On Low-Risk Heavy Hitters and Sparse Recovery Schemes

We study the heavy hitters and related sparse recovery problems in the low-failure probability regime. This regime is not well-understood, and has only been studied for non-adaptive schemes. The main previous work is on sparse recovery by Gilbert et al. (ICALP’13). We recognize an error in their analysis, improve their results, and contribute new non-adaptive and adaptive sparse recovery algorithms, as well as provide upper and lower bounds for the heavy hitters problem with low failure probability. Our results are summarized as follows: 1. (Heavy Hitters) We study three natural variants for finding heavy hitters in the strict turnstile model, where the variant depends on the quality of the desired output. For the weakest variant, we give a randomized algorithm improving the failure probability analysis of the ubiquitous Count-Min data structure. We also give a new lower bound for deterministic schemes, resolving a question about this variant posed in Question 4 in the IITK Workshop on Algorithms for Data Streams (2006). Finally, under the strongest and well-studied `∞/`2 variant, we provide the first randomized lower bound which is simultaneously optimal in the approximation factor , the universe size n, and the failure probability δ, for the full range of such parameters. Our lower bound shows that the classical Count-Sketch data structure is optimal in all parameters. 2. (Sparse Recovery Algorithms) For non-adaptive sparse-recovery, we give sublinear-time algorithms with low-failure probability, which improve upon Gilbert et al. (SICOMP’12) and Gilbert et al. (ICALP’13). In the adaptive case, we improve the failure probability from a constant by Indyk et al. (FOCS ’11) to e−k 0.99 , where k is the sparsity parameter. 3. (Optimal Average-Case Sparse Recovery Bounds) We give matching upper and lower bounds in all parameters on the measurement complexity of the `2/`2 sparse recovery problem in the spiked-covariance model, completely settling its complexity in this model.