Optimal Oblivious Subspace Embeddings with Near-optimal Sparsity

An oblivious subspace embedding is a random m× 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±ε factor. In this work, we give an oblivious subspace embedding with the optimal dimension m=Theta(d/ε2) that has a near-optimal sparsity of tilde O(1/ε) 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/ε6) 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.