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At least 19 records

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↗

Poisson-response Tensor-on-Tensor Regression and Applications

We introduce Poisson-response tensor-on-tensor regression (PToTR), a novel regression framework designed to handle tensor responses composed element-wise of random Poisson-distributed counts. Tensors, or multi-dimensional arrays, composed of counts are common data in fields such as inter national relations, social networks, epidemiology, and medical imaging, where events occur across multiple dimensions like time, location, and dyads. PToTR accommodates such tensor responses alongside tensor covariates, providing a versatile tool for multi dimensional data analysis. We propose algorithms for maximum likelihood estimation under a canonical polyadic (CP) structure on the regression coefficient tensor that satisfy the positivity of Poisson parameters and then provide an initial theoretical error analysis for PToTR estimators. We also demonstrate the utility of PToTR through three concrete applications: longitudinal data analysis of the Integrated Crisis Early Warning System database, positron emission tomography (PET) image reconstruction, and change-point detection of communication patterns in longitudinal dyadic data. These applications highlight the versatility of PToTR in addressing complex, structured count data across various domains.

97 MATHEMATICS AND COMPUTING↗

Compact video content representation for video coding using low multi-linear tensor rank approximation with dynamic core tensor order

Tensor is a multi-way representation of an entity that is widely used in many applications of mathematics and physics. The wide range of tensor algebra operations attract the researchers of graphical and vision field to identify unique methods for various applications such as segmentation, compression, indexing and so on. In this paper, low multi-linear rank approximation of a tensor is used for decomposition so as to perform compression on a video that is represented as a 4D tensor. Thus, the encoder has blocks for (1) multilinear rank approximation method so as to aid decomposition, (2) sparsity removal of residual data to reduce the required memory storage to a small level, (3) quantization on the sparse less residual data and (4) LZ77 dictionary coding for the video. The decoder uses a tensor reconstruction block along with residual error correction block that reduces loss/error in the reconstructed video. The size of core tensor is an important factor in rank approximation. It is identified at run time depending on the video contents. This size is determined adaptively using Tikhonov’s regularization method. The best value of core tensor size is identified with the corner of L-curve which preserves the video contents from heavy loss/error. Experimental results of the proposed work on comparison with H.264/AVC compression method indicate that the proposed method achieves up to 10% improvement in compression ratio with the least amount of loss and a maximum signal to noise ratio of 0.14. The proposed system gives better compression ratio than H.265 for videos with more number of frames without compromising quality.

97 MATHEMATICS AND COMPUTING↗

Nekrasov tensors and nonsingular H-tensors

Nekrasov matrices and nonsingular H-matrices are closely related. In this paper, Nekrasov tensors and S-Nekrasov tensors are proved to be nonsingular H-tensors. And tensors are generalized Nekrasov tensors if and only if they are nonsingular H-tensors. Furthermore, an iterative criterion for identifying nonsingular H-tensors is provided.

97 MATHEMATICS AND COMPUTING↗

Efficient tree decomposition of high-rank tensors [Efficient decomposition of high-rank tensors]

Tensors are a natural way to express correlations among many physical variables, but storing tensors in a computer naively requires memory which scales exponentially in the rank of the tensor. This is not optimal, as the required memory is actually set not by the rank but by the mutual information amongst the variables in question. Representations such as the tensor tree perform near-optimally when the tree decomposition is chosen to reflect the correlation structure in question, but making such a choice is non-trivial and good heuristics remain highly context-specific. In this work I present two new algorithms for choosing efficient tree decompositions, independent of the physical context of the tensor. The first is a brute-force algorithm which in most cases produces optimal decompositions but is generally impractical for high-rank tensors, as the number of possible choices grows exponentially in rank. Here, the second is a greedy algorithm, and while it is not optimal it performs extremely well in numerical experiments while having runtime which makes it practical even for tensors of very high rank.

97 MATHEMATICS AND COMPUTING↗

Generalized Tensor-on-Tensor Regression (GToTR)

SAND2026-23069O Generalized Tensor-on-Tensor Regression (GToTR) is a Python-based tool for conducting generalized tensor-on-tensor regression. It provides Canonical Polyadic (CP)-based generalized tensor regression models, support for generalized linear model-like families and links, alternating-optimization model fitting methods, and a standard statistics software interface. The tool supports tensor-valued responses and covariates using the open-source Python Tensor Toolbox (pyttb) software package. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Dunlavy, Daniel [Sandia National Lab. (SNL-CA), Li↗

The tensor t‐function: A definition for functions of third‐order tensors

Summary A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided.

