Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributions

TL;DR

This paper introduces Christoffel-DPS, a distribution-free sensor placement method for diffusion posterior sampling that outperforms traditional Gaussian-based approaches, especially in complex, non-Gaussian distributions.

Contribution

It proposes a novel, distribution-free sensor placement framework based on the Christoffel function, with theoretical guarantees and practical variants for generative models.

Findings

Christoffel-DPS outperforms Gaussian OSP baselines.

The method is effective on non-Gaussian benchmarks.

It provides non-asymptotic bounds on sensor numbers needed for recovery.

Abstract

State estimation is a critical task in scientific, engineering and control applications. Since the reliability of reconstructions depends on the number and position of sensors, optimal sensor placement (OSP) is essential in scenarios where measurements are sparse and expensive. Classical OSP approaches rely on Gaussian assumptions and are consequently unable to account for the complex distributions encountered in many real-world systems. Generative-model-based reconstruction using sensor guided diffusion posterior sampling (DPS) has emerged as a promising technique for reconstructing states from highly complex distributions. However, existing sensor-selection methods either require unrealistically many sensors or emulate classical OSP, creating a mismatch between modern recovery models with classical OSP tools motivating the need for fundamentally new ideas towards OSP that match the recent advances made in powerful recovery models. We introduce a distribution-free sensor placement framework based on the Christoffel function: a mathematical formulation of optimal sampling and recovery guarantees for posterior sampling with arbitrary sensors and signal distributions, from which we derive a new OSP strategy with non-asymptotic bounds on the number of sensors needed for recovery. We develop Christoffel-DPS, with offline and online variants, instantiating Christoffel sampling for generative models. Christoffel-DPS outperforms Gaussian OSP baselines and existing generative-model placement methods, validating that distribution-free sensing is both theoretically principled and practically superior. The framework is model-agnostic; we demonstrate its application to a range of unconditional DPS and flow-matching models on structurally non-Gaussian benchmarks, showing the efficacy of Christoffel-DPS in low sensor budget regimes.