Gauge optimization and duality

Transcription

1 1 / 54 Gauge optimization and duality Junfeng Yang Department of Mathematics Nanjing University Joint with Shiqian Ma, CUHK September, 2015

2 2 / 54 Outline Introduction Duality Lagrange duality Fenchel duality Gauge duality

3 3 / 54 Outline Introduction Duality Lagrange duality Fenchel duality Gauge duality

4 4 / 54 Gauge function Definition (gauge function) A function f : R n R {+ } is called a gauge function if 1. f is convex: for all x, y R n and α (0, 1), f (αx + (1 α)y) αf (x) + (1 α)f (y); 2. f is positively homogeneous: for all x R n and λ > 0, f (λx) = λf (x); 3. f is nonnegative: f (x) 0 for all x; 4. f vanishes at the origin: f (0) = 0. Norms and semi-norms are particular gauge functions.

5 5 / 54 The problem We are interested in the following gauge optimization min{κ(x) ρ(ax b) ε}, x κ and ρ are gauge functions, and ρ 1 (0) = 0; A, b and ε > 0 are given data.

6 Sparsity 6 / 54

7 7 / 54 Sparse solutions Problem: find a sparse solution x min{ x 0 Ax b}, x or, find x such that the residue is sparse min x Ax b 0. Drawback: Combinatorial optimization, difficult.

8 8 / 54 Sparse representation represent a signal y as a superposition of elementary signal atoms: y = i φ ix i or y = Φx; unique representation if the basis is orthonormal; overcomplete dictionaries have more flexibility, but representation is not unique; a good" transform leads to a fast decay in coefficients.

9 Sparse representation 9 / 54

10 Sparse representation 10 / 54

11 Sparse representation 11 / 54

12 Sparse representation 12 / 54

13 Sparse representation 13 / 54

14 Sparse representation 14 / 54

15 Sparse representation 15 / 54

16 Sparse representation 16 / 54

17 17 / 54 LASSO/BP/BPDN The LASSO model (Tibshirani, 1996) min{ Ax b 2 x 1 k}. x The BP/BPDN model (Donoho et al., 1998; Candes et al., 2006) min{ x 1 Ax b 2 ε}. x Joint sparsity (Malioutov-Cetin-Wilsky 2003) { } min X(:, j) 2 AX B 2 ε. X j The last two are gauge optimization.

18 18 / 54 Matrix completion Seek a matrix X R m n with some desired structure (e.g., low rank) that matches certain observations, possibly noisy. min{κ(x) AX b ε}, X where A is a linear mapping (e.g. observations of certain elements of X). Setting κ as the nuclear norm, i.e., sum of all singular values, promotes low rank solutions. Applications in recommender systems, e.g., Netflix, Amazon. (Recht et al., 2010)

19 19 / 54 Phase retrieval In phase retrieval, one recovers the phase information of a signal, e.g., an image, from magnitude-only measurements (Candes et al. 2012, Waldspurger 2015). Application in X-ray crystallography, used to image the molecular structure of a crystal (Harrison, 1993). unknown signal x C n ; measurements are given by b k = x, a k 2 for some vectors a k that encode the waveforms used to illuminate the signal, k = 1,..., m.

20 20 / 54 Phase Lift These measurements can be understood as b k = xx, a k a k = X, A k of the lifted signal X := xx, where A k := a k ak rank-1 measurement matrix. The Phase Lift convex model is the k-th lifted min X S n{ I, X + ι 0(X) AX b ε} also falls into the class of gauge optimization problems.

21 21 / 54 Image reconstruction In image reconstruction, one recovers an unknown image from a set of linear measurements b = Ax + noise. The total variation model (Rudin-Osher-Fatemi, 1992) min{tv(x) Ax b ε}, x where TV(x) := i D ix and A is identity (denoising) blurring operator (deconvolution) partial convolution/wavelet (inpainting) partial Fourier (MRI)

22 Ill-possed In deconvolution/deblurring, A is a convolution matrix (square and invertible). The least squares estimation (LSE) of x is x LSE = arg min Ax f x Blurry&Noisy Direct inverse Left: f ; Right: x LSE. 22 / 54

23 23 / 54 Ill-possed In MRI, observed data satisfy f = PF x + noise: F: 2D Fourier matrix P: projection matrix Left: k-space sampling (9.36%); Right: back projection.

