Related papers: Free Exponential Families as Kernel Families

Flexible models for probability distributions are an essential ingredient in many machine learning tasks. We develop and investigate a new class of probability distributions, which we call a Squared Neural Family (SNEFY), formed by squaring…

We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of…

We explicitly present expansions of the complex field which are models of the theories of green points in the multiplicative group case and in the case of an elliptic curve without complex multiplication defined over $\mathbb{R}$. In fact,…

This paper introduces a notion of 2-orthogonality for a sequence of polynomials to give extended versions of the Meixner and Feinsilver characterization results based on orthogonal polynomials. These new versions subsume the Letac-Mora…

We present exponential generating function analogues to two classical identities involving the ordinary generating function of the complete homogeneous symmetric functions. After a suitable specialization the new identities reduce to…

The purpose of this paper is to present a syatemic study of some familes of higher-order Euler numbers and polynomials. In particular, by using the basis property of higher-order Euler polynomials for the space of polynomials of degree less…

We extend the relation between random matrices and free probability theory from the level of expectations to the level of all correlation functions (which are classical cumulants of traces of products of the matrices). We introduce the…

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M…

A subfamily ${\cal F}'$ of a set family ${\cal F}$ is said to $q$-{\em represent} ${\cal F}$ if for every $A \in {\cal F}$ and $B$ of size $q$ such that $A \cap B = \emptyset$ there exists a set $A' \in {\cal F}'$ such that $A' \cap B =…

A prefixed polynomial equation is an equation of the form $P(t_1,\ldots,t_n) = 0$, where $P$ is a polynomial whose variables $t_1,\ldots,t_n$ range over the natural numbers, preceded by quantifiers over some, or all, of its variables. Here,…

Simplex-valued data appear throughout statistics and machine learning, for example in the context of transfer learning and compression of deep networks. Existing models for this class of data rely on the Dirichlet distribution or other…

We consider the question of learning the natural parameters of a $k$ parameter minimal exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the…

We construct spaces of quantum increasing sequences, which give quantum families of maps in the sense of Soltan. We then introduce a notion of quantum spreadability for a sequence of noncommutative random variables, by requiring their joint…

We exhibit some new infinite families of rational values of $\tau$, some of them squares of rationals, for which the group or even the semigroup generated by the matrices $\left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right)$…

We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.

The expression of the free energy density of a classical crystalline system as a gradient expansion in terms of a set of order parameters is developed using classical density functional theory. The goal here is to extend and complete an…

Families form the basis of society, and anthropologists have characterised various family systems. This study developed a multi-level evolutionary model of pre-industrial agricultural societies to simulate the evolution of family systems…

We evolve the binary black hole initial data family proposed by Bishop {\em et al.} in the limit in which the black holes are close to each other. We present an exact solution of the linearized initial value problem based on their proposal…

We propose extreme value analogues of natural exponential families and exponential dispersion models, and introduce the slope function as an analogue of the variance function. The set of quadratic and power slope functions characterize…

We define formal exponential maps for any graded manifold as maps from the formal tangent bundle (that we also define) into the graded manifold. We show that each such map uniquely determines and is determined by its associated Grothendieck…