Computational Information Geometry Wonderland

Please send me your favorite application of information geometry. Happy new year to all of you. Frank

Digital cameras are quickly merging with smart phones, and visual computing applications [1] that support computational photography and augmented reality applications are flourishing at a fast pace. By 2013, the annual worldwide IP traffic is predicted to be a zettabyte: 90% of consumer IP traffic and 60% of mobile IP traffic will be video.

How do we extract and use rich information from those massive data sets? As visual data abound, computer vision and computer graphics are increasingly relying on machine learning and information-theoretic methods. Computational Information Geometry is a novel paradigm to perform high-fidelity data analysis using the language and thinking of geometry.

Geometry allows us to map the data in space for efficient processing and retrieval of intrinsic information. Geometry is in essence coordinate-free and allows one to extract the very information from data.

Geometrization of statistics has provided novel algorithms for manipulating statistical mixture models such as Gaussian mixture models [2] that are commonly used in image processing: An image pixel at position (x,y) with color attributes (red, green, blue) is embedded into a 5D space so that a 2D color image is interpreted as a 5D spatial point cloud. We then seek for a compact generative statistical representation of the image point set. Such statistical methods are useful for explaning human cognitive and learning skills [3], and analyzing emerging phenomena of complex systems using hierarchical Bayesian models.

Geometry is well alive and continue to play a crucial role in natural sciences. For example, the propagation of seismic waves from an epicenter follows Fermat's principle of shortest paths (minimum arrival time). Since the Earth is made of anisotropic media such as the peridotite, shortest paths are not line segments: The geometry is not Euclidean. Seismic wave propagation is currently best modeled using Finsler geometry that extends Riemmanian geometry by taking into account the anisotropic direction of materials. In [4], we recently show how to aggregate and cluster information in such Finslerian spaces. Finsler geometry is also considered for advanced medical imaging of DT-MRI data-sets.

Last but not least, the theory of portfolio allocation has been traditionally carried out using the mean-variance method of Markowitz. Considering universal statistical distributions (exponential families) allows one to bypass the Gaussian assumption, and to derive the exact expression of the risk premium (a Bregman divergence) and certainty equivalent [5]. Moreover, we design an on-line learning algorithm with guaranteed lower bounds on its cumulated certainty equivalents [5]. It is interesting to note that portfolio theory has also been considered to explain robustness trade-offs of cells in biology [6] (bioeconomics).

REFERENCES:

• [1] Frank Nielsen: Visual Computing: Geometry, Graphics, and Vision; Charles River Media, ISBN: 1-58450-427-7, 2005.
• [2] Frank Nielsen, Sylvain Boltz: The Burbea-Rao and Bhattacharyya Centroids. IEEE Transactions on Information Theory 57(8): 5455-5466, 2011.
• [3] Joshua B. Tenenbaum, Charles Kemp, Thomas L. Griffiths, and Noah D. Goodman: How to Grow a Mind: Statistics, Structure, and Abstraction Science 331(6022):1279-1285, 2011.

• [4] Marc Arnaudon, Frank Nielsen: Medians and means in Finsler geometry, Cambridge LMS Journal of Computation and Mathematics, 2011.

• [5] Richard Nock, Brice Magdalou, Eric Briys and Frank Nielsen: On tracking portfolios with certainty equivalents on a generalization of Markowitz model: the Fool, the Wise and the Adaptive International Conference on Machine Learning, pp. 73-80, 2011.

• [6] Hiroaki Kitano: Violations of robustness trade-offs Mol Syst Biol. 2010; 6: 384. 10.1038/msb.2010.40