24 24 / 54 RGB image recovery Model: min{mtv(x) Kx f ε}. x SNR 6.25dB SNR: 17.53dB, CPU 2.78s, It: 10, Cross-channel blur, Gaussian noise,

25 25 / 54 Deconvolution with impulsive noise Model: min{mtv(x) Kx f 1 ε}. x Corruption: 60% SNR 11.74dB, CPU 18.06s, It 35, Blur: cross-channel; Noise: Salt & Pepper 60% ( ).

26 26 / 54 Wavelet inpainting Model: min{tv(x) PWx f ε}. x Back Projection. 50% data ADM. SNR: 29.1dB, CPU: 46.2s

27 27 / 54 MRI reconstruction Model: min x {TV(x) + τ Wx 1 PFx f ε}. Left: original ( ); Middle: k-space samples (9.36%, white noise 10 3 ); Right: recovered by ADM (relative error: 0.48%, CPU: 3s).

28 28 / 54 Recovery from incomplete data Model: min{tv(x) PKx f ε}. x Lena: ; Blur: Gaussian, 15 15; White noise: 10 3 ; SRs: 5%, 3%, 1%; SNRs: 13.32, 11.98, 6.14dB; CPU: 43, 46, 57s.

29 29 / 54 Robust linear regression Given a set of feature vectors a i R n and outcomes b i R, i = 1,..., m, find weights x that predict the outcome accurately, i.e., Ax b. The robust linear regression problem min x Ax b 1 is less sensitive to outliers. Equivalent gauge problem: min x,y { y 1 Ax + y = b}.

30 30 / 54 Semidefinite programming The semidefinite programming (SDP) problem min X { C, X AX = b, X 0}. can also be reformulated as gauge optimization. To see this, chose one dual feasible point y so that Ĉ := C A y 0. Then, this SDP is equivalent to a gauge optimization min { Ĉ, X + ι 0(X) AX = b}. X

31 31 / 54 Outline Introduction Duality Lagrange duality Fenchel duality Gauge duality

32 32 / 54 Lagrange duality Given a general nonlinear programming problem, referred to as primal problem, there exists another closely related nonlinear programming problem, named Lagrange dual problem. Under suitable regularity conditions, such as convexity and/or constraint qualifications, the primal and the dual problems have equal optimal objective values.

33 33 / 54 Lagrange duality Let f : R n R, h : R n R m, g : R n R q and X R n. The nonlinear constrained optimization problem: p := inf{f (x) h(x) = 0, g(x) 0, x X }. x The Lagrange function L(x, λ, µ) = f (x) + λ, h(x) + µ, g(x). The Lagrange dual problem d := sup {g(λ, µ) µ 0}, λ,µ where g(λ, µ) := inf x {L(x, λ, µ) x X} is the dual objective.

34 34 / 54 Lagrange duality The vectors λ and µ are the Lagrange multipliers; The Lagrange multipliers corresponding to inequality constraints are restricted to be nonnegative, and those corresponding to equality constraints are unrestricted; The dual problem is a concave maximization problem; Weak duality d p ; Strong duality d = p is in general not true. For convex problems (X convex, f and g convex, and h affine), if Slater s condition is satisfied, then duality gap vanishes; Different Lagrange dual problems can be derived depending on different formulations of the problem.

35 35 / 54 Example Consider l 1 -regularized least square problem Two equivalent formulations min x x Ax b 2. min x,z min x,z { x Az b 2 x = z}, { x z b 2 Ax = z}. Their Lagrange dual problems max y 1 max A T y 1 min z y, z Az b 2, b, y 1 2 y 2.

36 36 / 54 Fenchel duality Let f : R n R {+ }. Consider unconstrained optimization min f (x). x Any constraints must be incorporated into the objective function via the indicator function. The key ingredient of Fenchel duality is that any closed convex function is the pointwise supremum of the set of affine functions which minorize it. This leads to the Fenchel conjugate function: f (y) = sup{ y, x f (x)}, y. x

37 37 / 54 Fenchel duality Assume that φ(x, y) is a perturbation function of f (x), i.e., φ(x, 0) = f (x), x, and define h(y) := inf φ(x, y), y. x Then, minimizing f is equivalent to evaluating h(0). Can show that inf y φ (0, y) = h (0) h(0) = inf x φ(x, 0). The problem of evaluating h (0) is called the Fenchel dual problem. Strong duality h (0) = h(0) holds if and only if inf x sup y { x, y h(y)} = sup y inf x { x, y h(y)}.

