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Our nonconvex optimization formulation solves for a filter h on the unit sphere that produces sparse outputyi ~h.

In Section B, we provide some preliminaries. In Section C, we provide sparsity analysis. We show convergence analysis in Section D. In Section E, we show how to combine the sparsity, convergence, running time all together. In Section F, we show correlation between sparsity and spectral gap of Hessian in neural tangent kernel. In Section G, we discuss how to generalize our result to quantum setting.

Deep neural networks have achieved impressive performance in many areas. Designing a fast and provable method for training neural networks is a fundamental question in machine learning. The classical training method requires paying Ω(mnd) cost for both forward computation and backward computation, where m is the width of the neural network, and we are given n training points in d-dimensional space.

The goal of multi-task learning is to enable more efficient learning than single task learning by sharing model structures for a diverse set of tasks.

Neural transport augmented sampling, firstintroduced byParnoandMarzouk (2018),isageneral method for using normalizing flows to sample from a given densityπ. Thus, samples can be generated fromπ(θ)by running MCMC chain in theZ-space and pushing these samples onto theΘ-space usingT. Neural transport augmented samplers havebeen subsequently extended by Hoffman etal. In this paper, we proposed equivariant Stein variational gradient descent algorithm for sampling fromdensities thatareinvarianttosymmetry transformations. Another contributionofourworkis subsequently using this equivariant sampling method to efficiently train equivariant energy based models forprobabilistic modeling andinference.