Multilinear rank

Multilinear rank — one retained rank per mode, for example (2, 2, 3). Spanish: rango multilineal.

This is what Tucker gives you that PCA cannot: a separate budget for each axis. The taxi tensor starts at 2 directions for pickup borough, 2 for dropoff borough and 3 for hour of day. A long axis does not need a large rank. In compression golf the hour rank falls to 1, because the trips share one daily shape, and the error lives in the borough ranks: from (2, 2, 1), raising the hour rank to 3 buys 0.44 points of error, while raising both borough ranks to 3 buys 2.45.

A rank is a budget, not a subset. Rank 3 on the hour mode buys three directions to describe variation across all 24 hours — it does not keep three hours and throw away 21.