Optimal Oblivious Subspace Embeddings with Near-optimal Sparsity

• URL: http://arxiv.org/abs/2411.08773v1
• Date: Wed, 13 Nov 2024 16:58:51 GMT
• Title: Optimal Oblivious Subspace Embeddings with Near-optimal Sparsity
• Authors: Shabarish Chenakkod, Michał Dereziński, Xiaoyu Dong,
• Abstract summary: An oblivious subspace embedding is a random $mtimes n$ matrix $Pi$ that preserves the norms of all vectors in that subspace within a $1pmepsilon$ factor. We give an oblivious subspace embedding with the optimal dimension $m=Theta(d/epsilon2)$ that has a near-optimal sparsity of $tilde O (1/epsilon)$ non-zero entries per column of $Pi$.
• Score: 3.9657575162895196
• License:
• Abstract: An oblivious subspace embedding is a random $m\times n$ matrix $\Pi$ such that, for any $d$-dimensional subspace, with high probability $\Pi$ preserves the norms of all vectors in that subspace within a $1\pm\epsilon$ factor. In this work, we give an oblivious subspace embedding with the optimal dimension $m=\Theta(d/\epsilon^2)$ that has a near-optimal sparsity of $\tilde O(1/\epsilon)$ non-zero entries per column of $\Pi$. This is the first result to nearly match the conjecture of Nelson and Nguyen [FOCS 2013] in terms of the best sparsity attainable by an optimal oblivious subspace embedding, improving on a prior bound of $\tilde O(1/\epsilon^6)$ non-zeros per column [Chenakkod et al., STOC 2024]. We further extend our approach to the non-oblivious setting, proposing a new family of Leverage Score Sparsified embeddings with Independent Columns, which yield faster runtimes for matrix approximation and regression tasks. In our analysis, we develop a new method which uses a decoupling argument together with the cumulant method for bounding the edge universality error of isotropic random matrices. To achieve near-optimal sparsity, we combine this general-purpose approach with new traces inequalities that leverage the specific structure of our subspace embedding construction.

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