Searcharxiv⌕ Search

arXiv subjects

Sandeep Silwal

Publications and source records attributed to Sandeep Silwal.

At least 37 records · Page 2Linked to original sources
arXiv2025

Beyond Worst-Case Dimensionality Reduction for Sparse Vectors

We study beyond worst-case dimensionality reduction for $s$-sparse vectors. Our work is divided into two parts, each focusing on a different facet of beyond worst-case analysis: We first consider average-case guarantees. A folklore upper bound based on the birthday-paradox states: For any collection $X$ of $s$-sparse vectors in $\mathbb{R}^d$, there exists a linear map to $\mathbb{R}^{O(s^2)}$ which \emph{exactly} preserves the norm of $99\%$ of the vectors in $X$ in any $\ell_p$ norm (as opposed to the usual setting where guarantees hold for all vectors). We give lower bounds showing that this is indeed optimal in many settings: any oblivious linear map satisfying similar average-case guarantees must map to $Ω(s^2)$ dimensions. The same lower bound also holds for a wide class of smooth maps, including `encoder-decoder schemes', where we compare the norm of the original vector to that of a smooth function of the embedding. These lower bounds reveal a separation result, as an upper bound of $O(s \log(d))$ is possible if we instead use arbitrary (possibly non-smooth) functions, e.g., via compressed sensing algorithms. Given these lower bounds, we specialize to sparse \emph{non-negative} vectors. For a dataset $X$ of non-negative $s$-sparse vectors and any $p \ge 1$, we can non-linearly embed $X$ to $O(s\log(|X|s)/ε^2)$ dimensions while preserving all pairwise distances in $\ell_p$ norm up to $1\pm ε$, with no dependence on $p$. Surprisingly, the non-negativity assumption enables much smaller embeddings than arbitrary sparse vectors, where the best known bounds suffer exponential dependence. Our map also guarantees \emph{exact} dimensionality reduction for $\ell_{\infty}$ by embedding into $O(s\log |X|)$ dimensions, which is tight. We show that both the non-linearity of $f$ and the non-negativity of $X$ are necessary, and provide downstream algorithmic improvements.

cs.DS↗
arXiv2024

Optimal Algorithms for Augmented Testing of Discrete Distributions

We consider the problem of hypothesis testing for discrete distributions. In the standard model, where we have sample access to an underlying distribution $p$, extensive research has established optimal bounds for uniformity testing, identity testing (goodness of fit), and closeness testing (equivalence or two-sample testing). We explore these problems in a setting where a predicted data distribution, possibly derived from historical data or predictive machine learning models, is available. We demonstrate that such a predictor can indeed reduce the number of samples required for all three property testing tasks. The reduction in sample complexity depends directly on the predictor's quality, measured by its total variation distance from $p$. A key advantage of our algorithms is their adaptability to the precision of the prediction. Specifically, our algorithms can self-adjust their sample complexity based on the accuracy of the available prediction, operating without any prior knowledge of the estimation's accuracy (i.e. they are consistent). Additionally, we never use more samples than the standard approaches require, even if the predictions provide no meaningful information (i.e. they are also robust). We provide lower bounds to indicate that the improvements in sample complexity achieved by our algorithms are information-theoretically optimal. Furthermore, experimental results show that the performance of our algorithms on real data significantly exceeds our worst-case guarantees for sample complexity, demonstrating the practicality of our approach.

cs.LG↗
arXiv2024

Statistical-Computational Trade-offs for Density Estimation

We study the density estimation problem defined as follows: given $k$ distributions $p_1, \ldots, p_k$ over a discrete domain $[n]$, as well as a collection of samples chosen from a ``query'' distribution $q$ over $[n]$, output $p_i$ that is ``close'' to $q$. Recently~\cite{aamand2023data} gave the first and only known result that achieves sublinear bounds in {\em both} the sampling complexity and the query time while preserving polynomial data structure space. However, their improvement over linear samples and time is only by subpolynomial factors. Our main result is a lower bound showing that, for a broad class of data structures, their bounds cannot be significantly improved. In particular, if an algorithm uses $O(n/\log^c k)$ samples for some constant $c>0$ and polynomial space, then the query time of the data structure must be at least $k^{1-O(1)/\log \log k}$, i.e., close to linear in the number of distributions $k$. This is a novel \emph{statistical-computational} trade-off for density estimation, demonstrating that any data structure must use close to a linear number of samples or take close to linear query time. The lower bound holds even in the realizable case where $q=p_i$ for some $i$, and when the distributions are flat (specifically, all distributions are uniform over half of the domain $[n]$). We also give a simple data structure for our lower bound instance with asymptotically matching upper bounds. Experiments show that the data structure is quite efficient in practice.