Lund, Kathryn↗

CCUS 2024, Interpreting the strain tensor Larry Murdoch Interpreting strain tensor data to characterize and monitor reservoirs for CO2 storage and other applications

Recent advances in instrumentation have made it feasible to measure the transient strain tensor caused by small changes in fluid volume or pressure in the subsurface and this has opened the door to new opportunities for characterization and monitoring during CCUS. We have demonstrated this method by deploying strainmeters at shallow depths (30 to 40m) and then conducting injection well tests in an underlying reservoir at 530m depth. The resulting data indicated that the horizontal strain at shallow strainmeters was tensile and the vertical strain was compressive. The radial strain was less than the horizontal strain, and the strain rates decreased from 100 nanostrain/day to roughly 10 ne/d over a few days (1 nanostrain = 1 part per billion strain). We then used the strain data to estimate reservoir properties, geometry and pressure through inversion of poroelastic forward models using both numerical and novel analytical methods. The average horizontal strain in the caprock resembles the transient pressure in the underlying reservoir and classic type-curve methods from transient well testing can be used for preliminary interpretations of strain data. We have developed fast, closed-form analytical solutions to a pressurized poroelastic inclusion and inhomogeneity in a half-space. Numerical models developed using finite element methods allow more details of the subsurface to be included in the inversion, but they require much longer run times and this makes inversion cumbersome using standard methods. We have developed an inversion approach that uses a proxy model created using machine learning to do most of the forward calculations. This approach markedly reduces the computational requirements and makes it feasible to use Bayesian inversion with large numerical models. Bayesian inversion is important because it provides predictions with uncertainties, which makes the results useful for decision making. We have shown with field tests and simulations that the strain tensor in the caprock is sensitive to pressure in the reservoir, reservoir properties and boundaries, and pressure in the caprock caused by leaks. These results indicate that measuring and interpreting the shallow strain tensor could be a valuable tool for both initial reservoir characterization efforts and long-term monitoring during CCUS. Recent advances in instrumentation have made it feasible to measure the transient strain tensor caused by small changes in fluid volume or pressure in the subsurface and our objective was to evaluate opportunities for strain monitoring during characterization and monitoring for CCUS. Our approach was to deploy strainmeters at shallow depths (30 to 40m) and then conduct injection well tests in an underlying reservoir at 530m depth. The results indicate that the horizontal strain at shallow strainmeters was tensile and the vertical strain was compressive. The radial strain was less than the horizontal strain, and the strain rates decreased from 100 nanostrain/day to roughly 10 ne/d over a few days (1 nanostrain = 1 part per billion strain). We then used the strain data to estimate reservoir properties, geometry and pressure through inversion of poroelastic forward models using both numerical and novel analytical methods. The average horizontal strain in the caprock resembles the transient pressure in the underlying reservoir and classic type-curve methods from transient well testing can be used for preliminary interpretations of strain data. We have developed fast, closed-form analytical solutions to a pressurized poroelastic inclusion and inhomogeneity in a half-space. Numerical models developed using finite element methods allow more details of the subsurface to be included in the inversion, but they require much longer run times and this makes inversion cumbersome using standard methods. We have developed an inversion approach that uses a proxy model created using machine learning to do most of the forward calculations. This approach markedly reduces the computational requirements and makes it feasible to use Bayesian inversion with large numerical models. Bayesian inversion is important because it provides predictions with uncertainties, which makes the results useful for decision making. In conclusion, we have shown with field tests and simulations that the strain tensor in the caprock is sensitive to pressure in the reservoir, reservoir properties and boundaries, and pressure in the caprock caused by leaks. These results indicate that measuring and interpreting the shallow strain tensor could be a valuable tool for both initial reservoir characterization efforts and long-term monitoring during CCUS.

