Xiaoyu Dong

Shabarish Chenakkod, Michał Dereziński, Xiaoyu Dong et al.

A random $m\times n$ matrix $S$ is an oblivious subspace embedding (OSE) with parameters $ε>0$, $δ\in(0,1/3)$ and $d\leq m\leq n$, if for any $d$-dimensional subspace $W\subseteq R^n$, $P\big(\,\forall_{x\in W}\ (1+ε)^{-1}\|x\|\leq\|Sx\|\leq (1+ε)\|x\|\,\big)\geq 1-δ.$ It is known that the embedding dimension of an OSE must satisfy $m\geq d$, and for any $θ> 0$, a Gaussian embedding matrix with $m\geq (1+θ) d$ is an OSE with $ε= O_θ(1)$. However, such optimal embedding dimension is not known for other embeddings. Of particular interest are sparse OSEs, having $s\ll m$ non-zeros per column, with applications to problems such as least squares regression and low-rank approximation. We show that, given any $θ> 0$, an $m\times n$ random matrix $S$ with $m\geq (1+θ)d$ consisting of randomly sparsified $\pm1/\sqrt s$ entries and having $s= O(\log^4(d))$ non-zeros per column, is an oblivious subspace embedding with $ε= O_θ(1)$. Our result addresses the main open question posed by Nelson and Nguyen (FOCS 2013), who conjectured that sparse OSEs can achieve $m=O(d)$ embedding dimension, and it improves on $m=O(d\log(d))$ shown by Cohen (SODA 2016). We use this to construct the first oblivious subspace embedding with $O(d)$ embedding dimension that can be applied faster than current matrix multiplication time, and to obtain an optimal single-pass algorithm for least squares regression. We further extend our results to Leverage Score Sparsification (LESS), which is a recently introduced non-oblivious embedding technique. We use LESS to construct the first subspace embedding with low distortion $ε=o(1)$ and optimal embedding dimension $m=O(d/ε^2)$ that can be applied in current matrix multiplication time.

Shabarish Chenakkod, Michał Dereziński, Xiaoyu Dong

An oblivious subspace embedding is a random $m\times n$ matrix $Π$ such that, for any $d$-dimensional subspace, with high probability $Π$ preserves the norms of all vectors in that subspace within a $1\pmε$ factor. In this work, we give an oblivious subspace embedding with the optimal dimension $m=Θ(d/ε^2)$ that has a near-optimal sparsity of $\tilde O(1/ε)$ non-zero entries per column of $Π$. 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 $\tilde O(1/ε^6)$ 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.

Shabarish Chenakkod, Michał Dereziński, Xiaoyu Dong

We give a proof of the conjecture of Nelson and Nguyen [FOCS 2013] on the optimal dimension and sparsity of oblivious subspace embeddings, up to sub-polylogarithmic factors: For any $n\geq d$ and $ε\geq d^{-O(1)}$, there is a random $\tilde O(d/ε^2)\times n$ matrix $Π$ with $\tilde O(\log(d)/ε)$ non-zeros per column such that for any $A\in\mathbb{R}^{n\times d}$, with high probability, $(1-ε)\|Ax\|\leq\|ΠAx\|\leq(1+ε)\|Ax\|$ for all $x\in\mathbb{R}^d$, where $\tilde O(\cdot)$ hides only sub-polylogarithmic factors in $d$. Our result in particular implies a new fastest sub-current matrix multiplication time reduction of size $\tilde O(d/ε^2)$ for a broad class of $n\times d$ linear regression tasks. A key novelty in our analysis is a matrix concentration technique we call iterative decoupling, which we use to fine-tune the higher-order trace moment bounds attainable via existing random matrix universality tools [Brailovskaya and van Handel, GAFA 2024].