arXiv research

Local invertibility of higher rank tensor fields on curved manifolds proven.

Develops methods to estimate high rank tensors from noisy data.

TGCCA analyzes higher-order tensors using orthogonal rank-R CP decomposition.

New methods solve tensor-on-tensor regression with unknown rank, revealing benefits of over-parameterization.

Hidden symmetries of the Goryachev-Chaplygin and Kovalevskaya gyrostats spacetimes, as well as the Brdička-Eardley-Nappi-Witten pp-waves are studied. We find out that these spacetimes possess higher rank Stäckel-Killing tensors and that in the case of the pp-wave spacetimes the symmetry group of the Stäckel-Killing ten…

Low rank tensor learning, such as tensor completion and multilinear multitask learning, has received much attention in recent years. In this paper, we propose higher order matching pursuit for low rank tensor learning problems with a convex or a nonconvex cost function, which is a generalization of the matching pursuit…

Four algorithms improve sparse tensor BR1Approx with theoretical guarantees.

Model learns tensor representations from imperfect multimodal data.

Paper optimizes tensor deflation for non-orthogonal signals.

The higher-dimensional Kerr-NUT-de Sitter spacetime describes the general rotating asymptotically de Sitter black hole with NUT parameters. It is known that such a spacetime possesses a rank-2 closed conformal Killing-Yano (CKY) tensor as a ``hidden'' symmetry which provides the separation of variables for the geodesic…

Study uses random matrix theory to improve tensor approximation accuracy.

Efficient tensor decomposition for count data models achieves near-optimal multiway analysis.

HOTCAKE compresses CNNs by decomposing kernels into smaller parts.

New method estimates and completes tensors from ordinal data, improving accuracy and efficiency.

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 …

Robust tensor ring completion improves tensor recovery accuracy and efficiency.

Extends RRR to capture nonlinear interactions in multi-response regression.

Tensors play a central role in many modern machine learning and signal processing applications. In such applications, the target tensor is usually of low rank, i.e., can be expressed as a sum of a small number of rank one tensors. This motivates us to consider the problem of low rank tensor recovery from a class of lin…

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

Robust tensor CP decomposition involves decomposing a tensor into low rank and sparse components. We propose a novel non-convex iterative algorithm with guaranteed recovery. It alternates between low-rank CP decomposition through gradient ascent (a variant of the tensor power method), and hard thresholding of the resid…

New AMP algorithms reveal phase transitions in tensor recovery.

Unified approach tackles high-dimensional tensor bandits with convex optimization and weakly decomposable regularizers.

Proposes BHT-ARIMA for forecasting multiple short time series.

Unified model for tensor completion using low-rank and sparse Tucker decomposition.

We show that the spectral norm of a random $n_1\times n_2\times \cdots \times n_K$ tensor (or higher-order array) scales as $O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$ under some sub-Gaussian assumption on the entries. The proof is based on a covering number argument. Since the spectral norm is dual to the tensor…

New estimator reduces bias and variance in tensor and matrix denoising.

A new probabilistic BTD method for tensor data.

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

On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric ten…

Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations…

Tensor decompositions have rich applications in statistics and machine learning, and developing efficient, accurate algorithms for the problem has received much attention recently. Here, we present a new method built on Kruskal's uniqueness theorem to decompose symmetric, nearly orthogonally decomposable tensors. Unlik…

FasTR efficiently solves sparse and unit-rank tensor regression problems.

Proposes a method to enhance multi-view learning by maximizing higher order correlations.

GETF efficiently decomposes large-scale Boolean tensors.

The completion of tensors, or high-order arrays, attracts significant attention in recent research. Current literature on tensor completion primarily focuses on recovery from a set of uniformly randomly measured entries, and the required number of measurements to achieve recovery is not guaranteed to be optimal. In add…

Paper proposes a new tensor model for mixed memberships and provides error bounds.

In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of noisy tensor completion. Much of the attention has been focused on the fundamen…

The higher order singular value decomposition (HOSVD) of tensors is a generalization of matrix SVD. The perturbation analysis of HOSVD under random noise is more delicate than its matrix counterpart. Recently, polynomial time algorithms have been proposed where statistically optimal estimates of the singular subspaces …

Tensor Train layer improves BLEU scores in NMT models.

Rank-R FNN handles high-dimensional data efficiently.

Enhances stock movement prediction using Higher Order Transformers for multimodal time-series data.

In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a pr…

Revisits CP tensor decomposition for noisy, non-orthogonal data.

Tensor decomposition is an important technique for capturing the high-order interactions among multiway data. Multi-linear tensor composition methods, such as the Tucker decomposition and the CANDECOMP/PARAFAC (CP), assume that the complex interactions among objects are multi-linear, and are thus insufficient to repres…

A new tensor completion method handles missing data with missing not at random entries.

There has recently been considerable interest in completing a low-rank matrix or tensor given only a small fraction (or few linear combinations) of its entries. Related approaches have found considerable success in the area of recommender systems, under machine learning. From a statistical estimation point of view, the…

The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.

Interactive platform for knot invariant computation and identification.