A Multi-Scale Tensor Network Architecture for Classification and Regression

A Multi-Scale T ensor Network Architecture for Classification and Regression Justin Reyes 1 and E. Miles Stoudenmire 2 1 Department of Physics, University of Central Florida, 4000 Central Florida Blvd, Orlando, FL 32816, USA 2 Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New Y ork, NY 10010, USA (Dated: January 24, 2020) We present an algorithm for supervised learning using tensor networks, employing a step of preprocessing the data by coarse-graining through a sequence of wavelet transformations. We represent these transformations as a set of tensor network layers identical to those in a multi-scale entanglement renormalization ansatz (MERA) tensor network, and perform supervised learning and regression tasks through a model based on a matrix product state (MPS) tensor network acting on the coarse-grained data. Because the entire model consists of tensor contractions (apart from the initial nonlinear feature map), we can adaptively fine-grain the optimized MPS model backwards through the layers with essentially no loss in performance. The MPS itself is trained using an adaptive algorithm based on the density matrix renormalization group (DMRG) algorithm. We test our methods by performing a classification task on audio data and a regression task on temperature time-series data, studying the dependence of training accuracy on the number of coarse-graining layers and showing how fine-graining through the network may be used to initialize models with access to finer-scale features. I. INTRODUCTION Computational techniques developed across the machine learning and physics fields have consistently generated promising methods and applications in both areas of study. The application of well established machine learning architectures and optimization techniques has enriched the physics community with advances such as modeling and recognizing topological quantum states [1-3], optimizing quantum error correction codes [4], or classifying quantum walks [5]. Conversely, techniques known as tensor networks which model high-dimensional functions and are closely connected to physical principles have begun to be explored more in applied mathematics and machine learning [6-16].