Coffee + milk ≠ latte

Yesterday I wrote about the canonical example of how vector embeddings of words add:

king" - man" + woman" queen"

This should be interpreted as saying that the word vector forking, minus the word vector forman, plus the word vector forwoman, is in some sense close to the word vector forqueen.

This post will look at another example. Is the expression

coffee" + milk" latte"

true in some sense?

Notation

In this post I will use foo" to mean the vector embedding of the wordfoo.

Coffee + milk

The cosine similarity between coffee" + milk" and latte" is about 0.63. And for reasons given in the previous post, this is a large value of cosine similarity. But there are 11 words that are more similar to milk" + coffee" than latte". Here are the top 12 matches in order.

1. coffee
2. milk
3. tea
4. drink
5. chocolate
6. cream
7. breakfast
8. ice
9. beer
10. vanilla
11. starbucks
12. latte

There are two questions to resolve. First, why isn'tlatte one of the closest words? Second, why is the cosine similarity large even thoughlatte is not one of the best matches?

Concept arithmetic

When word vector arithmetic works, as in the king and queen example, the vectors combine concepts. If you replace the male gender component of king with a female component, you get a vector close to the vector forqueen.

But when you add the vectors formilk andcoffee, you're not adding concepts, you're adding ingredients.

The concepts ofmilk andcoffee are similar in that they're both common beverages, as are tea and even beer. A latte is a beverage, but it's not as common as milk, coffee, tea, or beer.

Extremely uneven distribution

If you divide word vectors by their norm, you get a point on a high-dimensional sphere. In the case of the glove-twitter-200 vector embedding, you get a point on a sphere in 200 dimensions. As explained in the earlier post, a fairly large cosine similarity corresponds to a tiny portion of the sphere's surface area.

In the example of king" - man" + woman", the vector queen" is the closest match (except for king" itself).

But there are a lot of words whose vectors are within a tiny region around coffee" + milk". And by tiny, I mean a region that accounts for a proportion of the sphere on the order of 10^-23.

The glove-twitter-200 vector list contains vectors for 1.2 million words. If these vectors were roughly evenly distributed on the sphere when normalized, you'd expect each patch representing 10^-6 of the sphere to contain about a word or two. You wouldn't expect a patch taking up 10^-12 of the sphere to contain more than one word, and you certainly wouldn't expect a patch taking up 10^-23 of the sphere to contain 12 words.

Rank order

Rank order based on cosine similarity is more robust than cosine similarity itself. This is an example of a phenomenon that occurs regularly: a metric whose values are dubious might still rank things well. Naive Bayes is another example. It naively computes probabilities in a way that is blatantly wrong, and yet ranking things by these spurious probabilities works well in some cases.

The cosine similarity between king" - man" + woman" and queen" is roughly the same as the cosine similarity between coffee" + milk" and latte." But in the former example, rank order picks outqueen as the best match; rank order works like you'd expect, because you're working with attributes that can be decomposed.

Dog + infant = puppy?

I wouldn't be surprised if the Anglo-Saxon word forpuppy was something likedogchild. The language was full of colorful compound words, such as hronrad (whale-road") for the sea and nosethyrl (nose-hole") for nostril.

Here are the top ten matches for dog" + infant" along with their cosine similarities.

1. dog, 0.819
2. infant, 0.809
3. toddler, 0.734
4. dogs, 0.697
5. puppy, 0.688
6. cat, 0.682
7. pet, 0.676
8. child, 0.671
9. newborn, 0.670
10. baby, 0.650

This shows that puppy" is close to dog" + infant", both in terms of cosine similarity and rank order, though it's not the closet.

This also shows that you have to take the addition of word vectors with a grain of salt. It's no surprise thatpuppy was a good match, but it's surprising thatcat is nearly as good.

The post Coffee + milk latte first appeared on John D. Cook.