User talk:Ahsh

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Problem Statement:

The underlying intelligent tools behind the webshoppers such as Amazon, Netflix, and Apple learn a suggestion function based on the the current user's and the others ratings in order to offer personalized recommendations. To this end, collaborative filtering provided a promising approach in which the rating patterns (of the products) by the current user and the others are used to estimate rates (or ranking) for unrated items. The task is more challenging once the user is unknown for the system (i.e., there is not any rating records from this user). Two different strategies might be incorporated for offering the recommendation list: rating or ranking. Ranking is different from rating in which the set of recommendations is obtaind directly, rather than first finding the rates and then sort them accordingly. For collaborative ratings, Maximum Marging Matrix Factorization (MMMF) had a promising result for estimating the unknown rates. This paper extends the use of MMMF for collaborative ranking. Since the top ranked items (products) are offered to the user, it is more important to predict what a user might like than what he/she dislikes. In other words, the items with higher rank should be ranked more accuratly than the last ones.

Objectives:

The algorithm should

1- directly optimize the ranking scores, 2- be adaptable to different scores, 3- not need any features extraction besides the actual ratings, 4- be scalable and parralizable with large number of items and users.

Definitions:

Polya-Littlewood-Hardy inequality

For any two vectors a, b , their inner product is maximized when a, b are sorted in the same order. That is [math]\displaystyle{ \lt a,b\gt \lt = \lt sort(a),sort(b)\gt }[/math]

Normalized Discounted Comulative Gain

Consider the rating vector [math]\displaystyle{ y\in\{1,...,r\}^{n} }[/math], and [math]\displaystyle{ \pi }[/math] the permutation of the rating vector. For the trucation threshold [math]\displaystyle{ k }[/math] (the number of recommendations), the Discounted Comulative Gains (DCG) score is defined as: [math]\displaystyle{ DCG@k(y,\pi)=\sum_{i=1}^{k}\frac{2^{y_{\pi_i}-1}}{log(i+2)} }[/math] Given the permutation [math]\displaystyle{ \pi_s }[/math] which sorts the [math]\displaystyle{ y }[/math] in decreasing order, the Normalized Discounted Comulative Gains (NDCG) score is given by:[math]\displaystyle{ NDCG@k(y,\pi)=\frac{DCG@k(y,\pi)}{DCG@k(y,\pi_{s})} }[/math]

The intution of the DCG score is that according to Polya-Littlewood-Hardy inequality, since the weighting [math]\displaystyle{ Insert formula here }[/math] is a monotinically decreaing function, we can get the highest score for the case in which the [math]\displaystyle{ y }[/math] is decreasingly ordered.