Manifold Warping: Manifold Alignment Over Time

Abstract

Knowledge transfer is computationally challenging, due in part to the curse of dimensionality, compounded by source and target domains expressed using different features (e.g., documents written in different languages). Recent work on manifold learning has shown that data collected in real-world settings often have high-dimensional representations, but lie on low-dimensional manifolds. Furthermore, data sets collected from similar generating processes often present different high-dimensional views, even though their underlying manifolds are similar. The ability to align these data sets and extract this common structure is critical for many transfer learning tasks. In this paper, we present a novel framework for aligning two sequentially-ordered data sets, taking advantage of a shared low-dimensional manifold representation. Our approach combines traditional manifold alignment and dynamic time warping algorithms using alternating projections. We also show that the previously-proposed canonical time warping algorithm is a special case of our approach. We provide a theoretical formulation as well as experimental results on synthetic and real-world data, comparing manifold warping to other alignment methods.

Introduction

The advent of large, often high-dimensional, digital data sets has made automated knowledge extraction a critical research focus in the field of machine learning. Often, we find real-world sequential data sets that encode the same information with disparate surface feature representations, such as sensor network data, activity and object recognition corpora, and audio/video streams. In these cases, an automated technique for discovering correlations between sets will allow easy transfer of knowledge from one domain to another, avoiding costly or infeasible re-learning. In this paper, we present a framework that combines manifold alignment \cite{ham2003MA,wang2009generalMAframework} and dynamic time warping (DTW) \cite{sakoe1978dtw} for aligning two such sequential data sets. We also show that the previously proposed method of canonical time warping (CTW) \cite{zhou2009ctw} is a special case of our approach.

Dynamic time warping has been used effectively for time-series alignment, but it requires an inter-set distance function, which usually implies that both input data sets must have the same dimensionality. DTW may also fail under arbitrary affine transformations of one or both inputs.