Murdoch, Larry↗

On the closedness and geometry of tensor network state sets

Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse of dimensionality is addressed by parametrizing the many-body state in terms of a network of partially contracted tensors. These tensors form a substantially reduced set of effective degrees of freedom. In practical algorithms, functionals like energy expectation values or overlaps are optimized over certain sets of TNS. Concerning algorithmic stability, it is important whether the considered sets are closed because, otherwise, the algorithms may approach a boundary point that is outside the TNS set and tensor elements diverge. Here we discuss the closedness and geometries of TNS sets, and we propose regularizations for optimization problems on non-closed TNS sets. We show that sets of matrix product states (MPS) with open boundary conditions, tree tensor network states, and the multiscale entanglement renormalization ansatz are always closed, whereas sets of translation-invariant MPS with periodic boundary conditions (PBC), heterogeneous MPS with PBC, and projected entangled pair states are generally not closed. The latter is done using explicit examples like the W state, states that we call two-domain states, and fine-grained versions thereof.

97 MATHEMATICS AND COMPUTING↗

Communication Lower Bounds and Optimal Algorithms for Multiple Tensor-Times-Matrix Computation

Multiple tensor-times-matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. Here, we establish communication lower bounds that determine how much data movement is required (under mild conditions) to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that, with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations when the input and output tensors vary greatly in size.

HBL-inequalities↗

Solving a class of infinite-dimensional tensor eigenvalue problems by translational invariant tensor ring approximations

Here, we examine a method for solving an infinite-dimensional tensor eigenvalue problem Hx = λx, where the infinite-dimensional symmetric matrix H exhibits a translational invariant structure. We provide a formulation of this type of problem from a numerical linear algebra point of view and describe how a power method applied to e -Ht is used to obtain an approximation to the desired eigenvector. This infinite-dimensional eigenvector is represented in a compact way by a translational invariant infinite Tensor Ring (iTR). Low rank approximation is used to keep the cost of subsequent power iterations bounded while preserving the iTR structure of the approximate eigenvector. We show how the averaged Rayleigh quotient of an iTR eigenvector approximation can be efficiently computed and introduce a projected residual to monitor its convergence. In the numerical examples, we illustrate that the norm of this projected iTR residual can also be used to automatically modify the time step to ensure accurate and rapid convergence of the power method.

97 MATHEMATICS AND COMPUTING↗

On the most general tensor B(sub ij) which is zero for i does not equal j, and the most general isotropic tensor I(sub ij)

It is shown that the most general second order tensor B sub ij which is zero for i not = j is proportional to the Kronecker delta (Delta sub ij). By a slight modification of that argument, the known result was obtained that the most general second order isotropic tensor is also proportional to Delta sub ij. These results are useful for instance in obtaining the stress tensor for a viscous fluid.

Deissler, Robert G.↗

de Sitter space as a tensor network: Cosmic no-hair, complementarity, and complexity

Here, we investigate the proposed connection between de Sitter spacetime and the multiscale entanglement renormalization ansatz (MERA) tensor network, and ask what can be learned via such a construction. We show that the quantum state obeys a cosmic no-hair theorem: the reduced density operator describing a causal patch of the MERA asymptotes to a fixed point of a quantum channel, just as spacetimes with a positive cosmological constant asymptote to de Sitter space. The MERA is potentially compatible with a weak form of complementarity (local physics only describes single patches at a time, but the overall Hilbert space is infinite dimensional) or, with certain specific modifications to the tensor structure, a strong form (the entire theory describes only a single patch plus its horizon, in a finite-dimensional Hilbert space). Furthermore, we suggest that de Sitter evolution has an interpretation in terms of circuit complexity, as has been conjectured for anti–de Sitter space.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Tensor Network Path Integral Study of Dynamics in B850 LH2 Ring with Atomistically Derived Vibrations

The recently introduced multisite tensor network path integral (MS-TNPI) allows simulation of extended quantum systems coupled to dissipative media. We use MS-TNPI to simulate the exciton transport and the absorption spectrum of a B850 bacteriochlorophyll (BChl) ring. The MS-TNPI network is extended to account for the ring topology of the B850 system. Accurate molecular-dynamics-based description of the molecular vibrations and the protein scaffold is incorporated through the framework of Feynman–Vernon influence functional. To relate the present work with the excitonic picture, an exploration of the absorption spectrum is done by simulating it using approximate and topologically consistent transition dipole moment vectors. Comparison of these numerically exact MS-TNPI absorption spectra are shown with second-order cumulant approximations. Finally, the effect of temperature on both the exact and the approximate spectra is also explored.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