Funded Grants

Tracking time-varying low-dimensional structure to uncover the building blocks of complex dynamics

Many processes across the life, physical and social sciences are thought of as dynamic complex systems, where sophisticated and unpredictable behavior sequences arise from interactions between connected components. For most of modern scientific history, one fundamental barrier to understanding these systems was our limited ability to record high-fidelity measurements of the phenomenon of interest. Fortunately, engineering advances in sensor technology over the last few decades have allowed observation of complex systems at an unprecedented scale, giving us novel views of many important phenomena.

While modern science is ingrained with the premise that higher fidelity observations of these dynamic phenomena can help us understand their underlying mechanisms and predict future behavior, the truth is that large-scale measurements are a mixed blessing. Increasing the volume and complexity of available data presents increasing challenges to extracting meaningful information about these systems. For example, there are now reports of sensors (e.g., the Large Hadron Collider at CERN) that produce data at a far higher rate than could possibly be processed or stored [1].

Despite the apparent complexity, a core belief of complex systems research is that these systems contain some type of underlying structure that will permit us to model them in a simpler form and gain functional insight [2]. The toolbox of the physicist (e.g., nonlinear dynamics, information theory, statistical mechanics) has traditionally been employed to seek this structure. In contrast, modern signal processing and machine learning have recently achieved unprecedented gains in extracting information from incomplete or corrupted high-dimensional data by assuming the information of interest can be efficiently described using a low-dimensional geometric structure. These approaches exploit a combination of tools, including statistical inference, convex optimization, unsupervised learning and differential geometry.

While this geometric approach has been extremely powerful for uncovering the fundamental informational building blocks of sensed data in many applications, these tools are generally designed for “static” data (e.g., an image) and contain no dynamic component that is necessary for the study of complex systems. In addition to contributing to the core of this research area, my research group has been extending these tools to develop fundamentally new approaches to tracking time- varying low-dimensional structure. This work has shown state-of-the-art results in uncovering the information in time-varying signals, and is the beginning of a framework for uncovering the building blocks of complex dynamics. Specifically, my lab is working to build a framework for principled methods to measure and track time-varying low-dimensional structure in complex systems. To aid in the development and testing of these general tools, we have developed collaborations with a number of application domain labs who study systems as varied as ocean dynamics, biomechanics and motor control for rehabilitation, and personalized learning. Working together, we will answer the question:

Can we uncover the building blocks of complex dynamics by tracking time-varying low-dimensional structure in complex systems?