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A Motivating Application: Sensor Array Signal Processing

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Inverse

A Motivating Application:Sensor Array Signal Processing- Goal: Estimate directions of arrival of acoustic sources using a microphone array

Data collection setup

Underlying “sparse” spatial spectrum f*

]

Underdetermined Linear Inverse Problems- Basic problem: find an estimate of , 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 is sparse (i.e. has few non-zero elements)?

Number of non-zero elements in f

Sparsity constraints- Prefer the sparsest solution:

- 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-based techniques produce solutions that look sparse
- l1 cost function can be optimized by linear programming!

l1-norm and sparsity – a simple example

A sparse signal

1.4142

2.0000

A non-sparse signal

0.5816

3.5549

- Goal: Rigorous characterization of the l1 – sparsity link

For these two signals f1 and f2 we have A*f1=A*f2 where A is a 16x128 DFT operator

Number of non-zero elements in f

- Thm. 1:
- What can we say about more tractable formulations like l1 ?

where

and K(A) is the largest integer such that any set of K(A) columns of A is linearly independent.

Unique l0solution

l0 uniqueness conditions- Prefer the sparsest solution:
- Let where
- When is ?

- is sparse enough exact solution by l1 optimization
- Can solve a combinatorial optimization problem by convex optimization!

where

(*) Donoho and Elad obtained a similar result concurrently.

l1solution = l0solution !

l1 equivalence conditions- Consider the l1 problem:
- Can we ever hope to get ?

where

lpsolution = l0solution !

- Smaller p more non-zero elements tolerated
- As p0 we recover the l0 condition, namely

Smaller p

lp (p ≤ 1) equivalence conditions- Consider the lp problem:
- How about ?

Number of non-zero elements in f

- 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 l0solution

l0 uniqueness conditions- Prefer the sparsest solution:
- Let
- When is ?

Definition: Maximum absolute dot product of columns

- Thm. 2(*):
- is sparse enough exact solution by l1 optimization
- Can solve a combinatorial optimization problem by convex optimization!

(*) Donoho and Elad obtained a similar result concurrently.

l1solution = l0solution !

l1 equivalence conditions- Consider the l1 problem:
- Can we ever hope to get ?

- Thm. 3:

Smaller p

lpsolution = l0solution !

- Smaller p more non-zero elements tolerated
- As p0 we recover the l0 condition, namely

- Consider the lp problem:
- How about ?

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