Related papers: Tensor Completion by Multi-Rank via Unitary Transf…
Tensor decomposition is an effective approach to compress over-parameterized neural networks and to enable their deployment on resource-constrained hardware platforms. However, directly applying tensor compression in the training process is…
We provide a nontrivial bound on the rank of any tensor $T$ over the quaternions $\mathbb{H}$ in the $n_1\times n_2\times n_3$ cases where $2\leq n_i\leq 3$. We describe a decomposition of $T$ into $3$ simple tensors in the $2\times 2\times…
We show that a generic tensor $T\in \mathbb{F}^{n\times n\times \dots\times n}$ of order $k$ and CP rank $d$ can be uniquely recovered from $n\log n+dn\log \log n +o(n\log \log n) $ uniformly random entries with high probability if $d$ and…
The block tensor of trifocal tensors provides crucial geometric information on the three-view geometry of a scene. The underlying synchronization problem seeks to recover camera poses (locations and orientations up to a global…
Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying…
Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation-originally developed for…
In this paper, a new definition of tensor p-shrinkage nuclear norm (p-TNN) is proposed based on tensor singular value decomposition (t-SVD). In particular, it can be proved that p-TNN is a better approximation of the tensor average rank…
This paper investigates the low-rank tensor completion problem, which is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix…
The recent low-rank prior based models solve the tensor completion problem efficiently. However, these models fail to exploit the local patterns of tensors, which compromises the performance of tensor completion. In this paper, we propose a…
We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative…
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.…
We show that finding rank-$R$ decompositions of a 3D tensor, for $R\le 4$, over a fixed finite field can be done in polynomial time. However, if some cells in the tensor are allowed to have arbitrary values, then rank-2 is NP-hard over the…
In a random unitary matrix model at large N, we study the properties of the expectation value of the character of the unitary matrix in the rank k symmetric tensor representation. We address the problem of whether the standard semiclassical…
This paper considers the problem of matrix completion when some number of the columns are completely and arbitrarily corrupted, potentially by a malicious adversary. It is well-known that standard algorithms for matrix completion can return…
Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high-dimensional data, achieving linear scaling with the input dimension…
We develop the first fast spectral algorithm to decompose a random third-order tensor over $\mathbb{R}^d$ of rank up to $O(d^{3/2}/\text{polylog}(d))$. Our algorithm only involves simple linear algebra operations and can recover all…
In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature…
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…
Understanding the nature of multipartite entanglement is a central mission of quantum information theory. To this end, we investigate the question of tripartite entanglement convertibility. We find that there exists no easy criterion to…
Uncertainty quantification based on stochastic spectral methods suffers from the curse of dimensionality. This issue was mitigated recently by low-rank tensor methods. However, there exist two fundamental challenges in low-rank tensor-based…