I have a training set with $N$ instances $\{I_1,…I_N\}$,
where each pair of instances is associated with a similarity score $S(I_x,I_y)\in [0,1]$ indicates if the two instances are similar or not.
I have developed $M$ similarity functions $\{S_1,…,S_M\}$, each of which is based on a different feature vector I extract from the two instances at the pair
$S_m(f_m(I_x), f_m(I_y))\in [0,1]$.
Note that these similarity functions are probably correlated in some way.
Given these functions and the my training set, I want to learn a unified similarity prediction function $P$ such that $P=\arg\min_P \|P(I_x,I_j)-S(I_x,I_j)\|^2$.
What is the best way to achieve such a $P$?
Best Answer
Welcome to the field of metric learning. If you use this as a google search query, you will get lots of material on your problem. Here is a quick idea on how you can do it.
One way is to find coefficients $\alpha_m$ for each of your similarity functions, and combine them into a global similarity: $S(I_x, I_y) = \frac{1}{M} \sum_m \alpha_m S_m(I_x, I_y)$. Given the squared error, this is a linear least squares problem.
One key issue with metric learning is that it the targets scale quadratically with the number of samples. This might be a hindrance for some least squares procedures, and you might have to resort to a stochastic gradient based optimization technique.