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We propose a new technique for constructing low-rank approximations of matrices that arise in kernel methods for machine learning. Our approach pairs a novel automatically constructed analytic expansion of the underlying kernel function…
Constrained optimization problems appear in a wide variety of challenging real-world problems, where constraints often capture the physics of the underlying system. Classic methods for solving these problems rely on iterative algorithms…
We are interested in solving linear systems arising from three applications: (1) kernel methods in machine learning, (2) discretization of boundary integral equations from mathematical physics, and (3) Schur complements formed in the…
Finite linear least squares is one of the core problems of numerical linear algebra, with countless applications across science and engineering. Consequently, there is a rich and ongoing literature on algorithms for solving linear least…
Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
Recently, convex formulations of low-rank matrix factorization problems have received considerable attention in machine learning. However, such formulations often require solving for a matrix of the size of the data matrix, making it…
Non-negative matrix factorization (NMF) is a prob- lem with many applications, ranging from facial recognition to document clustering. However, due to the variety of algorithms that solve NMF, the randomness involved in these algorithms,…
It has been found that stochastic algorithms often find good solutions much more rapidly than inherently-batch approaches. Indeed, a very useful rule of thumb is that often, when solving a machine learning problem, an iterative technique…
The computational cost of many signal processing and machine learning techniques is often dominated by the cost of applying certain linear operators to high-dimensional vectors. This paper introduces an algorithm aimed at reducing the…
Randomized iterative algorithms, such as the randomized Kaczmarz method, have gained considerable popularity due to their efficacy in solving matrix-vector and matrix-matrix regression problems. Our present work leverages the insights…
We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally embed structured outputs in a linear space. We exploit this fact to design…
Low-rank matrix regression is a fundamental problem in data science with various applications in systems and control. Nuclear norm regularization has been widely applied to solve this problem due to its convexity. However, it suffers from…
Graph representation learning plays an important role in many graph mining applications, but learning embeddings of large-scale graphs remains a problem. Recent works try to improve scalability via graph summarization -- i.e., they learn…
Efficiently representing real world data in a succinct and parsimonious manner is of central importance in many fields. We present a generalized greedy pursuit framework, allowing us to efficiently solve structured matrix factorization…
The aim of this paper is twofold: In the first part, we leverage recent results on scenario design to develop randomized algorithmsfor approximating the image set of a nonlinear mapping, that is, a (possibly noisy) mapping of a set via a…
We introduce a Bayesian perspective for the structured matrix factorization problem. The proposed framework provides a probabilistic interpretation for existing geometric methods based on determinant minimization. We model input data…
Randomized rounding is a standard method, based on the probabilistic method, for designing combinatorial approximation algorithms. In Raghavan's seminal paper introducing the method (1988), he writes: "The time taken to solve the linear…
Branes and defects in topological Landau-Ginzburg models are described by matrix factorisations. We revisit the problem of deforming them and discuss various deformation methods as well as their relations. We have implemented these…
We give sublinear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions…
Matrix Factorization has emerged as a widely adopted framework for modeling data exhibiting low-rank structures. To address challenges in manifold learning, this paper presents a subspace-constrained quadratic matrix factorization model.…