arXiv subjects

Archan Ray

Publications and source records attributed to Archan Ray.

10 recordsLinked to original sources

arXiv2025

A simple analysis of a quantum-inspired algorithm for solving low-rank linear systems

We describe and analyze a simple algorithm for sampling from the solution $\mathbf{x}^* := \mathbf{A}^+\mathbf{b}$ to a linear system $\mathbf{A}\mathbf{x} = \mathbf{b}$. We assume access to a sampler which allows us to draw indices proportional to the squared row/column-norms of $\mathbf{A}$. Our algorithm produces a compressed representation of some vector $\mathbf{x}$ for which $\|\mathbf{x}^* - \mathbf{x}\| < \varepsilon \|\mathbf{x}^* \|$ in $\widetilde{O}(\kappa_{\mathsf{F}}^4 \kappa^2 / \varepsilon^2)$ time, where $\kappa_{\mathsf{F}} := \|\mathbf{A}\|_{\mathsf{F}}\|\mathbf{A}^{+}\|$ and $\kappa := \|\mathbf{A}\|\|\mathbf{A}^{+}\|$. The representation of $\mathbf{x}$ allows us to query entries of $\mathbf{x}$ in $\widetilde{O}(\kappa_{\mathsf{F}}^2)$ time and sample proportional to the square entries of $\mathbf{x}$ in $\widetilde{O}(\kappa_{\mathsf{F}}^4 \kappa^6)$ time, assuming access to a sampler which allows us to draw indices proportional to the squared entries of any given row of $\mathbf{A}$. Our analysis, which is elementary, non-asymptotic, and fully self-contained, simplifies and clarifies several past analyses from literature including [Gily\'en, Song, and Tang; 2022, 2023] and [Shao and Montanaro; 2022].

cs.DS↗

arXiv2025

MetaTT: A Global Tensor-Train Adapter for Parameter-Efficient Fine-Tuning

We present MetaTT, a Tensor Train (TT) adapter framework for fine-tuning of pre-trained transformers. MetaTT enables flexible and parameter-efficient model adaptation by using a single shared TT to factorize transformer sub-modules. This factorization indexes key structural dimensions, including layer and matrix type, and can optionally incorporate heads and tasks. This design allows MetaTT's parameter count to scale with the sum, rather than the product, of the modes, resulting in a substantially more compact adapter. Our benchmarks compare MetaTT with LoRA along with recent state-of-the-art matrix and tensor decomposition based fine-tuning methods. We observe that when tested on single-task standard language modeling benchmarks, MetaTT achieves competitive parameter efficiency to accuracy tradeoff. We further demonstrate that MetaTT performs competitively when compared to state-of-the-art methods on multi-task learning. Finally, we leverage the TT decomposition to design a rank adaptive optimizer inspired by the DMRG method from many-body physics. Our results demonstrate that integrating this approach with AdamW enhances optimization performance for a specified target rank.

cs.LG↗

arXiv2025

A Unified Framework for Provably Efficient Algorithms to Estimate Shapley Values

Shapley values have emerged as a critical tool for explaining which features impact the decisions made by machine learning models. However, computing exact Shapley values is difficult, generally requiring an exponential (in the feature dimension) number of model evaluations. To address this, many model-agnostic randomized estimators have been developed, the most influential and widely used being the KernelSHAP method (Lundberg & Lee, 2017). While related estimators such as unbiased KernelSHAP (Covert & Lee, 2021) and LeverageSHAP (Musco & Witter, 2025) are known to satisfy theoretical guarantees, bounds for KernelSHAP have remained elusive. We describe a broad and unified framework that encompasses KernelSHAP and related estimators constructed using both with and without replacement sampling strategies. We then prove strong non-asymptotic theoretical guarantees that apply to all estimators from our framework. This provides, to the best of our knowledge, the first theoretical guarantees for KernelSHAP and sheds further light on tradeoffs between existing estimators. Through comprehensive benchmarking on small and medium dimensional datasets for Decision-Tree models, we validate our approach against exact Shapley values, consistently achieving low mean squared error with modest sample sizes. Furthermore, we make specific implementation improvements to enable scalability of our methods to high-dimensional datasets. Our methods, tested on datasets such MNIST and CIFAR10, provide consistently better results compared to the KernelSHAP library.

cs.LG↗

arXiv2025

GPU-Parallelizable Randomized Sketch-and-Precondition for Linear Regression using Sparse Sign Sketches

