Related papers: Relative $(p,\epsilon)$-Approximations in Geometry
Inducing-point-based sparse variational Gaussian processes have become the standard workhorse for scaling up GP models. Recent advances show that these methods can be improved by introducing a diagonal scaling matrix to the conditional…
In the work by M. C. Lee, A. Naber, and R. Neumayer a beautiful $\varepsilon$-regularity theorem is proved under small negative scalar curvature and entropy bounds. In that paper, the $d_p$ distance for Riemannian manifolds is introduced…
In this paper we introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula $\Phi$ when the…
It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$…
Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability…
A locally-optimal structure is a combinatorial structure such as a maximal independent set that cannot be improved by certain (greedy) local moves, even though it may not be globally optimal. It is trivial to construct an independent set in…
We study the problem of supervised learning a metric space under discriminative constraints. Given a universe $X$ and sets ${\cal S}, {\cal D}\subset {X \choose 2}$ of similar and dissimilar pairs, we seek to find a mapping $f:X\to Y$, into…
We survey results concerning sharp estimates on volumes of sections and projections of certain convex bodies, mainly $\ell_p$ balls, by and onto lower dimensional subspaces. This subject emerged from geometry of numbers several decades ago…
Rational approximations are introduced and studied in granular graded rough sets and generalizations thereof by the first author in recent research papers. The concept of rationality is determined by related ontologies and coherence between…
Approximate Message Passing (AMP), originally designed to solve high-dimensional linear inverse problems, has found broad applications in signal processing and statistical inference. Among its key variants, Vector Approximate Message…
Consider a geometric range space $(X,\c{A})$ where each data point $x \in X$ has two or more values (say $r(x)$ and $b(x)$). Also consider a function $\Phi(A)$ defined on any subset $A \in (X,\c{A})$ on the sum of values in that range e.g.,…
We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient…
Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm…
We introduce a new class of (not necessarily convex) bodies and show, among other things, that these bodies provide yet another link between convex geometric analysis and information theory. Namely, they give geometric interpretations of…
In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We…
This work proposes a view of probability as a relative measure rather than an absolute one. To demonstrate this concept, we focus on finite outcome spaces and develop three fundamental axioms that establish requirements for relative…
We consider the problem of deriving upper bounds on the parameters of sum-rank-metric codes, with focus on their dimension and block length. The sum-rank metric is a combination of the Hamming and the rank metric, and most of the available…
The nearest neighbor problem is defined as follows: Given a set $P$ of $n$ points in some metric space $(X,D)$, build a data structure that, given any point $q$, returns a point in $P$ that is closest to $q$ (its "nearest neighbor" in $P$).…
This paper introduces some inverse sequences of different polyhedra all based on finite approximations of a compact metric space so they can be used to capture the shape type of the original space. It is shown that they are HPol-expansions,…
We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global…