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The entropy computation of Gaussian mixture distributions with a large number of components has a prohibitive computational complexity. In this paper, we propose a novel approach exploiting the sphere decoding concept to bound and…
This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Fibonacci point set in the unit square. It was observed recently that this new characteristic allows us to obtain optimal rate…
We consider random instances of non-convex perceptron problems in the high-dimensional limit of a large number of examples $M$ and weights $N$, with finite load $\alpha = M/N$. We develop a formalism based on replica theory to predict the…
The averaging problem in cosmology is of considerable importance for the correct interpretation of cosmological data. A rigorous mathematical definition of averaging in a cosmological model is necessary. In general, a spacetime is…
The spherometer used for measuring radius of curvature of spherical surfaces is explicitly based on a geometric relation unique to circles and spheres. We present an alternate approach using coordinate geometry, which reproduces the…
Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a…
In this paper we prove an analogue in the discrete setting of \Bbb Z^d, of the spherical maximal theorem for \Bbb R^d. The methods used are two-fold: the application of certain "sampling" techniques, and ideas arising in the study of the…
Uncertainty quantification is essential in decision-making, especially when joint distributions of random variables are involved. While conformal prediction provides distribution-free prediction sets with valid coverage guarantees, it…
It is a common contention that it is an ``impossible mission'' to exactly determine the minimum sample size for the estimation of a binomial parameter with prescribed margin of error and confidence level. In this paper, we investigate such…
Fourier analysis and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry etc. in dealing with 2-dimensional…
This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of…
We investigate the averaging problem in cosmology as the problem of introducing a distance between spaces. We first introduce the spectral distance, which is a measure of closeness between spaces defined in terms of the spectra of the…
Calibration, the practice of choosing the parameters of a structural model to match certain empirical moments, can be viewed as minimum distance estimation. Existing standard error formulas for such estimators require a consistent estimate…
Sample coordination, where similar instances have similar samples, was proposed by statisticians four decades ago as a way to maximize overlap in repeated surveys. Coordinated sampling had been since used for summarizing massive data sets.…
The measurement of dispersion is one of the most fundamental and ubiquitous statistical concepts, in both applied and theoretical contexts. For dispersion measures, such as the standard deviation, to effectively capture the variability of a…
For the accurate representation and reconstruction of band-limited signals on the sphere, an optimal-dimensionality sampling scheme has been recently proposed which requires the optimal number of samples equal to the number of degrees of…
This paper introduces a kernel discrepancy-based framework for rerandomization to enhance the precision of causal inference in controlled experiments. We demonstrate that the kernel discrepancy is the key part of the variance upper bound…
The key issue in importance sampling is the choice of the alternative sampling distribution, which is often chosen from the exponential tilt family of the underlying distribution. However, when the problem exhibits certain kind of…
This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We…
Geometric discrepancies are standard measures to quantify the irregularity of distributions. They are an important notion in numerical integration. One of the most important discrepancy notions is the so-called \emph{star discrepancy}.…