Rank 1 Model

Bi-linear model:

Non-identifiability: scalar $\alpha \ne 0$: $\mathbf{u}\leftarrow \alpha \mathbf{u},\mathbf{v}\leftarrow \mathbf{(}\mathbf{1}\mathbf{/}\alpha \mathbf{)}\mathbf{v}$

Squared Error?!?

${\mathbf{uv}}^T$ is a rank 1 matrix:

• can be represented as an outer product of two verctors
• every pair of vectors defines a rank 1 matrix via their outer product

Scalar Problem

This induces following gradient field:

Minimas: hyperbola with two branches (red lines)
Use gradient descent.

Saddle point

Origin is a saddle point:

Characteristic Polynomial

The Hessian is indefinite at (0,0) as it has eigenvalues $\pm a$