Related papers: Barriers for Rank Methods in Arithmetic Complexity
Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic…
During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey…
Algebraic Branching Programs(ABPs) are standard models for computing polynomials. Syntactic multilinear ABPs (smABPs) are restrictions of ABPs where every variable is allowed to occur at most once in every path from the start to the…
This paper is concerned with the development and analysis of an iterative solver for high-dimensional second-order elliptic problems based on subspace-based low-rank tensor formats. Both the subspaces giving rise to low-rank approximations…
The so-called block-term decomposition (BTD) tensor model, especially in its rank-$(L_r,L_r,1)$ version, has been recently receiving increasing attention due to its enhanced ability of representing systems and signals that are composed of…
Decompositions of higher-order tensors into sums of simple terms are ubiquitous. We show that in order to verify that two tensors are generated by the same (possibly scaled) terms it is not necessary to compute the individual…
We consider the problem of finding the smallest or largest entry of a tensor of order N that is specified via its rank decomposition. Stated in a different way, we are given N sets of R-dimensional vectors and we wish to select one vector…
Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop…
Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are…
We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous…
Feasible interpolation is a general technique for proving proof complexity lower bounds. The monotone version of the technique converts, in its basic variant, lower bounds for monotone Boolean circuits separating two NP-sets to proof…
Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous…
This paper studies the problem of synthesizing (lexicographic) polynomial ranking functions for loops that can be described in polynomial arithmetic over integers and reals. While the analogous ranking function synthesis problem for linear…
Tensor methods have become a promising tool to solve high-dimensional problems in the big data era. By exploiting possible low-rank tensor factorization, many high-dimensional model-based or data-driven problems can be solved to facilitate…
In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for…
We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassen's algorithm for matrix multiplication. We build on a previous framework that establishes communication…
The lower and upper bound of any given algorithm is one of the most crucial pieces of information needed when evaluating the computational effectiveness for said algorithm. Here a novel method of Boolean Algebraic Programming for symbolic…
An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques when the degree of the polynomial is large enough. Applications to the syndrome computation…
We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over $\mathbb{R}$) of functions from some "simple" class ${\cal C}$. In particular, given ${\cal C}$ we are interested in finding…
Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding…