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Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix low rank approximation. Especially with today's applications involved in larger and…

Numerical Analysis · Computer Science 2016-09-09 Haishan Ye , Qiaoming Ye , Zhihua Zhang

An outstanding open question posed by Guha and Indyk in 2006 asks to characterize metric spaces in which distances can be estimated using efficient sketches. Specifically, we say that a sketching algorithm is efficient if it achieves…

Data Structures and Algorithms · Computer Science 2017-02-16 Alexandr Andoni , Robert Krauthgamer , Ilya Razenshteyn

A $(1 \pm \epsilon)$-sparsifier of a hypergraph $G(V,E)$ is a (weighted) subgraph that preserves the value of every cut to within a $(1 \pm \epsilon)$-factor. It is known that every hypergraph with $n$ vertices admits a $(1 \pm…

Data Structures and Algorithms · Computer Science 2024-07-08 Sanjeev Khanna , Aaron L. Putterman , Madhu Sudan

We initiate the study of trade-offs between sparsity and the number of measurements in sparse recovery schemes for generic norms. Specifically, for a norm $\|\cdot\|$, sparsity parameter $k$, approximation factor $K>0$, and probability of…

Data Structures and Algorithms · Computer Science 2015-04-07 Arturs Backurs , Piotr Indyk , Eric Price , Ilya Razenshteyn , David P. Woodruff

We provide efficient algorithms for overconstrained linear regression problems with size $n \times d$ when the loss function is a symmetric norm (a norm invariant under sign-flips and coordinate-permutations). An important class of…

Data Structures and Algorithms · Computer Science 2019-10-08 Zhao Song , Ruosong Wang , Lin F. Yang , Hongyang Zhang , Peilin Zhong

We initiate the study of sub-linear sketching and streaming techniques for estimating the output size of common dictionary compressors such as Lempel-Ziv '77, the run-length Burrows-Wheeler transform, and grammar compression. To this end,…

Data Structures and Algorithms · Computer Science 2024-08-20 Ruben Becker , Matteo Canton , Davide Cenzato , Sung-Hwan Kim , Bojana Kodric , Nicola Prezza

Sketching is a stochastic dimension reduction method that preserves geometric structures of data and has applications in high-dimensional regression, low rank approximation and graph sparsification. In this work, we show that sketching can…

Machine Learning · Statistics 2021-09-17 Zhishen Huang , Stephen Becker

We study sequential change-point detection procedures based on linear sketches of high-dimensional signal vectors using generalized likelihood ratio (GLR) statistics. The GLR statistics allow for an unknown post-change mean that represents…

Machine Learning · Computer Science 2018-05-01 Yang Cao , Andrew Thompson , Meng Wang , Yao Xie

We study the problem of sketching an input graph, so that given the sketch, one can estimate the weight of any cut in the graph within factor $1+\epsilon$. We present lower and upper bounds on the size of a randomized sketch, focusing on…

Data Structures and Algorithms · Computer Science 2014-11-11 Alexandr Andoni , Robert Krauthgamer , David P. Woodruff

Pairwise alignment of DNA sequencing data is a ubiquitous task in bioinformatics and typically represents a heavy computational burden. A standard approach to speed up this task is to compute "sketches" of the DNA reads (typically via…

Information Theory · Computer Science 2021-07-12 Ilan Shomorony , Govinda M. Kamath

Low-rank approximation of tensors has been widely used in high-dimensional data analysis. It usually involves singular value decomposition (SVD) of large-scale matrices with high computational complexity. Sketching is an effective data…

Numerical Analysis · Mathematics 2023-01-30 Wandi Dong , Gaohang Yu , Liqun Qi , Xiaohao Cai

Learning parameters from voluminous data can be prohibitive in terms of memory and computational requirements. We propose a "compressive learning" framework where we estimate model parameters from a sketch of the training data. This sketch…

Machine Learning · Computer Science 2017-05-08 Nicolas Keriven , Anthony Bourrier , Rémi Gribonval , Patrick Pérez

Matrix trace estimation is ubiquitous in machine learning applications and has traditionally relied on Hutchinson's method, which requires $O(\log(1/\delta)/\epsilon^2)$ matrix-vector product queries to achieve a $(1 \pm…

Data Structures and Algorithms · Computer Science 2021-11-02 Shuli Jiang , Hai Pham , David P. Woodruff , Qiuyi , Zhang

The problem of computing the edit-distance between a string and a finite automaton arises in a variety of applications in computational biology, text processing, and speech recognition. This paper presents linear-space algorithms for…

Formal Languages and Automata Theory · Computer Science 2009-04-30 Cyril Allauzen , Mehryar Mohri

We adapt a well known streaming algorithm for approximating item frequencies to the matrix sketching setting. The algorithm receives the rows of a large matrix $A \in \R^{n \times m}$ one after the other in a streaming fashion. It maintains…

Data Structures and Algorithms · Computer Science 2012-07-12 Edo Liberty

A longstanding observation, which was partially proven in \cite{LNW14,AHLW16}, is that any turnstile streaming algorithm can be implemented as a linear sketch (the reverse is trivially true). We study the relationship between turnstile…

Data Structures and Algorithms · Computer Science 2021-01-06 John Kallaugher , Eric Price

Matrix sketching is a powerful tool for reducing the size of large data matrices. Yet there are fundamental limitations to this size reduction when we want to recover an accurate estimator for a task such as least square regression. We show…

Data Structures and Algorithms · Computer Science 2024-05-10 Sachin Garg , Kevin Tan , Michał Dereziński

Given a persistence diagram with $n$ points, we give an algorithm that produces a sequence of $n$ persistence diagrams converging in bottleneck distance to the input diagram, the $i$th of which has $i$ distinct (weighted) points and is a…

Computational Geometry · Computer Science 2020-12-04 Donald R. Sheehy , Siddharth Sheth

We consider statistical and algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms. Prior results show that, from an \emph{algorithmic perspective}, when using sketching matrices…

Machine Learning · Statistics 2015-05-26 Garvesh Raskutti , Michael Mahoney

Given two strings $A[1..n]$ and $B[1..m]$, and a set of operations allowed to edit the strings, the edit distance between $A$ and $B$ is the minimum number of operations required to transform $A$ into $B$. Sequentially, a standard Dynamic…

Data Structures and Algorithms · Computer Science 2023-09-01 Xiangyun Ding , Xiaojun Dong , Yan Gu , Youzhe Liu , Yihan Sun