Related papers: On the Geometry of Maximum Entropy Problems
A well-known result across information theory, machine learning, and statistical physics shows that the maximum entropy distribution under a mean constraint has an exponential form called the Gibbs-Boltzmann distribution. This is used for…
We give new results for problems in computational and statistical machine learning using tools from high-dimensional geometry and probability. We break up our treatment into two parts. In Part I, we focus on computational considerations in…
In most data-scientific approaches, the principle of Maximum Entropy (MaxEnt) is used to a posteriori justify some parametric model which has been already chosen based on experience, prior knowledge or computational simplicity. In a…
Consider a rectangular matrix describing some type of communication or transportation between a set of origins and a set of destinations, or a classification of objects by two attributes. The problem is to infer the entries of the matrix…
Within the task of collaborative filtering two challenges for computing conditional probabilities exist. First, the amount of training data available is typically sparse with respect to the size of the domain. Thus, support for higher-order…
The need to estimate smooth probability distributions (a.k.a. probability densities) from finite sampled data is ubiquitous in science. Many approaches to this problem have been described, but none is yet regarded as providing a definitive…
For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We…
Combining quantum and Bayesian principles leads to optimality in metrology, but the optimisation equations involved are often hard to solve. This work mitigates this problem with a novel class of measurement strategies for quantities…
The topological entanglement entropy is used to measure long-range quantum correlations in the ground state of topological phases. Here we obtain closed form expressions for topological entropy of (2+1)- and (3+1)-dimensional loop gas…
The problem of packing equal circles in a circle is a classic and famous packing problem, which is well-studied in academia and has a variety of applications in industry. This problem is computationally challenging, and researchers mainly…
We consider the problem of parameter estimation from a generalized linear model with a random design matrix that is orthogonally invariant in law. Such a model allows the design have an arbitrary distribution of singular values and only…
In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic…
We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space $(E,\rho)$ up to an accuracy of…
We present a ${\Gamma}$-convergence approximation for the total anisotropic length of a partition. This theoretical result gives rise to a numerical method which allows the study of minimal partitions with respect to different anisotropies.…
We show that a solution of the optimal assignment problem can be obtained as the limit of the solution of an entropy maximization problem, as a deformation parameter tends to infinity. This allows us to apply entropy maximization algorithms…
We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in [Beskos et al., Stochastic Process. Appl., 2011]. This algorithm can be…
We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our…
Information geometry and optimal transport are two distinct geometric frameworks for modeling families of probability measures. During the recent years, there has been a surge of research endeavors that cut across these two areas and…
The time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions is proved in the entropy sense. The algebraic or exponential decay rates are computed…
Many numerical and learning algorithms rely on the solution of the Monge-Kantorovich problem and Wasserstein distances, which provide appropriate distributional metrics. While the natural approach is to treat the problem as an…