Difference between revisions of "Open Problems:62"
| Line 8: | Line 8: | ||
We can define a convex relaxation: | We can define a convex relaxation: | ||
| − | \[ \max Tr(WA) \text{ | + | \[ \max Tr(WA) \ \text{s.t}\ Tr(W) = 1, W \ge 0, W \succeq 0 \] |
| − | 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. | + | 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. | 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. | ||
Revision as of 21:30, 28 May 2014
| 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.
