We present an algorithm for supervised learning using tensor networks, employing a step of data pre-processing by coarse-graining through a sequence of wavelet transformations. These transformations are represented as a set of tensor network layers identical to those in a multi-scale entanglement renormalization ansatz tensor network. We perform supervised learning and regression tasks through a model based on a matrix product states (MPSs) acting on the coarse-grained data. Because the entire model consists of tensor contractions (apart from the initial non-linear 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 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 which access finer-scale features.

我们提出了一种使用张量网络进行监督学习的算法，采用通过一系列小波变换进行粗粒度的数据预处理步骤。这些变换表示为一组与多尺度纠缠重整化 ansatz 张量网络中的层相同的张量网络层。我们通过基于矩阵乘积状态 (MPS) 的模型执行监督学习和回归任务，该模型作用于粗粒度数据。因为整个模型由张量收缩组成（除了初始非线性特征图），我们可以通过层“向后”自适应地细粒度优化 MPS 模型，而基本上不会损失性能。MPS 本身使用基于密度矩阵重整化组算法的自适应算法进行训练。