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Model-Based Signal Processing with Factor Graphs

Factor graphs allow a unified approach to a wide variety of structured models and related algorithms. Much of my research, and of my PhD students' at ETH, was about further developing the factor graph approach to signal processing.

The original factor graphs were bipartite with variable nodes and factor nodes:

B.J. Frey, F.R. Kschischang, H.-A. L., and N. Wiberg,
Factor graphs and algorithms,
Proc. 35th Allerton Conf. on Communications, Control, and Computing, Monticello, Illinois, 1997, pp. 666-680.
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F.R. Kschischang, B.J. Frey, and H.-A. L.,
Factor graphs and the sum-product algorithm,
IEEE Trans. Inform. Theory, vol. 47, pp. 498-519, Feb. 2001.
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However, Forney's work (including "Codes on graphs: normal realizations", 2002, and "Codes on graphs: duality and MacWilliams identities", 2011) made it clear (to me) that variables should be represented by edges rather than nodes, which is the convention I have been using ever since. In fact, I have long been considering the earlier (bipartite) version of factor graphs as a deplorable blunder.

The primary early references on the factor graph approach to signal processing are

H.-A. L.,
An introduction to factor graphs,
IEEE Signal Proc. Mag, Jan. 2004, pp. 28-41.
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H.-A. L., J. Dauwels, J. Hu, S. Korl, Li Ping, and F. Kschischang,
The factor graph approach to model-based signal processing,
Proceedings of the IEEE, vol. 95, no. 6, pp. 1295-1322, June 2007.
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Much has been happening since; for example:

J. Dauwels, A. Eckford, S. Korl, and H.-A. L.,
Expectation maximization as message passing - Part I: principles and Gaussian messages,
arXiv:0910.2832, 2009.
Ch. Reller, M.V.R.S. Devarakonda, and H.-A. L.,
Glue factors, likelihood computation, and filtering in state space models,
Proc. 50th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Illinois, USA, Oct. 1-5, 2012, pp. 686-689.
IEEE Xplore / preprint
L. Bolliger, H.-A. L., and C. Vogel,
LMMSE estimation and interpolation of continuous-time signals from discrete-time samples using factor graphs,
arXiv:1301.4793, 2013.
H.-A. L. and Ch. Reller,
Signal processing with factor graphs: beamforming and Hilbert transform,
2013 Information Theory and Applications Workshop (ITA), San Diego, CA, Feb. 10-15, 2013.
IEEE Xplore / preprint
S. Maranò, D. Fäh, and H.-A. L.,
A state-space approach for the analysis of wave and diffusion fields,
40th IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP), Brisbane, Australia, April 19-24, 2015.
IEEE Xplore / Preprint
H.-A. L, L. Bruderer, H. Malmberg, F. Wadehn, and N. Zalmai,
On sparsity by NUV-EM, Gaussian message passing, and Kalman smoothing,
2016 Information Theory & Applications Workshop (ITA), La Jolla, CA, Jan. 31 - Feb. 5, 2016.
IEEE Xplore / Preprint / arXiv:1602.02673
N. Zalmai, C. Luneau, C. Stritt, and H.-A. L.,
Tomographic reconstruction using a new voxel-domain prior and Gaussian message passing,
2016 European Signal Processing Conference (EUSIPCO), Budapest, Hungary, Aug. 29 - Sept. 2, 2016.
Preprint / EURASIP / IEEE Xplore

The list above is very far from complete, but may give an idea of the topics. See also the PhD theses of Lukas Bolliger (2012), Christoph Reller (2012), Lukas Bruderer (2015), Nour Zalmai (2017), Carina Stritt (2017), Federico Wadehn (2019), Boxiao Ma (2021), Raphael Keusch (2022), and Elizabeth Ren (2023).

In recent years, there has been an emphasis on reducing nontrivial model-based inference to iterated linear-Gaussian estimation, e.g.:

R. Keusch and H.-A. L.,
Model-predictive control with NUP priors,
arXiv:2303.15806
H.-A. L.,
On NUP priors and Gaussian message passing,
IEEE Int. Workshop on Machine Learning for Signal Processing (MLSP), Sept. 2023.
IEEE Xplore / Preprint
Yun Peng Li and H.-A. L.,
Dual NUP representations and min-maximization in factor graphs,
2025 IEEE Int. Symp. on Information Theory (ISIT), June 2025.
IEEE Xplore / arXiv:2501.12113

See also the Lecture Notes of my course Model-Based Estimation and Signal Analysis (held at ETH until 2025).

However, many recent ideas and results have not yet been properly written up. I am planning to write a longer tutorial text on factor graphs for signal processing that will cover also these recent ideas.

Last modified: March 28, 2026