How can we find (approximatly) pricipal component without explicit calculating covariance matrix ?
- From: "shna" <nsh1979@xxxxxxxxxxxxx>
- Date: Wed, 5 Apr 2006 13:07:17 +0900
Hi, all.
Recently, I have been front of a problem which is formulated as follows .
Let v_1 and v_2 be given vectors.
C = v_1 v_1^t + v_2 v_2 ^ t
is semi-positive definite matrix.
We want to vector v to approximate well C
which decomposed into v v^t.
In other words,
v = argmin || C - v v ^t ||
where ||.|| is matrix norm (F-norm).
In fact, v will be the pricipal vector given C.
However, the problem is that feature vector v_1 and v_2 are very
high-dimensional matrix (more than 100,000).
So, C cannot be calculated effectively.
We want to approximate v without calculate C.
Do you have any idea or reference for this procedure ??
For example (it is only an example),
I can derive the following procedure for approximating v.
First, let's rewrite the above formula as follows.
C = v v^t ( Here, we used '=' instead of '~'. ) - (A)
v_1 v_1^t + v_2 v_2 ^ t = v v^t
If we multiply an arbitrary vector x on both side,
then
v_1 v_1^t x + v_2 v_2 ^ t x = v v^t x - (B)
From the above formula,we obtain a fixed point equation as like.
v <- ( v_1 (v_1^t x) + v_2 (v_2^t x) ) / (v^t x) - (*)
Then, this solution (if it can be one of solution) is calcuated without
explictly C.
My second question is, is there any theory how (*)-like equation is
effective???.
For third question, (It is somewhat related to the second question),
let v be a vector such that (B) is satisfied for all x.
Then, is there any meaning to find such v for obtaining real solution of
(A).
("Meaning" means that two solution is related with a strong inequility, ...
something like that).
Thank you.
From
Seung-Hoon Na
.
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