A litany of theoretical and numerical results have established the sketch-and-precondition paradigm as a powerful approach to solving large linear regression problems in standard computing environments. Perhaps surprisingly, much less work has been done on understanding how sketch-and-precondition performs on graphics processing unit (GPU) systems. We address this gap by benchmarking an implementation of sketch-and-precondition based on sparse sign-sketches on single and multi-GPU systems. In doing so, we describe a novel, easily parallelized, rejection-sampling based method for generating sparse sign sketches. Our approach, which is particularly well-suited for GPUs, is easily adapted to a variety of computing environments. Taken as a whole, our numerical experiments indicate that sketch-and-precondition with sparse sign sketches is particularly well-suited for GPUs, and may be suitable for use in black-box least-squares solvers.

cs.DS↗

arXiv2025

Provably faster randomized and quantum algorithms for $k$-means clustering via uniform sampling

The $k$-means algorithm (Lloyd's algorithm) is a widely used method for clustering unlabeled data. A key bottleneck of the $k$-means algorithm is that each iteration requires time linear in the number of data points, which can be expensive in big data applications. This was improved in recent works proposing quantum and quantum-inspired classical algorithms to approximate the $k$-means algorithm locally, in time depending only logarithmically on the number of data points (along with data dependent parameters) [q-means: A quantum algorithm for unsupervised machine learning, Kerenidis, Landman, Luongo, and Prakash, NeurIPS 2019; Do you know what $q$-means?, Cornelissen, Doriguello, Luongo, Tang, QTML 2025]. In this work, we describe a simple randomized mini-batch $k$-means algorithm and a quantum algorithm inspired by the classical algorithm. We demonstrate that the worst case guarantees of these algorithms can significantly improve upon the bounds for algorithms in prior work. Our improvements are due to a careful use of uniform sampling, which preserves certain symmetries of the $k$-means problem that are not preserved in previous algorithms that use data norm-based sampling.

quant-ph↗

arXiv2024

Improved Spectral Density Estimation via Explicit and Implicit Deflation

We study algorithms for approximating the spectral density of a symmetric matrix $A$ that is accessed through matrix-vector product queries. By combining a previously studied Chebyshev polynomial moment matching method with a deflation step that approximately projects off the largest magnitude eigendirections of $A$ before estimating the spectral density, we give an $\epsilon\cdot\sigma_\ell(A)$ error approximation to the spectral density in the Wasserstein-$1$ metric using $O(\ell\log n+ 1/\epsilon)$ matrix-vector products, where $\sigma_\ell(A)$ is the $\ell^{th}$ largest singular value of $A$. In the common case when $A$ exhibits fast singular value decay, our bound can be much stronger than prior work, which gives an error bound of $\epsilon \cdot ||A||_2$ using $O(1/\epsilon)$ matrix-vector products. We also show that it is nearly tight: any algorithm giving error $\epsilon \cdot \sigma_\ell(A)$ must use $\Omega(\ell+1/\epsilon)$ matrix-vector products. We further show that the popular Stochastic Lanczos Quadrature (SLQ) method matches the above bound, even though SLQ itself is parameter-free and performs no explicit deflation. This bound explains the strong practical performance of SLQ, and motivates a simple variant of SLQ that achieves an even tighter error bound. Our error bound for SLQ leverages an analysis that views it as an implicit polynomial moment matching method, along with recent results on low-rank approximation with single-vector Krylov methods. We use these results to show that the method can perform implicit deflation as part of moment matching.

cs.DS↗

arXiv2024

Adaptive and Robust Watermark for Generative Tabular Data

In recent years, watermarking generative tabular data has become a prominent framework to protect against the misuse of synthetic data. However, while most prior work in watermarking methods for tabular data demonstrate a wide variety of desirable properties (e.g., high fidelity, detectability, robustness), the findings often emphasize empirical guarantees against common oblivious and adversarial attacks. In this paper, we study a flexible and robust watermarking algorithm for generative tabular data. Specifically, we demonstrate theoretical guarantees on the performance of the algorithm on metrics like fidelity, detectability, robustness, and hardness of decoding. The proof techniques introduced in this work may be of independent interest and may find applicability in other areas of machine learning. Finally, we validate our theoretical findings on synthetic and real-world tabular datasets.

cs.CR↗

arXiv2023

Universal Matrix Sparsifiers and Fast Deterministic Algorithms for Linear Algebra

