 x n 1 x n L = L + S = x 1 x 2 x n 1 x n A x }{{} 2 x n 1 x n 1x Sparsity pattern will be assumed (uniform) random

59 Demixing by convex programming L unknown (rank unknown) M = L + S S unknown (# of entries 0, locations, magnitudes all unknown)

60 Demixing by convex programming L unknown (rank unknown) M = L + S S unknown (# of entries 0, locations, magnitudes all unknown) Recovery via SDP minimize ˆL + λ Ŝ 1 subject to ˆL + Ŝ = M See also Chandrasekaran, Sanghavi, Parrilo, Willsky ( 09) nuclear norm: L = i σ i(l) (sum of sing. values) l 1 norm: S 1 = ij S ij (sum of abs. values)

61 Exact recovery via SDP min ˆL + λ Ŝ 1 s. t. ˆL + Ŝ = M

62 Exact recovery via SDP min ˆL + λ Ŝ 1 s. t. ˆL + Ŝ = M Theorem L is n n of rank(l) ρ r n (log n) 2 and incoherent S is n n, random sparsity pattern of cardinality at most ρ s n 2 Then with probability 1 O(n 10 ), SDP with λ = 1/ n is exact: ˆL = L, Ŝ = S Same conclusion for rectangular matrices with λ = 1/ max dim

63 Exact recovery via SDP min ˆL + λ Ŝ 1 s. t. ˆL + Ŝ = M Theorem L is n n of rank(l) ρ r n (log n) 2 and incoherent S is n n, random sparsity pattern of cardinality at most ρ s n 2 Then with probability 1 O(n 10 ), SDP with λ = 1/ n is exact: ˆL = L, Ŝ = S Same conclusion for rectangular matrices with λ = 1/ max dim No tuning parameter! Whatever the magnitudes of L and S A A A A A A A A A A A A

64 Phase transitions in probability of success (a) PCP, Random Signs (b) PCP, Coherent Signs (c) Matrix Completion L = XY T is a product of independent n r i.i.d. N (0, 1/n) matrices

65 Missing and corrupted RPCA min ˆL + λ Ŝ 1 s. t. ˆLij + Ŝij = L ij + S ij (i, j) Ω obs A????? A?????????? A? A????? A??

66 Missing and corrupted RPCA min ˆL + λ Ŝ 1 s. t. ˆLij + Ŝij = L ij + S ij (i, j) Ω obs A????? A?????????? A? A????? A?? Theorem L as before Ω obs random set of size 0.1n 2 (missing frac. is arbitrary) Each observed entry corrupted with prob. τ τ 0 Then with prob. 1 O(n 10 ), PCP with λ = 1/ 0.1n is exact: ˆL = L Same conclusion for rectangular matrices with λ = 1/ 0.1max dim

67 Background subtraction

68 With noise With Li, Ma, Wright & Zhou ( 10) Z stochastic or deterministic perturbation Y ij = L ij + S ij + Z ij (i, j) Ω minimize ˆL + λ Ŝ 1 subject to (i,j) Ω (M ij ˆL ij Ŝij) 2 δ 2 When perfect (noiseless) separation occurs = noisy variant is stable

69 Story #3: Super-resolution Collaborator: C. Fernandez-Granda

70 Limits of resolution In any optical imaging system, diffraction imposes fundamental limit on resolution The physical phenomenon called diffraction is of the utmost importance in the theory of optical imaging systems (Joseph Goodman)

71 Bandlimited imaging systems (Fourier optics) f obs (t) = (h f)(t) h : point spread function (PSF) ˆf obs (ω) = ĥ(ω) ˆf(ω) ĥ : transfer function (TF) Bandlimited system ω > Ω ĥ(ω) = 0 ˆf obs (ω) = ĥ(ω) ˆf(ω) suppresses all high-frequency components

72 Bandlimited imaging systems (Fourier optics) f obs (t) = (h f)(t) h : point spread function (PSF) ˆf obs (ω) = ĥ(ω) ˆf(ω) ĥ : transfer function (TF) Bandlimited system ω > Ω ĥ(ω) = 0 ˆf obs (ω) = ĥ(ω) ˆf(ω) suppresses all high-frequency components Example: coherent imaging ĥ(ω) = 1 P (ω) indicator of pupil element TF PSF cross-section (PSF) Pupil Airy disk

73 Rayleigh resolution limit Lord Rayleigh

74 The super-resolution problem objective data Retrieve fine scale information from low-pass data Equivalent description: extrapolate spectrum (ill posed)

75 Random vs. low-frequency sampling Random sampling (CS) Low-frequency sampling (SR) Compressive sensing: spectrum interpolation Super-resolution: spectrum extrapolation

76 Super-resolving point sources This lecture Signal of interest is superposition of point sources Celestial bodies in astronomy Line spectra in speech analysis Fluorescent molecules in single-molecule microscopy Many applications Radar Spectroscopy Medical imaging Astronomy Geophysics...

