4 ms·
That was a really computation-intensive, brute-force derivation of grad log |det W|. I know there is a more elegant approach that makes better use of the deter
by civilized 4y ago
That was a really computation-intensive, brute-force derivation of grad log |det W|.
I know there is a more elegant approach that makes better use of the determinant's core properties, but it's been ages since I saw it.
- jarvist 4y agoWell write it up and put it on the ArXiV / send it to the author for version 2!
- yellowcake0 4y agoWriting det(W) in terms of an implicit parametrization of the eigenvalues is elegant, I think it's unlikely there's a better approach. Brute-forcing it would be to write down the multivariate polynomial in full generality, as there would be no way to break it up using the logarithm.
- kxyvr 4y agoThere is. My recollection is that it's easiest to use the adjugate like they do on this question: https://math.stackexchange.com/questions/38701/how-to-calculate-the-gradient-of-log-det-matrix-inverse https://math.stackexchange.com/questions/38701/how-to-calcul... Though, there are other ways to derive it. My personal opinion is that the vector and matrix calculus derivations in the book are too verbose, but this style may be more comfortable for some readers. My personal opinion is that the semidefinite and cone optimization communities have more concise ways of deriving these kind of derivatives and relationships. For example, this can be seen in Boyd and Vandenberghe's Convex Optimization or Ben-Tal and Nemirovski's Lectures on Modern Convex Optimization.
- civilized 4y agoYes! The top answer is basically "use the Jacobi formula", and the Jacobi formula can be derived incredibly elegantly using the chain rule: https://en.wikipedia.org/wiki/Jacobi's_formula#Via_Chain_Rule https://en.wikipedia.org/wiki/Jacobi's_formula#Via_Chain_Rul...