Let $\mathbf S \in \mathbb R^{n \times n}$ satisfy $\|\mathbf 1-\mathbf S\|_2\le\epsilon n$, where $\mathbf 1$ is the all ones matrix and $\|\cdot\|_2$ is the spectral norm. It is well-known that there exists such an $\mathbf S$ with just $O(n/\epsilon^2)$ non-zero entries: we can let $\mathbf S$ be the scaled adjacency matrix of a Ramanujan expander graph. We show that such an $\mathbf S$ yields a $universal$ $sparsifier$ for any positive semidefinite (PSD) matrix. In particular, for any PSD $\mathbf A \in \mathbb{R}^{n\times n}$ with entries bounded in magnitude by $1$, $\|\mathbf A - \mathbf A\circ\mathbf S\|_2 \le \epsilon n$, where $\circ$ denotes the entrywise (Hadamard) product. Our techniques also give universal sparsifiers for non-PSD matrices. In this case, letting $\mathbf S$ be the scaled adjacency matrix of a Ramanujan graph with $\tilde O(n/\epsilon^4)$ edges, we have $\|\mathbf A - \mathbf A \circ \mathbf S \|_2 \le \epsilon \cdot \max(n,\|\mathbf A\|_1)$, where $\|\mathbf A\|_1$ is the nuclear norm. We show that the above bounds for both PSD and non-PSD matrices are tight up to log factors. Since $\mathbf A \circ \mathbf S$ can be constructed deterministically, our result for PSD matrices derandomizes and improves upon known results for randomized matrix sparsification, which require randomly sampling ${O}(\frac{n \log n}{\epsilon^2})$ entries. We also leverage our results to give the first deterministic algorithms for several problems related to singular value approximation that run in faster than matrix multiplication time. Finally, if $\mathbf A \in \{-1,0,1\}^{n \times n}$ is PSD, we show that $\mathbf{\tilde A}$ with $\|\mathbf A - \mathbf{\tilde A}\|_2 \le \epsilon n$ can be obtained by deterministically reading $\tilde O(n/\epsilon)$ entries of $\mathbf A$. This improves the $1/\epsilon$ dependence on our result for general PSD matrices and is near-optimal.

cs.DS↗

arXiv2021

Sublinear Time Approximation of Text Similarity Matrices

We study algorithms for approximating pairwise similarity matrices that arise in natural language processing. Generally, computing a similarity matrix for $n$ data points requires $\Omega(n^2)$ similarity computations. This quadratic scaling is a significant bottleneck, especially when similarities are computed via expensive functions, e.g., via transformer models. Approximation methods reduce this quadratic complexity, often by using a small subset of exactly computed similarities to approximate the remainder of the complete pairwise similarity matrix. Significant work focuses on the efficient approximation of positive semidefinite (PSD) similarity matrices, which arise e.g., in kernel methods. However, much less is understood about indefinite (non-PSD) similarity matrices, which often arise in NLP. Motivated by the observation that many of these matrices are still somewhat close to PSD, we introduce a generalization of the popular Nystr\"{o}m method to the indefinite setting. Our algorithm can be applied to any similarity matrix and runs in sublinear time in the size of the matrix, producing a rank-$s$ approximation with just $O(ns)$ similarity computations. We show that our method, along with a simple variant of CUR decomposition, performs very well in approximating a variety of similarity matrices arising in NLP tasks. We demonstrate high accuracy of the approximated similarity matrices in the downstream tasks of document classification, sentence similarity, and cross-document coreference.

cs.LG↗

arXiv2021

Sublinear Time Eigenvalue Approximation via Random Sampling

We study the problem of approximating the eigenspectrum of a symmetric matrix $\mathbf A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|\mathbf A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that approximates all eigenvalues of $\mathbf{A}$ up to additive error $\pm \epsilon n$ using those of a randomly sampled $\tilde {O}\left (\frac{\log^3 n}{\epsilon^3}\right ) \times \tilde O\left (\frac{\log^3 n}{\epsilon^3}\right )$ principal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of $\pm \epsilon \sqrt{\text{nnz}(\mathbf{A})}$ and $\pm \epsilon \|\mathbf A\|_F$ when the rows of $\mathbf A$ can be sampled with probabilities proportional to their sparsities or their squared $\ell_2$ norms respectively. Here $\text{nnz}(\mathbf{A})$ is the number of non-zero entries in $\mathbf{A}$ and $\|\mathbf A\|_F$ is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS '20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of $\mathbf A$. We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice.

cs.DS↗