# Intuitive difference between cosine similarity and bilinear similarity

Given a pair of strings in vector form $(s_i,s_j)$, I can find cosine similarity of pairs as follows:

$cosine(s_i,s_j)=s_i.*s_j / (\|s_i\|\|s_j\|)$

Similarly, bilinear similarity is defined as:

$BilinearSimilarity(s_i,s_j) = s_i^TWs_j$

I want to know what property of strings does the matrix $W$ captures intuitively? Why Should I prefer Bilinear similarity compared to cosine similarity?

Lets say your vectors $s_i \in \mathbb{R}^2$. Imagine that the only first component of your vector is meaningful but the second component is noise. When you compare two vectors $s_i$ and $s_j$, you should be comparing only their first components. This can be captured through bilinear similarity using $W = \begin{bmatrix}1 & 0\\0&0\end{bmatrix}$, because $$s_i^T W s_j = s_{i1}W_{11}s_{j1} = s_{i1}s_{j1}.$$
Next, consider a case where similarity along the first co-ordinate should be weighted more than similarity along the second co-ordinate. This is captured using $W = \begin{bmatrix}2 & 0\\0&1\end{bmatrix}$.
If you ignore the normalization, cosine similarity is just bilinear similarity with $W = \begin{bmatrix}1 & 0\\0&1\end{bmatrix}$.
• I know bilinear similarity reduces to cosine similarity when $W$ is identity matrix. I wanted to know if the role of the matrix $W$ is more than just weighing different features? Another angle to look at it is bilinear map is just a projection over some space and then computing cosine in that map. What advantage does the projection bring here? – chandresh Sep 16 '18 at 5:21
• Yes, you can eigendecompose $W = U \Lambda U^T$ and think of bilinear similarity as a projection. The bilinear similarity is a more general inner product. The co-ordinates of $s$ may not be the correct space for an application, and you want to allow a transformed space. Imagine you have items with features $s_i$, and you've collected data by asking people "how similar is $i$ to $j$". You can fit a $W$ using this data and use it to estimate the similarity of two new items. – elexhobby Sep 16 '18 at 14:23