cs.DS↗
arXiv2024

Differentially Private Gomory-Hu Trees

Given an undirected, weighted $n$-vertex graph $G = (V, E, w)$, a Gomory-Hu tree $T$ is a weighted tree on $V$ such that for any pair of distinct vertices $s, t \in V$, the Min-$s$-$t$-Cut on $T$ is also a Min-$s$-$t$-Cut on $G$. Computing a Gomory-Hu tree is a well-studied problem in graph algorithms and has received considerable attention. In particular, a long line of work recently culminated in constructing a Gomory-Hu tree in almost linear time [Abboud, Li, Panigrahi and Saranurak, FOCS 2023]. We design a differentially private (DP) algorithm that computes an approximate Gomory-Hu tree. Our algorithm is $\varepsilon$-DP, runs in polynomial time, and can be used to compute $s$-$t$ cuts that are $\tilde{O}(n/\varepsilon)$-additive approximations of the Min-$s$-$t$-Cuts in $G$ for all distinct $s, t \in V$ with high probability. Our error bound is essentially optimal, as [Dalirrooyfard, Mitrović and Nevmyvaka, NeurIPS 2023] showed that privately outputting a single Min-$s$-$t$-Cut requires $Ω(n)$ additive error even with $(1, 0.1)$-DP and allowing for a multiplicative error term. Prior to our work, the best additive error bounds for approximate all-pairs Min-$s$-$t$-Cuts were $O(n^{3/2}/\varepsilon)$ for $\varepsilon$-DP [Gupta, Roth and Ullman, TCC 2012] and $O(\sqrt{mn} \cdot \text{polylog}(n/δ) / \varepsilon)$ for $(\varepsilon, δ)$-DP [Liu, Upadhyay and Zou, SODA 2024], both of which are implied by differential private algorithms that preserve all cuts in the graph. An important technical ingredient of our main result is an $\varepsilon$-DP algorithm for computing minimum Isolating Cuts with $\tilde{O}(n / \varepsilon)$ additive error, which may be of independent interest.

cs.DS↗
arXiv2024

A Bi-metric Framework for Fast Similarity Search

We propose a new "bi-metric" framework for designing nearest neighbor data structures. Our framework assumes two dissimilarity functions: a ground-truth metric that is accurate but expensive to compute, and a proxy metric that is cheaper but less accurate. In both theory and practice, we show how to construct data structures using only the proxy metric such that the query procedure achieves the accuracy of the expensive metric, while only using a limited number of calls to both metrics. Our theoretical results instantiate this framework for two popular nearest neighbor search algorithms: DiskANN and Cover Tree. In both cases we show that, as long as the proxy metric used to construct the data structure approximates the ground-truth metric up to a bounded factor, our data structure achieves arbitrarily good approximation guarantees with respect to the ground-truth metric. On the empirical side, we apply the framework to the text retrieval problem with two dissimilarity functions evaluated by ML models with vastly different computational costs. We observe that for almost all data sets in the MTEB benchmark, our approach achieves a considerably better accuracy-efficiency tradeoff than the alternatives, such as re-ranking.

cs.IR↗
arXiv2024

Efficiently Computing Similarities to Private Datasets

Many methods in differentially private model training rely on computing the similarity between a query point (such as public or synthetic data) and private data. We abstract out this common subroutine and study the following fundamental algorithmic problem: Given a similarity function $f$ and a large high-dimensional private dataset $X \subset \mathbb{R}^d$, output a differentially private (DP) data structure which approximates $\sum_{x \in X} f(x,y)$ for any query $y$. We consider the cases where $f$ is a kernel function, such as $f(x,y) = e^{-\|x-y\|_2^2/σ^2}$ (also known as DP kernel density estimation), or a distance function such as $f(x,y) = \|x-y\|_2$, among others. Our theoretical results improve upon prior work and give better privacy-utility trade-offs as well as faster query times for a wide range of kernels and distance functions. The unifying approach behind our results is leveraging `low-dimensional structures' present in the specific functions $f$ that we study, using tools such as provable dimensionality reduction, approximation theory, and one-dimensional decomposition of the functions. Our algorithms empirically exhibit improved query times and accuracy over prior state of the art. We also present an application to DP classification. Our experiments demonstrate that the simple methodology of classifying based on average similarity is orders of magnitude faster than prior DP-SGD based approaches for comparable accuracy.

