You now have embeddings for any pair of examples. A supervised similarity measure takes these embeddings and returns a number measuring their similarity. Remember that embeddings are vectors of numbers. To find the similarity between two vectors \(A = [a_1,a_2,...,a_n]\) and \(B = [b_1,b_2,...,b_n]\), choose one of these three similarity measures:
| Measure | Meaning | Formula | As similarity increases, this measure... |
|---|---|---|---|
| Euclidean distance | Distance between ends of vectors | \(\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+...+(a_N-b_N)^2}\) | Decreases |
| Cosine | Cosine of angle \(\theta\) between vectors | \(\frac{a^T b}{|a| \cdot |b|}\) | Increases |
| Dot product | Cosine multiplied by lengths of both vectors | \(a_1b_1+a_2b_2+...+a_nb_n\) \(=|a||b|cos(\theta)\) | Increases. Also increases with length of vectors. |
Choosing a similarity measure
In contrast to the cosine, the dot product is proportional to the vector length. This is important because examples that appear very frequently in the training set (for example, popular YouTube videos) tend to have embedding vectors with large lengths. If you want to capture popularity, then choose dot product. However, the risk is that popular examples may skew the similarity metric. To balance this skew, you can raise the length to an exponent \(\alpha\ < 1\) to calculate the dot product as \(|a|^{\alpha}|b|^{\alpha}\cos(\theta)\).
To better understand how vector length changes the similarity measure, normalize the vector lengths to 1 and notice that the three measures become proportional to each other.
- Euclidean distance = \(||a-b|| = \sqrt{||a||^2 + ||b||^2 - 2a^{T}b} = \sqrt{2-2\cos(\theta_{ab})}\).
- Dot product = \( |a||b| \cos(\theta_{ab}) = 1\cdot1\cdot \cos(\theta_{ab}) = cos(\theta_{ab})\).
- Cosine = \(\cos(\theta_{ab})\).
Review of similarity measures
A similarity measure quantifies the similarity between a pair of examples, relative to other pairs of examples. The two types, manual and supervised, are compared below:
| Type | How to create | Best for | Implications |
|---|---|---|---|
| Manual | Manually combine feature data. | Small datasets with features that are straightforward to combine. | Gives insight into results of similarity calculations. If feature data changes, you must manually update the similarity measure. |
| Supervised | Measure distance between embeddings generated by a supervised DNN. | Large datasets with hard-to-combine features. | Gives no insight into results. However, a DNN can automatically adapt to changing feature data. |