Difference between revisions of "Open Problems:62"

From Open Problems in Sublinear Algorithms
Jump to: navigation, search
(Created page with "{{Header |title=PCAs |source=bertinoro14 |who=Andrea Montanari }} ???")
 
m (Krzysztof Onak moved page Waiting:Andrea Montanari to Open Problems:62)
 
(8 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 
{{Header
 
{{Header
|title=PCAs
 
 
|source=bertinoro14
 
|source=bertinoro14
 
|who=Andrea Montanari
 
|who=Andrea Montanari
 
}}
 
}}
???
+
 
 +
Given a symmetric matrix $A$, we can think of Principal Component Analysis (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:
 +
 
 +
\[ \max \operatorname{Tr}(WA) \quad \text{s.t.} \quad \operatorname{Tr}(W) = 1,\ \ W \ge 0,\ \ W \succeq 0. \]
 +
 
 +
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 $A = v v^T + (\text{small amount of noise})$.

Latest revision as of 22:56, 13 June 2014

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

Given a symmetric matrix $A$, we can think of Principal Component Analysis (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:

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

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 $A = v v^T + (\text{small amount of noise})$.