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Forward. 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.

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A Motivating Application: Sensor Array Signal Processing

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Forward

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 ?

• Thm. 2(*):

• 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 ?

• Thm. 3:

where

lpsolution = l0solution !

• Smaller p  more non-zero elements tolerated

• As p0 we recover the l0 condition, namely

Smaller p

### lp (p ≤ 1) equivalence conditions

• Consider the lp problem:

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 ?

• Definition:

• Thm. 3:

Smaller p

lpsolution = l0solution !

• Smaller p  more non-zero elements tolerated

• As p0 we recover the l0 condition, namely

### lp (p ≤ 1) equivalence conditions

• Consider the lp problem: