Related papers: On tensor rank and commuting matrices
In this paper, we show that the low rank matrix completion problem can be reduced to the problem of finding the rank of a certain tensor.
Computing the reversal distances of signed permutations is an important topic in Bioinformatics. Recently, a new lower bound for the reversal distance was obtained via the plane permutation framework. This lower bound appears different from…
There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the…
The low displacement rank (LDR) framework for structured matrices represents a matrix through two displacement operators and a low-rank residual. Existing use of LDR matrices in deep learning has applied fixed displacement operators…
For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained…
This paper proposes low tensor-train (TT) rank and low multilinear (ML) rank approximations for de-speckling and compression of 3D optical coherence tomography (OCT) images for a given compression ratio (CR). To this end, we derive the…
In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted…
In intelligent transportation systems, traffic data imputation, estimating the missing value from partially observed data is an inevitable and challenging task. Previous studies have not fully considered traffic data's multidimensionality…
K be a field and let m and n be positive integers, where m does not exceed n. We say that a non-zero subspace of m x n matrices over K is a constant rank r subspace if each non-zero element of the subspace has rank r, where r is a positive…
We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale,…
Let $\textrm{Mat}_2(\mathbb{R})$ be the set of $2 \times 2$ matrices with real entries. For any $\varepsilon>0$ and any finitely--supported probability measure $\mu$ on $\textrm{Mat}_2(\mathbb{R})$, we prove that either \[ T(\mu) = \sum_{X,…
Arithmetic complexity is considered simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic…
We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th…
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…
In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor-train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix $A$, generated by the first column of the form…
An optimal rank-1 approximation of state transition tensors was developed as an efficient alternative to state transition tensors for nonlinear uncertainty quantification. While previous directional state transition tensors used the…
The set of all permutations with $n$ symbols is a symmetric group denoted by $S_n$. A transposition tree, $T$, is a spanning tree over its $n$ vertices $V_T=${$1, 2, 3, \ldots n$} where the vertices are the positions of a permutation $\pi$…
We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with…
In the first part of this paper we study a best approximation of a vector in Euclidean space R^n with respect to a closed semi-algebraic set C and a given semi-algebraic norm. Assuming that the given norm and its dual norm are…
As low-rank modeling has achieved great success in tensor recovery, many research efforts devote to defining the tensor rank. Among them, the recent popular tensor tubal rank, defined based on the tensor singular value decomposition…