cs.CR↗
arXiv2023

Improved Frequency Estimation Algorithms with and without Predictions

Estimating frequencies of elements appearing in a data stream is a key task in large-scale data analysis. Popular sketching approaches to this problem (e.g., CountMin and CountSketch) come with worst-case guarantees that probabilistically bound the error of the estimated frequencies for any possible input. The work of Hsu et al. (2019) introduced the idea of using machine learning to tailor sketching algorithms to the specific data distribution they are being run on. In particular, their learning-augmented frequency estimation algorithm uses a learned heavy-hitter oracle which predicts which elements will appear many times in the stream. We give a novel algorithm, which in some parameter regimes, already theoretically outperforms the learning based algorithm of Hsu et al. without the use of any predictions. Augmenting our algorithm with heavy-hitter predictions further reduces the error and improves upon the state of the art. Empirically, our algorithms achieve superior performance in all experiments compared to prior approaches.

cs.DS↗
arXiv2023

Constant Approximation for Individual Preference Stable Clustering

Individual preference (IP) stability, introduced by Ahmadi et al. (ICML 2022), is a natural clustering objective inspired by stability and fairness constraints. A clustering is $α$-IP stable if the average distance of every data point to its own cluster is at most $α$ times the average distance to any other cluster. Unfortunately, determining if a dataset admits a $1$-IP stable clustering is NP-Hard. Moreover, before this work, it was unknown if an $o(n)$-IP stable clustering always \emph{exists}, as the prior state of the art only guaranteed an $O(n)$-IP stable clustering. We close this gap in understanding and show that an $O(1)$-IP stable clustering always exists for general metrics, and we give an efficient algorithm which outputs such a clustering. We also introduce generalizations of IP stability beyond average distance and give efficient, near-optimal algorithms in the cases where we consider the maximum and minimum distances within and between clusters.

cs.DS↗
arXiv2023

A Near-Linear Time Algorithm for the Chamfer Distance

For any two point sets $A,B \subset \mathbb{R}^d$ of size up to $n$, the Chamfer distance from $A$ to $B$ is defined as $\text{CH}(A,B)=\sum_{a \in A} \min_{b \in B} d_X(a,b)$, where $d_X$ is the underlying distance measure (e.g., the Euclidean or Manhattan distance). The Chamfer distance is a popular measure of dissimilarity between point clouds, used in many machine learning, computer vision, and graphics applications, and admits a straightforward $O(d n^2)$-time brute force algorithm. Further, the Chamfer distance is often used as a proxy for the more computationally demanding Earth-Mover (Optimal Transport) Distance. However, the \emph{quadratic} dependence on $n$ in the running time makes the naive approach intractable for large datasets. We overcome this bottleneck and present the first $(1+ε)$-approximate algorithm for estimating the Chamfer distance with a near-linear running time. Specifically, our algorithm runs in time $O(nd \log (n)/\varepsilon^2)$ and is implementable. Our experiments demonstrate that it is both accurate and fast on large high-dimensional datasets. We believe that our algorithm will open new avenues for analyzing large high-dimensional point clouds. We also give evidence that if the goal is to \emph{report} a $(1+\varepsilon)$-approximate mapping from $A$ to $B$ (as opposed to just its value), then any sub-quadratic time algorithm is unlikely to exist.

cs.DS↗
arXiv2023

Data Structures for Density Estimation

We study statistical/computational tradeoffs for the following density estimation problem: given $k$ distributions $v_1, \ldots, v_k$ over a discrete domain of size $n$, and sampling access to a distribution $p$, identify $v_i$ that is "close" to $p$. Our main result is the first data structure that, given a sublinear (in $n$) number of samples from $p$, identifies $v_i$ in time sublinear in $k$. We also give an improved version of the algorithm of Acharya et al. (2018) that reports $v_i$ in time linear in $k$. The experimental evaluation of the latter algorithm shows that it achieves a significant reduction in the number of operations needed to achieve a given accuracy compared to prior work.

cs.DS↗
arXiv2023

Learned Interpolation for Better Streaming Quantile Approximation with Worst-Case Guarantees

An $\varepsilon$-approximate quantile sketch over a stream of $n$ inputs approximates the rank of any query point $q$ - that is, the number of input points less than $q$ - up to an additive error of $\varepsilon n$, generally with some probability of at least $1 - 1/\mathrm{poly}(n)$, while consuming $o(n)$ space. While the celebrated KLL sketch of Karnin, Lang, and Liberty achieves a provably optimal quantile approximation algorithm over worst-case streams, the approximations it achieves in practice are often far from optimal. Indeed, the most commonly used technique in practice is Dunning's t-digest, which often achieves much better approximations than KLL on real-world data but is known to have arbitrarily large errors in the worst case. We apply interpolation techniques to the streaming quantiles problem to attempt to achieve better approximations on real-world data sets than KLL while maintaining similar guarantees in the worst case.

cs.DS↗
arXiv2023

Robust Algorithms on Adaptive Inputs from Bounded Adversaries

We study dynamic algorithms robust to adaptive input generated from sources with bounded capabilities, such as sparsity or limited interaction. For example, we consider robust linear algebraic algorithms when the updates to the input are sparse but given by an adversary with access to a query oracle. We also study robust algorithms in the standard centralized setting, where an adversary queries an algorithm in an adaptive manner, but the number of interactions between the adversary and the algorithm is bounded. We first recall a unified framework of [HKM+20, BKM+22, ACSS23] for answering $Q$ adaptive queries that incurs $\widetilde{\mathcal{O}}(\sqrt{Q})$ overhead in space, which is roughly a quadratic improvement over the naïve implementation, and only incurs a logarithmic overhead in query time. Although the general framework has diverse applications in machine learning and data science, such as adaptive distance estimation, kernel density estimation, linear regression, range queries, and point queries and serves as a preliminary benchmark, we demonstrate even better algorithmic improvements for (1) reducing the pre-processing time for adaptive distance estimation and (2) permitting an unlimited number of adaptive queries for kernel density estimation. Finally, we complement our theoretical results with additional empirical evaluations.

cs.DS↗
arXiv2023

Improved Space Bounds for Learning with Experts

We give improved tradeoffs between space and regret for the online learning with expert advice problem over $T$ days with $n$ experts. Given a space budget of $n^δ$ for $δ\in (0,1)$, we provide an algorithm achieving regret $\tilde{O}(n^2 T^{1/(1+δ)})$, improving upon the regret bound $\tilde{O}(n^2 T^{2/(2+δ)})$ in the recent work of [PZ23]. The improvement is particularly salient in the regime $δ\rightarrow 1$ where the regret of our algorithm approaches $\tilde{O}_n(\sqrt{T})$, matching the $T$ dependence in the standard online setting without space restrictions.

cs.DS↗
arXiv2023

Optimal Algorithms for Linear Algebra in the Current Matrix Multiplication Time

We study fundamental problems in linear algebra, such as finding a maximal linearly independent subset of rows or columns (a basis), solving linear regression, or computing a subspace embedding. For these problems, we consider input matrices $\mathbf{A}\in\mathbb{R}^{n\times d}$ with $n > d$. The input can be read in $\text{nnz}(\mathbf{A})$ time, which denotes the number of nonzero entries of $\mathbf{A}$. In this paper, we show that beyond the time required to read the input matrix, these fundamental linear algebra problems can be solved in $d^ω$ time, i.e., where $ω\approx 2.37$ is the current matrix-multiplication exponent. To do so, we introduce a constant-factor subspace embedding with the optimal $m=\mathcal{O}(d)$ number of rows, and which can be applied in time $\mathcal{O}\left(\frac{\text{nnz}(\mathbf{A})}α\right) + d^{2 + α}\text{poly}(\log d)$ for any trade-off parameter $α>0$, tightening a recent result by Chepurko et. al. [SODA 2022] that achieves an $\exp(\text{poly}(\log\log n))$ distortion with $m=d\cdot\text{poly}(\log\log d)$ rows in $\mathcal{O}\left(\frac{\text{nnz}(\mathbf{A})}α+d^{2+α+o(1)}\right)$ time. Our subspace embedding uses a recently shown property of stacked Subsampled Randomized Hadamard Transforms (SRHT), which actually increase the input dimension, to "spread" the mass of an input vector among a large number of coordinates, followed by random sampling. To control the effects of random sampling, we use fast semidefinite programming to reweight the rows. We then use our constant-factor subspace embedding to give the first optimal runtime algorithms for finding a maximal linearly independent subset of columns, regression, and leverage score sampling. To do so, we also introduce a novel subroutine that iteratively grows a set of independent rows, which may be of independent interest.

cs.DS↗
arXiv2022

Exponentially Improving the Complexity of Simulating the Weisfeiler-Lehman Test with Graph Neural Networks

Recent work shows that the expressive power of Graph Neural Networks (GNNs) in distinguishing non-isomorphic graphs is exactly the same as that of the Weisfeiler-Lehman (WL) graph test. In particular, they show that the WL test can be simulated by GNNs. However, those simulations involve neural networks for the 'combine' function of size polynomial or even exponential in the number of graph nodes $n$, as well as feature vectors of length linear in $n$. We present an improved simulation of the WL test on GNNs with \emph{exponentially} lower complexity. In particular, the neural network implementing the combine function in each node has only a polylogarithmic number of parameters in $n$, and the feature vectors exchanged by the nodes of GNN consists of only $O(\log n)$ bits. We also give logarithmic lower bounds for the feature vector length and the size of the neural networks, showing the (near)-optimality of our construction.

