# SVD++

(Redirected from SVDPlusPlus)

SVD++ refers to a matrix factorization model 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:

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

where $N(u)$ 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

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

Here $Ufeed(u)$ 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). $\alpha_i$ is a feature weight associates with the user feedback information. With the most two common choices: (1) $\frac{1}{\sqrt{|N(u)|}}$ for implicit feedback, (2) $\frac{r_{uj} - b_u}{\sqrt{|R(u)|}}$ for explicit feedback.

## Learning

SVD++ can be trained using ALS.

It is 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.  describes efficient training with user feedback information in section 4