Learning User Preferences for Sets of Objects

by Marie desJardins, Eric Eaton, Kiri L. Wagstaff

In the setting of the proposed article, we only have positive samples. The difficulty of so-called preference learning problems lies on the portfolio effect, which says putting the best together won't necessarily generates the best group. The authors proposed a language DD-PREF, which takes this effect into consideration.

The language is a tuple of four factors, P=(q, d, w, alpha). q is called depth, namely the degree that the user prefers on several features, which can be estimated with kernel density estimators(KDEs) using the positive samples. The total depth of a given set s is the weighted sum of average feature depth:

V_dep(s|P) = sum_{over f} w_f * sum_{over x in s} q_f(x^f) /|s|

And there is another factor called diversity. d is used to specify the desired diversity. In real computation, it is simply the average diversity over the positive set. The total diversity is:

V_div(s|P) = sum_{over f} w_f * ( 1 - (d_f - div_f(s) )² )

And the final preferences is ranked according to the weighted(alpha) sum of V_dep and V_div.

Then the learning procedure requires q, d, w while alpha is set according to application. Find w requires an optimization technique called BFGS.

Here we have to learn more about the following techniques:
KDE, two references, one is Duda's Pattern Classification, the other is Representing data distributions with kernel density estimates.
BFGS, I think it is included in Neumeric Optimization.