Sparse Matrix Factorization Problem

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Aqeel Anwar
Aqeel Anwar 2016-6-27
回答： sufyan masood 2018-8-18
I am trying to solve the following problem
Is their a name for such sparse factorization or something similar. Can anyone please guide me or refer some text from where I can find methods of solving such problem, or may be modify an existing problem to get this one.
Any help in this regard will be highly appreciated. Thanks.
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Chris Turnes
Chris Turnes 2016-6-29
What is the norm on
||AXB-C||?
Is this the operator norm, or the Frobenius norm?

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John D'Errico
John D'Errico 2016-6-29
So you want to solve the problem
A*X*B == C
Subject to minimizing an L_0 norm on X?
You have stated that A and B are full rank. n and m are apparently not equal, but that does not really matter a lot. Assume that
n > m
for no good reason. We have essentially n*m unknowns in X. And we have n*m equations.
This is a linear system in the unknowns, but with full rank matrices. So we have no degrees of freedom to play with.
X = pinv(A)*C*pinv(B);
In general, there will probably be no zero elements in X. That you wish to minimize the L_0 norm, so you wish to minimize the number of non-zeros, (or maximize the number of zeros) is not relevant. As long as A and B are full rank, there will be no flexibility in the solution that I can see.
I wrote the above using pinv. I could also have written it using slash.
X = A\C/B;
In either case, there is simply no flexibility. A and B are full rank, square matrices. For example:
X0 = randn(5,3)
X0 =
-0.58903 1.6555 0.79142
-0.29375 0.30754 -1.332
-0.84793 -1.2571 -2.3299
-1.1201 -0.86547 -1.4491
2.526 -0.17653 0.33351
A = randn(5);
B = randn(3);
C = A*X0*B;
X = pinv(A)*C*pinv(B)
X =
-0.58903 1.6555 0.79142
-0.29375 0.30754 -1.332
-0.84793 -1.2571 -2.3299
-1.1201 -0.86547 -1.4491
2.526 -0.17653 0.33351
X = A\C/B
X =
-0.58903 1.6555 0.79142
-0.29375 0.30754 -1.332
-0.84793 -1.2571 -2.3299
-1.1201 -0.86547 -1.4491
2.526 -0.17653 0.33351
As you can see, in either case, X recovers the original X0.
If it were true that A and(or) B were NOT square matrices, or not full rank, then some things would change. As you have stated it though, there is no way to solve this problem otherwise.
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John D'Errico
John D'Errico 2016-6-29
编辑：John D'Errico 2016-6-29
That changes things, though it was not stated that way, it makes more sense with some flexibility given.
Aqeel Anwar
Aqeel Anwar 2016-6-29
Yes I dont need an exact solution rather constrained solution as mentioned by Torsten. I guess I forgot to put the constrained when typing in LaTeX.
Also i need the matrix X to be sparse, and the measure of sparsity being controlled by the lambda parameter

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sufyan masood
sufyan masood 2018-8-18
hello can any one tell about sparsify subband signal and subband modification of subband signals coeffiects in compressive signal. kindly guide me about it. i will be highly thankful to all for help me counter this problem.

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