Optimal Mass Transport: Signal processing and machine-learning applications

Transport-based techniques for signal and data analysis have recently received increased interest. Given their ability to provide accurate generative models for signal intensities and other data distributions, they have been used in a variety of applications, including content-based retrieval, cancer detection, image superresolution, and statistical machine learning, to name a few, and they have been shown to produce state-of-the-art results. Moreover, the geometric characteristics of transport-related metrics have inspired new kinds of algorithms for interpreting the meaning of data distributions. Here, we provide a practical overview of the mathematical underpinnings of mass transport-related methods, including numerical implementation, as well as a review, with demonstrations, of several applications. Software accompanying this article is available from [43].

Purposes for Optimal Mass Transport

Numerous applications in science and technology depend on effective modeling and information extraction from signal and image data. Examples include being able to distinguish between benign and malignant tumors in medical images; learning models (e.g., dictionaries) for solving inverse problems; identifying people from images of faces, voice profiles, or fingerprints; and many others. Techniques based on the mathematics of optimal mass transport, also known as Earth Mover's Distance in engineering-related fields, have received significant attention recently given their ability to incorporate spatial (in addition to intensity) information when comparing signals, images, and other data sources, thus giving rise to different geometric interpretations of data distributions. These techniques have been used to simplify and augment the accuracy of numerous pattern recognition-related problems. Some examples covered in this article include image retrieval [32], [44], signal and image representation [25], [27], [40], [50], inverse problems [30], cancer detection [4], [39], texture and color modeling [18], [41], shape and image registration [22], [29], and machine learning [12], [17], [19], [28], [36], [42], to name a few. This article is meant to serve as an introductory guide to those wishing to familiarize themselves with these emerging techniques. Specifically, we ■

provide a brief overview of key mathematical concepts related to optimal mass transport

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describe recent advances in transport-related methodology and theory

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provide a practical overview of their applications in modem signal analysis, modeling, and learning problems.