arXiv research

Geometrically, tensors of fixed rank form a minimal submanifold.

Minimizing the nuclear norm of a matrix has been shown to be very efficient in reconstructing a low-rank sampled matrix. Furthermore, minimizing the sum of nuclear norms of matricizations of a tensor has been shown to be very efficient in recovering a low-Tucker-rank sampled tensor. In this paper, we propose to recover…

Model learns tensor representations from imperfect multimodal data.

Paper develops inference methods for low-rank tensors without debiasing.

Proposes tensor Q-rank for better tensor rank recovery in complex data.

Deterministic tensor completion using hypergraph expanders with linear sample complexity.

New algorithm improves tensor completion performance.

New tensor completion method reduces impact of outliers.

Optimizes tensor rank selection for neural network compression.

Tensor completion and robust principal component analysis have been widely used in machine learning while the key problem relies on the minimization of a tensor rank that is very challenging. A common way to tackle this difficulty is to approximate the tensor rank with the $\ell_1-$norm of singular values based on its …

New method for tensor completion from specific mode observations.

The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the d…

A new method for decomposing non-negative tensors using energy-based modeling.

New tensor recovery method improves efficiency under strict complementarity.

Introduces TT-NF for more compact neural field representations.

The paper improves tensor completion bounds using spectral gap.

New tensor completion method converges linearly and is highly practical.

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 aim to approximate the unknown rank based on the location of sampled entries an…

Unified algorithm for tensor decomposition supports multiple loss functions and models.

The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.

A new method for filling in missing traffic data improves accuracy over existing techniques.

We study the problem of low-rank tensor factorization in the presence of missing data. We ask the following question: how many sampled entries do we need, to efficiently and exactly reconstruct a tensor with a low-rank orthogonal decomposition? We propose a novel alternating minimization based method which iteratively …

We analyze low rank tensor completion (TC) using noisy measurements of a subset of the tensor. Assuming a rank-$r$, order-$d$, $N \times N \times \cdots \times N$ tensor where $r=O(1)$, the best sampling complexity that was achieved is $O(N^{\frac{d}{2}})$, which is obtained by solving a tensor nuclear-norm minimizatio…

In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model possibilities grows exponentially with the tensor order, which makes it rather ch…

Gradient descent promotes low-rank solutions in tensor completion.

New algorithm completes noisy tensors quickly and accurately.

In this paper we focus on the problem of completion of multidimensional arrays (also referred to as tensors) from limited sampling. Our approach is based on a recently proposed tensor-Singular Value Decomposition (t-SVD) [1]. Using this factorization one can derive notion of tensor rank, referred to as the tensor tubal…

New model fills in missing traffic data efficiently.

A new method for traffic data imputation considering spatiotemporal correlations.

This paper conducts a rigorous analysis for provable estimation of multidimensional arrays, in particular third-order tensors, from a random subset of its corrupted entries. Our study rests heavily on a recently proposed tensor algebraic framework in which we can obtain tensor singular value decomposition (t-SVD) that …

Recovering a low-rank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recov…

Tree tensor networks balance model complexity and empirical risk for high-dimensional function approximation.

New method reduces uncertainty in high-dimensional circuits by automatically determining tensor rank and adaptive sampling.

The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an interesting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011]. It plays a similar role as the matrix nuclear norm which is the convex surrogate of the matrix rank. Considering that the TNN based Tensor Robust PCA [Lu et al., 2…

A new algorithm completes rank-1 tensors with minimal samples and time.

Paper proves sufficient conditions for tensor recovery using t-RIP with random measurements.

A novel regularizer of the PARAFAC decomposition factors capturing the tensor's rank is proposed in this paper, as the key enabler for completion of three-way data arrays with missing entries. Set in a Bayesian framework, the tensor completion method incorporates prior information to enhance its smoothing and predictio…

In this paper, we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion. We introduce a class of tensor norms by allowing for different levels of coherence, which allows us to leverage the incoherence of a tensor. In particular, we show that …

In this paper, we investigate the sample size requirement for exact recovery of a high order tensor of low rank from a subset of its entries. We show that a gradient descent algorithm with initial value obtained from a spectral method can, in particular, reconstruct a ${d\times d\times d}$ tensor of multilinear ranks $…

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To overcome these difficulties, existing approaches often proceed by unfolding tensors i…

We study stability and local minimizing properties of $L^p$- norms of Riemannian curvature tensor denoted by $\mathcal{R}_p$ by variational methods. We compute the Hessian of $\mathcal{R}_p$ at compact rank 1 symmetric spaces and prove that they are stable for $\mathcal{R}_p$ for certain values of p > 2. A similar resu…

DeepTensor uses deep networks to efficiently decompose tensors with improved performance and robustness.

We solve linear equations with tensors of any rank.

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial…

Paper proposes a new tensor imputation method for spatiotemporal traffic data with missing patterns.

E$^2$M optimizes tensor density estimation by relaxing $α$-divergence to KL-divergence.

Develops methods to estimate high rank tensors from noisy data.

Efficiently reduces tensor ranks using mean-field approximation.