[jira] [Created] (MXNET-1350) Cholesky factorization: matrix inversion, determinant, logdeterminant

["Zach Boldyga (JIRA)" <ji...@apache.org>]

Zach Boldyga created MXNET-1350:
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             Summary: Cholesky factorization: matrix inversion, determinant, logdeterminant
                 Key: MXNET-1350
                 URL: https://issues.apache.org/jira/browse/MXNET-1350
             Project: Apache MXNet
          Issue Type: New Feature
          Components: Apache MXNet Backend
            Reporter: Zach Boldyga


MXNet already includes a routine for Cholesky factorization, and allows for matrix inversion using this factorization.

The determinant and log determinant are straightforward to calculate using the Cholesky factorization, but MXNet does not provide a way to compute the determinant or log determinant. 

This issue was reported and upvoted by the community: [https://github.com/apache/incubator-mxnet/issues/14360] . A reference was made to Torch, which includes routines for the inverse, determinant, and log determinant.

I'd like to implement the following changes:
 # Provide a single, clear API endpoint for calculating matrix inversion. e.g. mxnet.ndarray.linalg.inverse. This can be in addition to the existing endpoints, which may still be useful to some users. It will be simple to implement; a merger of the two existing ops. But currently it's confusing for users to discover that MXNet actually supports matrix inversion, and it's a two step process.
 # Add an API endpoint for calculating the matrix determinant. e.g. mxnet.ndarray.linalg.det. This would run Cholesky factorization internally, then use the resultant triangular matrix to simplify the determinant calculation.
 # Add an API endpoint for calculating the log determinant of the input matrix. e.g. mxnet.ndarray.linalg.logdet. Similar to #2, but it may be as simple as combining cholesky factorization and sumlogdiag internally. 

Overall, this is just piecing together of other existing linear algebra operations and clarifying the documentation so that users can access these operations without having to have a deeper understanding of linear algebra.

I can open a PR as soon as someone reviews this, just let me know if a different approach is preferred or if there are any specifics I should be careful about in regards to implementation. Thanks!



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