The Maximum Inner Product Search (MIPS) problem, prevalent in matrix factorization-based recommender systems, scales linearly with the number of objects to score. Recent work has shown that clever post-processing steps can turn the MIPS problem into a nearest neighbour one, allowing sublinear retrieval time either through Locality Sensitive Hashing or various tree structures that partition the Euclidian space. This work shows that instead of employing post-processing steps, substantially faster retrieval times can be achieved for the same accuracy when inference is not decoupled from the indexing process. By framing matrix factorization to be natively indexable, so that any solution is immediately sublinearly searchable, we use the machinery of Machine Learning to best learn such a solution. We introduce Indexable Probabilistic Matrix Factorization (IPMF) to shift the traditional post-processing complexity into the training phase of the model. Its inference procedure is based on Geodesic Monte Carlo, and adds minimal additional computational cost to standard Monte Carlo methods for matrix factorization. By coupling inference and indexing in this way, we achieve more than a 50% improvement in retrieval time against two state of the art methods, for a given level of accuracy in the recommendations of two large-scale recommender systems.