He, Xiangnan, et al. “Neural collaborative filtering.” Proceedings of the 26th international conference on world wide web. 2017.

This paper points out that despite the effectiveness of Matrix Factorization for collaborative filtering, its performance can be hindered by the simple choice of the interaction function — inner product which simply combines the multiplication of latent features linearly.

So, it presents a general framework named NCF, short for Neural network-based Collaborative Filtering, by replacing the inner product with a neural architecture that can learn an arbitrary function from data.

And it shows significant improvements of proposed NCF framework over the state-of-the-art methods by experiments on two real-world datasets(movielens, pinterest).

1. Learning from Implicit Data

$\text{Let }M = \text{the number of users}$, $N = \text{the number of items}$

  • Define the user-item interaction matrix from users’ implicit feedback as :
\[\mathbf{Y} = [y_{ui}] \in \mathbb{R}^{M \times N}\\ y_{ui} = \begin{cases} 1, \quad \mbox{if interaction (user u and item i) is observed}\\ 0, \quad \mbox{o.w} \end{cases}\]
  • Predictive model
\[\hat{y}_{ui} = f(u, i \vert \Theta)\]

where $\Theta$ denotes model parameters and $f$ denotes the function that maps model parameter to the predicted score(termed as an interaction function in this paper).

2. Matrix Factorization

MF can be deemed as a linear model of latent factors. See FM post here.

  • Predictive model
\[\text{Let } \mathbf{p}_u = \text{the latent vector for user }u, \mathbf{q}_i = \text{the latent vector for item }i\\ \hat{y}_{ui} = f(u, i \vert \mathbf{p}_u, \mathbf{q}_i) = \mathbf{p}^T_u \mathbf{q}_i = \underset{k=1}{\sum^K}p_{uk}q_{ik}\\ \text{where }K=\text{the dimension of the latent space}\]
  • How the inner product function can limit the expressiveness of MF?

limitation example

From data matrix (a), $u_4$ is most similar to $u_1$, followed by $u_3$, and lastly $u_2$. (jaccard coefficient similarity : $s_{41}(0.6) > s_{43}(0.4) >s_{42}(0.2)$ ). However, in the latent space (b), placing $\mathbf{p}_4$ closest to $\mathbf{p}_1$ makes $\mathbf{p}_4$ closer to $\mathbf{p}_2$ than $\mathbf{p}_3$, incurring a large ranking loss.

  • How to resolve the issue?
    • How about to use a large number of latent factors $K$?
      $\rightarrow$ it may adversely hurt the generalization of the model (e.g. overfitting the data)
    • This paper address the limitation by learning the interaction function using DNNs from data.

3. Neural Collaborative Filtering

3.1 General Framework

general framework

Predictive Model

\[\begin{align*} \hat{y}_{ui} &= f(\mathbf{P}^T\mathbf{v}^U_u, \mathbf{Q}^T\mathbf{v}^I_i \vert \mathbf{P}, \mathbf{Q}, \Theta_f)\\ f(\mathbf{P}^T\mathbf{v}^U_u, \mathbf{Q}^T\mathbf{v}^I_i) &= \phi_{out}(\phi_{X}(\cdots \phi_{2}(\phi_{1}(\mathbf{P}^T\mathbf{v}^U_u, \mathbf{Q}^T\mathbf{v}^I_i)))) \end{align*}\]

where $\mathbf{P} \in \mathbb{R}^{M \times K}, \mathbf{Q} \in \mathbb{R}^{N \times K}$ denotes the latent factor matrix for users and items, respectively
$\Theta_f$ denotes the model parameters of the interaction function $f$.

Learning NCF

Probabilistic approach for learning the pointwise NCF that pays special attention to the binary property of implicit data. By employing a probabilistic treatment for NCF, can address recommendation with implicit feedback as a binary classification problem.

\[p(\mathcal{Y}, \mathcal{Y}^{-} \vert \mathbf{P}, \mathbf{Q}, \Theta_f) = \underset{(u, i) \in \mathcal{Y}}{\prod}\hat{y}_{ui} \underset{(u, i) \in \mathcal{Y}^{-}}{\prod}\left(1- \hat{y}_{ui} \right)\] \[\begin{align*} L &= -\underset{(u, i) \in \mathcal{Y}}{\sum}\log \hat{y}_{ui} - \underset{(u, j) \in \mathcal{Y}^{-}}{\sum}\log (1-\hat{y}\_{uj})\\ &=-\underset{(u, i) \in \mathcal{Y}\cup\mathcal{Y}^{-}}{\sum}y_{ui} \log \hat{y}_{ui} + (1-y\_{ui})\log (1-\hat{y}\_{ui}) \end{align*}\]

where $\mathcal{Y}$ = the set of observed interactions in $\mathbf{Y}$, $\mathcal{Y^{-}}$ = the set of negative instances, all(or sampled from) unobserved interactions.

