On data structures and asymmetric communication complexity
Journal of Computer and System Sciences
Finding Frequent Items in Data Streams
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Data streams: algorithms and applications
Foundations and Trends® in Theoretical Computer Science
Block-sparsity: Coherence and efficient recovery
ICASSP '09 Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing
Approximate sparse recovery: optimizing time and measurements
Proceedings of the forty-second ACM symposium on Theory of computing
Model-based compressive sensing
IEEE Transactions on Information Theory
Lower bounds for sparse recovery
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
The Gelfand widths of lp-balls for 0
Journal of Complexity
Efficient sketches for the set query problem
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
IEEE Transactions on Information Theory
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The goal of sparse recovery is to recover the (approximately) best k-sparse approximation x of an n-dimensional vector x from linear measurements Ax of x. We consider a variant of the problem which takes into account partial knowledge about the signal. In particular, we focus on the scenario where, after the measurements are taken, we are given a set S of size s that is supposed to contain most of the "large" coefficients of x. The goal is then to find x such that ||x - x||p ≤ C min k-sparse x′ supp(x′)⊆S ||x-x′||q. We refer to this formulation as the sparse recovery with partial support knowledge problem (SRPSK). We show that SRPSK can be solved, up to an approximation factor of C = 1+ε, using O((k/ε) log(s/k)) measurements, for p = q = 2. Moreover, this bound is tight as long as s = O(εn/ log(n/ε)). This completely resolves the asymptotic measurement complexity of the problem except for a very small range of the parameter s. To the best of our knowledge, this is the first variant of (1 + ε)- approximate sparse recovery for which the asymptotic measurement complexity has been determined.