# SVD++

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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. [1] describes efficient training with user feedback information in section 4