77 Single molecule imaging Microscope receives light from fluorescent molecules Problem Resolution is much coarser than size of individual molecules (low-pass data) Can we beat the diffraction limit and super-resolve those molecules? Higher molecule density faster imaging

78 Mathematical model Signal x = j a jδ τj a j C, τ j T [0, 1] Data y = F n x: n = 2f lo + 1 low-frequency coefficients (Nyquist sampling) y(k) = 1 0 e i2πkt x(dt) = j a j e i2πkτj k Z, k f lo Resolution limit: (λ lo /2 is Rayleigh distance) 1/f lo = λ lo

79 Mathematical model Signal x = j a jδ τj a j C, τ j T [0, 1] Data y = F n x: n = 2f lo + 1 low-frequency coefficients (Nyquist sampling) y(k) = 1 0 e i2πkt x(dt) = j a j e i2πkτj k Z, k f lo Resolution limit: (λ lo /2 is Rayleigh distance) 1/f lo = λ lo Question Can we resolve the signal beyond this limit? Swap time and frequency spectral estimation

80 Can you find the spikes? Low-frequency data about spike train

81 Can you find the spikes? Low-frequency data about spike train

82 Recovery by minimum total-variation Recovery by cvx prog. min x TV subject to F n x = y x TV = x(dt) is continuous analog of l 1 norm x = j a j δ τj = x TV = j a j With noise min 1 2 y F n x 2 l 2 + λ x TV

83 Recovery by convex programming y(k) = 1 0 e i2πkt x(dt) k f lo

84 Recovery by convex programming y(k) = 1 0 e i2πkt x(dt) k f lo Theorem (C. and Fernandez Granda (2012)) If spikes are separated by at least 2 /f lo := 2 λ lo then min TV solution is exact! For real-valued x, a min dist. of 1.87λ lo suffices

85 Recovery by convex programming y(k) = 1 0 e i2πkt x(dt) k f lo Theorem (C. and Fernandez Granda (2012)) If spikes are separated by at least 2 /f lo := 2 λ lo then min TV solution is exact! For real-valued x, a min dist. of 1.87λ lo suffices Infinite precision!

86 Recovery by convex programming y(k) = 1 0 e i2πkt x(dt) k f lo Theorem (C. and Fernandez Granda (2012)) If spikes are separated by at least 2 /f lo := 2 λ lo then min TV solution is exact! For real-valued x, a min dist. of 1.87λ lo suffices Infinite precision! Whatever the amplitudes

87 Recovery by convex programming y(k) = 1 0 e i2πkt x(dt) k f lo Theorem (C. and Fernandez Granda (2012)) If spikes are separated by at least 2 /f lo := 2 λ lo then min TV solution is exact! For real-valued x, a min dist. of 1.87λ lo suffices Infinite precision! Whatever the amplitudes Can recover (2λ lo ) 1 = f lo /2 = n/4 spikes from n low-freq. samples

88 Recovery by convex programming y(k) = 1 0 e i2πkt x(dt) k f lo Theorem (C. and Fernandez Granda (2012)) If spikes are separated by at least 2 /f lo := 2 λ lo then min TV solution is exact! For real-valued x, a min dist. of 1.87λ lo suffices Infinite precision! Whatever the amplitudes Can recover (2λ lo ) 1 = f lo /2 = n/4 spikes from n low-freq. samples Cannot go below λ lo

89 Recovery by convex programming y(k) = 1 0 e i2πkt x(dt) k f lo Theorem (C. and Fernandez Granda (2012)) If spikes are separated by at least 2 /f lo := 2 λ lo then min TV solution is exact! For real-valued x, a min dist. of 1.87λ lo suffices Infinite precision! Whatever the amplitudes Can recover (2λ lo ) 1 = f lo /2 = n/4 spikes from n low-freq. samples Cannot go below λ lo Essentially same result in higher dimensions

