Problem with minimizing the matrix norm

In summary, the problem discussed is finding the entries of a complex matrix X that minimize the functional Tr((A - B.X)(A - B.X)+), where A and B are complex matrices and + denotes Hermitian conjugate or transpose conjugate. The conversation suggests using Einstein notation and matrix calculus identities to simplify the problem and arrive at a solution.
  • #1
mnb96
715
5
Hello,

I have to to find the entries of a matrix [itex]X\in \mathbb{R}^{n\times n}[/itex] that minimize the functional: [itex]Tr \{ (A-XB)(A-XB)^* \}[/itex], where Tr denotes the trace operator, and * is the conjugate transpose of a matrix. The matrices A and B are complex and not necessarily square.

I tried to reformulate the problem with Einstein notation, then take the partial derivatives with respect to each [itex]a^{i}_{j}[/itex] and set them all to zero. The expression becomes pretty cumbersome and error-prone.

I was wondering if there is an easier and/or known solution for this problem.
Thanks.
 
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  • #2
In effect, minimizing Tr((A - B.X).(A - B.X)+) where the + means Hermitian conjugate or transpose conjugate. First one expands it:

Tr(A.A+) - Tr(A+.B.X) - Tr(X+.B+.A) + Tr(B+.B.X.X+)

Next, consider how X varies: X = Xr + i*Xi where Xr and Xi are the real and imaginary parts.

The Hermitian conjugate X+ = (Xr - i*Xi)T and it is evident that one can treat X and X+ as separate variables, since they are different linear combinations of Xr and Xi. Differentiating by X+ and X yields

B+.B.X= B+.A
X+.B+.B = A+.B
 
  • #3
Thanks lpetrich,

I actually got the same result few hours before I found your reply, though you actually showed how to solve a more general problem where also X is complex.
What I did was basically to use Einstein notation to derive some useful matrix calculus identities, and then by using them it was kind of easy to arrive at the final formula(s) that you wrote.
It was interesting to see how using a different notation made the problem more manageable.
 

FAQ: Problem with minimizing the matrix norm

1. What is the matrix norm and why is it important?

The matrix norm is a measure of the size or magnitude of a matrix. It is important because it provides a way to quantify the error or distance between two matrices, which is useful in many applications such as optimization and machine learning.

2. What is the problem with minimizing the matrix norm?

The problem with minimizing the matrix norm is that it can be a computationally expensive task, especially for large matrices. This is because it involves finding the minimum value of a function over a potentially infinite set of matrices.

3. How can the matrix norm be minimized?

The matrix norm can be minimized using various optimization techniques, such as gradient descent or Newton's method. These methods involve iteratively updating the parameters of the matrix to eventually reach the minimum norm.

4. Can minimizing the matrix norm lead to overfitting?

Yes, minimizing the matrix norm can lead to overfitting in certain situations. This is because the norm only measures the error between the predicted and actual values, but it does not take into account the complexity of the model. This can result in a model that fits the training data very well, but performs poorly on new, unseen data.

5. Are there any alternatives to minimizing the matrix norm?

Yes, there are alternative methods for solving problems that involve minimizing the matrix norm. These include using regularized or constrained optimization techniques, which can help prevent overfitting by penalizing complex models. Additionally, other performance metrics, such as mean squared error, can also be used to evaluate and optimize a model's performance.

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