Related papers: Approximation by partial isometries and symmetric …

In this paper we give an explicit solution to the rank constrained matrix approximation in Frobenius norm, which is a generalization of the classical approximation of an m by n matrix A by a matrix of rank k at most.

In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation…

In multi-objective optimization, computing the entire non-dominated set (also known as the Pareto front or the Pareto frontier) is often intractable. However, for any multiplicative factor greater than one, an approximation set can be…

We propose a general convex optimization problem for computing regularized geodesic distances. We show that under mild conditions on the regularizer the problem is well posed. We propose three different regularizers and provide analytical…

We derive, similar to Lau and Riha, a matrix formulation of a general best approximation theorem of Singer for the special case of spectral approximations of a given matrix from a given subspace. Using our matrix formulation we describe the…

We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular…

Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We…

Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this…

We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…

The low-rank matrix approximation problem is ubiquitous in computational mathematics. Traditionally, this problem is solved in spectral or Frobenius norms, where the accuracy of the approximation is related to the rate of decrease of the…

Examining the extent to which measurements of rotation matrices are close to each other is challenging due measurement noise. To overcome this, data is typically smoothed and Riemannian and Euclidean metrics are applied. However, if…

Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…

We develop two adaptive discretization algorithms for convex semi-infinite optimization, which terminate after finitely many iterations at approximate solutions of arbitrary precision. In particular, they terminate at a feasible point of…

We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric…

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions.…

This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…

Solving the inverse kinematics problem is a fundamental challenge in motion planning, control, and calibration for articulated robots. Kinematic models for these robots are typically parametrized by joint angles, generating a complicated…

We solve the convergence case of the generalized Baker-Schmidt problem for simultaneous approximation on affine subspaces, under natural diophantine type conditions. In one of our theorems, we do not require monotonicity on the…

Finite unit norm tight frames provide Parseval-like decompositions of vectors in terms of redundant components of equal weight. They are known to be exceptionally robust against additive noise and erasures, and as such, have great potential…

We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a…