[Paper Review] Scaffolding Simulations with Deep Learning for High-dimensional Deconvolution

Anders Andreassen, Patrick Komiske+4 more|arXiv (Cornell University)|May 10, 2021

Gaussian Processes and Bayesian Inference16 references20 citations

TL;DR

OmniFold is a deep learning-based, unbinned deconvolution method that leverages likelihood-ratio estimation via binary classifiers to remove high-dimensional detector distortions from experimental data. It extends the Richardson-Lucy algorithm to handle noise, acceptance, and efficiency effects without binning, enabling accurate, high-dimensional truth reconstruction in scientific simulations with state-of-the-art precision and convergence in few iterations.

ABSTRACT

A common setting for scientific inference is the ability to sample from a high-fidelity forward model (simulation) without having an explicit probability density of the data. We propose a simulation-based maximum likelihood deconvolution approach in this setting called OmniFold. Deep learning enables this approach to be naturally unbinned and (variable-, and) high-dimensional. In contrast to model parameter estimation, the goal of deconvolution is to remove detector distortions in order to enable a variety of down-stream inference tasks. Our approach is the deep learning generalization of the common Richardson-Lucy approach that is also called Iterative Bayesian Unfolding in particle physics. We show how OmniFold can not only remove detector distortions, but it can also account for noise processes and acceptance effects.

Motivation & Objective

• To address the limitations of traditional binned deconvolution methods like Richardson-Lucy, which require fixed binning and struggle with high-dimensional or variable-dimensional data.
• To develop a simulation-based maximum likelihood deconvolution framework that operates directly on unbinned, high-dimensional data using deep learning.
• To incorporate realistic experimental effects—such as noise processes, detector acceptance, and efficiency thresholds—into the deconvolution process.
• To enable accurate reconstruction of truth-level distributions from detector-level data without relying on analytical models of the response function.
• To provide a scalable, differentiable, and statistically principled method for scientific inference in high-energy physics and other data-intensive fields.

Proposed method

• OmniFold uses iterative likelihood-ratio estimation via deep neural networks to approximate the inverse of the detector response, replacing traditional matrix inversion with end-to-end learning.
• It formulates deconvolution as a series of binary classification tasks: distinguishing between truth and data-level samples, with class-conditional densities estimated via neural networks.
• The method employs weighted classifiers to estimate the likelihood ratio $ p_{S|G} / p_{S|G}^{ ext{prior}} $, which drives the iterative unfolding process.
• It handles noise by training classifiers to distinguish signal from background, and accounts for acceptance and efficiency via dummy values ($\emptyset$) and weighted training examples.
• The algorithm uses a two-step iterative process: Step I estimates the truth-level distribution from data and simulation, and Step II refines the signal reconstruction using likelihood ratios.
• The framework is implemented using fully connected neural networks with ReLU activations and sigmoid outputs, trained via Adam optimization with early stopping.

Experimental results

Research questions

• RQ1 Can deep learning enable unbinned, high-dimensional deconvolution in the absence of analytical response models?
• RQ2 How does the inclusion of auxiliary features (e.g., noise, acceptance, efficiency) affect the accuracy and convergence of deconvolution?
• RQ3 Can a differentiable, iterative framework based on likelihood-ratio estimation outperform traditional binned methods like Richardson-Lucy in high-dimensional settings?
• RQ4 To what extent can the method handle complex detector effects such as non-uniform acceptance and efficiency thresholds?
• RQ5 Does the use of neural networks for likelihood-ratio estimation preserve statistical consistency and maximum likelihood properties?

Key findings

• OmniFold achieves near-convergence in just three iterations, with results stable and consistent across multiple runs in the one-dimensional Gaussian example.
• In the five-dimensional Gaussian test case, using all available features ($X_D, X_S, Z_0, ..., Z_4$) reduced the mean absolute deviation to $3.1 \times 10^{-2}$ with a standard deviation of $3.1 \times 10^{-2}$, approaching the true value of 30.00.
• The method maintains high accuracy even when the number of features increases, demonstrating robustness to high-dimensional input spaces.
• Including auxiliary variables like noise components ($Z_i$) significantly improves precision, with the standard deviation of the mean decreasing from 8.4(5) to 3.1(2) over iterations.
• The algorithm successfully reconstructs the true distribution with minimal bias, as evidenced by the mean of the unfolded distribution being within $2\sigma$ of the true value across all scenarios.
• The framework is extensible to complex data types, such as permutation-invariant, variable-length inputs, as demonstrated by prior work using deep sets in collider physics.