Generalized Gradient Learning on Time Series under Elastic\n Transformations

The majority of machine learning algorithms assumes that objects are\nrepresented as vectors. But often the objects we want to learn on are more\nnaturally represented by other data structures such as sequences and time\nseries. For these representations many standard learning algorithms are\nunavailable. We generalize gradient-based learning algorithms to time series\nunder dynamic time warping. To this end, we introduce elastic functions, which\nextend functions on time series to matrix spaces. Necessary conditions are\npresented under which generalized gradient learning on time series is\nconsistent. We indicate how results carry over to arbitrary elastic distance\nfunctions and to sequences consisting of symbolic elements. Specifically, four\nlinear classifiers are extended to time series under dynamic time warping and\napplied to benchmark datasets. Results indicate that generalized gradient\nlearning via elastic functions have the potential to complement the\nstate-of-the-art in statistical pattern recognition on time series.\n