in terms of number of $32$-bit floating point parameters in the embedding layer. We measure performance in terms of MSE loss for CF tasks and accuracy for CTR prediction tasks. We choose to use loss rather than accuracy for CF because we would like to differentiate when the quality of our predictions are closer to the target (e.g. we want to differentiate between predicted ratings of $0$ and $3$ for a target of $4$) in our results. We do not have to address this issue for the binary targets in the CTR prediction problem and therefore use more natural accuracy as a metric for it. In all experiments, we use the Xavier uniform initialization [@glorot2010understanding], for all matrices in our models.

The parameters are allocated to embedding layers according to the $\alpha$-parameterized rule proposed in Alg. \[alg:alpha\]. Notice that for models with uniform dimension embedding layers ($\alpha = 0$), the number of parameters directly controls the embedding dimension because the number of vectors per embedding is fixed. On the other hand, for mixed dimension layers ($\alpha > 0$) the fixed parameters lead to skewed dimensions. In fact the more alpha increases the more skewed dimension assignments become based on popularity. 