Gradient estimation for particle flow induced by log-homotopy for nonlinear filters

We study 17 distinct methods to approximate the gradient of the log-homotopy for nonlinear filters. This is a challenging problem because the data are given as function values at random points in high dimensional space. This general problem is important in optimization, financial engineering, quantum chemistry, chemistry, physics and engineering. The best general method that we have developed so far uses a simple idea borrowed from geology combined with a fast approximate k-NN algorithm. Extensive numerical experiments for five classes of problems shows that we get excellent performance.

[1]  Nando de Freitas,et al.  The Unscented Particle Filter , 2000, NIPS.

[2]  Fred Daum,et al.  Particle flow for nonlinear filters with log-homotopy , 2008, SPIE Defense + Commercial Sensing.

[3]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[4]  Piotr Indyk,et al.  Nearest Neighbors in High-Dimensional Spaces , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[5]  F. Daum,et al.  A Split-Step Solution of the Fokker-Planck Equation for the Conditional Density , 2006, 2006 Fortieth Asilomar Conference on Signals, Systems and Computers.

[6]  James McNames,et al.  A Fast Nearest-Neighbor Algorithm Based on a Principal Axis Search Tree , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  Fred Daum,et al.  Nonlinear filters with particle flow induced by log-homotopy , 2009, Defense + Commercial Sensing.

[8]  Erich Novak,et al.  Simple Monte Carlo and the Metropolis algorithm , 2007, J. Complex..

[9]  K.C. Chang,et al.  K-nearest neighbor particle filters for dynamic hybrid Bayesian networks , 2008, IEEE Transactions on Aerospace and Electronic Systems.

[10]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[11]  David J. C. MacKay,et al.  Information Theory, Inference, and Learning Algorithms , 2004, IEEE Transactions on Information Theory.

[12]  T. H. Meyer,et al.  Gradient Estimation from Irregularly Spaced Data Sets , 2001 .

[13]  A. J. Noushin,et al.  Some interesting observations regarding the initialization of unscented and extended Kalman filters , 2008, SPIE Defense + Commercial Sensing.

[14]  Alexandr Andoni,et al.  Near-Optimal Hashing Algorithms for Approximate Nearest Neighbor in High Dimensions , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[15]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[16]  Branko Ristic,et al.  Beyond the Kalman Filter: Particle Filters for Tracking Applications , 2004 .

[17]  F. Daum.  Nonlinear filters: beyond the Kalman filter , 2005, IEEE Aerospace and Electronic Systems Magazine.

[18]  J. Huang,et al.  Curse of dimensionality and particle filters , 2003, 2003 IEEE Aerospace Conference Proceedings (Cat. No.03TH8652).

[19]  Arnaud Doucet,et al.  A survey of convergence results on particle filtering methods for practitioners , 2002, IEEE Trans. Signal Process..

[20]  Persi Diaconis,et al.  The Markov chain Monte Carlo revolution , 2008 .