Near-Optimal Sparse Recovery in the L1 Norm

Abstract— We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a highdimensional vector x ∈ Rn from its lower-dimensional sketch Ax ∈ Rm . Specifically, we focus on the sparse recovery problem in the ℓ1 norm: for a parameter k, given the sketch Ax, compute an approximation ˆx of x such that the ℓ1 approximation error x − ˆx 1 is close to minx′ x − x′ 1, where x′ ranges over all vectors with at most k terms. The sparse recovery problem has been subject to extensive research over the last few years. Many solutions to this problem have been discovered, achieving different trade-offs between various attributes, such as the sketch length, encoding and recovery times. A recent paper [IR08] provided a sparse recovery scheme which achieved close to optimal performance on virtually all attributes (see Figure 1). In particular, this was the first recovery scheme that guaranteed O(k log(n/k)) sketch length, and nearlinear O(n lo...

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Piotr Indyk, Milan Ruzic
• Keywords: Approximate Sparse Recovery, FOCS 2008, Sketch Length, Sparse Recovery Problems, Theoretical Computer Science