Problem 62: Principal Component Analysis with Nonnegativity Constraints

Suggested by	Andrea Montanari
Source	Bertinoro 2014
Short link	https://sublinear.info/62

Given a symmetric matrix $A$, we can think of PCA as maximizing $x^\top A x$ subject to $\|x\|=1$. If we also add the condition $x \ge 0$, this problem becomes NP-hard. We can define a convex relaxation:

\[ \max Tr(WA) \ \text{s.t}\ Tr(W) = 1, W \ge 0, W \succeq 0 \]

Suppose that $A$ is a random matrix: in particular, set $A_{ij}$ to be i.i.d $N(0,1)$. Then empirical results show that the resulting $W$ is a rank-1 matrix, which means that we recover the optimal $x$ exactly.

Is this true in general ? Note that we can prove that the solution is rank 1 if $A = v v^T + $ small amounts of noise.