arXiv subjects

Shabarish Chenakkod

Publications and source records attributed to Shabarish Chenakkod.

6 recordsLinked to original sources

arXiv2026

Accelerating Power Method with Fast Sketching for Stronger Low-Rank Approximation

The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.

math.NA↗

arXiv2026

Well-Conditioned Oblivious Perturbations in Linear Space

Perturbing a deterministic $n$-dimensional matrix with small Gaussian noise is a cornerstone of smoothed analysis of algorithms [Spielman and Teng, JACM 2004], as it reduces the condition number of the input to $O(n)$, and with it the complexity of many matrix algorithms. However, when deployed algorithmically, these perturbations are expensive due to the cost of generating and storing $n^2$ Gaussian random variables. We propose a perturbation that requires generating and storing $O(n)$ random numbers in $O(\log n)$ bits of precision, and reduces the condition number of any deterministic matrix to $O(n)$, matching Gaussian perturbations. Our result in particular implies a better complexity for the perturbed conjugate gradient algorithm, showing that we can solve an $n\times n$ linear system in linear space to within an arbitrarily small constant backward error using $O(n)$ matrix-vector products. In our construction, we introduce the concept of a pattern matrix, which is a dense deterministic matrix that maps all sparse vectors into dense vectors, and we combine it with a sparse perturbation whose entries are dependent and located in a non-uniform fashion. In order to analyze this construction, we develop new techniques for lower bounding the smallest singular value of a random matrix with dependent entries.

cs.DS↗

arXiv2025

Optimal Subspace Embeddings: Resolving Nelson-Nguyen Conjecture Up to Sub-Polylogarithmic Factors

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].

cs.DS↗

arXiv2025

Optimal Oblivious Subspace Embeddings with Near-optimal Sparsity

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.

cs.DS↗

arXiv2024

Optimal Embedding Dimension for Sparse Subspace Embeddings

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.

cs.DS↗

arXiv2021

Translation surfaces and periods of meromorphic differentials

Let $S$ be an oriented surface of genus $g$ and $n$ punctures. The periods of any meromorphic differential on $S$, with respect to a choice of complex structure, determine a representation $χ:Γ_{g,n} \to\mathbb C$ where $Γ_{g,n}$ is the first homology group of $S$. We characterize the representations that thus arise, that is, lie in the image of the period map $\textsf{Per}:Ω\mathcal{M}_{g,n}\to \textsf{Hom}(Γ_{g,n},\mathbb{C})$. This generalizes a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on $S$ with the prescribed holonomy $χ$. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.

math.GT↗