Large Language Models are transformer neural networks which are trained to produce a probability distribution on the possible next words to given texts in a corpus, in such a way that the most likely word predicted is the actual word in the training text. In this paper we find what is the mathematical structure defined by such conditional probability distributions of text extensions. Changing the view point from probabilities to -log probabilities we observe that the subtext order is completely encoded in a metric structure defined on the space of texts $\mathcal{L}$, by -log probabilities. We then construct a metric polyhedron $P(\mathcal{L})$ and an isometric embedding (called Yoneda embedding) of $\mathcal{L}$ into $P(\mathcal{L})$ such that texts map to generators of certain special extremal rays. We explain that $P(\mathcal{L})$ is a $(\min,+)$ (tropical) linear span of these extremal ray generators. The generators also satisfy a system of $(\min+)$ linear equations. We then show that $P(\mathcal{L})$ is compatible with adding more text and from this we derive an approximation of a text vector as a Boltzmann weighted linear combination of the vectors for words in that text. We then prove a duality theorem showing that texts extensions and text restrictions give isometric polyhedra (even though they look a priory very different). Moreover we prove that $P(\mathcal{L})$ is the lattice closure of (a version of) the so called, Isbell completion of $\mathcal{L}$ which turns out to be the $(\max,+)$ span of the text extremal ray generators. All constructions have interpretations in category theory but we don't use category theory explicitly. The categorical interpretations are briefly explained in an appendix. In the final appendix we describe how the syntax to semantics problem could fit in a general well known mathematical duality.

Continuous-valued word embeddings learned by neural language models have recently been shown to capture semantic and syntactic information about words very well, setting performance records on several word similarity tasks. The best results are obtained by learning high-dimensional embeddings from very large quantities of data, which makes scalability of the training method a critical factor. We propose a simple and scalable new approach to learning word embeddings based on training log-bilinear models with noise-contrastive estimation. Our approach is simpler, faster, and produces better results than the current state-of-theart method. We achieve results comparable to the best ones reported, which were obtained on a cluster, using four times less data and more than an order of magnitude less computing time. We also investigate several model types and find that the embeddings learned by the simpler models perform at least as well as those learned by the more complex ones.

Continuous-valued word embeddings learned by neural language models have recently been shown to capture semantic and syntactic information about words very well, setting performance records on several word similarity tasks. The best results are obtained by learning high-dimensional embeddings from very large quantities of data, which makes scalability of the training method a critical factor. We propose a simple and scalable new approach to learning word embeddings based on training log-bilinear models with noise-contrastive estimation. Our approach is simpler, faster, and produces better results than the current state-of-the art method of Mikolov et al. (2013a). We achieve results comparable to the best ones reported, which were obtained on a cluster, using four times less data and more than an order of magnitude less computing time. We also investigate several model types and find that the embeddings learned by the simpler models perform at least as well as those learned by the more complex ones.