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Results for “Canonical Polyadic Tensor Decomposition”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Streaming Generalized Canonical Polyadic Tensor Decompositions

In this paper, we develop a method which we call OnlineGCP for computing the Generalized Canonical Polyadic (GCP) tensor decomposition of streaming data. GCP differs from traditional canonical polyadic (CP) tensor decompositions as it allows for arbitrary objective functions which the CP model attempts to minimize. This approach can provide better fits and more interpretable models when the observed tensor data is strongly non-Gaussian. In the streaming case, tensor data is gradually observed over time and the algorithm must incrementally update a GCP factorization with limited access to prior data. In this work, we extend the GCP formalism to the streaming context by deriving a GCP optimization problem to be solved as new tensor data is observed, formulate a tunable history term to balance reconstruction of recently observed data with data observed in the past, develop a scalable solution strategy based on segregated solves using stochastic gradient descent methods, describe a software implementation that provides performance and portability to contemporary CPU and GPU architectures and integrates with Matlab for enhanced usability, and demonstrate the utility and performance of the approach and software on several synthetic and real tensor data sets.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Tensor decompositions for count data that leverage stochastic and deterministic optimization

There is growing interest to extend low-rank matrix decompositions to multi-way arrays, or tensors. One fundamental low-rank tensor decomposition is the canonical polyadic decomposition (CPD). The challenge of fitting a low-rank, nonnegative CPD model to Poisson-distributed count data is of particular interest. Several popular algorithms use local search methods to approximate the maximum likelihood estimator (MLE) of the Poisson CPD model. Here, this work presents two new algorithms that extend state-of-the-art local methods for Poisson CPD. Hybrid GCP-CPAPR combines Generalized Canonical Decomposition (GCP) with stochastic optimization and CP Alternating Poisson Regression (CPAPR), a deterministic algorithm, to increase the probability of converging to the MLE over either method used alone. Restarted CPAPR with SVDrop uses a heuristic based on the singular values of the CPD model unfoldings to identify convergence toward optimizers that are not the MLE and restarts within the feasible domain of the optimization problem, thus reducing overall computational cost when using a multi-start strategy. We provide empirical evidence that indicates our approaches outperform existing methods with respect to converging to the Poisson CPD MLE.

CPAPR↗
DOE OSTI2020

Stochastic Gradients for Large-Scale Tensor Decomposition

Tensor decomposition is a well-known tool for multiway data analysis. This work proposes using stochastic gradients for efficient generalized canonical polyadic (GCP) tensor decomposition of large-scale tensors. GCP tensor decomposition is a recently proposed version of tensor decomposition that allows for a variety of loss functions such as Bernoulli loss for binary data or Huber loss for robust estimation. Here, the stochastic gradient is formed from randomly sampled elements of the tensor and is efficient because it can be computed using the sparse matricized-tensor times Khatri--Rao product tensor kernel. For dense tensors, we simply use uniform sampling. For sparse tensors, we propose two types of stratified sampling that give precedence to sampling nonzeros. Numerical results demonstrate the advantages of the proposed approach and its scalability to large-scale problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Parameter Sensitivity Analysis of the SparTen High Performance Sparse Tensor Decomposition Software (Extended Analysis)

Tensor decomposition models play an increasingly important role in modern data science applications. One problem of particular interest is fitting a low-rank Canonical Polyadic (CP) tensor decomposition model when the tensor has sparse structure and the tensor elements are nonnegative count data. SparTen is a high-performance C++ library which computes a low-rank decomposition using different solvers: a first-order quasi-Newton or a second-order damped Newton method, along with the appropriate choice of runtime parameters. Since default parameters in SparTen are tuned to experimental results in prior published work on a single real-world dataset conducted using MATLAB implementations of these methods, it remains unclear if the parameter defaults in SparTen are appropriate for general tensor data. Furthermore, it is unknown how sensitive algorithm convergence is to changes in the input parameter values. This report addresses these unresolved issues with large-scale experimentation on three benchmark tensor data sets. Experiments were conducted on several different CPU architectures and replicated with many initial states to establish generalized profiles of algorithm convergence behavior.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Zero-Truncated Poisson Tensor Decomposition for Sparse Count Data

We propose a novel statistical inference paradigm for zero-inflated multiway count data that dispenses with the need to distinguish between true and false zero counts. Our approach ignores all zero entries and applies zero-truncated Poisson regression on the positive counts. Inference is accomplished via tensor completion that imposes low-rank structure on the Poisson parameter space. Our main result shows that an $\textit{N}$-way rank-R parametric tensor 𝓜 ϵ (0, ∞) $I$Χ∙∙∙Χ$I$ generating Poisson observations can be accurately estimated from approximately $IR^2 \text{log}^2_2(I)$ non-zero counts for a nonnegative canonical polyadic decomposition. Several numerical experiments are presented demonstrating that our zero-truncated paradigm is comparable to the ideal scenario where the locations of false zero counts are known $\textit{a priori}$.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Zero-truncated Poisson regression for sparse multiway count data corrupted by false zeros

