ConvNCF

prior research

• MF Koren, Y., Bell, R., & Volinsky, C. (2009). Matrix factorization techniques for recommender systems. Computer, 42(8), 30-37.
  • 행렬 분해(matrix factorization)는 쌍선형 가정 하에 엔티티의 사상 공간만을 모델링할 뿐(representation learning) 매칭 함수(matching function)를 모델링하지 않아 고차 상호작용을 포착하기 어려움
• NCF He, X., Liao, L., Zhang, H., Nie, L., Hu, X., & Chua, T. S. (2017, April). Neural collaborative filtering. In Proceedings of the 26th international conference on world wide web (pp. 173-182).
  • mlp(multi-layer perceptron) 기반 매칭 함수(matching function)는 사전 가정의 제약을 받지 않아 데이터에 적합한 여러 매칭 함수를 근사할 수 있지만 그 자유도로 인하여 과적합 위험이 있음

idea

• ConvNCF(Convolutional Neural Collaborative Filtering): 외적(Outer Product)으로 사용자와 아이템의 차원 간 2차 교차 상호작용(pairwise cross-dimensional interaction)을 명시적으로 생성하고, 합성곱 신경망(Convolution Neural Networks)으로 상호작용의 국소적 패턴(higher-order interaction patterns)을 모델링함으로써 자유도(파라미터 수)를 개선하는 단일 모형

  

notation

• $u=1,2,\cdots,M$: user idx
• $i=1,2,\cdots,N$: item idx
• $\mathbf{X}^{\mathrm{(user)}} \in \mathbb{R}^{M \times M}$: user one-hot matrix
• $\mathbf{X}^{\mathrm{(item)}} \in \mathbb{R}^{N \times N}$: item one-hot matrix
• $\mathbf{Y} \in \mathbb{R}^{M \times N}$: user-item interaction matrix
• $\mathbf{p}_{u} \in \mathbb{R}^{K}$: user latent factor vector
• $\mathbf{q}_{i} \in \mathbb{R}^{K}$: item latent factor vector
• $\mathbf{E}_{u,i} \in \mathbb{R}^{K \times K}$: interaction map of user $u$ and item $i$
• $\mathbf{z}_{u,i} \in \mathbb{R}^{K}$: predictive vector of user $u$ and item $i$
• $\hat{y}_{u,i}$: interaction probability of user $u$ and item $i$

function

• index embedding:

  \[\begin{aligned} \mathbf{p}_{u} &=\mathbf{W}\cdot\mathbf{X}_{u*}^{\mathrm{(user)}}\\ \mathbf{q}_{i} &=\mathbf{W}\cdot\mathbf{X}_{i*}^{\mathrm{(item)}} \end{aligned}\]

• outer product of user $u$ and item $i$:

  \[\begin{aligned} \mathbf{E}_{u,i} &=\mathbf{p}_{u}\otimes\mathbf{q}_{i} \end{aligned}\]

• capture interdimensional high-level interaction of user $u$ and item $i$:

  

  \[\begin{aligned} \mathbf{z}_{u,i} &= \mathrm{flatten}\left[\mathrm{conv}_{\mathrm{ReLU}}\left(\mathbf{E}_{u,i}\right)\right] \end{aligned}\]
  • \(\mathcal{W} \in \mathbb{R}^{2 \times 2}\): kernel window dimension
  • the number of filters per kernel size is $32$ (empirical args)
  • dimension of feature map is reduced $K \times K, K/2 \times K/2, \cdots, 1 \times 1$

• predict interaction probability of user $u$ and item $i$:

  \[\begin{aligned} \hat{y}_{u,i} &=\sigma\left(\mathbf{W}\cdot\mathbf{z}_{u,i}\right) \end{aligned}\]