# Language, trees, and geometry in neural networks

Language is made of discrete structures, yet neural networks operate on continuous data: vectors in high-dimensional space. A successful language-processing network must translate this symbolic information into some kind of geometric representation—but in what form? Word embeddings provide two well-known examples: distance encodes semantic similarity, while certain directions correspond to polarities (e.g. male vs. female).

A fresh and interesting finding suggests a completely new form of representation. The syntactic structure of a sentence is an important piece of linguistic information. This structure may be represented as a tree, with nodes corresponding to sentence words. In A structural probe for discovering syntax in word representations, Hewitt and Manning demonstrate that various language-processing networks build geometric replicas of such syntax trees. Words are assigned places in a high-dimensional space, and the Euclidean distance between these sites translates to tree distance (through a transformation).

This finding, however, is accompanied with an intriguing dilemma. The relationship between tree distance and Euclidean distance is not a straight line. Hewitt and Manning discovered that tree distance equals the square of Euclidean distance. They wonder whether squaring distance is required and if there are alternative potential translations.

Squared-distance mappings of trees are particularly intuitive from a mathematical standpoint. Certain randomized tree embeddings will also follow an approximate squared-distance law. Furthermore, simply understanding the squared-distance connection enables us to provide a straightforward, clear description of the overall form of a tree embedding.

Any two Pythagorean embeddings of the same tree are isometric—and related by a rotation or reflection—since distances between all pairs of points are the same in both. So we may speak of the Pythagorean embedding of a tree, and this theorem tells us exactly what it looks like.

Many people have studied these embeddings to see what sort of information they might contain. In our terminology, the context embeddings approximate a Pythagorean embedding of a sentence’s dependency parse tree. That means we have a good idea—simply from the squared-distance property and Theorem 1.1—of the overall shape of the tree embedding.

To begin, the context embeddings must be transformed using a specific matrix B, known as a structural probe. However, the square of the Euclidean distance between the context embeddings of two words approximates the parse tree distance between the two words.

“Here is where the math in the previous section pays off. In our terminology, the context embeddings approximate a Pythagorean embedding of a sentence’s dependency parse tree. That means we have a good idea—simply from the squared-distance property and Theorem 1.1—of the overall shape of the tree embedding.”

The precise form is unknown because the embedding is only roughly Pythagorean. However, the disparity between the desired form and the actual shape has the potential to be quite fascinating. Systematic discrepancies between empirical embeddings and their mathematical idealization may give further information about how BERT analyzes language.