Related papers: Exponential families and affine Grassmannians

Gaussian graphical models have become a well-recognized tool for the analysis of conditional independencies within a set of continuous random variables. From an inferential point of view, it is important to realize that they are composite…

An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional…

We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition…

We resolve a conjecture about a class of binomial initial ideals of $I_{2,n}$, the ideal of the Grassmannian, Gr$(2,\mathbb{C}^n$), which are associated to phylogenetic trees. For a weight vector $\omega$ in the tropical Grassmannian,…

Given a linear group G over a field k, we define a notion of index and residue of an element g of G(k((t)). This provides an alternative proof of Gabber's theorem stating that G has no subgroups isomorphic to the additive or the commutative…

We study the maximum Hamming distance (or rather, the complementary notion of "minimum approximability") of a general function on a finite group $G$ to either of the sets $\operatorname{End}(G)$ and $\operatorname{Aff}(G)$, of group…

We consider a possible framework to categorify the exponential map exp(-f) given the categorification of a generator f of $\frak{sl}_2$ by Lauda. In this setup the Taylor expansions of exp(-f) and exp(f) turn into complexes built out of…

We describe convolutional deep exponential families (CDEFs) in this paper. CDEFs are built based on deep exponential families, deep probabilistic models that capture the hierarchical dependence between latent variables. CDEFs greatly reduce…

We prove that if two normed-algebra-valued cosine families indexed by a single Abelian group, of which one is bounded and comprised solely of scalar elements of the underlying algebra, differ in norm by less than $1$ uniformly in the…

This paper discusses the family of distributions on the Grassmannian of the linear span of r central gaussian vectors parametrized by the covariance matrix. Our main result is an existence and uniqueness criterion for the maximum likelihood…

We introduce random matrix ensembles that correspond to the infinite families of irreducible Riemannian symmetric spaces of type I. In particular, we recover the Circular Orthogonal and Symplectic Ensembles of Dyson, and find other families…

We generalize the concept of Sato Grassmannians of locally linearly compact topological vector spaces (Tate spaces) to the category limA of the "locally compact objects" of an exact category A, and study some of their properties. This…

Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on…

Guo and Xu determined the maximum size of intersecting families over finite affine spaces and showed that any family reaches maximum size must be trivial. In this paper, we characterize non-trivial intersecting family with maximum size.

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with…

In this paper we work in o-minimal structures with definable Skolem functions and show that a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is proper morphism in…

We tackle several problems related to a finite irreducible crystallographic root system $\Phi$ in the real vector space $\mathbb E$. In particular, we study the combinatorial structure of the subsets of $\Phi$ cut by affine subspaces of…

Let $\mu$ be a probability measure on $\mathbb{R}^n$ with a bounded density $f$. We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there…

Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(\alpha,\beta)$-parameterized system of orthogonal polynomials of the…

Exponential families form the backbone of modern statistics and machine learning, but textbooks seldom derive them from first principles in an accessible way. Although minimal sufficiency and the principle of maximum entropy, originating in…