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Found 24 papers, 2 papers with code
Stochastic Rounding Implicitly Regularizes Tall-and-Thin Matrices
Motivated by the popularity of stochastic rounding in the context of machine learning and the training of large-scale deep neural network models, we consider stochastic nearness rounding of real matrices $\mathbf{A}$ with many more rows than columns.
Feature Space Sketching for Logistic Regression
We present novel bounds for coreset construction, feature selection, and dimensionality reduction for logistic regression.
Low-Rank Updates of Matrix Square Roots
Given a low rank perturbation to a matrix, we argue that a low-rank approximate correction to the (inverse) square root exists.
Faster Randomized Infeasible Interior Point Methods for Tall/Wide Linear Programs
Linear programming (LP) is used in many machine learning applications, such as $\ell_1$-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc.
Approximation Algorithms for Sparse Principal Component Analysis
To improve the interpretability of PCA, various approaches to obtain sparse principal direction loadings have been proposed, which are termed Sparse Principal Component Analysis (SPCA).
Randomized Iterative Algorithms for Fisher Discriminant Analysis
When the number of predictor variables greatly exceeds the number of observations, one of the alternatives for conventional FDA is regularized Fisher discriminant analysis (RFDA).
An Iterative, Sketching-based Framework for Ridge Regression
Ridge regression is a variant of regularized least squares regression that is particularly suitable in settings where the number of predictor variables greatly exceeds the number of observations.
Constructing Compact Brain Connectomes for Individual Fingerprinting
Recent neuroimaging studies have shown that functional connectomes are unique to individuals, i. e., two distinct fMRIs taken over different sessions of the same subject are more similar in terms of their connectomes than those from two different subjects.
Lectures on Randomized Numerical Linear Algebra
This chapter is based on lectures on Randomized Numerical Linear Algebra from the 2016 Park City Mathematics Institute summer school on The Mathematics of Data.
Structural Conditions for Projection-Cost Preservation via Randomized Matrix Multiplication
Projection-cost preservation is a low-rank approximation guarantee which ensures that the cost of any rank-$k$ projection can be preserved using a smaller sketch of the original data matrix.
A Randomized Rounding Algorithm for Sparse PCA
We present and analyze a simple, two-step algorithm to approximate the optimal solution of the sparse PCA problem.
Feature Selection for Ridge Regression with Provable Guarantees
We introduce single-set spectral sparsification as a deterministic sampling based feature selection technique for regularized least squares classification, which is the classification analogue to ridge regression.
Approximating Sparse PCA from Incomplete Data
We show that for a wide class of optimization problems, if the sketch is close (in the spectral norm) to the original data matrix, then one can recover a near optimal solution to the optimization problem by using the sketch.
Recovering PCA from Hybrid-$(\ell_1,\ell_2)$ Sparse Sampling of Data Elements
This paper addresses how well we can recover a data matrix when only given a few of its elements.
Feature Selection for Linear SVM with Provable Guarantees
In the unsupervised setting, we also provide worst-case guarantees of the radius of the minimum enclosing ball, thereby ensuring comparable generalization as in the full feature space and resolving an open problem posed in Dasgupta et al. We present extensive experiments on real-world datasets to support our theory and to demonstrate that our method is competitive and often better than prior state-of-the-art, for which there are no known provable guarantees.
Identifying Influential Entries in a Matrix
For any matrix A in R^(m x n) of rank \rho, we present a probability distribution over the entries of A (the element-wise leverage scores of equation (2)) that reveals the most influential entries in the matrix.
Random Projections for Linear Support Vector Machines
Let X be a data matrix of rank \rho, whose rows represent n points in d-dimensional space.
The Fast Cauchy Transform and Faster Robust Linear Regression
We provide fast algorithms for overconstrained $\ell_p$ regression and related problems: for an $n\times d$ input matrix $A$ and vector $b\in\mathbb{R}^n$, in $O(nd\log n)$ time we reduce the problem $\min_{x\in\mathbb{R}^d} \|Ax-b\|_p$ to the same problem with input matrix $\tilde A$ of dimension $s \times d$ and corresponding $\tilde b$ of dimension $s\times 1$.
Near-optimal Coresets For Least-Squares Regression
We study (constrained) least-squares regression as well as multiple response least-squares regression and ask the question of whether a subset of the data, a coreset, suffices to compute a good approximate solution to the regression.
Sparse Features for PCA-Like Linear Regression
Principal Components Analysis~(PCA) is often used as a feature extraction procedure.
Randomized Dimensionality Reduction for k-means Clustering
On the other hand, two provably accurate feature extraction methods for $k$-means clustering are known in the literature; one is based on random projections and the other is based on the singular value decomposition (SVD).
Random Projections for k-means Clustering
This paper discusses the topic of dimensionality reduction for $k$-means clustering.
Effective Resistances, Statistical Leverage, and Applications to Linear Equation Solving
Our first and main result is a simple algorithm to approximate the solution to a set of linear equations defined by a Laplacian (for a graph $G$ with $n$ nodes and $m \le n^2$ edges) constraint matrix.
Numerical Analysis
Unsupervised Feature Selection for the k-means Clustering Problem
We present a novel feature selection algorithm for the $k$-means clustering problem.
