Related papers: A note on numerical ranges of tensors
We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the…
Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this…
The results of Strassen and Raz show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we…
Higher-order tensor datasets arise commonly in recommendation systems, neuroimaging, and social networks. Here we develop probable methods for estimating a possibly high rank signal tensor from noisy observations. We consider a generative…
We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a…
Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex…
This paper studies a high-dimensional inference problem involving the matrix tensor product of random matrices. This problem generalizes a number of contemporary data science problems including the spiked matrix models used in sparse…
We obtain a far-reaching generalization (in several directions) of the theorem of A. Lambert on the existence of the projective tensor product of operator sequence spaces. This result is obtained in the context of spaces, generalizing…
The analytic rank of a tensor, first defined by Gowers and Wolf in the context of higher-order Fourier analysis, is defined to be the logarithm of the bias of the tensor. We prove that it is a subadditive measure of rank: that is, the…
This is the second part in a series of papers presenting a theory of tensor products for module categories for a vertex operator algebra. In Part I (hep-th/9309076), the notions of $P(z)$- and $Q(z)$-tensor product of two modules for a…
We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on $k$ vertices. For $k\geq4$, we show that the exponent per edge is at most 0.77, outperforming…
Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and…
We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a…
We consider a generalization of the Bauer maximum principle. We work with tensorial products of convex measures sets, that are non necessarily compact but generated by their extreme points. We show that the maximum of a quasi-convex lower…
The asymptotic restriction problem for tensors is to decide, given tensors $s$ and $t$, whether the nth tensor power of $s$ can be obtained from the $(n+o(n))$th tensor power of t by applying linear maps to the tensor legs (this we call…
This educational article highlights the geometric and algebraic complexities that distinguish tensors from matrices, to supplement coverage in advanced courses on linear algebra, matrix analysis, and tensor decompositions. Using the case of…
We study a family of inequalities on pairs of measure spaces involving functions defined on product domains. Our main result establishes a Jensen-type inequality under a general product-measure framework, extending classical inequalities…
The weighted numerical radius of a Hilbert space operator has been defined recently. This article explores other properties and uses this newly defined numerical radius to obtain several new interesting inequalities for the weighted…
Linear methods are ubiquitous for control and estimation problems. In this work, we present a number of tensor operator norms as a means to approximately bound the error associated with linear methods and determine the situations in which…
In this paper we derive an explicit formula for the numerical range of (non-self-adjoint) tridiagonal random operators. As a corollary we obtain that the numerical range of such an operator is always the convex hull of its spectrum, this…