Related papers: On the Minimum of a Positive Definite Quadratic Fo…
We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights…
Motivated by quantum thermodynamics we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action…
We study the problem of estimating an unknown deterministic signal that is observed through an unknown deterministic data matrix under additive noise. In particular, we present a minimax optimization framework to the least squares problems,…
The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed…
We consider the problem of computing sample points in each connected component of a semi-algebraic set defined by the non-vanishing or the positivity of an n-variate polynomial of degree d, with rational coefficients of bit size bounded by…
$D$-optimal designs originate in statistics literature as an approach for optimal experimental designs. In numerical analysis points and weights resulting from maximal determinants turned out to be useful for quadrature and interpolation.…
Consider the random matrix $\Sigma = D^{1/2} X \widetilde D^{1/2}$ where $D$ and $\widetilde D$ are deterministic Hermitian nonnegative matrices with respective dimensions $N \times N$ and $n \times n$, and where $X$ is a random matrix with…
Positive semidefinite rank (PSD-rank) is a relatively new quantity with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for…
Networked systems usually face different random uncertainties that make the performance of the least-squares (LS) linear filter decline significantly. For this reason, great attention has been paid to the search for other kinds of…
We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially…
Recently, Pagh presented a randomized approximation algorithm for the multiplication of real-valued matrices building upon work for detecting the most frequent items in data streams. We continue this line of research and present new {\em…
In optimization problems involving smooth functions and real and matrix variables, that contain matrix semidefiniteness constraints, consider the following change of variables: Replace the positive semidefinite matrix $X \in \mathbb{S}^d$,…
We study the problem of testing whether a symmetric $d \times d$ input matrix $A$ is symmetric positive semidefinite (PSD), or is $\epsilon$-far from the PSD cone, meaning that $\lambda_{\min}(A) \leq - \epsilon \|A\|_p$, where $\|A\|_p$ is…
In linear regression we wish to estimate the optimum linear least squares predictor for a distribution over $d$-dimensional input points and real-valued responses, based on a small sample. Under standard random design analysis, where the…
We study the harmonic measure (i.e. the limit of the hitting distribution of a simple random walk starting from a distant point) on three canonical two-dimensional lattices: the square lattice $\mathbb{Z}^2$, the triangular lattice…
This work studies applications and generalizations of a simple estimation technique that provides exponential concentration under heavy-tailed distributions, assuming only bounded low-order moments. We show that the technique can be used…
Symmetric positive definite~(SPD) matrices have shown important value and applications in statistics and machine learning, such as FMRI analysis and traffic prediction. Previous works on SPD matrices mostly focus on discriminative models,…
We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^d$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that…
We analyze a new random algorithm for numerical integration of $d$-variate functions over $[0,1]^d$ from a weighted Sobolev space with dominating mixed smoothness $\alpha\ge 0$ and product weights $1\ge\gamma_1\ge\gamma_2\ge\cdots>0$, where…
In this paper, we further develop the approach, originating in [14 (arXiv:1311.6765),20 (arXiv:1604.02576)], to "computation-friendly" hypothesis testing and statistical estimation via Convex Programming. Specifically, we focus on…