Recovery Guarantees for Posterior Sampling of One-Bit Compressed Sensing

summary

The gist

The gist: The authors prove that posterior sampling achieves accurate recovery for noisy one-bit compressed sensing when measurement counts scale with the logarithm of the approximate covering

In short

The authors investigate reliable recovery from noisy one-bit compressed sensing when signals follow arbitrary prior distributions. They prove that posterior sampling achieves accurate recovery if measurement counts scale with the logarithm of the approximate covering number, establishing this as a necessary information-theoretic lower bound for this problem.

Key concepts

Approximate Covering Number: This is a geometric measure used to quantify the complexity or 'size' of a high probability region in signal space. In this context, it helps determine how many distinct signal patterns are needed to cover the relevant area of possibilities, which directly relates to the required sample size for reliable recovery.
Posterior Sampling: This is a method used to estimate the true underlying signal distribution by sampling from a posterior distribution. The authors extend this technique from continuous measurements to the challenging setting of noisy one-bit compressed sensing, analyzing its performance under various conditions.
Sample Complexity: This refers to the minimum number of measurements required for a reliable recovery method to succeed. The paper establishes bounds (both upper and lower) on this complexity, showing that it is fundamentally linked to the geometric complexity of the signal's prior distribution.
Noisy One-Bit Compressed Sensing: This scenario involves reconstructing a signal from extremely limited measurements where each measurement is quantized into only one bit. The paper analyzes recovery guarantees in this severe form of compression, considering how noise and quantization affect the ability to recover the original signal.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 25 episodes
Retrieval-Augmented Generation (RAG) — explained in 18 episodes
Federated Learning — explained in 17 episodes
Quantum Fisher Information (QFI) — explained in 16 episodes
Flow Matching — explained in 15 episodes
Interpretability — explained in 15 episodes

This episode discusses

The paper

Recovery Guarantees for Posterior Sampling of One-Bit Compressed Sensing · Read on arXiv

University of Electronic Science and Technology of China

We study the sample complexity of noisy one-bit compressed sensing for signals drawn from a prior distribution. By characterizing the effective distributional complexity of the prior via its approximate covering number, we prove that posterior sampling achieves accurate recovery with high probability when the number of measurements scales with the logarithm of the approximate covering number, up to a one-bit separation gap factor. This upper bound is robust to learned prior mismatch. Specifically, we show that posterior sampling with an approximate prior remains reliable, provided that the learned prior distribution is sufficiently close to the true signal distribution in Wasserstein distance. In addition, we establish a sample complexity lower bound for any reliable method of noisy one-bit compressed sensing, showing that our upper bound is nearly matched in its main prior dependent term. To approximate the ideal posterior sampling process for real world scenarios, we instantiate posterior sampling through a plug-and-play algorithm with diffusion priors. Experiments on the FFHQ and ImageNet datasets demonstrate the effectiveness of our proposed approach.

Transcript

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Recovery Guarantees for Posterior Sampling of One-Bit Compressed Sensing".

Tom: The gist: The authors prove that posterior sampling achieves accurate recovery for noisy one-bit compressed sensing when measurement counts scale with the logarithm of the approximate covering number,

Jane: First, who's behind it and why it matters.

Paper summary: Jane: So, looking at "Recovery Guarantees for Posterior Sampling of One-Bit Compressed Sensing" by Jing Ma and Yujia Wu and Zhaoqiang Liu, what’s the big picture here?

Tom: It really boils down to this: we have a precise way to measure the complexity of the prior distribution using its approximate covering number <ref:2610.11834#pg2>. This complexity dictates exactly how many measurements you need for reliable recovery in one-bit compressed sensing <ref:2610.11834#pg3>.

Lu: It’s about establishing a nearly matching information-theoretic necessary condition, which is something we needed for this specific setting <ref:2610.11834#pg3>.

Meng: For the practical person, it means we can use diffusion priors with an algorithm like PnP-OneBit to get good results on data like FFHQ or ImageNet, even when compression is really aggressive <ref:2610.11834#pg3>.

Lalam: And from a culture perspective, if this helps us understand the limits of what we can recover with limited measurements in compressed sensing, it gives us a better framework for designing next-generation signal processing models <ref:2610.11834#pg2>.

Tom: So, the paper shows that reliable recovery under severe nonlinear quantization can be precisely characterized by the high probability geometric complexity of the signal prior <ref:2610.11834#pg2>. That’s what they proved.

Conclusion: Tom: So we're wrapping up this deep dive into "Recovery Guarantees for Posterior Sampling of One-Bit Compressed Sensing."

Jane: It’s about how much data you actually need to get a signal back when your measurements are super noisy, one bit at a time.

Lu: The authors are basically showing that the complexity of the prior distribution—specifically this approximate covering number—is the real limit here.

Meng: So, instead of just guessing how much data you need, they give you a hard number based on the shape of your data.

Lalam: It establishes a necessary condition for recovery; without that much information complexity, you just can't reliably recover the signal.

Tom: It’s like saying this is the minimum cost to get a reliable result in this specific kind of compression.

Jane: And they did this by looking at both the best possible upper bound and a lower bound for how many measurements are required.

Lu: The key result is that that approximate covering number term shows up as unavoidable in any reliable recovery method you try to use.

Meng: For practical terms, it means if your prior distribution has high geometric complexity, you need more measurements to be sure you get the right signal back.

Lalam: This shifts the focus from just making algorithms work to understanding the fundamental information limits of this sensing problem.

Tom: It’s a really clean way of characterizing recovery under these kinds of severe nonlinear quantization issues.

Jane: And they even showed how a specific algorithm, PnP-OneBit, actually performs well on big image datasets like FFHQ and ImageNet with this framework.

Lu: That practical validation is important because it connects the theory to what we can actually build in practice with diffusion priors.

Meng: So, the world changes because now we have a clear way to predict if our limited sensing setup will fail before we even start the experiment.

Lalam: This pushes us toward designing sensing schemes that match the inherent geometric complexity of the data they are trying to capture.

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