Input Layer

  • Consists of two feature vectors $\mathbf{v}^U_u$ and $\mathbf{v}^I_i$ that describe user $u$ and item $i$, respectively.
  • Transforming it to a binarized sparse vector with one-hot encoding.

Embedding Layer

  • Fully connected layer that projects the sparse representation to a dense vector
  • The obtained user(item) embedding can be seen as the latent vector for user(item) in the context of latent factor model.

Neural CF Layers

  • The user embedding and item embedding are fed into a multi-layer neural architecture “Neural CF Layers”
  • The dimension of the last hidden layer $X$ determines the model’s capability.

Output Layer

  • The predicted score $\hat{y}_{ui}$
  • Training is performed by minimizing the pointwise loss between $\hat{y}_{ui}$ and its target value $y_{ui}$.

3.2 Generalized Matrix Factorization

  • Show how MF can be interpreted as a special case of NCF framework

Let the user latent vector $\mathbf{p}_u$ be $\mathbf{P}^T\mathbf{v}^U_u$ and item latent vector $\mathbf{q}_i$ be $\mathbf{Q}^T\mathbf{v}^I_i$

Define the mapping function of the first neural CF layer as :

\[\phi_1 (\mathbf{p}_u, \mathbf{q}_i) = \mathbf{p}_u \odot \mathbf{q}_i\]

then project the vector to the output layer :

\[\hat{y}_{ui} = a_{out}(\mathbf{h}^T(\mathbf{p}_u \odot \mathbf{q}_i))\]

This paper implement a generalized version of MF under NCF that uses the sigmoid function as $a_{out}$ and learns $\mathbf{h}$ from data with the log loss.

3.3 Multi-Layer Perceptron

  • MLP model under NCF framework is defined as :
\[\begin{align*} \mathbf{z}_1 &= \phi_{1}(\mathbf{p}_u, \mathbf{q}_i) = \begin{bmatrix} \mathbf{p}_u \\ \mathbf{q}_i \end{bmatrix}\\ \phi_{2}(\mathbf{z}_1) &= a_2(\mathbf{W}^T_2 \mathbf{z}_1 + \mathbf{b}_2)\\ \vdots\\ \phi_{L}(\mathbf{z}_{L-1}) &= a_{L}(\mathbf{W}^T_{L}\mathbf{z}_{L-1} + \mathbf{b}_{L})\\ \hat{y}_{ui} &= \sigma(\mathbf{h}^T \phi_{L}(\mathbf{z}_{L-1})) \end{align*}\]

where $\mathbf{W}_x, \mathbf{b}_x$ and $a_x$ denote the weight matrix, bias vector, and activation function for the $x$-th layer’s perceptron.

3.4 Fusion of GMF and MLP

ncf model

  • GMF : applies a linear kernel to model the latent feature interactions.
  • MLP : uses a non-linear kernel to learn the interaction function from data.
  • Allow GMF and MLP to learn separate embeddings, and combine the 2 models by concatenating their last hidden layer.
\[\phi^{GMF} = \mathbf{p}^G_u \odot \mathbf{q}^G_i\\ \phi^{MLP} = a_L(\mathbf{W}^T_L (a_{L-1}(\cdots a_{2}(\mathbf{W}^T_{2} \begin{bmatrix} \mathbf{p}^M_u \\ \mathbf{q}^M_i \end{bmatrix} + \mathbf{b}_2 )\cdots)) + \mathbf{b}_L)\\ \hat{y}_{ui} = \sigma(\mathbf{h}^T \begin{bmatrix} \phi^{GMF}\\ \phi^{MLP} \end{bmatrix} )\]

4. Experiments

Dataset

  • Movielens(1m) ;transformed each entry as 0 or 1 along whether the user has rated the item
  • Pinterest ;each interaction denotes whether the user has pinned the image to own board

Evaluaion metrics - performance of a ranked list

  • HR(Hit Ratio) : measures whether the test item is present on the top-10 list.
  • NDCG(Normalized Discounted Cumulative Gain) : accounts for the position of the hit by assigning higher scores to hits at top ranks.

Result

result

Summary

  1. It present a neural network architecture to model latent features of users and items and devise a general framework NCF for collaborative filtering based on neural networks.

  2. It show that MF can be interpreted as a specialization of NCF and utilize a multi-layer perceptron to endow NCF modelling with a high level of non-linearities.

  3. Extensive experiments on two real-world datasets show significant improvements of NCF framework over the state-of-the-art methods.

Reference

[1] He, Xiangnan, et al. “Neural collaborative filtering.” Proceedings of the 26th international conference on world wide web. 2017.