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Description:
This work shows how the performance of sparse random embeddings depends on the Renyi entropy-like property of data, improving upon recent works from NIPS'18 and NIPS'19. While the prior works relied on involved combinatorics, the novel approach is simpler and modular. As the building blocks, it develops the following probabilistic facts of general interest: b) a comparison inequality between the linear and quadratic chaos c) a comparison inequality between heterogenic and homogenic linear chaos d) a simpler proof of Latala’s strong result on estimating distributions of IID sums e) sharp bounds for binomial moments in all parameter regimes.