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Planted directed polymer: Inferring a random walk from noisy images.

Sun Woo P Kim¹, Austen Lamacraft²

¹King's College London, Department of Physics, Strand, London WC2R 2LS, United Kingdom.

Physical Review. E

|March 19, 2025

Summary

This summary is machine-generated.

We introduce a planted directed polymer model for inferring random walker paths from noisy data. On a Cayley tree, a phase transition occurs where path inference becomes impossible as noise increases.

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Area of Science:

Statistical Physics
Machine Learning
Information Theory

Background:

The directed polymer model is a fundamental concept in statistical physics.
Bayesian inference is crucial for reconstructing hidden states from noisy observations.
Hidden Markov models provide a framework for sequential data analysis.

Purpose of the Study:

To introduce and analyze the planted directed polymer model.
To investigate the behavior of random walkers inferred from noisy image data.
To explore phase transitions in Bayesian inference for hidden Markov models.

Main Methods:

Formulation as a nonlinear Bayesian inference problem.
Generalization of the directed polymer problem in statistical physics.
Numerical investigations and analytical arguments for one-dimensional walkers.
Application of directed polymer methods to Cayley tree structures.

Main Results:

No phase transition was observed for a one-dimensional walker.
A phase transition was identified on a Cayley tree with decreasing signal-to-noise ratio.
Inference becomes impossible at the transition, with path overlap dropping from one to zero.

Conclusions:

The planted directed polymer model offers new insights into hidden state inference.
The presence or absence of phase transitions depends on the underlying structure (1D vs. Cayley tree).
Decreasing signal-to-noise ratio critically impacts the reliability of path inference.