Related papers: On the Approximation Theory of Linear Variational …
In this note, we develop fast and deterministic dimensionality reduction techniques for a family of subspace approximation problems. Let $P\subset \mathbbm{R}^N$ be a given set of $M$ points. The techniques developed herein find an $O(n…
Real-world problems of operations research are typically high-dimensional and combinatorial. Linear programs are generally used to formulate and efficiently solve these large decision problems. However, in multi-period decision problems, we…
Linear subspace representations of appearance variation are pervasive in computer vision. This paper addresses the problem of robustly matching such subspaces (computing the similarity between them) when they are used to describe the scope…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…
The effectiveness of dimensionality reduction with quadratic manifolds hinges on the choice of a reduced basis and the associated quadratic correction terms. Existing approaches typically rely on subspaces spanned by the leading principal…
The merit of projecting data onto linear subspaces is well known from, e.g., dimension reduction. One key aspect of subspace projections, the maximum preservation of variance (principal component analysis), has been thoroughly researched…
Modern day engineering problems are ubiquitously characterized by sophisticated computer codes that map parameters or inputs to an underlying physical process. In other situations, experimental setups are used to model the physical process…
Sliced inverse regression is a popular tool for sufficient dimension reduction, which replaces covariates with a minimal set of their linear combinations without loss of information on the conditional distribution of the response given the…
The real-life data have a complex and non-linear structure due to their nature. These non-linearities and the large number of features can usually cause problems such as the empty-space phenomenon and the well-known curse of dimensionality.…
Subset selection in multiple linear regression aims to choose a subset of candidate explanatory variables that tradeoff fitting error (explanatory power) and model complexity (number of variables selected). We build mathematical programming…
We propose novel randomized optimization methods for high-dimensional convex problems based on restrictions of variables to random subspaces. We consider oblivious and data-adaptive subspaces and study their approximation properties via…
We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates…
We consider algorithmic approaches to the D-optimality problem for cases where the input design matrix is large and highly structured, in particular implicitly specified as a full quadratic or linear response-surface model in several levels…
We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most…
In recent years, various subspace algorithms have been developed to handle large-scale optimization problems. Although existing subspace Newton methods require fewer iterations to converge in practice, the matrix operations and full…
Embedded boundary methods alleviate many computational challenges, including those associated with meshing complex geometries and solving problems with evolving domains and interfaces. Developing model reduction methods for computational…
Large-scale unconstrained optimization is a fundamental and important class of, yet not well-solved problems in numerical optimization. The main challenge in designing an algorithm is to require a few storage locations or very inexpensive…
The optimization of high-dimensional black-box functions is a challenging problem. When a low-dimensional linear embedding structure can be assumed, existing Bayesian optimization (BO) methods often transform the original problem into…