SVD++

SVD++ refers to the matrix factorization algorithm which makes use of implicit feedback information. In general, implicit feedback can refer to any kinds of users' history information that can help indicate users' preference.

Model Formalization

The SVD++ model is formally described as following equation:

<math>r_{ui} = \mu + b_u + b_i + \left(p_u + \frac{1}{\sqrt{|N(u)|}}\sum_{j\in N(u)} y_j \right)^T q_i</math>

Where <math>N(u)</math> is the set of implicit information( the set of items user u rated ).

General Formalization for User Feedback Information

A more general form of utilizing implicit/explicit information as user factor can be described in following equation

<math>r_{ui} = \mu + b_u + b_i + \left(p_u + \sum_{i\in Ufeed(u)} \alpha_i y_i \right)^T q_i</math>

Here <math>Ufeed(u)</math> is the set of user feedback information( e.g: the web pages the user clicked, the music on users' favorite list, the movies user watched, any kinds of information that can be used to describe the user). <math> \alpha_i </math> is a feature weight associates with the user feedback information. With the most two common choices: (1) <math>\frac{1}{\sqrt{|N(u)|}}</math> for implicit feedback, (2) <math>\frac{r_{uj} - b_u}{\sqrt{|R(u)|}}</math> for explicit feedback.

Model Learning

• SVD++ can be trained using ALS.
• It's slow to train a SVD++ style model using stochastic gradient descent due to the size of user feedback information, however, an efficient SGD training algorithm can be used.

Efficient SGD Training for SVD++

http://arxiv.org/abs/1109.2271 describes efficient training with user feedback information in section 4

Literature

• Yehuda Koren: Factorization meets the neighborhood: a multifaceted collaborative filtering model, KDD 2008, http://portal.acm.org/citation.cfm?id=1401890.1401944

Implementations

• GraphLab Collaborative Filtering Library has implemented SVD++ for multicore: http://graphlab.org/pmf.html
• SVDFeature is a toolkit designed for feature-based matrix factorization, can be used to implement SVD++ and it's extensions.