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Interpreting the predictions of neural network classification based on a Taylor Coefficient Analysis (TCA)

Markus Klute, Artur Monsch, Lars Sowa, Roger Wolf

arXiv:2609.12801v1cs.AIphysics.data-an

TL;DR

Signal-inference analyses in enriched phase space can lack control regions for benchmarking, heightening the importance of interpreting neural-network predictions. The paper introduces TCA, a Taylor-based analysis of input-feature influence, and concludes that properties most influential in representative LHC-complexity tasks are usually captured through second order.

  • Problem

    Signal-inference analyses in an S-enriched phase space lack a control region by construction, making trust in neural-network predictions increasingly important.

    Source: §1 · p. 1

  • Method

    TCA expands a neural network's output in its input space and treats Taylor coefficients, including higher-order terms, as generalized features of prediction.

    Source: §1 · p. 2, §2 · p. 2

  • Results

    The study supports the assumption that TCA up to second order usually captures the input properties most influential for tasks of typical CERN LHC complexity.

    Source: §1 · p. 2

  • Takeaways & Limitations

    TCA offers a way to identify which input-space properties contribute to a given neural-network prediction.

    Source: §1 · p. 2

    With ReLU activations, TCA provides no information beyond first order about hidden layers, an intrinsic property of that network model.

    Source: §4.4.2 · p. 15

Abstract

from arXiv · show

We introduce a rigid and comprehensive taxonomy and paradigm for characterizing the influence of the input feature space $X$ on the predictions $\hat{y}$ of a neural network (NN) used for event classification, based on a Taylor expansion of $\hat{y}$ in $X$. The complete process of introspection we refer to as Taylor Coefficient Analysis (TCA). Based on two simplistic example tasks, which can be easily understood and bencmarked, we illustrate the power of the TCA when it comes to revealing, what properties of $X$ have led to what value of $\hat{y}$, of a given NN model, building up intuition for the method. A more complex application is meant to represent $X$ of a typical classification task at a CERN LHC experiment. Based on this application, we play through the different levels of introspection that the TCA offers and discuss a number of practical aspects for a TCA application typical for the analysis of CERN LHC data. We conclude with a study to support the assumption that those properties of $X$ most relevant for tasks of the complexity typical for a CERN LHC experiment, are usually caught by a TCA up to the second order.

1 Introduction

Signal-inference analyses can lack the control regions used to benchmark object-identification algorithms, increasing the importance of trust and interpretability. The paper presents Taylor Coefficient Analysis (TCA) as a framework for examining how input properties shape neural-network predictions, with examples and a CERN LHC-like application.

1 Introduction

  • Signal-inference analyses in an S-enriched phase space lack a control region by construction, motivating trust in algorithmic predictions.Object-identification benchmarks in control regions can reveal sizeable corrections to identification efficiencies and misidentification rates.Source: §1 · p. 1
  • High-energy particle physics offers large IID datasets, realistic event simulators, and a few tens of preprocessed observables, making it a setting for NN introspection.The paper notes that the observables are rich in non-trivial correlations.Source: §1 · p. 1
  • TCA decomposes a neural network's output by Taylor-expanding it over input features to examine how those features influence predictions.The paper develops a taxonomy, illustrates interpretation with two simple tasks, and applies the method to a more realistic CERN LHC analysis.Source: §1 · pp. 1–2
  • The paper's LHC-like study supports the assumption that TCA up to second order usually captures the input properties most influential on predictions for tasks of representative complexity.The authors also discuss practical aspects of applying TCA to CERN LHC data.Source: §1 · p. 2

2 Foundation of the method

The method treats Taylor coefficients of the network output as generalized input features, extending ordinary single-observable features to interactions and higher orders. It aggregates these coefficients over samples and situates TCA among feature-attribution and introspection methods.

2 Foundation of the method

  • TCA estimates the influence of input observables on the network output through a Taylor expansion, considered here for binary classification.The setup associates background with output approaching 0 and signal with output approaching 1.Source: §2 · p. 2
  • A generalized feature α is associated with Taylor coefficient t_α, with its order given by the number of observables in α.This extends the ordinary feature concept to include higher-order properties of the input space.Source: §2 · p. 2
  • First-order features correspond to individual observables, while second-order features include output curvature and pairwise correlations between observables.The Taylor expansion can extend to features of still higher order.Source: §2 · p. 2
  • For a peaked signal over a non-peaking background, the curvature coefficient for invariant mass can encode the peak's position and its importance relative to the background shape.The paper illustrates this with a new particle decaying to two reconstructed τ leptons.Source: §2 · p. 2
  • The paper aggregates Taylor coefficients over samples using their absolute values, since signs can cancel when outputs have symmetries across input subspaces.Other aggregation methods, including higher-dimensional summary statistics, are also possible.Source: §2 · pp. 2–3
  • Evaluating aggregated coefficients on validation data estimates expected feature influence under testing conditions, while evaluations across training checkpoints track its development.Evaluating coefficients over pairs of observables can identify the input subspaces from which that influence arises.Source: §2 · p. 3

