Abstract：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.
Cardona A, Saalfeld S, Preibisch S, Schmid B, Cheng A, Pulokas J, Tomancak P, and Hartenstein V 2010 PLOS Biol.8 e1000502
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Here we avoid to use Dirac's symbols for readers who are not familiar with quantum physics. We use a bold letter to represent a tensor (or matrix, vector, such as ${\boldsymbol\varPsi}$), and the same normal letter with lower indexes to represent the tensor elements (such as $\varPsi_{s_1}\ldots{s_{_{\scriptstyle M}}}$).
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Pérez-García D, Verstraete F, Wolf M M, and Cirac J I 2007 Quantum Inf. Comput.7 401
2022, Vol. 39 
