Related papers: The geometry of rank-one tensor completion
We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we…
Differential geometries derived from tensor decompositions have been extensively studied and provided the foundations for a variety of efficient numerical methods. Despite the practical success of the tensor ring (TR) decomposition, its…
A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of…
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…
A formula for the partial trace of a full symmetrizer is obtained. The formula is used to provide an inductive proof of the well-known formula for the dimension of a full symmetry class of tensors.
In this paper, one of our main purposes is to prove the boundedness of solution set of tensor complementarity problem with B tensor such that the specific bounds only depend on the structural properties of tensor. To achieve this purpose,…
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…
We prove a lower bound on the rank of tensors constructed from families of linear maps that `expand' the dimension of every subspace. Such families, called {\em dimension expanders} have been studied for many years with several known…
In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of…
The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we mainly study low rank third-order tensor completion problems by using Riemannian optimization methods on…
This paper studies the problem of finding best rank-1 approximations for both symmetric and nonsymmetric tensors. For symmetric tensors, this is equivalent to optimizing homogeneous polynomials over unit spheres; for nonsymmetric tensors,…
Due to the multi-linearity of tensors, most algorithms for tensor optimization problems are designed based on the block coordinate descent method. Such algorithms are widely employed by practitioners for their implementability and…
In the realm of tensor optimization, the low-rank Tucker decomposition is crucial for reducing the number of parameters and for saving storage. We explore the geometry of Tucker tensor varieties -- the set of tensors with bounded Tucker…
Recently, fundamental conditions on the sampling patterns have been obtained for finite completability of low-rank matrices or tensors given the corresponding ranks. In this paper, we consider the scenario where the rank is not given and we…
We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of $G$-invariant tensors. Here $G$ is the sub- group of the real orthogonal…
In this work, a tensor completion problem is studied, which aims to perfectly recover the tensor from partial observations. The existing theoretical guarantee requires the involved transform to be orthogonal, which hinders its applications.…
In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be…
The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able…
A tensor is a multi-way array that can represent, in addition to a data set, the expression of a joint law or a multivariate function. As such it contains the description of the interactions between the variables corresponding to each of…
Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications…