Related papers: Computing linear sections of varieties: quantum en…
We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as…
Learning intersections of halfspaces is a central problem in Computational Learning Theory. Even for just two halfspaces, it remains a major open question whether learning is possible in polynomial time with respect to the margin $\gamma$…
The groundbreaking performance of deep neural networks (NNs) promoted a surge of interest in providing a mathematical basis to deep learning theory. Low-rank tensor decompositions are specially befitting for this task due to their close…
The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in…
Let V be a smooth equidimensional quasi-affine variety of dimension r over the complex numbers $C$ and let $F$ be a $(p\times s)$-matrix of coordinate functions of $C[V]$, where $s\ge p+r$. The pair $(V,F)$ determines a vector bundle $E$ of…
We prove a generalization to Jennrich's uniqueness theorem for tensor decompositions in the undercomplete setting. Our uniqueness theorem is based on an alternative definition of the standard tensor decomposition, which we call…
Max-cut, clustering, and many other partitioning problems that are of significant importance to machine learning and other scientific fields are NP-hard, a reality that has motivated researchers to develop a wealth of approximation…
Current quantum computers can only solve optimization problems of a very limited size. For larger problems, decomposition methods are required in which the original problem is broken down into several smaller sub-problems. These are then…
Many problems in information theory can be reduced to optimizations over matrices, where the rank of the matrices is constrained. We establish a link between rank-constrained optimization and the theory of quantum entanglement. More…
Variational quantum algorithms have emerged as a powerful tool for harnessing the potential of near-term quantum devices to address complex challenges across quantum science and technology. Yet, the robust and scalable quantification of…
In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems)…
We consider the problem of factorizing a structured 3-way tensor into its constituent Canonical Polyadic (CP) factors. This decomposition, which can be viewed as a generalization of singular value decomposition (SVD) for tensors, reveals…
We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. We develop…
We give a new approach to the dictionary learning (also known as "sparse coding") problem of recovering an unknown $n\times m$ matrix $A$ (for $m \geq n$) from examples of the form \[ y = Ax + e, \] where $x$ is a random vector in $\mathbb…
For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained…
We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete…
We study here several variants of the covariates fine balance problem where we generalize some of these problems and introduce a number of others. We present here a comprehensive complexity study of the covariates problems providing…
Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. In the abstract form of the problem, where no noise or other corruptions are present, the data are assumed to lie in general position…
Different hybrid quantum-classical algorithms have recently been developed as a near-term way to solve linear systems of equations on quantum devices. However, the focus has so far been mostly on the methods, rather than the problems that…
A problem that is frequently encountered in a variety of mathematical contexts, is to find the common invariant subspaces of a single, or set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea…