Welcome to MathOverflow

Question

0

Hi.

I need to do max. likelihood for an objective likelihood function L (minimize it), and the target is a matrix. ie:

$$min_KL(K)$$

For example: K is, let's say, of size 3x3 and with initial value of ones ($k_{i,j}=1∀i,j$). L is $L=∥\nabla(K)∥$ or $L=∥K∥^{1.1}$.

I know how to do gradient descend etc., but here I need to minimize the function L by iterating over K and I don't really know how to approach it. I'd expect something of this sort: $K:=K-f(\nabla(L))$, but I don't know what.

*note: It might have something to do with the Euler-Largange method ($L_x-L_t\left(L_{x'}\right)=0$) but I'd have to do it iteratively if any...

Appreciate any help.

edited Jan 10 2012 at 11:36

asked Jan 10 2012 at 10:50

id0
31●3

	I think you will have to give significantly more information to get a useful answer. Optimization is a large field and you haven't restricted your problem very much so it is hard to say what methods would apply. Is $K$ constrained? If not, your second objective looks like it would be optimized at $K=0$. As for the first objective, it is not clear to me what $\nabla K$ means for a single matrix $K$. Finally, saying that you "need to" solve the problem in a certain way suggests homework help, which is well-received at math.stackexchange.com but generally not at MO as per the FAQ. – Noah Stein Jan 10 2012 at 13:47
	Your notation is quite unclear. By $\\| K \\|^{1.1}$, do you mean the $p$ norm of $K$ with $p=1.1$?, or do you mean the $2$-norm of $K$ raised to the power 1.1? In the latter case, raising the norm to the power 1.1 has no effect on the minimum. By $\nabla K$, do you mean the kind of image gradient often used in image processing? It seems odd that you'd be applying that to a 3x3 matrix. Or, do you mean something else? – Brian Borchers Jan 11 2012 at 6:20
	This is not a HW question, I didn't specify any other terms because I'm not sure about what I want to do yet. Obviously K=0 is a solution but since I do it iteratively I may not get there. The general form of L would be rather the sum of both L's I specified, so the exponents have a meaning. Also, I didn't decide yet whether I want the Frobenius or L2 norm. My question is more technical - Say I have a matrix K of size 3x3, how do I iteratively modify it so that I minimize L. Thanks. – id0 Jan 11 2012 at 7:11

Optimization of a matrix with an objective function (for ML)

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.