相关论文: VC bounds on the cardinality of nearly orthogonal …
In this work, we investigate the convergence of numerical approximations to coercivity constants of variational problems. These constants are essential components of rigorous error bounds for reduced-order modeling; extension of these…
In many high-impact applications, it is important to ensure the quality of output of a machine learning algorithm as well as its reliability in comparison with the complexity of the algorithm used. In this paper, we have initiated a…
Many fundamental machine learning tasks can be formulated as a problem of learning with vector-valued functions, where we learn multiple scalar-valued functions together. Although there is some generalization analysis on different specific…
Machine learning models with inputs in a Euclidean space $\mathbb{R}^d$, when implemented on digital computers, generalize, and their generalization gap converges to $0$ at a rate of $c/N^{1/2}$ concerning the sample size $N$. However, the…
This paper studies approximate solutions of a linear fractional vector optimization problem without requiring boundedness of the constraint set. We establish necessary and sufficient conditions for approximating weakly efficient points of…
We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.
We derive two upper bounds for the probability of deviation of a vector-valued Lipschitz function of a collection of random variables from its expected value. The resulting upper bounds can be tighter than bounds obtained by a direct…
We present an extensive analysis of relative deviation bounds, including detailed proofs of two-sided inequalities and their implications. We also give detailed proofs of two-sided generalization bounds that hold in the general case of…
We derive novel anti-concentration bounds for the difference between the maximal values of two Gaussian random vectors across various settings. Our bounds are dimension-free, scaling with the dimension of the Gaussian vectors only through…
We present two algorithms for constructing orthonormal bases of rational function vectors with respect to a discrete inner product, and discuss how to use them for a rational approximation problem. Building on the pencil-based formulation…
We study range spaces, where the ground set consists of either polygonal curves in $\mathbb{R}^d$ or polygonal regions in the plane that may contain holes and the ranges are balls defined by an elastic distance measure, such as the…
A Vec-variety is a suitable functor from finite-dimensional vector spaces to finite-dimensional varieties. Most varieties in the geometry of tensors, e.g. the variety of d-way tensors of slice rank at most r, are of this form. We prove that…
This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only…
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…
I discuss possible definitions of categories of vector spaces enriched with a notion of formal infinite linear combination in the likes of the formal infinite linear combinations one has in the context of generalized power series, I call…
We augment the method of Wooley (2015) by some new ideas and in a series of results, improve his metric bounds on the Weyl sums and the discrepancy of fractional parts of real polynomials with partially prescribed coefficients. We also…
We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the…
This work advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite…
Variational inference (VI) is widely used for approximate inference in Bayesian machine learning. In addition to this practical success, generalization bounds for variational inference and related algorithms have been developed, mostly…
We introduce a technique to compute probably approximately correct (PAC) bounds on precision and recall for matching algorithms. The bounds require some verified matches, but those matches may be used to develop the algorithms. The bounds…