Discrete Time Elastic Vector Spaces

TL;DR

This paper introduces a novel framework for constructing time elastic inner products that embed non-uniformly sampled multivariate time series into vector spaces, generalizing Euclidean inner products.

Contribution

It presents a recursive framework for time elastic inner products applicable to multivariate time series of varying lengths, with proofs of existence and initial applications.

Findings

Inner products exist within the proposed framework.

A simple instance generalizes Euclidean inner product.

Initial toy applications demonstrate the framework's utility.

Abstract

We propose in this paper a framework dedicated to the construction of what we call time elastic inner products that allows embedding sets of non-uniformly sampled multivariate time series of varying lengths into vector 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 framework and show how a simple instance of this inner product class operates on some toy applications, while generalizing the Euclidean inner product.