Shortcut Matrix Product States and its applications

Abstract

Matrix Product States (MPS), also known as Tensor Train (TT) decomposition in mathematics, has been proposed originally for describing an (especially one-dimensional) quantum system, and recently has found applications in various applications such as compressing high-dimensional data, supervised kernel linear classifier, and unsupervised generative modeling. However, when applied to systems which are not defined on one-dimensional lattices, a serious drawback of the MPS is the exponential decay of the correlations, which limits its power in capturing long-range dependences among variables in the system. To alleviate this problem, we propose to introduce long-range interactions, which act as shortcuts, to MPS, resulting in a new model \textit{ Shortcut Matrix Product States} (SMPS). When chosen properly, the shortcuts can decrease significantly the correlation length of the MPS, while preserving the computational efficiency. We develop efficient training methods of SMPS for various tasks, establish some of their mathematical properties, and show how to find a good location to add shortcuts. Finally, using extensive numerical experiments we evaluate its performance in a variety of applications, including function fitting, partition function calculation of $2-$d Ising model, and unsupervised generative modeling of handwritten digits, to illustrate its advantages over vanilla matrix product states.

arXiv comment15pages, 11 figures

References in corpus (14)

• The density-matrix renormalization group
• Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
• Tensor renormalization group approach to 2D classical lattice models
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• Coarse-graining renormalization by higher-order singular value decomposition
• Tensor Network Renormalization
• The computational complexity of PEPS
• Tensor Ring Decomposition
• Unsupervised Generative Modeling Using Matrix Product States
• Perfect Sampling with Unitary Tensor Networks
• Computations in Quantum Tensor Networks
• Analysis and Design of Convolutional Networks via Hierarchical Tensor Decompositions
• A generalized Lanczos method for systematic optimization of tensor network states
• Tensor network language model

Cited by in corpus (5)

• Tree Tensor Networks for Generative Modeling
• Modeling Sequences with Quantum States: A Look Under the Hood
• Quantum-Classical Machine learning by Hybrid Tensor Networks
• Infinitely Wide Tensor Networks as Gaussian Process
• SchrödingeRNN: Generative Modeling of Raw Audio as a Continuously Observed Quantum State