Solve matrix equation in matlab

问题:

I have an equation of the type c = Ax + By where c, x and y are vectors of dimensions say 50,000 X 1, and A and B are matrices with dimensions 50,000 X 50,000.

Is there any way in Matlab to find matrices A and B when c, x and y are known?

I have about 100,000 samples of c, x, and y. A and B remain the same for all.

回答1:

Let X be the collection of all 100,000 xs you got (such that the i-th column of X equals the x_i-th vector).
In the same manner we can define Y and C as 2D collections of ys and cs respectively.

What you wish to solve is for A and B such that

C = AX + BY

You have 2 * 50,000^2 unknowns (all entries of A and B) and numel(C) equations.

So, if the number of data vectors you have is 100,000 you have a single solution (up to linearly dependent samples). If you have more than 100,000 samples you may seek for a least-squares solution.

Re-writing:

C = [A B] * [X ; Y]  ==>  [X' Y'] * [A';B'] = C'

So, I suppose

[A' ; B'] = pinv( [X' Y'] ) * C'

In matlab:

ABt = pinv( [X' Y'] ) * C';
A = ABt(1:50000,:)';
B = ABt(50001:end,:)';

Correct me if I'm wrong...

EDIT:
It seems like there is quite a fuss around dimensionality here. So, I'll try and make it as clear as possible.

Model: There are two (unknown) matrices A and B, each of size 50,000x50,000 (total 5e9 unknowns).
An observation is a triplet of vectors: (x,y,c) each such vector has 50,000 elements (total of 150,000 observed points at each sample). The underlying model assumption is that an observation is generated by c = Ax + By in this model.
The task: given n observations (that is n triplets of vectors { (x_i, y_i, c_i) }_i=1..n) the task is to uncover A and B.

Now, each sample (x_i,y_i,c_i) induces 50,000 equations of the form c_i = Ax_i + By_i in the unknown A and B. If the number of samples n is greater than 100,000, then there are more than 50,000 * 100,000 ( > 5e9 ) equations and the system is over constraint.

To write the system in a matrix form I proposed to stack all observations into matrices:

• A matrix X of size 50,000 x n with its i-th column equals to observed x_i
• A matrix Y of size 50,000 x n with its i-th column equals to observed y_i
• A matrix C of size 50,000 x n with its i-th column equals to observed c_i

With these matrices we can write the model as:

C = A*X + B*Y

I hope this clears things up a bit.

Thank you @Dan and @woodchips for your interest and enlightening comments.

EDIT (2):
Submitting the following code to octave. In this example instead of 50,000 dimension I work with only 2, instead of n=100,000 observations I settled for n=100:

n = 100;
A = rand(2,2);
B = rand(2,2);
X = rand(2,n);
Y = rand(2,n);
C = A*X + B*Y + .001*randn(size(X)); % adding noise to observations 
ABt = pinv( [ X' Y'] ) * C';

Checking the difference between ground truth model (A and B) and recovered ABt:

ABt - [A' ; B']

Yields

  ans =

   5.8457e-05   3.0483e-04
   1.1023e-04   6.1842e-05
  -1.2277e-04  -3.2866e-04
  -3.1930e-05  -5.2149e-05

Which is close enough to zero. (remember, the observations were noisy and solution is a least-square one).