An enriched category theory of language: from syntax to semantics

Given a piece of text, the ability to generate a coherent extension of it implies some sophistication, including a knowledge of grammar and semantics. In this paper, we propose a mathematical framework for passing from probability distributions on extensions of given texts to an enriched category containing semantic information. Roughly speaking, we model probability distributions on texts as a category enriched over the unit interval. Objects of this category are expressions in language and hom objects are conditional probabilities that one expression is an extension of another. This category is syntactical: it describes what goes with what. We then pass to the enriched category of unit interval-valued copresheaves on this syntactical category to find semantic information.

Tensor Networks for Language Modeling

The tensor network formalism has enjoyed over two decades of success in modeling the behavior of complex quantum-mechanical systems, but has only recently and sporadically been leveraged in machine learning. Here we introduce a uniform matrix product state (u-MPS) model for probabilistic modeling of sequence data. We identify several distinctive features of this recurrent generative model, notably the ability to condition or marginalize sampling on characters at arbitrary locations within a sequence, with no need for approximate sampling methods. Despite the sequential architecture of u-MPS, we show that a recursive evaluation algorithm can be used to parallelize its inference and training, with a string of length n only requiring parallel time $\mathcal{O}(\log n)$ to evaluate. Experiments on a context-free language demonstrate a strong capacity to learn grammatical structure from limited data, pointing towards the potential of tensor networks for language modeling applications.

Modeling Sequences with Quantum States: A Look Under the Hood

Classical probability distributions on sets of sequences can be modeled using quantum states. Here, we do so with a quantum state that is pure and entangled. Because it is entangled, the reduced densities that describe subsystems also carry information about the complementary subsystem. This is in contrast to the classical marginal distributions on a subsystem in which information about the complementary system has been integrated out and lost. A training algorithm based on the density matrix renormalization group (DMRG) procedure uses the extra information contained in the reduced densities and organizes it into a tensor network model. An understanding of the extra information contained in the reduced densities allow us to examine the mechanics of this DMRG algorithm and study the generalization error of the resulting model. As an illustration, we work with the even-parity dataset and produce an estimate for the generalization error as a function of the fraction of the dataset used in training.

Probabilistic Modeling with Matrix Product States

Inspired by the possibility that generative models based on quantum circuits can provide a useful inductive bias for sequence modeling tasks, we propose an efficient training algorithm for a subset of classically simulable quantum circuit models. The gradient-free algorithm, presented as a sequence of exactly solvable effective models, is a modification of the density matrix renormalization group procedure adapted for learning a probability distribution. The conclusion that circuit-based models offer a useful inductive bias for classical datasets is supported by experimental results on the parity learning problem.

Language as a matrix product state

We propose a statistical model for natural language that begins by considering language as a monoid, then representing it in complex matrices with a compatible translation invariant probability measure. We interpret the probability measure as arising via the Born rule from a translation invariant matrix product state.