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We solve the problem of best approximation by Parseval frames to an arbitrary frame in a subspace of an infinite dimensional Hilbert space. We explicitly describe all the solutions and we give a criterion for uniqueness. This best…
This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…
Bogoliubov's 1947 approximation, originally developed in the microscopic theory of superfluidity, laid the foundation for solving previously intractable quantum models and later became part of "quantum mathematics". Regarding mathematically…
Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using…
The classical theorem of Bishop-Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension of the theorem gives a quantitative description of just how dense the norm-achieving…
Metrizable spaces are studied in which every closed set is an $\alpha$-limit set for some continuous map and some point. It is shown that this property is enjoyed by every space containing sufficiently many arcs (formalized in the notion of…
We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval $[a,b]$. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
Deep learning researchers commonly suggest that converged models are stuck in local minima. More recently, some researchers observed that under reasonable assumptions, the vast majority of critical points are saddle points, not true minima.…
Definition. A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists…
In recent years, the success of deep learning has inspired many researchers to study the optimization of general smooth non-convex functions. However, recent works have established pessimistic worst-case complexities for this class…
Consider a sum of convex functions, where the only information known about each individual summand is the location of a minimizer. In this work, we give an exact characterization of the set of possible minimizers of the sum. Our results…
In this paper the problem of maximizing the distance to a given fixed point over an intersection of balls is considered. It is known that this problem is NP complete in the general case, since any subset sum problem can be solved upon…
Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is…
On a convex bounded open set, we prove that Poincar\'e-Sobolev constants for functions vanishing at the boundary can be bounded from below in terms of the norm of the distance function in a suitable Lebesgue space. This generalizes a result…
Constructing or learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation problem, regularized learning…
The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range…
The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a feasible point does not exist. However,…
Finding the minimum and the minimizers of convex functions has been of primary concern in convex analysis since its conception. It is well-known that if a convex function has a minimum, then that minimum is global. The minimizers, however,…
Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of…