Toeplitz matrices arise naturally in machine learning whenever a stationary kernel is evaluated on equally spaced indices; one common example is the use of Gaussian processes to model time series data. Unfortunately, the usefulness of kernel methods involving Toeplitz matrices is limited by both the quadratic memory requirements and the cubic runtime complexity of key matrix operations such as determinants and linear solves.
Toeblitz is a MATLAB/Octave package for operations on positive definite Toeplitz matrices. It uses existing findings from linear algebra to perform the following for any positive definite Toeplitz matrix of size n by n:
(i) solve Toeplitz systems in nlogn time and linear memory (ii) compute Toeplitz matrix inverses (with free log determinant) in quadratic time and memory (iii) compute Toeplitz log determinants (without inverses) in quadratic time and linear memory (iv) compute traces of matrix products for any matrix A and a Toeplitz matrix T, in minimal quadratic time and memory.
- Changes to previous version:
Initial Announcement on mloss.org.
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