38 38 / 54 Fenchel duality Theorem (Fenchel duality theorem) Let f : R n R {+ } and g : R n R {+ } be closed proper convex functions. Then, under regularity conditions it holds that inf x {f (x) g(x)} = sup {g (y) f (y)}, y where f is the convex conjugate of f and g is the concave conjugate of g, i.e., f (y) := sup{ y, x f (x)}, x g (y) := inf{ y, x g(x)}. x

39 39 / 54 Geometric interpretation 150 [slop, dual obj] = [-2.05, ]

40 40 / 54 Geometric interpretation 150 [slop, dual obj] = [-1.60,-73.00]

41 41 / 54 Geometric interpretation 150 [slop, dual obj] = [-1.10,-38.62]

42 42 / 54 Geometric interpretation 150 [slop, dual obj] = [-0.60,-10.50]

43 43 / 54 Geometric interpretation 150 [slop, dual obj] = [-0.10,11.38]

44 44 / 54 Geometric interpretation 150 [slop, dual obj] = [1.40,39.50]

45 45 / 54 Geometric interpretation 150 [slop, dual obj] = [1.90,36.38]

46 46 / 54 Geometric interpretation 150 [slop, dual obj] = [2.40,27.00]

47 47 / 54 Gauge optimization Definition (gauge optimization) Minimizing a closed gauge function κ over a closed convex set C is called a gauge optimization. That is Definition (feasible point) min x {κ(x) x C}. (GP) A point x C is called a feasible point of GP; If x C and κ(x) = +, then x is essentially infeasible; If x C and κ(x) < +, then x is strongly feasible; Similar definitions for the gauge dual problem defined below.

48 48 / 54 Gauge optimization Definition (antipolar set) The antipolar of a closed convex set C is given by C = {y R n x, y 1 for all x C}. Definition (polar function) The polar function of a gauge κ is defined as κ (y) = sup{ x, y κ(x) 1}. In particular, the polar function of a norm is its dual norm. Definition (gauge dual) The nonlinear gauge dual problem of (GP) is defined by min y {κ (y) y C }. (GD)

49 49 / 54 Gauge duality Theorem (weak duality, Freund 1987) Let p and d be respectively the optimal values for GP and GD. 1. If x and y are, resp., strongly feasible for GP and GD, then κ(x)κ (y) 1, and hence p d If p = 0, then d = +, i.e., GD is essentially infeasible; 3. If d = 0, then p = +, i.e., GP is essentially infeasible.

50 50 / 54 Gauge duality Theorem (strong duality, Freund 1987) Let p and d be respectively the optimal values for GP and GD. If (ri C) (ri dom κ) and (ri C ) (ri dom κ ), then p d = 1, and each program attains its optimum.

51 51 / 54 Gauge duality The Fenchel dual of GP is given by d f = max{ ι y C ( y) k (y) 1}. (FD) Theorem (weak duality, Friedlander et al., 2014) Suppose that dom k C. Then Furthermore, p d f = 1/d > if y solves FD, then y /d f solves GD; 2. if y solves GD and d > 0, then y /d solves FD.

52 52 / 54 Gauge duality Theorem (strong duality, Friedlander et al., 2014) 1. Suppose that dom k C and (ri dom κ) (ri C), then p d = 1 and GD attains its optimal value. 2. Suppose that (ri dom κ) (ri C) and (ri dom κ ) (ri C ), then p d = 1 and both GP and GD attain their optimal values.

53 53 / 54 Gauge duality Consider the gauge optimization Lagrange dual Gauge dual min{κ(x) ρ(ax b) ε}. x max{b T y ερ (y) κ (A T y) 1} y min y {κ (A T y) b T y ερ (y) 1} In general, problems with simper constraints and complex objective are more preferred than those with simpler objective yet complex constraints.

54 54 / 54 Acknowledgements Some pictures are taken from Michael P. Friedlander s SIAM Optimization talk in 2011.