Hi,
Sorry for this basic question, but I am a bit confused regarding the proof of a unique global minima of a function. A function has a unique global minima if it is strictly convex. I am aware that the Hessian needs to be psd for the function to be convex. Therefore, my question is what are the requirements of a Hessian for the function to be strictly convex?
Thank you for your help
Respectively, I am confused why for this question here, it is enough to show that the matrix XWX is invertible to proof that beta has a unique solution:
I am aware that the Hessian needs to be psd for the function to be convex. Therefore, my question is what are the requirements of a Hessian for the function to be strictly convex?
The Hessian should be pd (positive definite) to be strictly convex.
Specifically regarding the screenshot you've posted, you can verify that the hessian is \(X^\top W X\), and we need it to be positive definite. (And this is a stricter condition than it being invertible, but given the limited context I cannot say more about it.)
Where does this problem come from ? I need more context because I don;t see obvious reason why \(W\) is positive definite.
Relation strict convex and Hessian
Hi,
Sorry for this basic question, but I am a bit confused regarding the proof of a unique global minima of a function. A function has a unique global minima if it is strictly convex. I am aware that the Hessian needs to be psd for the function to be convex. Therefore, my question is what are the requirements of a Hessian for the function to be strictly convex?
Thank you for your help
1
Respectively, I am confused why for this question here, it is enough to show that the matrix XWX is invertible to proof that beta has a unique solution:
1
Hi,
The Hessian should be pd (positive definite) to be strictly convex.
Specifically regarding the screenshot you've posted, you can verify that the hessian is \(X^\top W X\), and we need it to be positive definite. (And this is a stricter condition than it being invertible, but given the limited context I cannot say more about it.)
Where does this problem come from ? I need more context because I don;t see obvious reason why \(W\) is positive definite.
1
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