Sascha Korl

The Factor Graph Approach to Model-Based Signal Processing

The message-passing approach to model-based signal processing is developed with a focus on Gaussian message passing in linear state-space models, which includes recursive least squares, linear minimum-mean-squared-error estimation, and Kalman filtering algorithms. Tabulated message computation rules for the building blocks of linear models allow us to compose a variety of such algorithms without additional derivations or computations. Beyond the Gaussian case, it is emphasized that the message-passing approach encourages us to mix and match different algorithmic techniques, which is exemplified by two different approaches - steepest descent and expectation maximization - to message passing through a multiplier node.

Particle Methods as Message Passing

It is shown how particle methods can be viewed as message passing on factor graphs. In this setting, particle methods can readily be combined with other message-passing techniques such as the sum-product and max-product algorithm, expectation maximization, iterative conditional modes, steepest descent, Kaiman filters, etc. Generic message computation rules for particle-based representations of sum-product messages are formulated. Various existing particle methods are described as instances of those generic rules, i.e., Gibbs sampling, importance sampling, Markov-chain Monte Carlo methods (MCMC), particle filtering, and simulated annealing

Expectation maximization as message passing

Based on prior work by Eckford, it is shown how expectation maximization (EM) may be viewed, and used, as a message passing algorithm in factor graphs

Signal processing with factor graphs: examples

Graphical models such as factor graphs allow to model complex systems and help to derive practical detection/estimation algorithms as message passing in the graph. In this paper, we outline three examples of on-going work of this type. For an introduction to factor graphs, we refer to F. R. Kschischang (Feb. 2001) and H. A. Loeliger (Jan. 2004). We use the notation of H. A. Loeliger (Jan. 2004).

Localizing, forgetting, and likelihood filtering in state-space models

The context of this paper are cycle-free factor graphs such as hidden Markov models or linear state space models. The paper offers some observations and suggestions on “localizating” such models and their likelihoods. First, it is suggested that a localized version of the model likelihood, which is easily computed by forward sum-product message passing, may be useful for feature extraction and detection. Second, the notion of a “local” model (local factor graph) is introduced. A first class of local models arises from exponential message damping and scale factors as in recursive least squares. A second class of local models arises from the problem of estimating the moment of a model switch from some known model A to some known model B. This problem can be solved by forward sum-product message passing in model A and backward sum-product message passing in model B. It is pointed out that this method is applicable to pulse position estimation for any pulse with a (deterministic or stochastic) state space model.

A Numerical Method to Compute Cramer-Rao-Type Bounds for Challenging Estimation Problems

A numerical algorithm is proposed to compute Cramer-Rao-type bounds. The Cramer-Rao-type bounds are derived from information matrices of marginals of the joint pdf of the system at hand. The key ingredient is message-passing on a factor graph of the system. The method can be applied to a wide class of estimation problems. As an illustration, the problem of estimating the parameters of an AR model is considered