Melbourne School of Engineering
Nonlinear Signal Processing Laboratory

GOALS

NSP Lab project structure enables researchers to achieve scale and focus in their research by working as a team on a small number of strategically chosen long-term challenges. The structure also delineates how researchers work towards the overarching aims of the Lab.

 

Geometric Design of Numerical Algorithms for Signal Processing
    1. Optimisation
    2. Root finding
    3. Filtering

The last century saw considerable effort devoted to finding closed-form solutions to problems of interest. Numerical solutions were often seen as second-best, but this is essentially only the case if the numerical solution takes too long to compute or its accuracy cannot be guaranteed. It is highly likely then that this century will see the flourishing of efficient numerical solutions with guaranteed performance to a broad range of problems.

Consistent with the overarching aims of NSP Lab, our approach is to understand and use the geometry (and topology) of a problem to inform the development of numerical algorithms for its solution. The virtue of studying the geometry of a problem is exemplified by the classical problem of computing the roots of a polynomial equation. Galois theory links the symmetries of the roots of a polynomial to properties of algorithms for computing the roots. This can be used to prove a general quintic equation cannot be solved algebraically, and on a more positive note, it has provided insight into how quintic and higher-order equations can be solved using elliptic functions. Recent work by Steve Smale "The fundamental theorem of algebra and complexity theory" and "On the topology of algorithms, I" takes the story considerably further. In a beautiful application of pure mathematics to an applied problem, Smale used algebraic topology to obtain bounds on the difficulty of finding the roots of a polynomial. He also presented an algorithm for numerically finding the roots and, importantly, he gave a rigorous analysis of the algorithm's performance. (Extensions of this work can be found in the 1997 book by Blum, Cucker, Shub and Smale titled,"Complexity and Real Computation".) We wish to apply analogous ideas to other classes of numerical problems.

A key innovation is to generalise from working in Euclidean space to working on differentiable manifolds. Numerous problems are naturally formulated as problems on manifolds and to force them into a Euclidean formulation is generally not helpful. Furthermore, the extra generality means more problems can be found that have fast and accurate numerical solutions. Problems involving compact manifolds are prime candidates in this respect because boundary points and non-compactness can complicate algorithm design and in Euclidean space, all compact subsets have boundary points. Working at a higher level of generality also brings insight by suppressing irrelevant details such as Euclidean coordinates which would otherwise distract attention from the intrinsic properties of the problem at hand.

Information geometry is an important ingredient in our work. In addition to the obvious connection --- information geometry studies the (differential) geometry of estimation problems --- we also use it in a loose sense as a bridge between deterministic problems (optimisation and root finding) and stochastic problems (estimation and filtering).

 

Multi-scale Computational Modelling of Biological Systems
  1. Computational Neuroscience
    1.1 Auditory pathways in the human brain

    1.2 C. Elegans

  2. Systems Biology
    2.1 Tumor cells in the human colon

We use multi-scale computational modelling as a hub for coordinating our research efforts on understanding how biological networks compute. Importantly, our models are top-down, multi-scale, based on biophysical principles and have experimentally validated predictive power. These are precisely the attributes that climate models have; we believe that fifty years of climate model development helps justify this approach for understanding complex systems. Furthermore, being top-down and focused on end-to-end systems means the full gamut of mathematical frameworks can be brought to bear. (“We need to celebrate the equally vital contribution of those who dare to take what I call ‘a crude look at the whole’.” — Murray Gell-Mann, Nobel Laureate in Physics, 1994.)

The overarching goal is to discover the mathematical principles behind how networks of neurons and other biological networks compute. Biological networks are extremely energy efficient compared with engineered systems. The human brain uses only 20 watts of power yet for myriad tasks it outperforms super-computers consuming mega-watts of power. Learning the secrets of how nature computes will therefore revolutionise not only health and medicine but also the field of engineering.

We seek to extend the mathematical frameworks of signal processing, control and information theory rather than optimistically apply concepts outside their understood scope. (Without extended frameworks, it is open to debate what the meaning is of having calculated the entropy measure of a neuron, for example.) By extending mathematical frameworks we aim to provide a unified and consistent theoretical treatment of biological processes at multiple levels of analysis. Without extended frameworks, mathematical analysis of biological processes can at best be fragmentary and incomplete.

 

Transdisciplinary Collaboration and Real-World Applications

NSP Lab is interested in the full spectrum of research, from fundamental to applied. Fundamental and applied research drive each other, with inspiration for fundamental research generated by working on applied problems, and conversely, efficient and effective solutions to applied problems generally only being found after the underlying fundamental principles have been well understood.

We are particularly interested in transdisciplinary collaborations. As Norbert Wiener eloquently explains, it is the ``boundary regions of science which offer the richest opportunities to the qualified investigator. They are at the same time the most refractory to the accepted techniques of mass attack and the division of labor'' --- page 2 of Cybernetics: or Control and Communication in the Animal and Machine, 1948, by Norbert Wiener.

In addition to the multi-scale computational modelling of several different biological systems, other recent examples of transdisciplinary collaboration and real-world applications include developing image processing algorithms capable of assessing objectively the severity of skin disorders, and the development of algorithms for processing the raw stream of pulses from a radiation detector.