Symposium session

Thursday, Dec. 11, Regency Ballroom A/B at the Vancouver Hyatt
13.30 - 14.00 Risi Kondor: Non-commutative harmonic analysis
14.05 - 14.35 Guy Lebanon: Modeling distributions on permutations and partial ranking
14.40 - 15.10 Jason Morton: Algebraic models for multilinear dependence
15.15 - 15.25 coffee break
15.25 - 15.55 Yanxi Liu: Symmetry Group-based Learning for Regularity Discovery from Real World Patterns
16.00 - 16.30 Marina Meila: Estimation and model selection in stagewise ranking: a representation story

Non-commutative harmonic analysis

Risi Kondor, Gatsby Unit, UCL

Fourier analysis is one of the central pillars of applied mathematics. Representation theory makes it possible to generalize Fourier transformation to non-commutative groups, such as permutations and 3D rotations. This talk will survey new applications of this theory in machine learning for problems such as identity management in multi-object tracking, transformation invariant representations of images, and similarity measures between graphs. The talk is intended for a wide audience, no background in group theory or representation theory will be assumed.


Modeling distributions on permutations and partial ranking

Guy Lebanon, Georgia Institute of Technology

We explore several probabilistic models over the symmetric group of permutations and its cosets - representing partial rankings. We will cover both traditional statistical models such as the Luce-Plackett and Bradley Terry models, as well as more modern ones. Special attention will be given to non-parametric models and their use in modeling and visualizing preference data.


Algebraic models for multilinear dependence

Jason Morton, Stanford University

We discuss a new statistical technique inspired by research in tensor geometry and making use of cumulants, the higher order tensor analogs of the covariance matrix. For non-Gaussian data not derived from independent factors, tensor decomposition techniques for factor analysis such as Principal Component Analysis and Independent Component Analysis are inadequate. Seeking a closed space of models which is computable and captures higher-order dependence leads to a proposed extension of PCA and ICA, Principal Cumulant Component Analysis (PCCA). Estimation is performed by maximization over a Grassmannian. Joint work with L.-H. Lim.


Symmetry Group-based Learning for Regularity Discovery from Real World Patterns

Yanxi Liu, Penn State

We explore a formal and computational characterization of real world regularity using discrete symmetry groups (hierarchy) as a theoretical basis, embedded in a well-defined Bayesian framework. Our existing work on ``Near-regular texture analysis and manipulation'' (SIGGRAPH 2004) and ``A Lattice-based MRF Model for Dynamic Near-regular Texture Tracking'' (TPAMI 2007) already demonstrate the power of such a formalization on a diverse set of real problems, such as texture analysis, synthesis, tracking, perception and manipulation in terms of regularity. Symmetry and symmetry group detection from real world data turns out to be a very challenging problem that has been puzzling computer vision researchers for the past 40 years. Our novel formalization will lead the way to a more robust and comprehensive algorithmic treatment of the whole regularity spectrum, from regular (perfect symmetry), near-regular (approximate symmetry), to various types of irregularities. The proposed method will be justified by several real world applications such as gait recognition, grid-cell clustering, symmetry of dance, automatic geo-tagging and image de-fencing.


Estimation and model selection in stagewise ranking - a representation story

Marina Meila, University of Washington

This talk is another example of the well-known statement that ``representation matters''. We describe the code of a permutation, first used in statistics by Fligner and Verducci to define stagewise raking models. The code represents a permutation as a sequence of $n-1$ independent numbers. This property makes statistical models based on the code have singular advantages w.r.t other probabilistic models over the symmetric group. In particular, the parameters can be better understood and can be estimated more easily. We illustrate this by comparisons to other exponential models of the symmetric group and by describing a suite of algorithms that allow one to estimate stagewise ranking models based on the code under a variety of missing data scenarios. Joint work with Bhushan Madhani and Kobi Abayomi.