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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 ksparse 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 kmore »Free, publiclyaccessible full text available January 1, 2023

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 kstep 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 amore »Free, publiclyaccessible full text available January 1, 2023

The CSGM framework (BoraJalalPriceDimakis'17) has shown that deepgenerative priors can be powerful tools for solving inverse problems.However, to date this framework has been empirically successful only oncertain datasets (for example, human faces and MNIST digits), and itis known to perform poorly on outofdistribution samples. In thispaper, we present the first successful application of the CSGMframework on clinical MRI data. We train a generative prior on brainscans from the fastMRI dataset, and show that posterior sampling viaLangevin dynamics achieves high quality reconstructions. Furthermore,our experiments and theory show that posterior sampling is robust tochanges in the groundtruth distribution and measurement process.Our codemore »