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Statistics Optimization & Information Computing

Randomized density matrix renormalization group algorithm for low-rank tensor train decomposition

Huilin Jiang, Zhongming Chen, Gaohang Yu

Published May 14, 2024• 1 citation
Publisher

Abstract

Tensor train decomposition is a powerful tool for processing high-dimensional data. Density matrix renormalization group (DMRG) is an alternating scheme for low-rank tensor train decomposition of large tensors. However, it may suffer from the curse of dimensionality due to the large scale of subproblems. In this paper, we proposed a novel randomized proximal DMRG algorithm for low-rank tensor train decomposition by using TensorSketch to alleviate the curse of dimensionality. Numerical experiments on synthetic and real-world data also demonstrate the effectiveness and efficiency of the proposed algorithm.

DOI
10.19139/soic-2310-5070-2030

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Tensor train decomposition is a powerful tool for processing high-dimensional data. Density matrix renormalization group (DMRG) is an alternating scheme for low-rank tensor train decomposition of large tensors. However, it may suffer from the curse of dimensionality due to the large scale of subproblems. In this paper, we proposed a novel randomized proximal DMRG algorithm for low-rank tensor train decomposition by using TensorSketch to alleviate the curse of dimensionality. Numerical experiments on synthetic and real-world data also demonstrate the effectiveness and efficiency of the proposed algorithm.
Tensor train is one of the modern decompositions used as low-rank tensor approximations of multidimensional arrays. If the original data is nonnegative we sometimes want the approximant to keep this property. In this work new methods for nonnegative tensor train factorization are proposed. Low-rank approximation approach helps to speed up the computations whereas DMRG technique allows to adapt nonnegative TT ranks for better accuracy. The performance analysis of the proposed algorithms as well as comparison with other nonnegative TT factorization method are presented.
The simulation and analysis of high-dimensional problems is often infeasible due to the curse of dimensionality. In this thesis, we investigate the potential of tensor decompositions for mitigating this curse when considering systems from several application areas. Using tensor-based solvers, we directly compute numerical solutions of master equations associated with Markov processes on extremely large state spaces. Furthermore, we exploit the tensor-train format to approximate eigenvalues and corresponding eigentensors of linear tensor operators. In order to analyze the dominant dynamics of high- dimensional stochastic processes, we propose several decomposition techniques for highly diverse problems. These include tensor representations for operators based on nearest-neighbor interactions, construction of pseudoinverses for tensor-based reformulations of dimensionality reduction methods, and the approximation of transfer operators of dynamical systems. The results show that the tensor-train format enables us to compute low-rank approximations for various numerical problems as well as to reduce the memory consumption and the computational costs compared to classical approaches significantly. We demonstrate that tensor decompositions are a powerful tool for solving high-dimensional problems from various application areas.
A simple nonrecursive form of the tensor decomposition in d dimensions is presented. It does not inherently suffer from the curse of dimensionality, it has asymptotically the same number of parameters as the canonical decomposition, but it is stable and its computation is based on low-rank approximation of auxiliary unfolding matrices. The new form gives a clear and convenient way to implement all basic operations efficiently. A fast rounding procedure is presented, as well as basic linear algebra operations. Examples showing the benefits of the decomposition are given, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.
Tensors arise naturally in high-dimensional problems in chemistry, financial mathematics, and many other areas. The numerical treatment of such problems is difficult due to the curse of dimensionality: the number of unknowns and the computational complexity grow exponentially with the dimension of the problem. To break the curse of dimensionality, low-parametric representations, or formats, have to be used. In this paper we make use of the TT-format (tensor-train format) which is one of the most effective stable representations of high-dimensional tensors. Basic linear algebra operations in the TT-format are now well developed. Our goal is to provide a “black-box” type of solver for linear systems where both the matrix and the right-hand side are in the TT-format. An efficient DMRG (density matrix renormalization group) method is proposed, and several tricks are employed to make it work. The numerical experiments confirm the effectiveness of our approach.
Abstract The tensor renormalization group (TRG) is a highly effective method for calculating the free energy of lattice models and has demonstrated exceptional accuracy for several low-dimensional systems. However, its application to higher-dimensional models is often hampered by a large memory footprint and high computational cost. In this paper, we propose a new algorithm, “tt-lattice”, which approximates the tensor at each lattice vertex using a tensor train (tt) decomposition. The algorithm is designed to maintain the tt-format throughout the TRG computation, which significantly reduces both memory usage and computational cost. For a uniform lattice model, the computational cost of tt-lattice scales as O(d2log⁡(L)χ6)\mathcal {O}(d^2\log (L)\chi ^6)O(d2log(L)χ6), with the largest intermediate tensor size of O(χ4)\mathcal {O}(\chi ^4)O(χ4), where d is the lattice dimension, L is the linear size of each lattice axis, and χ\chiχ is the cutoff of the number of lattice edge states. To validate our algorithm, we conducted numerical experiments on a Gaussian field model, for which the exact free energy is computable in any lattice dimensions. The results demonstrate that the tt-lattice method can calculate the free energy with reasonable accuracy even for a 10D lattice model with modern computational architectures.
Tensors with unit Frobenius norm are fundamental objects in many fields, including scientific computing and quantum physics, which are able to represent normalized eigenvectors and pure quantum states. While the tensor train decomposition provides a powerful low-rank format for tackling high-dimensional problems, it does not intrinsically enforce the unit-norm constraint. To address this, we introduce the normalized tensor train (NTT) decomposition, which aims to approximate a tensor by unit-norm tensors in tensor train format. The low-rank structure of NTT decomposition not only saves storage and computational cost but also preserves the underlying unit-norm structure. We prove that the set of fixed-rank NTT tensors forms a smooth manifold, and the corresponding Riemannian geometry is derived, paving the way for geometric methods. We propose NTT-based methods for low-rank tensor recovery, high-dimensional eigenvalue problem, estimation of stabilizer rank, and calculation of the minimum output Rényi 2-entropy of quantum channels. Numerical experiments demonstrate the superior efficiency and scalability of the proposed NTT-based methods.