[Paper Review] CaloMan: Fast generation of calorimeter showers with density estimation on learned manifolds

Jesse C. Cresswell, Brendan Leigh Ross+4 more|arXiv (Cornell University)|Nov 23, 2022

Particle physics theoretical and experimental studies40 references36 citations

TL;DR

CaloMan learns a low-dimensional manifold of calorimeter showers with a density estimator on that manifold, enabling fast, conditioned generation of realistic showers. It achieves substantial speedups over prior methods while maintaining realism.

ABSTRACT

Precision measurements and new physics searches at the Large Hadron Collider require efficient simulations of particle propagation and interactions within the detectors. The most computationally expensive simulations involve calorimeter showers. Advances in deep generative modelling - particularly in the realm of high-dimensional data - have opened the possibility of generating realistic calorimeter showers orders of magnitude more quickly than physics-based simulation. However, the high-dimensional representation of showers belies the relative simplicity and structure of the underlying physical laws. This phenomenon is yet another example of the manifold hypothesis from machine learning, which states that high-dimensional data is supported on low-dimensional manifolds. We thus propose modelling calorimeter showers first by learning their manifold structure, and then estimating the density of data across this manifold. Learning manifold structure reduces the dimensionality of the data, which enables fast training and generation when compared with competing methods.

Motivation & Objective

• Motivate fast calorimeter shower simulation to reduce computational cost in LHC detector studies.
• Present a two-step approach that first learns a manifold for calorimeter showers and then estimates density on that manifold.
• Demonstrate that manifold-based density estimation can provide fast generation with accurate high-level features.

Proposed method

• Use a generalized autoencoder (a variational autoencoder) to learn a low-dimensional manifold of calorimeter showers (latent dimension ~20).
• Train a conditional normalizing flow (neural spline flow) on the latent space to estimate p(z|E_inc) and thus p_M(x|E_inc) on the manifold.
• Condition the density estimator on incident energy while keeping the autoencoder unconditioned to the energy for clustering purposes.
• Estimate intrinsic dimensionality via the Levina-Bickel estimator (k nearest neighbors) to set latent dimension.
• Preprocess data by normalizing voxel energies and appending E_dep as a feature; decode via VAE decoder and apply preprocessing undo steps for generation.

Experimental results

Research questions

• RQ1 How can calorimeter showers be represented on a low-dimensional learned manifold?
• RQ2 Can density be effectively estimated on the learned manifold to generate realistic showers?
• RQ3 What is the impact of intrinsic dimensionality on training efficiency and generation speed?
• RQ4 How does CaloMan compare to existing fast shower generators in realism and speed?

Key findings

• Intrinsic dimension of the photon shower data is about 20 (d ≈ 20) based on the k=10 Levina-Bickel estimator.
• A two-step model with a VAE on the manifold and a conditional normalizing flow for p(z|E_inc) achieves realistic high-level features.
• Binary classifier distinguishing real vs generated showers achieves AUC of 0.78, below 1.0 but competitive with prior methods (CaloScore reported 0.98).
• Generation speed is significantly faster than prior methods, with amortized times as low as 0.02 ms per shower for large batches (e.g., 50k/100k).
• On the 1k and 10k batch settings, sample times range from ~0.02 to ~0.09 ms per shower, demonstrating substantial speedups over CaloScore (40 ms) and CaloFlow II (0.08 ms).
• Generated showers match deposited energy distributions and centers of energy; widths require further improvement, and conditioning on E_inc could improve layer-wise energy allocations.