  The majority of machine learning algorithms assumes that objects are
represented as vectors. But often the objects we want to learn on are more
naturally represented by other data structures such as sequences and time
series. For these representations many standard learning algorithms are
unavailable. We generalize gradient-based learning algorithms to time series
under dynamic time warping. To this end, we introduce elastic functions, which
extend functions on time series to matrix spaces. Necessary conditions are
presented under which generalized gradient learning on time series is
consistent. We indicate how results carry over to arbitrary elastic distance
functions and to sequences consisting of symbolic elements. Specifically, four
linear classifiers are extended to time series under dynamic time warping and
applied to benchmark datasets. Results indicate that generalized gradient
learning via elastic functions have the potential to complement the
state-of-the-art in statistical pattern recognition on time series.
