02-09-fully-observed-rank-one-model

#eth/cil/theory

Fully observed Rank 1 Model

Useful identiy:

| | R | |_{F}^{2} = tr (R^{T} R)

We can then rewrite the objective:

l (u, v) = \frac{1}{2} | | A - u v^{T} | |_{F}^{2}

Directionality of $u, v$ is purely determined by the last term:

(u, v) \to max {u^{T} A v}, s.t. | | u | | = | | v | | = 1

This can be solved with Lagrange multipliers:

L = u^{T} A v - λ u \cdot u - μ v \cdot v

First order optimality condition:

\nabla_{u} L = A v - 2 λ u \overset{!}{=} 0 \Rightarrow u = \frac{A v}{| | A v | |}