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arXiv:2111.08014 · Physical Review Research

Tensor network to learn the wave function of data

Anatoly Y. Dymarsky, Kirill Pavlenko

Published Nov 16, 2022• 6 citations
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Abstract

Tensor network architectures have emerged recently as a promising approach to various tasks of machine learning, both supervised and unsupervised. In this work we introduce a matrix product state-based network that simultaneously accomplishes the following two tasks: classification (discrimination) and sampling of visual data. We train the network using binary (black and white) version of MNIST, a data set of handwritten digits, to recognize as well as to sample images of a particular digit. We show our trained network is qualitatively representing the indicator function of the ``full set'' of all possible images of a given format depicting the particular digit. While the notion of the full set is difficult to define from the first principles, our construction provides a working definition, and we show that different ways to build and train the network lead to similar results. We emphasize, this means the trained network learns the ``wave function of data,'' i.e., can be used to characterize the data itself, providing a novel tool to study global properties of the data sets of interest. First, using quantum mechanical interpretation we characterize the full set by calculating its entanglement entropy. Then we study its geometric properties such as mean Hamming distance, effective dimension, and size. The latter is the total number of images in binary black and white MNIST format which would be recognized as depicting a particular digit. Alternatively, it is the number of images of a given digit one would need to sample before the probability of sampling the same image twice would be of order one. While this number cannot be defined completely rigorously, we show its logarithm is largely independent of the way the network is defined and trained. We find that for different digits this number varies dramatically, from ${2}^{22}$ for digit 1 to ${2}^{92}$ for digit 8.

DOI
10.1103/physrevresearch.4.043111

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Given an image of a white shoe drawn on a blackboard, how are the white pixels deemed (say by human minds) to be informative for recognizing the shoe without any labeling information on the pixels? Here we investigate such a “white shoe” recognition problem from the perspective of tensor network (TN) machine learning and quantum entanglement. Utilizing a generative TN that captures the probability distribution of the features as quantum amplitudes, we propose an unsupervised recognition scheme of informative features with variations of entanglement entropy (EE) caused by designed measurements. In this way, a given sample, where the values of its features are statistically meaningless, is mapped to the variations of EE that statistically characterize the gain of information. We show that the EE variations identify the features that are critical to recognize this specific sample, and the EE itself reveals the information distribution of the probabilities represented by the TN model. The signs of the variations further reveal the entanglement structures among the features. We test the validity of our scheme on a toy dataset of strip images, the MNIST dataset of hand-drawn digits, the fashion-MNIST dataset of the pictures of fashion articles, and the images of nerve cord. Our scheme opens the avenue to the quantum-inspired and interpreted unsupervised learning, which can be applied to, e.g., image segmentation and object detection.
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