Related papers: Barriers for Rank Methods in Arithmetic Complexity
In recent years much effort has been concentrated towards achieving polynomial time lower bounds on algorithms for solving various well-known problems. A useful technique for showing such lower bounds is to prove them conditionally based on…
Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems…
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…
In this paper, we propose new lower and upper bounds on the linear extension complexity of regular $n$-gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size $r$ of a polytope $P$, and…
Novel reinforcement learning algorithms, or improvements on existing ones, are commonly justified by evaluating their performance on benchmark environments and are compared to an ever-changing set of standard algorithms. However, despite…
We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as $O(n^{\omega+\epsilon})$ time matrix multiplication, and in…
We present a general method for converting any family of unsatisfiable CNF formulas that is hard for one of the simplest proof systems, tree resolution, into formulas that require large rank in any proof system that manipulates polynomials…
We show that the border support rank of the tensor corresponding to two-by-two matrix multiplication is seven over the complex numbers. We do this by constructing two polynomials that vanish on all complex tensors with format…
In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be…
The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the…
Engineering and applied sciences use models of increasing complexity to simulate the behaviour of manufactured and physical systems. Propagation of uncertainties from the input to a response quantity of interest through such models may…
Modeling of multidimensional signal using tensor is more convincing than representing it as a collection of matrices. The tensor based approaches can explore the abundant spatial and temporal structures of the mutlidimensional signal. The…
We present a topological barrier to efficient computation, revealed by comparing the geometry of 2 SAT and 3 SAT solution spaces. Viewing the set of satisfying assignments as a cubical complex within the Boolean hypercube, we prove that…
Low Rank Approximation (LRA) of a matrix is a hot research subject, fundamental for Matrix and Tensor Computations and Big Data Mining and Analysis. Computations with low rank matrices can be performed at sublinear cost -- by using much…
We construct a reduction which proves that the fooling set number and the determinantal rank of a Boolean matrix are NP-hard to compute.
We analyze rank decompositions of the $3\times 3$ matrix multiplication tensor over $\mathbb{Z}/2\mathbb{Z}$. We restrict our attention to decompositions of rank $\le 21$, as only those decompositions will yield an asymptotically faster…
In many high-dimensional problems,polynomial-time algorithms fall short of achieving the statistical limits attainable without computational constraints. A powerful approach to probe the limits of polynomial-time algorithms is to study the…
We show that under some widely believed assumptions, there are no higher-order algorithms for basic tasks in computational mathematics such as: Computing integrals with neural network integrands, computing solutions of a Poisson equation…
Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…
A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two…