This is a longer version of my recent Language Corner post. Language Corner is a feature I write regularly on LinkedIn. It is focused on linguistics, so I cut most of the quantitative material and moved it here. I motivated that post with a reference to Semantle, a popular online word game, and showed how the game uses cosine similarity.
Cosine Similarity
The target word that day was “inmate.” The nearest word to “inmate,” with a similarity score of 81, is “inmates,” plural. The nearest word after that is “prisoner,” a synonym, with a score of 72. These scores can be a little misleading because they’re not a straight scale. They are “cosine similarity” in the Word2Vec pretrained embedding model.
Most of my coursework in linguistics focused on syntax – why words and word-parts are where they are in a sentence. The state of the art in semantics was observing that words reside in “semantic fields.” Today, thanks to AI language models, we can study semantic distance with precision.
For instance, “gold” denotes both a metal and a currency. It occupies the intersection of at least two different semantic fields. One, it shares with iron and tungsten. The other, with bitcoin and chalcedony beads. Instead of sets intersecting, as in a Venn diagram, I always pictured them as axes crossing.
Gold sits at the corner of metals and currency. Gold is also a color, so we’re going to need at least three dimensions. Word2Vec uses 300 – which is small, as embeddings go.
Word Embeddings
The great success of Large Language Models (LLMs) owes to their training data being self-labeled. Training a neural network generally requires tons of pre-labeled data. You feed in images from ImageNet, your neural net tries to guess the subject, and the labels provide feedback. By the way, I wrote an explainer on deep neural networks back in 2021, before the LLM mania took over.
The task assigned to an LLM is simply to predict the next word in a text. So, each word is a test case. The context leading up to the word is the input, and the word itself is the label. Word embeddings arise as a byproduct of this activity.
Inputs to a neural network need to be encoded in some way. Quantitative variables are straightforward, but others are not. An embedding is a layer in the neural network that allows it to learn the most suitable representation for a given input. Here is an example from a project where I needed to accept car models as input:
carmodel_inputs = keras.layers.Input(shape=[], dtype="string") carmodel_indices = keras.layers.Lambda( lambda carmodel: table.lookup(carmodel) )(carmodel_inputs) models_embed = keras.layers.Embedding(input_dim=table_len, output_dim=10)( carmodel_indices ) all_inputs = keras.layers.concatenate([num_scaled, cats_encoded, models_embed])
This is an efficient way to handle the encoding problem, and it has the advantage that you can save the embedding for later use. The developers of Word2Vec ran their text-completion training including an embedding with output_dim = 300, and then saved the embedding.
It’s important to note that word embeddings are built on syntactic substitutability. The word nearest to “north” is “south.”
The Word Globe
Semantle calculates similarity as the cosine (times 100) of the angle between the two vectors that represent the words. Imagine if all the words in the language were arranged on the globe, in three dimensions instead of 300, with word-vectors radiating from the center of the Earth.
The cosine of 44 degrees is 0.72, the similarity between “inmate” and “prisoner.” An arc of 44 degrees, on planet Earth, would be roughly 3,000 miles – the distance from Boston to Dublin. If that seems like a long flight to reach the next synonym, think of how many words are totally unrelated (cos = 0) and how much room they have to spread out in 300 dimensions.
A single vector in 300-D space has an entire 299-D subspace (its orthogonal complement) of words totally unrelated to it. Below is the distribution of similarity scores relative to “inmate,” calculated on the Word2Vec embedding. Most words are near zero. Those above 0.60 are too few to see on the chart.
Nonetheless, this word-globe is key to understanding how to beat Semantle. The first guess gives the absolute distance from “giant.” That’s a circle. The second guess gives the absolute distance from “enemy.” That’s another circle (actually, they’re 298-dimensional hyperspheres, but you get the idea).
You can now load the Word2Vec embedding into a short Python script and calculate where the two circles cross.
Principal Component Analysis
Now that Semantle is licked, we come back to the semantic fields containing “gold.” Gold is the textbook example because it participates in a few distinct fields. I chose a handful of words, all within striking distance, and projected them down to three dimensions – from fifteen hundred. For this exercise, I have switched to OpenAi’s text-embedding-3-small, which I have written about previously.
Readers of this blog are probably familiar with Principal Component Analysis (PCA). This algorithm finds the one dimension that captures the most variance across the sample, and then the second most, etc., finally projecting the points onto this subspace.
I feel good about this projection. The three principal components explain a respectable 35% of the total variance:
- PC1: 16.2%
- PC2: 11.0%
- PC3: 7.4%
PC1 is the color-wealth axis, with ordinary colors on the right, “money” and “wealth” on the left, and “gold” in the middle. It’s not surprising that PC1 captures most of the variance. Apart from “gold,” colors have nothing to do with wealth.
PC2 captures the “metal” concept omitted by PC1. Industrial metals like “iron” and “palladium” are on the right. Lastly, PC3 is the wealth-metal axis, with “bullion” on the left and “iron” on the right. Colors sit this one out, with values near zero.
