arcadiaphy opened a new pull request #15007: Add matrix determinant operator in linalg URL: https://github.com/apache/incubator-mxnet/pull/15007 ## Description ## The second PR on linalg enhancement. Three operators regarding matrix determinant is added: det, logdet, slogdet. Something worth mentioning in this PR: 1. Matrix inversion code is refactored a little bit to remove duplicated workspace query in getri, and make it more readable and easy to use in det operators. 2. Log(det(x)) in logdet follows the log operator, the grad is passed backwards even when det(x) < 0. 3. Sign in slogdet follows the sign operator, the grad on it is ignored in backward pass since it's not properly defined. 4. The determinant calculation uses [LU factorization with partial pivoting](https://en.wikipedia.org/wiki/LU_decomposition#Computing_the_determinant). 5. The grad of determinant is derived from [Jacobi's formula](https://en.wikipedia.org/wiki/Jacobi%27s_formula), which has a pretty friendly closed form solution for numerical computing when input matrix A invertible. The non-invertible case is not easy to implement since it involves adjugate matrix. In tensorflow, this case is ignored; while pytorch uses SVD to compute the grad. In this PR, it's left for future work, now as a temporary method, no grad is passed backwards when det = 0. My inclination is to re-use LU instead of SVD for non-invertible case since it's already calculated. ## Checklist ## ### Essentials ### Please feel free to remove inapplicable items for your PR. - [ ] The PR title starts with [MXNET-$JIRA_ID], where $JIRA_ID refers to the relevant [JIRA issue](https://issues.apache.org/jira/projects/MXNET/issues) created (except PRs with tiny changes) - [ ] Changes are complete (i.e. I finished coding on this PR) - [ ] All changes have test coverage: - Unit tests are added for small changes to verify correctness (e.g. adding a new operator) - Nightly tests are added for complicated/long-running ones (e.g. changing distributed kvstore) - Build tests will be added for build configuration changes (e.g. adding a new build option with NCCL) - [ ] Code is well-documented: - For user-facing API changes, API doc string has been updated. - For new C++ functions in header files, their functionalities and arguments are documented. - For new examples, README.md is added to explain the what the example does, the source of the dataset, expected performance on test set and reference to the original paper if applicable - Check the API doc at http://mxnet-ci-doc.s3-accelerate.dualstack.amazonaws.com/PR-$PR_ID/$BUILD_ID/index.html - [ ] To the my best knowledge, examples are either not affected by this change, or have been fixed to be compatible with this change ### Changes ### - [ ] Feature1, tests, (and when applicable, API doc) - [ ] Feature2, tests, (and when applicable, API doc) ## Comments ## - If this change is a backward incompatible change, why must this change be made. - Interesting edge cases to note here
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