  This paper proposes a framework dedicated to the construction of what we call
discrete elastic inner product allowing one to embed sets of non-uniformly
sampled multivariate time series or sequences of varying lengths into inner
product space structures. This framework is based on a recursive definition
that covers the case of multiple embedded time elastic dimensions. We prove
that such inner products exist in our general framework and show how a simple
instance of this inner product class operates on some prospective applications,
while generalizing the Euclidean inner product. Classification experimentations
on time series and symbolic sequences datasets demonstrate the benefits that we
can expect by embedding time series or sequences into elastic inner spaces
rather than into classical Euclidean spaces. These experiments show good
accuracy when compared to the euclidean distance or even dynamic programming
algorithms while maintaining a linear algorithmic complexity at exploitation
stage, although a quadratic indexing phase beforehand is required.
