Related papers: Estimates for Interpolation Projectors and Related…
We study the problem of transfer learning, observing that previous efforts to understand its information-theoretic limits do not fully exploit the geometric structure of the source and target domains. In contrast, our study first…
This paper considers binary classification of high-dimensional features under a postulated model with a low-dimensional latent Gaussian mixture structure and non-vanishing noise. A generalized least squares estimator is used to estimate the…
This paper gives a general interpretation of Linear Prediction (LP) by interpolation framework different from the perspective of statistics. This interpretation is proved to be useful by several following results, such as: The mechanism of…
This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix…
The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions -- computing the size of projections of high dimensional polytopes…
Along this work we study an indefinite abstract smoothing problem. After establishing necessary and sufficient conditions for the existence of solutions to this problem, the set of admissible parameters is discussed in detail. Then, its…
This paper outlines a methodology for constructing a geometrically smooth interpolatory curve in $\mathbb{R}^d$ applicable to oriented and flattenable points with $d\ge 2$. The construction involves four essential components: local…
The Euclidean Algorithm is the often forgotten key to rational approximation techniques, including Taylor, Lagrange, Hermite, osculating, cubic spline, Chebyshev, Pade and other interpolation schemes. A unified view of these various…
In this paper, we introduce and study the Parallel Polyhedral Projection Method (3PM) and the Approximate Parallel Polyhedral Projection Method (A3PM) for finding a point in the intersection of finitely many closed convex sets. Each…
The main contributions of this paper are the proposition and the convergence analysis of a class of inertial projection-type algorithm for solving variational inequality problems in real Hilbert spaces where the underline operator is…
An important yet challenging problem in numerical linear algebra is finding a principal submatrix with maximum determinant from a given symmetric positive semidefinite matrix. This problem arises in experimental design, statistics, and…
We study schemes for interpolating functions that take values in the special orthogonal group $SO(n)$. Our focus is on interpolation schemes obtained by embedding $SO(n)$ in a linear space, interpolating in the linear space, and mapping the…
We present a nodal interpolation method to approximate a subdivision model. The main application is to model and represent curved geometry without gaps and preserving the required simulation intent. Accordingly, we devise the technique to…
Approximate solutions of linear and nonlinear integral equations using methods related to an interpolatory projection involve many integrals which need to be evaluated using a numerical quadrature formula. In this paper, we consider…
Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be written: \[ \int h_A^\beta d\mu \ge 2 \mu(A)(1-\mu(A)). \] Here $\mu$ is uniform measure on $V=\{0,1\}^n$ ($=V(Q_n)$); $\beta=\log_2(3/2)$; and, for $S\subseteq V$…
We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere $\mathbb{S}^q$ in $\mathbb{R}^{q+1}$, with $q\ge 2$. Like any other polynomial projection, the study concerns the growth, as the…
Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example computer vision and quantum control, there is a growing…
It can be shown that Stable Diffusion has a permutation-invariance property with respect to the rows of Contrastive Language-Image Pretraining (CLIP) embedding matrices. This inspired the novel observation that these embeddings can…
The basic tool for solving problems in metric geometry and isotonic regression is the metric projection onto closed convex cones. Isotonicity of these projections with respect to a given order relation can facilitate finding the solutions…
The classical isoperimetric inequality can be extended to a general normed plane. In the Euclidean plane, the defect in the isoperimetric inequality can be calculated in terms of the signed areas of some singular sets. In this paper we…