When the rank of $M$ is a known priori, it can be used to set the dimension of the embedding layer, otherwise it is treated as a hyperparameter.

### Neural Collaborative Filtering

Although NCF lacks the nice theoretical guarantees offered by MF, it empirically performs mildly better on real-world datasets [@he2017neural]. Since we are primarily concerned with the embedding layer, we adopt the simplest NCF model where user and item embeddings are concatenated to form an input to the following multilayer perceptron (MLP).

In NCF we define $\mathcal{A}$ as solving the following optimization problem

$$\min_{W, V, \theta} \sum_{i,j \in \Omega} (F([\textbf{w}_i,  \textbf{v}_j], \theta) - M_{ij})^2$$

where $W$ and $V$ are embeddings, while $\theta$ denotes additional parameters of the MLP, denoted by function $F$.

Click-Through Rate Prediction Problem
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CTR prediction tasks can be interpreted as event-probability prediction problem or as context-based CF with binary ratings. Compared to canonical CF, these tasks also include a large amount of context which can be used to better predict user and item interaction events. Therefore, this problem can be viewed as restricting targets $M_{ij} \in \{0, 1\}$, while allowing user and items to be represented by multiple features, often expressed through sets of indices 