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We explore algorithms and limitations for sparse optimization problems such as sparse linear regression and robust linear regression. The goal of the sparse linear regression problem is to identify a small number of key features, while the goal of the robust linear regression problem is to identify a small number of erroneous measurements. Specifically, the sparse linear regression problem seeks a k-sparse vector x ∈ Rd to minimize ‖Ax − b‖2, given an input matrix A ∈ Rn×d and a target vector b ∈ Rn, while the robust linear regression problem seeks a set S that ignores at most k rows and a vector x to minimize ‖(Ax − b)S ‖2. We first show bicriteria, NP-hardness of approximation for robust regression building on the work of [OWZ15] which implies a similar result for sparse regression. We further show fine-grained hardness of robust regression through a reduction from the minimum-weight k-clique conjecture. On the positive side, we give an algorithm for robust regression that achieves arbitrarily accurate additive error and uses runtime that closely matches the lower bound from the fine-grained hardness result, as well as an algorithm for sparse regression with similar runtime. Both our upper and lower bounds rely on a general reduction from robust linear regression to sparse regression that we introduce. Our algorithms, inspired by the 3SUM problem, use approximate nearest neighbor data structures and may be of independent interest for solving sparse optimization problems. For instance, we demonstrate that our techniques can also be used for the well-studied sparse PCA problem. 

The random order graph streaming model has received significant attention recently, with problems such as matching size estimation, component counting, and the evaluation of bounded degree constant query testable properties shown to admit surprisingly space efficient algorithms. The main result of this paper is a space efficient single pass random order streaming algorithm for simulating nearly independent random walks that start at uniformly random vertices. We show that the distribution of k-step walks from b vertices chosen uniformly at random can be approximated up to error ∊ per walk using  words of space with a single pass over a randomly ordered stream of edges, solving an open problem of Peng and Sohler [SODA '18]. Applications of our result include the estimation of the average return probability of the k-step walk (the trace of the kth power of the random walk matrix) as well as the estimation of PageRank. We complement our algorithm with a strong impossibility result for directed graphs.