Discussion, Acknowledgments, and References

Table of Links

Abstract and 1. Introduction

2. Unfolding

   2.1 Posing the Unfolding Problem

   2.2 Our Unfolding Approach

3. Denoising Diffusion Probabilistic Models

   3.1 Conditional DDPM

4. Unfolding with cDDPMs

5. Results

   5.1 Toy models

   5.2 Physics Results

6. Discussion, Acknowledgments, and References

Appendices

A. Conditional DDPM Loss Derivation

B. Physics Simulations

C. Detector Simulation and Jet Matching

D. Toy Model Results

E. Complete Physics Results

6 Discussion

Our results demonstrate that a single cDDPM can successfully unfold detector effects on particle jets from a variety of physics processes, including those not seen during training. The distinguishing feature of our method is its non-iterative and flexible posterior sampling approach, which exhibits a strong inductive bias that allows the cDDPM to generalize to unseen processes without explicitly assuming the underlying physics distribution, setting it apart from other unfolding techniques so far.

The cDDPM’s ability to unfold data from combined processes presents a new opportunity to handle detector data prior to background subtraction, streamlining the unfolding process. These results are consistent for both the DELPHES CMS and the analytical data-driven detector simulations, indicating that the cDDPM’s performance is not drastically limited by the degree of detector smearing. We expect this approach to be applicable to other particles, detector-level observables, and event-wise quantities, enabling the reconstruction of full events after unfolding.

Several open questions remain regarding the implementation of the conditioning on the moments. These include optimal selection of priors and the number of moments required for the best unfolding performance. Further investigation is needed to determine the extent of the cDDPM’s inductive bias and its tolerance to variations in the underlying physics processes. Understanding these aspects will be crucial for refining the method and ensuring its robustness across a wide range of scenarios.

While this approach shows promise, improvements such as uncertainty estimation, accounting for systematic and experimental uncertainties, and handling particles falling outside detector thresholds are necessary to fully realize its potential. Incorporating these enhancements will be essential for making the cDDPM a reliable and comprehensive unfolding tool in high-energy physics. We leave these developments for future work.

Acknowledgments

This work has been made possible thanks to the support of the Department of Energy Office of Science through the Grant DE-SC0023964. Shuchin Aeron and Taritree Wonhjirad would also like to acknowledge support by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi.org/).

References

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A Conditional DDPM Loss Derivation

In the proposed conditional DDPM, the forward process is a Markov chain that gradually adds Gaussian noise to the data according to a variance schedule β.

Training is performed by optimizing the variational bound on negative log likelihood:

Following the similar derivation provided in [13], this loss can then be rewritten using the KL-divergence

and the reverse process posterior as

B Physics Simulations

C Detector Simulation and Jet Matching

Our results present the unfolding of detector effects from two different detector simulation frameworks: DELPHES and a data-driven approach. DELPHES is a framework developed for the simulation of multipurpose detectors for physics studies [12]. Specifically, the DELPHES CMS configuration is frequently used as the detector simulation of choice in recent machine-learning based unfolding studies.

D Toy Model Results

The full unfolding results of the toy model tests are shown here, including the η, ϕ, and E distributions of the data 4-vectors. In fig. 9 we show the full results of multidimensional unfolding tests as well as the tests on the cDDPM dependence on the training prior when learning the posterior. In fig. 10 the full results for the moments-based unfolding are shown.

In fig. 10 we present the complete unfolding results for the multidimensional toy model tests using a class of exponential functions. The successful unfolding of all particle properties for the test datasets underscores the cDDPM’s capacity to generalize beyond the specific distributions seen during training.

E Complete Physics Results

In fig. 11 and fig. 12 we present the unfolding results for the remaining particle properties (η, ϕ, E, px, py, pz) that were not shown in the main text. These additional plots demonstrate the cDDPM’s ability to successfully unfold the full particle vector, providing a comprehensive view of its performance across all dimensions.

The tables in this appendix (table 3 and table 4) provide a detailed breakdown of the unfolding performance metrics for each particle property and dataset. Furthermore, these metrics (Energy distance, Wasserstein distance between histograms, and KL divergence between histograms) offer a more in-depth perspective on the unfolding quality.

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