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We study the loss surface of a feed-forward neural network with ReLU non-linearities, regularized with weight decay. We show that the regularized loss function is piecewise strongly convex on an important open set which contains, under some…
A linear functional of an object from a convex symmetric set can be optimally estimated, in a worst-case sense, by a linear functional of observations made on the object. This well-known fact is extended here to a nonlinear setting: other…
This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this…
This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices $\text{SO}(n)$. Such problems are nonconvex due to the constraint $X \in \text{SO}(n)$. Nonetheless, we show…
Constraint satisfaction problems (CSPs) consist of a set of variables taking values from some finite domain and a set of local constraints on these variables. The objective is to find an assignment to the variables that maximizes the…
Optimal recovery is a mathematical framework for learning functions from observational data by adopting a worst-case perspective tied to model assumptions on the functions to be learned. Working in a finite-dimensional Hilbert space, we…
Let $H$ be a real Hilbert space. In this short note, using some of the properties of bounded linear operators with closed range defined on $H$, certain bounds for a specific convex subset of the solution set of infinite linear…
Convex geometries form a subclass of closure systems with unique criticals, or $UC$-systems. We show that the $F$-basis introduced in [1] for $UC$-systems, becomes optimum in convex geometries, in two essential parts of the basis: right…
We study the Cauchy problem for the cubic fractional nonlinear Schr\"odinger equation (fNLS) on the real line and on the circle. In particular, we prove global well-posedness of the cubic fNLS with all orders of dispersion higher than the…
In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common…
We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension…
As objects of study in functional analysis, Hilbert spaces stand out as special objects of study as do nuclear spaces in view of a rich geometrical structure they possess as Banach and Frechet spaces, respectively. On the other hand, there…
The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection…
Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of…
We deal with linear programming problems involving absolute values in their formulations, so that they are no more expressible as standard linear programs. The presence of absolute values causes the problems to be nonconvex and nonsmooth,…
Recently the theory of widths of Kolmogorov (especially of Gelfand widths) has received a great deal of interest due to its close relationship with the newly born area of Compressed Sensing. It has been realized that widths reflect properly…
Given pointwise samples of an unknown function belonging to a certain model set, one seeks in Optimal Recovery to recover this function in a way that minimizes the worst-case error of the recovery procedure. While it is often known that…
The optimization problem concerning the determination of the minimizer for the sum of convex functions holds significant importance in the realm of distributed and decentralized optimization. In scenarios where full knowledge of the…
This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal…
In recent years, the nuclear norm minimization (NNM) problem has been attracting much attention in computer vision and machine learning. The NNM problem is capitalized on its convexity and it can be solved efficiently. The standard nuclear…