Related papers: The G-stable rank for tensors

For any finite group G, we show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor,…

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…

Chen and Ye recently proved that the analytic rank of tensors is stable under field extensions, assuming a fixed base field. Using a more careful analysis, we show that this assumption is unnecessary.

We establish a general result about extending a right invertible row over a Banach algebra to an invertible matrix. This is applied to the computation of right topological stable rank of a split exact sequence. We also introduce a…

Grassmann tensors arise from classical problems of scene reconstruction in computer vision. Trifocal Grassmann tensors, related to three projections from a projective space of dimension k onto view-spaces of varying dimensions are studied…

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…

In this paper we study the robustness of strong stability of a discrete semigroup on a Hilbert space under bounded finite rank perturbations. As the main result we characterize classes of perturbations preserving the strong stability of the…

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove…

We introduce a generalized version of the famous Stable Marriage problem, now based on multi-modal preference lists. The central twist herein is to allow each agent to rank its potentially matching counterparts based on more than one…

We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible…

We consider a class of discretionary stopping problems within the $G$-framework. We first establish the well-definedness of the stopping problem under the $G$-expectation, by showing the quasi-continuity of the stopped process. We then…

We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between…

We prove the existence of an open set of $n_1\times n_2 \times n_3$ tensors of rank $r$ on which a popular and efficient class of algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically…

In this paper we examine a symmetric tensor decomposition problem, the Gramian decomposition, posed as a rank minimization problem. We study the relaxation of the problem and consider cases when the relaxed solution is a solution to the…

We give a new numerical criterion in the spirit of GIT for existence of solutions to inverse Hessian equations, including in particular the J-equation. Our criterion is formulated in terms of stability of pairs in the sense of Paul. To that…

We give a short proof for a well-known formula for the rank of a $G$-crossed braided extension of a modular tensor category.

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…

In the present work we suggest a general covariant theory which can be used to study the stability of any physical system treated geometrically. Stability conditions are connected to the magnitude of the deviation vector. This theory is a…

We give a method for verifying, by a symbolic calculation, the stability or semistability with respect to a linearization of fixed, possibly small, degree $m$, of the Hilbert point of a scheme $X \in {\mathbb P}(V)$ having a suitably large…

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric…