memory budgets there is an $\alpha > 0$ that produces mixed embeddings that are equivalent or better than uniform dimension embeddings ($\alpha = 0$).

Finally, we point out that the optimal $\alpha$ is dependant on the total memory budget. Thus, at any given budget, one should tune $\alpha$ as a hyperparameter. *Ultimately, we are able to achieve approximately 0.02 lower MSE while using approximately 4$\times$ fewer parameters by using mixed dimension embeddings.*

Neural Collaborative Filtering
------------------------------

NCF models represent more modern approaches for CF. From the perspective of this work, NCF models are interesting because they add a moderate degree of realism and show how the presence of the non-linearity in neural layers affects the results.

In our experiments we use NCF with a 3-layer MLP with dimension 128. We train at learning rate $10^{-3}$ using Amsgrad optimizer for 100 epochs, taking model that scored the lowest validation loss. We still use the same dataset and run 3 random seeds for each point in the experiments.

In Fig. \[fig:ncf\_train\_loss\_sample\], we report learning curves for the NCF model based on three different embedding layers, plotting the number of training epochs against the validation loss for each epoch. We plot the learning 