Two Empty Rows

A marketplace serves recommendations from a latent factor model, $\hat{r}_{ui} = q_i^{\top} p_u$ , retrained nightly on the interaction log and nothing else. This morning it must rank items for a user who signed up an hour ago, and it must decide whether to show a product listed an hour ago.

Which assessment is correct?

Initialising both new vectors to the mean of the existing user and item factors resolves it exactly, since the mean is the optimum under the regulariser given no observations.

Neither $p_u$ nor $q_i$ appears in any term of the training loss, so both stay untrained: score the new item from content attributes (category, title text, images) and the new user from a popularity prior or a short onboarding, since both fixes bring in a signal from outside the interaction matrix.

Only the new item is a problem: the new user can be scored immediately because all the item factors $q_i$ are already trained from other users' interactions.

Raising the number of latent factors and training more epochs resolves both, because a higher-capacity model extrapolates the empty rows from the rest of the matrix.