Abstract We propose a novel statistical inference methodology for multiway count data that is corrupted by false zeros that are indistinguishable from true zero counts. Our approach consists of zero-truncating the Poisson distribution to neglect all zero values. This simple truncated approach dispenses with the need to distinguish between true and false zero counts and reduces the amount of data to be processed. Inference is accomplished via tensor completion that imposes low-rank tensor structure on the Poisson parameter space. Our main result shows that an $N$-way rank-$R$ parametric tensor $\boldsymbol{\mathscr{M}}\in (0,\infty )^{I\times \cdots \times I}$ generating Poisson observations can be accurately estimated by zero-truncated Poisson regression from approximately $IR^2\log _2^2(I)$ non-zero counts under the nonnegative canonical polyadic decomposition. Our result also quantifies the error made by zero-truncating the Poisson distribution when the parameter is uniformly bounded from below. Therefore, under a low-rank multiparameter model, we propose an implementable approach guaranteed to achieve accurate regression in under-determined scenarios with substantial corruption by false zeros. Several numerical experiments are presented to explore the theoretical results.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

CP decomposition for tensors via alternating least squares with QR decomposition

The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

A Flexible Forwarding Scheme to Improve Latency-Bound Irregular P2P Communication in MPI

We propose an algorithm to efficiently perform latency-bound communication scenarios that consist of many small messages. In these parallel scenarios, processes typically pass around a lot of small-sized messages of a few KBs of size. Performing communication operations with P2P MPI routines or collective MPI routines (including neighborhood collectives) in such scenarios may not always yield the optimal results and may not resolve the latency bottleneck. To this end, we develop a regular structure called virtual process topology (VPT) on which the messages can be communicated in a structured and controlled manner. Using parameters of this topology, one can tune the rate of aggression in tackling the latency costs. We demonstrate that our communication algorithm is preferable to MPI P2P and collective routines for latency-bound communication and it can easily be adapted only by replacing calls to MPI routines in a parallel application. We show how to adapt existing topology-aware mapping heuristics to address the volume overhead due to communicating messages on the VPT. Moreover, we propose a novel swap-based mapping heuristic to address this overhead by optimizing the maximum volume handled by a process. Experiments on synthetic communication graphs as well as real-world applications such as parallel Canonical Polyadic sparse tensor decomposition and parallel sparse matrix-dense matrix multiplication show that our approach is a powerful way of overcoming the bottlenecks posed by sparse and latency-bound irregular communication.

communication algorithm↗
DOE OSTI2024

The Average Spectrum Norm and Near-Optimal Tensor Completion

We propose the average spectrum norm to study the minimum number of measurements required to approximate a multidimensional array (i.e., sample complexity) via low-rank tensor recovery. Our focus is on the tensor completion problem, where the aim is to estimate a multiway array using a subset of tensor entries corrupted by noise. Our average spectrum norm-based analysis provides near-optimal sample complexities, exhibiting dependence on the ambient dimensions and rank that do not suffer from exponential scaling as the order increases.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Multi-Area Model-Free State Estimation via Distributed Tensor Decomposition

This paper proposes a model-free method for distribution system state estimation based on tensor completion using canonical polyadic decomposition. In particular, we consider a setting where the network is divided into multiple areas. The measured physical quantities at buses located in the same area are processed by an area controller. A third-order tensor is constructed to collect these measured quantities. The measurements are analyzed locally to recover the full state information of the network. A closed-form iterative algorithm based on the alternating direction method of multipliers is developed to obtain the low-rank factors of the whole network state tensor where information exchange happens only between neighboring areas. To demonstrate the efficacy of the developed algorithm, numerical simulations are carried out using an IEEE test system.

alternating direction method of multipliers↗
DOE OSTI2023

Tensor Decompositions for Count Data that Leverage Stochastic and Deterministic Optimization

There is growing interest to extend low-rank matrix decompositions to multi-way arrays, or tensors. One fundamental low-rank tensor decomposition is the canonical polyadic decomposition (CPD). The challenge of fitting a low-rank, nonnegative CPD model to Poisson-distributed count data is of particular interest. Several popular algorithms use local search methods to approximate the global maximum likelihood estimator from local minima. Simultaneously, a recent trend in theoretical computer science and numerical linear algebra leverages randomization to solve very large, hard problems. The typical approach is to use randomization for a fast approximation and determinism for refinement to yield effective algorithms with theoretical guarantees. Two popular algorithms for Poisson CPD reflect that emergent dichotomy: CP Alternating Poisson Regression is a deterministic algorithm and Generalized Canonical Polyadic decomposition makes use of stochastic algorithms in several variants. This work extends recent work to develop two new methods that leverage randomized and deterministic algorithms for improved accuracy and performance.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Multiarea Distribution System State Estimation via Distributed Tensor Completion

Here, this paper proposes a model-free distribution system state estimation method based on tensor completion using canonical polyadic decomposition. In particular, we consider a setting where the network is divided into multiple areas. The measured physical quantities at buses located in the same area are processed by an area controller. A three-way tensor is constructed to collect these measured quantities. The measurements are analyzed locally to recover the full state information of the network. A distributed closed-form iterative algorithm based on the alternating direction method of multipliers is developed to obtain the low-rank factors of the whole network state tensor where information exchange happens only between neighboring areas. The convergence properties of the distributed algorithm and the sufficient conditions on the number of samples for each smaller network that guarantee the identifiability of the factors of the state tensor are presented. To demonstrate the efficacy of the proposed algorithm and to check the identifiability conditions, numerical simulations are carried out using the IEEE 123-bus system and a large-scale real utility feeder.

24 POWER TRANSMISSION AND DISTRIBUTION↗