Multilinear transformation on matrices

Notes on hadamard products of matrices - Linear and Multilinear Algebra

Musings on Multilinear Fitting

We show that the problems of approximating tensors and multivariate functions as a sums of (tensor) products of vectors/functions can be considered in a unified framework, thus exposing their common multilinear structure. We study the alternating least squares algorithm within this framework from the orthogonal projection and gradient perspectives. We then use these perspectives to study its co...

On the Concentration of Random Multilinear Forms and the Universality of Random Block Matrices

The circular law asserts that if Xn is a n×n matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of 1 √ n Xn converges almost surely to the uniform distribution on the unit disk as n tends to infinity. Answering a question of Tao, we prove the circular law for a general class of random block matrices with dependent entries. The proof relies o...

Multilinear-NC1 6= Multilinear-NC2

An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit example for a polynomial f(x1, ..., xn), with coefficients in {0, 1}, such that over any field: 1. f can be computed by a polynomial-size multilinear circuit of depth O(log n). 2. Any multilinear formula for f is of size nΩ(logn). This gives a super-polynomial gap ...

Multilinear Analysis on Metric Spaces

The multilinear Calderón–Zygmund theory is developed in the setting of RD-spaces, namely, spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel–Lizorkin spaces in the full range of exponents are among the main...