90 About separation: sparsity is not enough! CS: sparse signals are away from null space of sampling operator Super-res: this is not the case Signal Spectrum

91 About separation: sparsity is not enough! CS: sparse signals are away from null space of sampling operator Super-res: this is not the case Signal Spectrum

92 Analysis via prolate spheroidal functions David Slepian If distance between spikes less than λ lo /2 (Rayleigh), problem hopelessly ill posed

93 The super-resolution factor (SRF) Have data at resolution λ lo Super-resolution factor Wish resolution λ hi SRF = λ lo λ hi

94 The super-resolution factor (SRF): frequency viewpoint Observe spectrum up to f lo Wish to extrapolate up to f hi Super-resolution factor SRF = f hi f lo

95 With noise y = F n x + noise F n x = 1 0 e i2πkt x(dt) k f lo At native resolution (ˆx x) ϕ λlo TV noise level

96 With noise y = F n x + noise F n x = 1 0 e i2πkt x(dt) k f lo At finer resolution λ hi = λ lo /SRF, convex programming achieves (ˆx x) ϕ λhi TV SRF 2 noise level

97 With noise y = F n x + noise F n x = 1 0 e i2πkt x(dt) k f lo At finer resolution λ hi = λ lo /SRF, convex programming achieves (ˆx x) ϕ λhi TV SRF 2 noise level Modulus of continuity studies for super-resolution: Donoho ( 92)

98 Formulation as a finite-dimensional problem Primal problem min x TV s. t. F n x = y Dual problem max Re y, c s. t. F nc 1 Infinite-dimensional variable x Finitely many constraints Finite-dimensional variable c Infinitely many constraints (Fn c)(t) = c k e i2πkt k f lo

99 Formulation as a finite-dimensional problem Primal problem min x TV s. t. F n x = y Dual problem max Re y, c s. t. F nc 1 Infinite-dimensional variable x Finitely many constraints Finite-dimensional variable c Infinitely many constraints (Fn c)(t) = c k e i2πkt k f lo Semidefinite representability (F n c)(t) 1 for all t [0, 1] equivalent to (1) there is Q Hermitian s. t. [ ] Q c c 0 1 (2) Tr(Q) = 1 (3) sums along superdiagonals vanish: n j i=1 Q i,i+j = 0 for 1 j n 1

100 SDP formulation Dual as an SDP maximize Re y, c subject to [ ] Q c c 0 1 n j i=1 Q i,i+j = δ j 0 j n 1 Dual solution c: coeffs. of low-pass trig. polynomial k c ke i2πkt interpolating the sign of the primal solution

101 SDP formulation Dual as an SDP maximize Re y, c subject to [ ] Q c c 0 1 n j i=1 Q i,i+j = δ j 0 j n 1 Dual solution c: coeffs. of low-pass trig. polynomial k c ke i2πkt interpolating the sign of the primal solution To recover spike locations (1) Solve dual (2) Check when polynomial takes on magnitude 1

102 Noisy example SNR: 14 db Measurements

103 Noisy example SNR: 14 db Measurements Low res signal

104 Noisy example SNR: 14 db Measurements High res signal

105 Noisy example Average localization error: High res Signal Estimate

106 Summary Three important problems with missing data Phase retrieval Matrix completion/rpca Super-resolution Three simple and model-free recovery procedures via convex programming Three near-perfect solutions

107 Apologies: things I have not talked about Algorithms Applications Avalanche of related works

108 A small sample of papers I have greatly enjoyed Phase retrieval Netrapalli, Jain, Sanghavi, Phase retrieval using alternating minimization ( 13) Waldspurger, d Aspremont, Mallat, Phase recovery, MaxCut and complex semidefinite programming ( 12) Robust PCA Gross, Recovering low-rank matrices from few coefficients in any basis ( 09) Chandrasekaran, Parrilo and Willsky, Latent variable graphical model selection via convex optimization ( 11) Hsu, Kakade and Zhang, Robust matrix decomposition with outliers ( 11) Super-resolution Kahane, Analyse et synthèse harmoniques ( 11) Slepian, Prolate spheroidal wave functions, Fourier analysis, and uncertainty. V - The discrete case ( 78)