2.1 Related work

  • Leave One Out removes each observable and trains a reduced-input model, but does not directly inspect the original model being interpreted.The method assesses each observable's contribution by monitoring task performance across the resulting models.Source: §2.1 · p. 3
  • SHAP estimates an observable's additive contribution across coalitions while marginalizing other variables, with exponential complexity in input dimension.Unlike first-order saliency maps, TCA explicitly addresses higher-order derivatives and correlated inputs in manageable-dimensional spaces.Source: §2.1 · p. 3

3 Instructive Examples

Across two binary Gaussian classification examples, Taylor Coefficient Analysis (TCA) tracks how input properties contribute to neural-network predictions. The examples show first-order location information and second-order correlations shaping classification, with the learned influence changing over training in the correlated example.

3 Instructive Examples

  • Both examples use 2D Gaussian S and B samples and the same one-hidden-layer network; Example-1 has zero within-class covariance, while Example-2 uses correlations of 0.5 and −0.5.Each class has 100,000 training and testing events, and each update uses the full dataset as one epoch.Source: §3 · pp. 3–4

3.1 Example-1

  • In Example-1, first- and second-order coefficients converge after an initial transition, and an ensemble of 100 random initializations shows small final spreads.The loss curves for training and validation are nearly indistinguishable, which the authors interpret as indicating no overfitting and excellent generalization.Source: §3.1 · p. 4
  • In Example-1, location features and x1–x2 correlation have the largest post-training influence, despite zero off-diagonal covariance in the generating Gaussians.The authors attribute the cross-term to an intrinsic correlation in the samples, as S and B occupy opposite corners of the feature plane.Source: §3 · p. 3, §3.1 · p. 4
  • Example-1’s largest coefficient contributions lie near the decision boundary, where classification is unclear; first-order terms encode location, while second-order terms encode correlation and feature confinement.The cross-term exceeds the diagonal second-order terms in this example.Source: §3.1 · pp. 5–6

3.2 Example-2

  • After about 700 epochs in Example-2, the x1–x2 cross-term reliably exceeds every first-order feature, while its maps show correlation effects contributing where S and B overlap most.The overlap region is where the off-diagonal covariance elements are most important for classification.Source: §3.2 · p. 7
  • In Example-2, training first identifies positional features, then reduces their influence as second-order features—especially the cross-term—gain importance and the loss falls from 0.5 to about 0.15.The authors describe this as learning coarse, obvious features before subtler ones; they suggest coefficient trajectories can help monitor training progress.Source: §3.2 · pp. 6–7
  • After 120 epochs, Example-2’s coefficient values remain close to Example-1’s, but after 380 further epochs its decision plane bends to reflect the changed class overlap and correlations.The cross-term receives contributions from transition regions and from the region of greatest S–B overlap.Source: §3.2 · p. 7

4.1 Datasets

The study models H→ττ as signal and semileptonic tt production as background, using simulated events and object selections to build the classification dataset.

4.1 Datasets

  • The simulated signal is ggH with H→ττ, while the background is semileptonic tt production; samples use leading-order QCD generation and detector simulation without pileup.The simulation uses MadGraph5_aMC@NLO, Pythia, and DELPHES.Source: §4.1 · pp. 7, 9
  • Events require exactly one light lepton and at least one τh; when multiple τh candidates occur, the one with highest Djet forms the pair, with ΔR(ℓ, τh) > 0.5.The τh selection corresponds roughly to 50% identification efficiency and 0.5% light-jet misidentification.Source: §4.1 · p. 9
  • The selected samples contain 358,000 events each for signal and background, and enter training and evaluation without process-prevalence scaling.Source: §4.1 · p. 9
  • Expected distinctions include a signal mass peak near 125 GeV and lower jet multiplicity, versus at least four background jets and typically two b jets.The b-tagging observable β uses loose, medium, or tight working points with stated light-jet misidentification rates of approximately 10%, 1%, and 0.1%.Source: §4.1 · p. 9

4.2 NN task

A 20-dimensional DNN separates the simulated signal and background using event observables; marginal prediction plots illustrate how selected inputs relate to the classification.

4.2 NN task

  • The task uses a 20-input DNN with one 100-node tanh hidden layer and sigmoid output, trained on 60% of balanced samples with 40% reserved for validation.The output associates background with 0 and signal with 1; the setup aims for high separation power and interpretable TCA features.Source: §4.2 · pp. 9–10
  • Inputs include object kinematics, lepton azimuth, b-tag observables, missing transverse momentum, angular and spatial separations, jet multiplicity, and τ-pair mass variables.Source: §4.2 · p. 9
  • Tail-bearing inputs are trimmed at their 97.5% quantiles and log-transformed where specified; all inputs are standardized to mean zero and unit standard deviation.Source: §4.2 · p. 10
  • Marginal plots show mvis peaking for signal but also background, while mcol modes are stretched for both; predictions can saturate near signal-density maxima when multiple observables contribute.Source: §4.2 · p. 10
  • pmiss_T has similar unimodal distributions with shifted maxima; Δϕ(pmiss_T) and βj2 separate signal and background, whereas φℓ is flat and non-discriminating.Source: §4.2 · p. 10

4.3 Assessment of the training success

The network trains successfully, with low converged loss and high signal-background separation, while TCA tracks which first- and second-order features shape its predictions. The analysis shows how feature influence evolves during training and maps prediction changes and curvature across the mvis–pmiss T plane.

