14 · CP Factorization and the Rank-1 View

Deep dive 14 of Tensors for Machine Learning. Section 11 fitted CP and compared its storage with Tucker’s. This deep dive asks the other question:

What is a single CP component, and what can you read off it?

One component is one vector per axis: a profile over its own axis, with peaks you can point at. A Tucker core mixes every basis vector with every other, so there is no component to read. That is the reason people reach for CP.

You fit it by hand. The full problem is not convex, but fix every factor except one and it becomes the least squares of section 07. That is alternating least squares, and it is the whole algorithm. Then non-negativity, and a real tensor: Tamara Kolda’s demo of 43 neurons × 200 time steps × 88 trials, recorded while a monkey moved a cursor to one of four targets. The trial factor sorts the trials by target — a label the method never saw.

In the six-hour workshop this deep dive is a 20-minute live lab: the predict-first question and Exercise 1, a rank-1 tensor and its lam. Alternating least squares, non-negative CP and the hidden label go home.

Study here, then run the real thing: notebook 14 on Colab · the notebooks · one rank-1 term, interactive

Built from the workshop’s own material by Ravi Kalia and Sebastian Laverde Chunza, CC BY 4.0.