Representation Theorem for Matrix Product States
This provides foundational insights into the expressive power of MPS for quantum and classical machine learning, though it is incremental in connecting MPS to established neural network frameworks.
The paper tackled the universal representation capacity of Matrix Product States (MPS) by showing they can realize arbitrary boolean functions and are dense in continuous function spaces, and demonstrated equivalence to one-hidden-layer neural networks with kernels like polynomial kernels.
In this work, we investigate the universal representation capacity of the Matrix Product States (MPS) from the perspective of boolean functions and continuous functions. We show that MPS can accurately realize arbitrary boolean functions by providing a construction method of the corresponding MPS structure for an arbitrarily given boolean gate. Moreover, we prove that the function space of MPS with the scale-invariant sigmoidal activation is dense in the space of continuous functions defined on a compact subspace of the $n$-dimensional real coordinate space $\mathbb{R^{n}}$. We study the relation between MPS and neural networks and show that the MPS with a scale-invariant sigmoidal function is equivalent to a one-hidden-layer neural network equipped with a kernel function. We construct the equivalent neural networks for several specific MPS models and show that non-linear kernels such as the polynomial kernel which introduces the couplings between different components of the input into the model appear naturally in the equivalent neural networks. At last, we discuss the realization of the Gaussian Process (GP) with infinitely wide MPS by studying their equivalent neural networks.