Related papers: Random tensor isomorphism under orthogonal and uni…
We consider the problem of decomposing higher-order moment tensors, i.e., the sum of symmetric outer products of data vectors. Such a decomposition can be used to estimate the means in a Gaussian mixture model and for other applications in…
The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of…
A key problem in approximation theory is the recovery of high-dimensional functions from samples. In many cases, the functions of interest exhibit anisotropic smoothness, and, in many practical settings, the nature of this anisotropy may be…
We construct a matrix $M\in R^{m\otimes d^c}$ with just $m=O(c\,\lambda\,\varepsilon^{-2}\text{poly}\log1/\varepsilon\delta)$ rows, which preserves the norm $\|Mx\|_2=(1\pm\varepsilon)\|x\|_2$ of all $x$ in any given $\lambda$ dimensional…
We present an $O^*(|\mathbb{F}|^{\min\left\{R,\ \sum_{d\ge 2} n_d\right\} + (R-n_0)(\sum_{d\ne 0} n_d)})$-time algorithm for determining whether the rank of a concise tensor $T\in\mathbb{F}^{n_0\times\dots\times n_{D-1}}$ is $\le R$,…
This paper tackles the problem of recovering a low-rank signal tensor with possibly correlated components from a random noisy tensor, or so-called spiked tensor model. When the underlying components are orthogonal, they can be recovered…
Random operators constitute fundamental building blocks of models of complex systems yet are far from fully understood. Here, we explain an asymmetry emerging upon repeating identical isotropic (uniformly random) operations. Specifically,…
Random tensors are the natural generalization of random matrices to higher order objects. They provide generating functions for random geometries and, assuming some familiarity with random matrix theory and quantum field theory, we discuss…
We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as $O(n^{\omega+\epsilon})$ time matrix multiplication, and in…
Tensor networks have been an important concept and technique in many research areas, such as quantum computation and machine learning. We study the exponential complexity of contracting tensor networks on two special graph structures:…
This paper presents an algorithmic method for generating random orthogonal matrices \(A\) that satisfy the property \(A^t S A = S\), where \(S\) is a fixed real invertible symmetric or skew-symmetric matrix. This method is significant as it…
We study tensor completion in the agnostic setting. In the classical tensor completion problem, we receive $n$ entries of an unknown rank-$r$ tensor and wish to exactly complete the remaining entries. In agnostic tensor completion, we make…
There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this…
The symmetry-constrained response tensors on transport, optical, and electromagnetic effects are of central importance in condensed matter physics because they can guide experimental detections and verify theoretical calculations. These…
Since the work of Ornstein and Weiss in 1987 (J. Analyse Math. 48 (1987)) it has been understood that the natural category for classical ergodic theory would be probability measure preserving actions of discrete amenable groups. A…
We study planar random motions with finite velocities, of norm $c>0$, along orthogonal directions and changing at the instants of occurrence of a non-homogeneous Poisson process with rate function $\lambda(t),\ t\ge0$. We focus on the…
Many idealized problems in signal processing, machine learning and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing…
We study the symmetric outer product decomposition which decomposes a fully (partially) symmetric tensor into a sum of rank-one fully (partially) symmetric tensors. We present iterative algorithms for the third-order partially symmetric…
A tensor network is a diagram that specifies a way to "multiply" a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although…
Many models for chaotic systems consist of joining two integrable systems with incompatible constants of motion. The quantum counterparts of such models have a propagator which factorizes into two integrable parts. Each part can be…