Related papers: Fixed-parameter debordering of Waring rank
Convolutional neural networks (CNNs) handle the case where filters extend beyond the image boundary using several heuristics, such as zero, repeat or mean padding. These schemes are applied in an ad-hoc fashion and, being weakly related to…
We discuss efficient methods for unranking derangements and m\'enage permutations. That is, we will provide an algorithm to efficiently extract the $k$-th earliest such permutation under the lexicographic ordering. We will show that this…
A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…
In this paper we introduce the open Waring rank of a form of degree d in n variables and prove the that this rank in bounded from above by \binom{n+d-2}{d-1} - \binom{n+d-6}{d-3} whenever n, d >= 3. This proves the same upper bound for the…
Thresholding--the pruning of nodes or edges based on their properties or weights--is an essential preprocessing tool for extracting interpretable structure from complex network data, yet existing methods face several key limitations.…
We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be…
The rigidity of a matrix measures how many of its entries need to be changed in order to reduce its rank to some value. Good lower bounds on the rigidity of an explicit matrix would imply good lower bounds for arithmetic circuits as well as…
Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the…
In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most $1$. His paper concludes with the suggestion that…
The weight decay regularization term is widely used during training to constrain expressivity, avoid overfitting, and improve generalization. Historically, this concept was borrowed from the SVM maximum margin principle and extended to…
We develop new techniques for proving lower bounds on the least singular value of random matrices with limited randomness. The matrices we consider have entries that are given by polynomials of a few underlying base random variables. This…
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…
We thoroughly study a novel but basic combinatorial matrix completion problem: Given a binary incomplete matrix, fill in the missing entries so that every pair of rows in the resulting matrix has a Hamming distance within a specified range.…
VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form $A_0 + \sum_{i=1}^n A_ix_i$ where the size of each $A_i$ is polynomial in the number of variables (equivalently, computable by…
Constrained low-rank matrix approximations have been known for decades as powerful linear dimensionality reduction techniques to be able to extract the information contained in large data sets in a relevant way. However, such low-rank…
We study the matrix completion problem when the observation pattern is deterministic and possibly non-uniform. We propose a simple and efficient debiased projection scheme for recovery from noisy observations and analyze the error under a…
Rank-constrained matrix problems appear frequently across science and engineering. The convergence analysis of iterative algorithms developed for these problems often hinges on local error bounds, which correlate the distance to the…
Branes and defects in topological Landau-Ginzburg models are described by matrix factorisations. We revisit the problem of deforming them and discuss various deformation methods as well as their relations. We have implemented these…
The low-complexity assumption in linear systems can often be expressed as rank deficiency in data matrices with generalized Hankel structure. This makes it possible to denoise the data by estimating the underlying structured low-rank…
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…