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- W4320854279 abstract "The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $mathbb{R}^d$ into $m=O(varepsilon^{-2} lg n)$ dimensions while preserving all pairwise distances to within $1 pm varepsilon$. Each input point $x$ is embedded to $Ax$, where $A$ is an $m times d$ matrix having $s$ non-zeros per column, allowing for an embedding time of $O(s |x|_0)$. Since the sparsity of $A$ governs the embedding time, much work has gone into improving the sparsity $s$. The current state-of-the-art by Kane and Nelson (JACM'14) shows that $s = O(varepsilon ^{-1} lg n)$ suffices. This is almost matched by a lower bound of $s = Omega(varepsilon ^{-1} lg n/lg(1/varepsilon))$ by Nelson and Nguyen (STOC'13). Previous work thus suggests that we have near-optimal embeddings. In this work, we revisit sparse embeddings and identify a loophole in the lower bound. Concretely, it requires $d geq n$, which in many applications is unrealistic. We exploit this loophole to give a sparser embedding when $d = o(n)$, achieving $s = O(varepsilon^{-1}(lg n/lg(1/varepsilon)+lg^{2/3}n lg^{1/3} d))$. We also complement our analysis by strengthening the lower bound of Nelson and Nguyen to hold also when $d ll n$, thereby matching the first term in our new sparsity upper bound. Finally, we also improve the sparsity of the best oblivious subspace embeddings for optimal embedding dimensionality." @default.
- W4320854279 created "2023-02-16" @default.
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- W4320854279 date "2023-02-13" @default.
- W4320854279 modified "2023-09-29" @default.
- W4320854279 title "Sparse Dimensionality Reduction Revisited" @default.
- W4320854279 doi "https://doi.org/10.48550/arxiv.2302.06165" @default.
- W4320854279 hasPublicationYear "2023" @default.
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