Optimal sparse representations in general overcomplete bases Dmitry M. Malioutov, Müjdat Çetin, and Alan S. Willsky LIDS, MIT This work was supported by the Army Research Office under Grant DAAD19-00-1-0466, and the Office of Naval Research under Grant N00014-00-1-0089. May 20, 2004

Outline of the presentation Sparse signal reconstruction problem l0, l1, and lp measures of sparsity Uniqueness and equivalence conditions for l0, l1, and lp Sign patterns of exact solutions Sparsity under a transformation Numerical optimization of l1 and lp objective functions

Underdetermined Linear Inverse Problems Basic problem: find an estimate of x , where Underdetermined -- non-uniqueness of solutions Additional information/constraints needed for a unique solution A typical approach is the min-norm solution: What if we know x is sparse (i.e. has few non-zero elements)? [ ]

A motivating application: sensor array source localization Goal: Estimate directions of arrival of acoustic sources using a microphone array Data collection setup Underlying “sparse” spatial spectrum x Forward Inverse

Sparsity constraints Prefer the sparsest solution: Number of non-zero elements in x Can be viewed as finding a sparse representation of the signal y in an overcomplete dictionary A Intractable combinatorial optimization problem Are there tractable alternatives that might produce the same result? Empirical observation: l1- norm and lp-norm based techniques produce solutions that look sparse l1 cost function can be optimized by linear programming!

l1 and lp-norms and sparsity – an example A sparse signal 2.000 2.000 2.000 A non-sparse signal 0.3382 3.5549 84.8104 See lp_norm_example.m, which produces these plots as well as plots of lp-norm vs p for these two signals Goal: Rigorous characterization of the l1, lp - sparsity link For these two signals x1 and x2 we have Ax1=Ax2 where A is a 16x128 DFT operator

l0 uniqueness conditions Prefer the sparsest solution: Let When is ? Number of non-zero elements in x Definition: The index of ambiguity K(A) of A is the largest integer such that any set of K(A) columns of A is linearly independent. Thm. 1: What can we say about more tractable formulations like l1 ? Unique l0 solution

l0 uniqueness conditions (continued) The measure K(A) is not continuous in entries of A New Measure of well-separatedness of an overcomplete basis: Definition: Maximum absolute dot product of columns Thm. 2: Our proof is based on the optimality of the regular simplex for line packing

l1 equivalence conditions Consider the l1 problem: Can we ever hope to get ? Thm. 3(*): is sparse enough  exact solution by l1 optimization Can solve a combinatorial optimization problem by convex optimization! l1 solution = l0 solution ! (*) Donoho and Huo proved this for pairs of orthogoanl bases. We extended this result to general overcomplete bases. Independently, Donoho and Elad, Gribonval and Nielsen, and Fuchs made this extension.

lp (p ≤ 1) equivalence conditions Consider the lp problem: How about ? Definition: Thm. 4: lp solution = l0 solution ! Smaller p Smaller p  more non-zero elements tolerated As p0 we recover the l0 condition, namely

Sign patterns of exact solutions Additional characterization is possible when the sufficient conditions for equivalence of l0 and l1 problems are not met. The support and the sign pattern of an l0-optimal solution x determine whether the solution is also l1-optimal. A is 10x40. Left: correct l1 solutions. Right: wrong l1 solutions.

Sparsity under a transformation Consider a more general problem: where 0<p·1, and D is a given full-row rank linear mapping Let N = Null(D), and let F = A N. Project y onto 1) range space of F and 2) its orthogonal complement Define z = Dx, then the problem reduces to:

Sparsity under a transformation (continued) Example: total variation (TV) reconstruction of a piecewise-constant signal. D is a 39x40 pairwise difference opearator, A is 10x40. l2 blurs the edges l1 recovers the original signal

Numerical optimization for l1 l1 problem: Solution by linear programming: Now the problem becomes :

Numerical optimization: noisy complex data Handling noise: For some applications the data is complex – we use second order cone programming (SOC) Efficient solution by an interior point implementation

Numerical optimization for lp half-quadratic regularization Noisy lp formulation: Smooth approximation to lp: Iterative half-quadratic regularization algorithm:

Applications Source localization: Radar imaging: Other applications: subset and feature selection, denoising, object recognition