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Mixture models are traditionally represented and learned by adding several distributions as components. Allowing mixtures to subtract probability mass or density can drastically reduce the number of components needed to model complex…
In this work, we present an algorithmic treatment of the representation theory of the algebra of partially transposed permutation operators, denoted by $\mathcal{A}^d_{p,p}$, which is a matrix representation of the abstract walled Brauer…
We propose a new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels. The algorithms are based on projecting the code on its cosets, recursively decoding the projected codes (which are…
We develop the first stochastic incremental method for calculating the Moore-Penrose pseudoinverse of a real matrix. By leveraging three alternative characterizations of pseudoinverse matrices, we design three methods for calculating the…
This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian…
Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The corresponding equivariant map algebra is the Lie algebra M of equivariant regular maps from X to g. We classify the irreducible finite-dimensional…
The reconstruction of X-rays CT images from sparse or limited-angle geometries is a highly challenging task. The lack of data typically results in artifacts in the reconstructed image and may even lead to object distortions. For this…
We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural…
We introduce a new constructive recognition algorithm for finite special linear groups in their natural representation. Given a group $G$ generated by a set of $d\times d$ matrices over a finite field $\mathbb{F}_q$, known to be isomorphic…
Logarithms of determinants of large positive definite matrices appear ubiquitously in machine learning applications including Gaussian graphical and Gaussian process models, partition functions of discrete graphical models, minimum-volume…
A fully tensorial theoretical framework for hypercomplex-valued neural networks is presented. The proposed approach enables neural network architectures to operate on data defined over arbitrary finite-dimensional algebras. The central…
Many applications using large datasets require efficient methods for minimizing a proximable convex function subject to satisfying a set of linear constraints within a specified tolerance. For this task, we present a proximal projection…
In geometric representation theory, one often wishes to describe representations realized on spaces of invariant functions as trace functions of equivariant perverse sheaves. In the case of principal series representations of a connected…
We give an algorithm for computing Segre classes of subschemes of arbitrary projective varieties by computing degrees of a sequence of linear projections. Based on the fact that Segre classes of projective varieties commute with…
This paper generalizes the Maurer--Pontil framework of finite-dimensional lossy coding schemes to the setting where a high-dimensional random vector is mapped to an element of a compact set of latent representations in a lower-dimensional…
We introduce a new compile-time notion of type subsumption based on type simulation. We show how to apply this static subsumption relation to support a more intuitive, object oriented approach to generic programming of reusable, high…
Given an arbitrary matrix $A\in\mathbb{R}^{n\times n}$, we consider the fundamental problem of computing $Ax$ for any $x\in\mathbb{R}^n$ such that $Ax$ is $s$-sparse. While fast algorithms exist for particular choices of $A$, such as the…
The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to…
A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some…
Random projections or sketching are widely used in many algorithmic and learning contexts. Here we study the performance of iterative Hessian sketch for least-squares problems. By leveraging and extending recent results from random matrix…