[Paper Review] Machine learning as an instrument for data unfolding

A. Glazov|arXiv (Cornell University)|Jan 1, 2017

Machine Learning and Data Classification2 references17 citations

TL;DR

This paper proposes a machine learning-based unfolding method that corrects detector smearing effects by treating unfolding as a categorical classification problem, using a sequential neural network with categorical cross-entropy loss. The method achieves stable, bias-free unfolding on event-by-event basis using multiple reconstructed variables, demonstrating convergence and improved statistical precision in toy simulations, particularly when combining variables with different resolution characteristics.

ABSTRACT

A method for correcting for detector smearing effects using machine learning techniques is presented. Compared to the standard approaches the method can use more than one reconstructed variable to infere the value of the unsmeared quantity on event by event basis. The method is implemented using a sequential neural network with a categorical cross entropy as the loss function. It is tested on a toy example and is shown to satisfy basic closure tests. Possible application of the method for analysis of the data from high energy physics experiments is discussed.

Motivation & Objective

• Address the challenge of detector smearing in high-energy physics data unfolding without relying on binning in reconstructed variables.
• Develop a machine learning approach that treats unfolding as a categorization problem, assigning truth bins to each event based on reconstructed observables.
• Ensure the method is unbiased and convergent through iterative refinement using closure tests and prior distribution updates.
• Demonstrate the advantage of incorporating multiple reconstructed variables—especially those with complementary resolution—into a single unfolding framework.
• Provide a scalable, data-driven alternative to traditional matrix inversion or regularization-based unfolding techniques, with potential for integration into complex HEP analyses.

Proposed method

• Formulates the unfolding problem as a multi-class classification task, where each event is assigned to the most probable truth bin based on reconstructed variables.
• Employs a three-layer sequential neural network: input layer with N_inp neurons (1 or 2 variables), a dense hidden layer with κ×N_bin² ReLU-activated neurons (κ≈8), and an output layer with N_bin softmax-activated neurons.
• Uses categorical cross-entropy loss: H(p, q) = −Σ_e Σ_i p_e,i log q_e,i, where p_e,i is the one-hot truth label and q_e,i is the predicted probability.
• Applies ADADELTA optimization with dynamically adjusted learning rates for efficient training.
• Implements an iterative unfolding procedure: train on initial prior (e.g. flat or truth-based), apply to data, update prior using predicted probabilities, retrain, and repeat until convergence.
• Reduces statistical uncertainty by averaging results over 20 independent training sessions with different random seeds.

Experimental results

Research questions

• RQ1 Can a deep learning model effectively perform event-by-event unfolding without binning in reconstructed variables?
• RQ2 Does the iterative ML unfolding procedure converge to the true underlying distribution without introducing bias?
• RQ3 How does incorporating multiple reconstructed variables—especially with different smearing models—improve unfolding accuracy and statistical precision?
• RQ4 Can the method maintain stability and avoid bin-to-bin anticorrelations in the unfolded distribution, especially when the number of iterations increases?
• RQ5 To what extent does the method reduce statistical uncertainty compared to single-variable unfolding, particularly when one variable has poor resolution?

Key findings

• The ML unfolding method successfully passes all closure tests, including zero-level and first-level tests, confirming that the model architecture and training strategy are sufficient and unbiased.
• The iterative procedure converges rapidly, with stable results by the 10th iteration, and shows oscillatory behavior beyond that, indicating overfitting and bin-to-bin anticorrelations.
• When using a flat prior, the method achieves a prediction accuracy (PA) of ~65% with a single reconstructed variable, improving to ~70% when a second variable with log-normal smearing is added.
• The inclusion of a second reconstructed variable reduces the statistical uncertainty of the unfolded result, as shown in Figure 6, right, demonstrating improved precision.
• The method is insensitive to miscalibration between truth and reconstructed variables and automatically determines effective binning through the network’s learned boundaries.
• Training one epoch takes ~5 seconds on a GPU-accelerated laptop, but full training with convergence and uncertainty estimation is computationally intensive due to multiple training sessions and iterations.