0
$\begingroup$

Define $f(A): \mathbb{R}^{p\times m} \to \mathbb{R}$ as follows: $$ f(A) = \frac{1}{2}\|Y-XAB\|_F^2 = \frac{1}{2}\text{tr}\{(Y-XAB)^T(Y-XAB)\}, $$ where matrices $Y\in\mathbb{R}^{n\times q}, X\in\mathbb{R}^{n\times p},$ and $ B\in\mathbb{R}^{m\times q}$ are given.

Now, I would like to know whether $\nabla f$ is Lipschitz continuous or not, and what the Lipschitz constant is.

I know that $g(A) = \nabla f = -X^TYB^T + X^TXABB^T$. But then, how are the Lipschitz continuity and the Lipschitz constant defined for the function $g: \mathbb{R}^{p\times m} \to \mathbb{R}^{p\times q}$?

$\endgroup$
1
$\begingroup$

Using any norm, you have $$ \|g(A)-g(A')\|=\|X^TX(A-A')BB^T\|\le\|X^TX\|\cdot \|BB^T\|\cdot \|A-A'\| $$ and so $g$ is Lipschitz. The actual Lipschitz constant is of course at most $\|X^TX\|\cdot \|BB^T\|$ but it will depend on what is your induced norm and on the actual matrices $X$ and $B$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.