About

This is a really barbones toy project to illustrate how to compute a Gram matrix for the one-sided mean alignment kernel.

It reads a set of sequences from file example_data.txt, computes a Gram matrix of these sequences using the one-sided mean alignment kernel, computes the eigenvalues of this matrix and displays them. As the one-sided mean alignment kernel is provably positive definite, the eigenvalues are all positive.

The project also contains code to generate random dummy time series.

Usage

The project doesn't need any dependency so you can just execute it:

python mean_alignment.py

About the one-sided mean alignment kernel

Well-known kernel methods such as for example SVM and Kernel PCA rely on a kernel which is used computes a Gram matrix which can generally be interpreted as a matrix of similarity values between any pair of samples in the dataset. These algorithms require that the kernel be positive definite, which guarantees that the resulting optimization programs are convex whatever the samples.

When dealing with vector data the most sensible choice is generally a Gaussian kernel, but when the samples are time-series (or more generally sequences) custom kernels must be used. There are not many sensible choices, as merely using a classic dynamic time warping distance does not lead to a positive definite kernel.

To this end we propose the one-sided mean kernel, which has many advantages:

• Provably positive definite,
• Faster than competing techniques with a time complexity of O(l × (m - l)) instead of O(l × m) for sequences of length l < m,
• Consistent with a vector kernel when time series are of equal length,
• Does not suffer from issues of diagonal dominance like the global alignment kernel for example.

An implementation in Haskell is available here.

References

• N. Chrysanthos, P. Beauseroy, H. Snoussi, E. Grall-Maës, Theoretical properties and implementation of the one-sided mean kernel for time series, in: Neurocomputing 169 (2015) 196–204
• M. Cuturi, J.-P. Vert, O. Birkenes, T. Matsui, A kernel for time series based on global alignments, in: IEEE International Conference on Acoustics, Speech and Signal Processing, 2007
• J.-P. Vert, The Optimal Assignment Kernel is not Positive Definite in: arXiv:0801. 4061