Abstract
The resemblance between the methods used in quantum-many body physics and in machine learning has drawn considerable attention. In particular, tensor networks (TNs) and deep learning architectures bear striking similarities to the extent that TNs can be used for machine learning. Previous results used one-dimensional TNs in image recognition, showing limited scalability and flexibilities. In this work, we train two-dimensional hierarchical TNs to solve image recognition problems, using a training algorithm derived from the multi-scale entanglement renormalization ansatz. This approach introduces mathematical connections among quantum many-body physics, quantum information theory, and machine learning. While keeping the TN unitary in the training phase, TN states are defined, which encode classes of images into quantum many-body states. We study the quantum features of the TN states, including quantum entanglement and fidelity. We find these quantities could be properties that characterize the image classes, as well as the machine learning tasks.
Highlights
Malization ansatz (MERA) approach [22,23,24,25] and hierarchical representation that is known as the tree tensor networks (TNs) (TTN) [26, 27]
Tensor networks (TNs) and deep learning architectures bear striking similarities to the extent that TNs can be used for machine learning
We train two-dimensional hierarchical TNs to solve image recognition problems, using a training algorithm derived from the multi-scale entanglement renormalization ansatz
Summary
We obtain the TTN not by decomposing but by training, i.e., by minimizing the cost function This is essentially different from the existing works: (1) our aim is to perform classification, instead of approximating a given tensor; (2) the tensors in the TTN are not restricted to a special form (like delta tensors) but are updated by the proposed MERA algorithm. The sequence of convolutional and pooling layers in the feature extraction part of a deep learning network is known to arrive at higher and higher levels of abstraction that helps separate the classes in a discriminative learner [13] This is often visualized by embedding the representation in two dimensions using t-distributed stochastic neighbor embedding (t-SNE), which is a method for dimensionality reduction that is well suited for the visualization of high-dimensional datasets [34]. The samples incline to gather in different regions to higher and higher layers, and are separated into two curves (1-Dimensional manifold) after the top layer
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