Sparse Dimensionality Reduction Revisited

• URL: http://arxiv.org/abs/2302.06165v1
• Date: Mon, 13 Feb 2023 08:01:25 GMT
• Title: Sparse Dimensionality Reduction Revisited
• Authors: Mikael M{\o}ller H{\o}gsgaard, Lion Kamma, Kasper Green Larsen, Jelani Nelson, Chris Schwiegelshohn
• Abstract summary: The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. We revisit sparse embeddings and identify a loophole in the lower bound. We also improve the sparsity of the best oblivious subspace embeddings for optimal embedding dimensionality.
• Score: 13.170012290527017
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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