1. 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 DAAD ,
and the Office of Naval Research under Grant N
May 20, 2004

2. 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

3. 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)? [ ]

4. 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

5. 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!

6. 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
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

7. 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

8. 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

9. 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.

10. 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

11. 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.

12. 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:

13. 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

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

15. 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

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

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