Related papers: A Generalization of Varnavides's Theorem

In dictionary learning, also known as sparse coding, the algorithm is given samples of the form $y = Ax$ where $x\in \mathbb{R}^m$ is an unknown random sparse vector and $A$ is an unknown dictionary matrix in $\mathbb{R}^{n\times m}$…

Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = \delta_{ij} \xi_{ij}$ for $i \leq j$. $\delta_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq…

Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times…

We describe two algorithms for computing a sparse solution to a least-squares problem where the coefficient matrix can have arbitrary dimensions. We show that the solution vector obtained by our algorithms is close to the solution vector…

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic…

Consider the $n$-dimensional vector $y=X\be+\e$, where $\be \in \R^p$ has only $k$ nonzero entries and $\e \in \R^n$ is a Gaussian noise. This can be viewed as a linear system with sparsity constraints, corrupted by noise. We find a…

A hypergraph G with n vertices and m hyperedges with d endpoints each is (k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a linearly…

A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the arithmetic thickness of $E$, if $E$ is uniformly…

In subset selection we search for the best linear predictor that involves a small subset of variables. From a computational complexity viewpoint, subset selection is NP-hard and few classes are known to be solvable in polynomial time. Using…

A linear inverse problem is proposed that requires the determination of multiple unknown signal vectors. Each unknown vector passes through a different system matrix and the results are added to yield a single observation vector. Given the…

Finding the sparse representation of a signal in an overcomplete dictionary has attracted a lot of attention over the past years. This paper studies ProSparse, a new polynomial complexity algorithm that solves the sparse representation…

Recently, sparsity has become a key concept in various areas of applied mathematics, computer science, and electrical engineering. One application of this novel methodology is the separation of data, which is composed of two (or more)…

We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices…

Let $H=(V,E)$ be an $r$-uniform hypergraph. For each $1 \leq s \leq r-1$, an $s$-path ${\mathcal P}^{r,s}_n$ of length $n$ in $H$ is a sequence of distinct vertices $v_1,v_2,\ldots,v_{s+n(r-s)}$ such that $\{v_{1+i(r-s)},\ldots,…

Checking whether a system of linear equations is consistent is a basic computational problem with ubiquitous applications. When dealing with inconsistent systems, one may seek an assignment that minimizes the number of unsatisfied…

An "oblivious subspace embedding (OSE)" given some parameters eps,d is a distribution D over matrices B in R^{m x n} such that for any linear subspace W in R^n with dim(W) = d it holds that Pr_{B ~ D}(forall x in W ||B x||_2 in (1 +/-…

The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem…

The problem of sparse linear regression is relevant in the context of linear system identification from large datasets. When data are collected from real-world experiments, measurements are always affected by perturbations or low-precision…

Compressed sensing is a relatively new mathematical paradigm that shows a small number of linear measurements are enough to efficiently reconstruct a large dimensional signal under the assumption the signal is sparse. Applications for this…

We show that if a subset A of {1,...,N} does not contain any solutions to the equation x+y+z=3w with the variables not all equal, then A has size at most exp(-c(log N)^{1/7}) N, where c > 0 is some absolute constant. In view of Behrend's…