Tensor tree learns hidden relational structures in data to construct generative models

TL;DR

The paper tackles learning generative models that uncover hidden relational structure by marrying Born machines with a flexible tensor-tree topology. It introduces the adaptive tensor tree (ATT) framework, guided by bond mutual information, and a branch-reconnection algorithm to jointly optimize tensors and network structure while controlling information flow. Across artificial patterns, QMNIST, Bayesian networks, and stock-market data, ATT yields improved negative log-likelihood and interpretable topologies that reflect actual data relationships. This approach provides a scalable, physics-inspired pathway to structure-aware generative modeling with potential benefits for quantum circuit design and data-analysis pipelines.

Abstract

Based on the tensor tree network with the Born machine framework, we propose a general method for constructing a generative model by expressing the target distribution function as the amplitude of the quantum wave function represented by a tensor tree. The key idea is dynamically optimizing the tree structure that minimizes the bond mutual information. The proposed method offers enhanced performance and uncovers hidden relational structures in the target data. We illustrate potential practical applications with four examples: (i) random patterns, (ii) QMNIST handwritten digits, (iii) Bayesian networks, and (iv) the pattern of stock price fluctuation pattern in S&P500. In (i) and (ii), the strongly correlated variables were concentrated near the center of the network; in (iii), the causality pattern was identified; and in (iv), a structure corresponding to the eleven sectors emerged.

Figures (3)

• Figure 1: Procedures to be iterated for optimizing the tensors and network structure in the adaptive tensor tree (ATT) method. (a) Three candidate decompositions of a combined tensor with four legs. Before computing the bond mutual information (BMI) for each configuration, the component tensors (red) are improved. (b) For a target bond (the thick black line), we obtain a bigger tensor with four legs by contracting it. We choose the one with the smallest BMI among the three possible decompositions.
• Figure 2: Results of random binary patterns and images of handwritten digits. (a) Tensor train with open branches corresponding to random bit variables, represented by rainbow-colored circles. Bits with dotted open branches in a middle region are fixed at 0. (b) Negative log-likelihood in learning processes for ten random binary patterns. The initial network is a tensor train. (c) Optimized network structure for ten random binary patterns. Colored circles represent random bit variables. The color of an edge indicates the amplitude of the bond mutual information. (d) Converged values of negative log-likelihood for the images of the handwritten digits vs. bond dimension. (e) Optimized network structure for the images of the handwritten digits. The color of an edge indicates the amplitude of the bond mutual information. The initial network is a random tensor tree. (f) Ranking of pixels according to their distance from the center of the tensor tree network.
• Figure 3: (a-c) Target Bayesian networks of random binary variables and corresponding tensor network structures: (a) single dependency with no branching, (b) single dependency with branching, and (c) multiple dependencies. In each diagram, the lower layer is the target Bayesian network, and the upper layer is the corresponding tensor network. All tensor trees are not schematic diagrams but actual solutions obtained by the method regardless of the initial network configurations. (d,e) The result of the adaptive tensor tree method applied to the stock price fluctuation patterns in S&P 500 index: (d) Bond-dimension dependency of the negative log-likelihood, and (e) sample of generated tree structure at the bond dimension of 5. Companies are colored according to the sector to which they belong.