Related papers: Best rank k approximation for binary forms
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also…
We show that if a reproducing kernel Hilbert space $H_K,$ consisting of functions defined on ${\bf E},$ enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number $n,$ and…
We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…
We give a new proof of Fejes T\'oth's zone conjecture: for any sequence $v_1,v_2,...,v_n$ of unit vectors in a real Hilbert space $\mathcal{H}$, there exists a unit vector $v$ in $\mathcal{H}$ such that \begin{equation*} |\langle v_k,v…
We study the $\ell_0$-Low Rank Approximation Problem, where the goal is, given an $m \times n$ matrix $A$, to output a rank-$k$ matrix $A'$ for which $\|A'-A\|_0$ is minimized. Here, for a matrix $B$, $\|B\|_0$ denotes the number of its…
Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames,…
A basic problem of approximation theory, the approximation of functions from the Sobolev space W_p^r([0,1]^d) in the norm of L_q([0,1]^d), is considered from the point of view of quantum computation. We determine the quantum query…
In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained…
Suppose $k\nmid n$ and $H$ is an $n$-vertex $k$-uniform hypergraph. A near perfect matching in $H$ is a matching of size $\lfloor n/k\rfloor$. We give a divisibility barrier construction that prevents the existence of near perfect matchings…
Many binary classification problems minimize misclassification above (or below) a threshold. We show that instances of ranking problems, accuracy at the top or hypothesis testing may be written in this form. We propose a general framework…
In the numerical linear algebra community, it was suggested that to obtain nearly optimal bounds for various problems such as rank computation, finding a maximal linearly independent subset of columns (a basis), regression, or low-rank…
We study approximation of embeddings between finite dimensional L_p spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The…
The classical low rank approximation problem is to find a rank $k$ matrix $UV$ (where $U$ has $k$ columns and $V$ has $k$ rows) that minimizes the Frobenius norm of $A - UV$. Although this problem can be solved efficiently, we study an…
The Bin Packing Problem is one of the most important Combinatorial Optimization problems in optimization and has a lot of real-world applications. Many approximation algorithms have been presented for this problem because of its NP-hard…
A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the…
The weighted low-rank approximation problem is a fundamental numerical linear algebra problem and has many applications in machine learning. Given a $n \times n$ weight matrix $W$ and a $n \times n$ matrix $A$, the goal is to find two…
The k-means problem consists of finding k centers in the d-dimensional Euclidean space that minimize the sum of the squared distances of all points in an input set P to their closest respective center. Awasthi et. al. recently showed that…
We propose a method for low-rank semidefinite programming in application to the semidefinite relaxation of unconstrained binary quadratic problems. The method improves an existing solution of the semidefinite programming relaxation to…
Binary embedding is the problem of mapping points from a high-dimensional space to a Hamming cube in lower dimension while preserving pairwise distances. An efficient way to accomplish this is to make use of fast embedding techniques…
We give sufficient conditions on a symmetric tensor S in S^dF^n to satisfy the equality: the symmetric rank of S, denoted as srank(S), is equal to the rank of S, denoted as rank(S). This is done by considering the rank of the unfolded S…