Related papers: Exploring the Meta Flip Graph for Matrix Multiplic…
Duan, Wu and Zhou (FOCS 2023) recently obtained the improved upper bound on the exponent of square matrix multiplication $\omega<2.3719$ by introducing a new approach to quantify and compensate the ``combination loss" in prior analyses of…
For each object in a tensor triangulated category, we construct a natural continuous map from the object's support---a closed subset of the category's triangular spectrum---to the Zariski spectrum of a certain commutative ring of…
A triangulation of a point configuration is regular if it can be given by a height function, that is every point gets lifted to a certain height and projecting the lower convex hull gives the triangulation. Checking regularity of a…
Tensor computations, with matrix multiplication being the primary operation, serve as the fundamental basis for data analysis, physics, machine learning, and deep learning. As the scale and complexity of data continue to grow rapidly, the…
Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured…
Reinforcement learning has gained wide popularity as a technique for simulation-driven approximate dynamic programming. A less known aspect is that the very reasons that make it effective in dynamic programming can also be leveraged for…
We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed…
Graphs emerge in almost every real-world application domain, ranging from online social networks all the way to health data and movie viewership patterns. Typically, such real-world graphs are big and dynamic, in the sense that they evolve…
We propose a transfer matrix algorithm for the enumeration of alternating link diagrams with external legs, giving a weight $n$ to each connected component. Considering more general tetravalent diagrams with self-intersections and…
In a paper published in 1981, Sch\"onhage showed that large total matrix multiplications can be reduced to powers of partial matrix multiplication tensors, which correspond to the bilinear computation task of multiplying matrices with some…
We obtain new uniform upper bounds for the (non necessarily symmetric) tensor rank of the multiplication in the extensions of the finite fields $\F_q$ for any prime or prime power $q\geq2$; moreover these uniform bounds lead to new…
We show that every triangulation (maximal planar graph) on $n\ge 6$ vertices can be flipped into a Hamiltonian triangulation using a sequence of less than $n/2$ combinatorial edge flips. The previously best upper bound uses $4$-connectivity…
We establish new upper bounds about symmetric bilinear complexity in any extension of finite fields. Note that these bounds are not asymptotical but uniform. Moreover we give examples of Shimura curves that do not descend over their field…
Motivated by the search for a deeper understanding of tensor rank, in view of its computational complexity applications, we investigate a possible path to determine the maximum symmetric rank in given degree and dimension. We work in terms…
In this paper we study the rank of planar rigidity matrix of 4-valent graphs, both in case of generic realizations and configurations in general position, under various connectivity assumptions on the graphs. For each case considered, we…
We propose a new class of multiplier bootstraps for count functionals, ranging from a fast, approximate linear bootstrap tailored to sparse, massive graphs to a quadratic bootstrap procedure that offers refined accuracy for smaller, denser…
We introduce a tensor-based model of shared representation for meta-learning from a diverse set of tasks. Prior works on learning linear representations for meta-learning assume that there is a common shared representation across different…
We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
Motivated by the PageRank model for graph partitioning, we develop an extension of PageRank for partitioning uniform hypergraphs. Starting from adjacency tensors of uniform hypergraphs, we establish the multi-linear pseudo-PageRank (MLPPR)…