Sparse Recovery ( Using Sparse Matrices)

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Sparse Recovery (Using Sparse Matrices) Piotr Indyk MIT

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Sparse Recovery ( Using Sparse Matrices). Piotr Indyk MIT. Heavy Hitters. Also called frequent elements and elephants Define HH p φ ( x ) = { i : |x i | ≥ φ || x|| p } L p Heavy Hitter Problem: Parameters: φ and φ ’ (often φ ’ = φ - ε ) - PowerPoint PPT Presentation

Transcript of Sparse Recovery ( Using Sparse Matrices)

Sparse recovery using sparse random matrices

Sparse Recovery (Using Sparse Matrices)Piotr IndykMIT1Heavy HittersAlso called frequent elements and elephantsDefineHHp (x) = { i: |xi| ||x||p }Lp Heavy Hitter Problem:Parameters: and (often = -)Goal: return a set S of coordinates s.t.S contains HHp (x) S is included in HHp (x) Lp Point Query Problem:Parameter: Goal: at the end of the stream, given i, report x*i=xi ||x||p

Can solve L2 point query problem, with parameter and failure probability P by storing O(1/2 log(1/P)) numbers sketches [A-Gibbons-M-S99], [Gilbert-Kotidis-Muthukrishnan-Strauss01]

Sparse Recovery(approximation theory, learning Fourier coeffs, linear sketching, finite rate of innovation, compressed sensing...)Setup:Data/signal in n-dimensional space : x E.g., x is an 256x256 image n=65536Goal: compress x into a sketch Ax , where A is a m x n sketch matrix, m 1/2Assume |xi1| |xim| and let S={i1ik} (elephants)When is xi* > xi Err/k ?

Event 1: S and i collide, i.e., h(i) in h(S-{i}) Probability: at most k/(4/)k = /4