4.3 Assessment of the training success

  • L converges to ≈0.1 on training and validation samples, with no indication of overtraining; the ROC AUC is 0.99.The loss evolution and selected TCA features are shown in Fig. 7.Source: §4.3 · p. 10, Figure 7 · p. 12
  • ∆ϕ(pmiss T, pτhℓ T) is the most influential second-order feature, while βj2, ∆ϕ(pmiss T, pτhℓ T), and mvis lead the first-order features.The second-order ranking is followed by pj2; mvis and pmiss T rank sixth and seventh among first-order features, and mcol ranks eighth.Source: §4.3 · p. 10
  • mvis and mcol rank lower than their distinctive signal might suggest because their prediction values saturate near signal-density maxima and other features provide redundancy.Their squared-feature ranks are 27 and 76, respectively, among 210 second-order features.Source: §4.3 · p. 10
  • βj2 influence is identified quickly, but ∆ϕ(pmiss T, pτhℓ T) takes the lead from epoch 300, while mvis approaches βj2 by epoch 600.The dominant second-order feature continues to rise after 600 epochs despite stable loss; the authors interpret this as continued learning without indication of overtraining.Source: §4.3 · p. 12
  • In the mvis–pmiss T plane, the signal-enriched region forms a ridge, and the mvis directional derivative changes from maximal near mvis ≲50 GeV to minimal near 150 GeV.The pmiss T directional derivative rises near mvis ≈50 GeV and drops for mvis ≳100 GeV; second-order structure includes localized positive curvature near mvis and pmiss T ≲50 GeV.Source: §4.3 · pp. 12–13
  • The two-dimensional feature planes are harder to interpret because point-to-point fluctuations are amplified when predictions depend strongly on multiple observables.The paper notes that these fluctuations are not caused only by limited training-sample size.Source: §4.3 · p. 13

4.4 Practical considerations

The section sets out practical ways to manage TCA’s feature-analysis workload, explains how ReLU limits higher-order information, tests the relevance of Taylor orders, and extends the method to more complex architectures.

4.4.1 Analysis strategy

  • When feature dimension is moderate, evaluate all first- and second-order features; for larger analyses, prioritize first-order features and targeted second- and third-order checks.The authors caution that directing analysis toward presumed high-power features may introduce human selection bias.Source: §4.4.1 · p. 14
  • The cumulative count figure ranks observables by the cumulative appearance of their features, with area under each curve providing an influence score.∆ϕ(pmiss T, pτhℓ) leads with 0.83, followed by mvis (0.61), mcol (0.56), and ϕℓ (0.36).Source: §4.4.1 · p. 14, Figure 9 · p. 14

4.4.2 Neural networks with ReLU activation

  • With ReLU hidden activations, higher-order Taylor coefficients encode output-layer information, while hidden-layer contributions are limited to first order.The paper characterizes this as an intrinsic property of the chosen network model rather than a flaw in TCA.Source: §4.4.2 · p. 15

4.4.3 Relevance of higher orders of the TCA

  • For tasks of typical LHC complexity, the study supports that second-order TCA usually captures the input properties most relevant to classification.The approximation study used Taylor series expanded through orders one, two, and three; the model was retrained to stop after exceeding AUC 0.95 to reduce sensitivity to approximation errors.Source: §4.4.3 · pp. 15–16, Figure 10 · p. 16

4.4.4 TCA of more complex NN architectures

  • TCA can be applied to multiclass outputs and to graph-level or node-level predictions by expanding around selected outputs, nodes, or edges.For complex architectures, interpretation depends on a sound understanding of how the architecture addresses the physics task.Source: §4.4.4 · p. 16

5 Summary

The paper presents Taylor Coefficient Analysis (TCA) as a framework for examining how input-space properties shape neural-network event-classification predictions. Its examples and LHC-like application demonstrate the approach, while a study supports second-order analysis for tasks of typical LHC complexity.

5 Summary

  • TCA uses a Taylor expansion of neural-network predictions in the input space to characterize how input properties influence event-classification predictions.The authors position it as a taxonomy and paradigm for model introspection.Source: §5 · p. 17
  • Two simple tasks illustrate TCA, and an LHC-like classification task demonstrates its levels of introspection and practical application considerations.The simple tasks are described as readily understandable and benchmarkable.Source: §5 · p. 17
  • The study supports that TCA up to second order usually captures the input properties most relevant to tasks of typical LHC complexity.The authors say the framework grounds prior applications and provides a basis for future work, with code in Ref..Source: §5 · p. 17

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