Systems and methods of applying matrix product states to machine learning

A system for applying non-linear regression on a set of N datapoints to get an estimate, the system comprising at least one processor configured to:

• receive the set of N datapoints, each of the datapoints having associated input values;
• separate the set of N datapoints into Nb batches, each of the Nb batches having M=N/Nb datapoints;
• receive a family of fitting functions, each function comprising a dot product of a feature map Φ with a weight vector W, where the feature map Φ is local and represented by two-dimensional feature vectors;
• and
• minimize a cost function for each of the Nb batches by training a matrix product state (MPS) model to minimize loss over all the Nb batches, the MPS model providing an MPS representation of the weight vector W and comprising M+D tensors,
• wherein D corresponds to a dimension of a target function, and
• wherein each of the tensors have respective coefficients, such that:
• all tensors except D correspond to one datapoint i in each of the Nb batches;
• D extra tensors in the MPS have a physical dimension of size M corresponding to a number of possible outputs for a given batch from the Nb batches;
• and
• the coefficients of the tensors in the MPS model minimize the cost function sequentially over all the Nb batches.