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A Motivating Application: Sensor Array Signal Processing

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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  1. 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*

  2. [ ] 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)?

  3. 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!

  4. 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

  5. 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 ?

  6. 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 ?

  7. 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: • How about ?

  8. 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 ?

  9. 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 ?

  10. 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: • How about ?

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