Grantee: ** Johns Hopkins University**, Baltimore, MD, USA

Researcher: Noah Cowan, Ph.D.

Grant Title: Decoding Complex Animal Behavior Via Sparsity

https://doi.org/10.37717/220020323

Program Area: Studying Complex Systems

Grant Type: Scholar Award

Amount: $450,000

Year Awarded: 2012

Duration: 6 years

**Decoding Complex Animal Behavior Via Sparsity**

Do complex systems require high-dimensional descriptions? In other words, is there a sensible notion of “dimensionality reduction” for complex systems? Surely ‘low-dimensional’ and ‘complex’ are antithetical, right? I suspect not.

About 5 years ago, Terrence Tao received a Fields Medal for groundbreaking work on the representation of signals using a concept called sparsity. Sparsity embodies the notion that quite often very complex (and high-bandwidth) signals can be represented as a combination of surprisingly few basis vectors. This observation enables one to completely side-step the well-known Nyquist Sampling Theorem, which states that, in order to be able to reproduce a signal from discretely sampled data, one must sample the signal at a rate that is at least twice that of the highest frequency present in the signal.

But does this potentially transformative theory work in practice? Indeed: sparsity has been concretely instantiated through the creation of the new field of Compressive Sensing (Candes and Wakin, 2008), which is built around the notion of sparsity. The idea was popularized by the invention of a “single pixel camera” – a concept developed by researchers at Rice University, in which they built a digital camera that takes a surprisingly small number of random one dimensional (i.e. single pixel) projections of the scene. Using these “one pixel images”, the researchers demonstrated the successful recovery of a photographic image. This achievement required solving a seemingly ill-posed inverse problem, rendered solvable by using the Tao's sparsity constraint. This and other applications to medical image reconstruction, computer vision and machine learning have demonstrated the extraordinary applicability of the theory of sparse representations to real-world problems and data.

What does sparsity have to do with dynamical systems in general, and complex systems in particular? To date, almost nothing: on the one hand, sparsity has been applied almost exclusively to static linear problems, while on the other hand, the complexity of dynamical systems is typically characterized in terms of the number of independent variables in a nonlinear model. With help from my collaborator Rene Vidal, I propose a research program to advance the theory of sparse representations in the context of dynamics as a new approach to dimensionality reduction in complex systems. This will lead to a new theory of sparse representation of dynamical systems, predicated on the existence of unknown but sparse (and switched) inputs to a dynamical system.

While the principal aims of this work are theoretical in nature, I find it essential to ground this theory by testing it against data from natural systems. Specifically, I choose animal behavior (including human behavior) for empirical testing. My goal is to create a program at the intersection of dynamical systems theory and experimental animal behavior. I will achieve this research program by building upon existing collaborations with experimental biologists, and branching into the realm of cooperative (social) animal behavior. Through these collaborations I will have access to wide- ranging behavioral data sets which I can use to test the basic theories that I will develop under a James S. McDonnell Scholar Award.

My goal is to develop a new theory of sparse dynamical systems, develop statistical techniques to identify their structure from data, and apply this new theory to decipher some of the most complex systems in existence: naturally behaving animals.

- Candes, E. and Wakin, M. (2008). An Introduction To Compressive Sampling. IEEE Signal Proc. Mag. 25,21-30.