cs.LG↗
arXiv2022

Sub-quadratic Algorithms for Kernel Matrices via Kernel Density Estimation

Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is efficiency -- given $n$ input points, most kernel-based algorithms need to materialize the full $n \times n$ kernel matrix before performing any subsequent computation, thus incurring $Ω(n^2)$ runtime. Breaking this quadratic barrier for various problems has therefore, been a subject of extensive research efforts. We break the quadratic barrier and obtain $\textit{subquadratic}$ time algorithms for several fundamental linear-algebraic and graph processing primitives, including approximating the top eigenvalue and eigenvector, spectral sparsification, solving linear systems, local clustering, low-rank approximation, arboricity estimation and counting weighted triangles. We build on the recent Kernel Density Estimation framework, which (after preprocessing in time subquadratic in $n$) can return estimates of row/column sums of the kernel matrix. In particular, we develop efficient reductions from $\textit{weighted vertex}$ and $\textit{weighted edge sampling}$ on kernel graphs, $\textit{simulating random walks}$ on kernel graphs, and $\textit{importance sampling}$ on matrices to Kernel Density Estimation and show that we can generate samples from these distributions in $\textit{sublinear}$ (in the support of the distribution) time. Our reductions are the central ingredient in each of our applications and we believe they may be of independent interest. We empirically demonstrate the efficacy of our algorithms on low-rank approximation (LRA) and spectral sparsification, where we observe a $\textbf{9x}$ decrease in the number of kernel evaluations over baselines for LRA and a $\textbf{41x}$ reduction in the graph size for spectral sparsification.

cs.LG↗
arXiv2022

Faster Linear Algebra for Distance Matrices

The distance matrix of a dataset $X$ of $n$ points with respect to a distance function $f$ represents all pairwise distances between points in $X$ induced by $f$. Due to their wide applicability, distance matrices and related families of matrices have been the focus of many recent algorithmic works. We continue this line of research and take a broad view of algorithm design for distance matrices with the goal of designing fast algorithms, which are specifically tailored for distance matrices, for fundamental linear algebraic primitives. Our results include efficient algorithms for computing matrix-vector products for a wide class of distance matrices, such as the $\ell_1$ metric for which we get a linear runtime, as well as an $Ω(n^2)$ lower bound for any algorithm which computes a matrix-vector product for the $\ell_{\infty}$ case, showing a separation between the $\ell_1$ and the $\ell_{\infty}$ metrics. Our upper bound results, in conjunction with recent works on the matrix-vector query model, have many further downstream applications, including the fastest algorithm for computing a relative error low-rank approximation for the distance matrix induced by $\ell_1$ and $\ell_2^2$ functions and the fastest algorithm for computing an additive error low-rank approximation for the $\ell_2$ metric, in addition to applications for fast matrix multiplication among others. We also give algorithms for constructing distance matrices and show that one can construct an approximate $\ell_2$ distance matrix in time faster than the bound implied by the Johnson-Lindenstrauss lemma.

cs.DS↗
arXiv2022

Learning-Augmented Algorithms for Online Linear and Semidefinite Programming

Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist known impossibility results for approximating the optimal solution when constraints for covering SDPs arrive in an online fashion. In this paper, we study online covering linear and semidefinite programs in which the algorithm is augmented with advice from a possibly erroneous predictor. We show that if the predictor is accurate, we can efficiently bypass these impossibility results and achieve a constant-factor approximation to the optimal solution, i.e., consistency. On the other hand, if the predictor is inaccurate, under some technical conditions, we achieve results that match both the classical optimal upper bounds and the tight lower bounds up to constant factors, i.e., robustness. More broadly, we introduce a framework that extends both (1) the online set cover problem augmented with machine-learning predictors, studied by Bamas, Maggiori, and Svensson (NeurIPS 2020), and (2) the online covering SDP problem, initiated by Elad, Kale, and Naor (ICALP 2016). Specifically, we obtain general online learning-augmented algorithms for covering linear programs with fractional advice and constraints, and initiate the study of learning-augmented algorithms for covering SDP problems. Our techniques are based on the primal-dual framework of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and can be further adjusted to handle constraints where the variables lie in a bounded region, i.e., box constraints.

cs.DS↗