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Anomaly Detection with Density Estimation
Benjamin Nachman, David Shih
TL;DR
The paper targets the need for more model-agnostic searches because specialized searches cannot cover every possible signal model or unexplored LHC phase-space region. It introduces ANODE, a neural-density-estimation method demonstrated on a simulated large-radius dijet search using the LHC Olympics 2020 R&D dataset.
Problem
Specialized LHC searches cannot cover every signal model, motivating more model-agnostic methods for unexplored phase space.
Method
ANODE uses neural density estimation to learn data and background densities in signal and sideband regions, forming their ratio and estimating background efficiency.
Results
ANODE is demonstrated using a simulated large-radius dijet search based on the LHC Olympics 2020 R&D dataset.
Takeaways & Limitations
The learned density ratio provides a classifier and direct background estimate, while ANODE remains robust against correlations that can affect CWoLa.
Takeaways & Limitations
The presented results are a proof of concept, with broader applications and alternative density-estimation algorithms left for future study.
Abstract
from arXiv · showhide
We leverage recent breakthroughs in neural density estimation to propose a new unsupervised anomaly detection technique (ANODE). By estimating the probability density of the data in a signal region and in sidebands, and interpolating the latter into the signal region, a likelihood ratio of data vs. background can be constructed. This likelihood ratio is broadly sensitive to overdensities in the data that could be due to localized anomalies. In addition, a unique potential benefit of the ANODE method is that the background can be directly estimated using the learned densities. Finally, ANODE is robust against systematic differences between signal region and sidebands, giving it broader applicability than other methods. We demonstrate the power of this new approach using the LHC Olympics 2020 R\&D Dataset. We show how ANODE can enhance the significance of a dijet bump hunt by up to a factor of 7 with a 10\% accuracy on the background prediction. While the LHC is used as the recurring example, the methods developed here have a much broader applicability to anomaly detection in physics and beyond.
1 Introduction
The paper motivates model-agnostic anomaly searches because existing LHC searches cannot cover every possible signal model or unexplored phase-space region. It introduces ANODE, a neural density-estimation method demonstrated on a dijet search, with potential advantages over CWoLa for correlated features and data-driven background estimation.
- Existing LHC searches have found no convincing evidence for new phenomena despite extensive searches, while theoretical and experimental motivations for new physics remain.
- Because specially optimized searches cannot cover every model or unexplored LHC phase-space region, the search program needs more model-agnostic methods.
- ANODE uses recent neural density-estimation advances to learn densities for signal-region and sideband data and construct a likelihood ratio sensitive to anomalies.
- Background estimation is essential for claiming discovery, motivating data-driven approaches alongside anomaly-sensitive discriminants.
- ANODE is demonstrated on the LHC Olympics 2020 R&D dataset through a large-radius dijet search using jet-substructure features to enhance an invariant-mass bump hunt.
- The method is compared with CWoLa hunting, which identifies differences between signal regions and sidebands but requires independence between discriminating and resonant features.
2 An Overview of Model (In)dependent Searches
The section organizes collider searches by their dependence on signal and background models, then compares common background-estimation strategies. ANODE occupies the model-agnostic landscape while directly learning a background density, though accurate density estimation is demanding.
- Overview: Searches must provide both sensitivity to new phenomena and an estimate of the Standard Model background under the null hypothesis.
- 2.1 BSM sensitivity: LHC search strategies span simulation-optimized searches, signal- or background-agnostic approaches, and newer methods seeking independence from both models.
- 2.1 BSM sensitivity: Autoencoders make few model assumptions but lack an optimality guarantee, whereas CWoLa assumes an uncorrelated feature set and approaches the likelihood ratio asymptotically.
- 2.1 BSM sensitivity: ANODE estimates background-only density from sidebands and compares it with density in a signal-sensitive region, providing broad resonant sensitivity and asymptotic optimality.
- 2.2 Background estimation: Common background estimates use simulation, control regions, ABCD regions, or smooth sideband fits interpolated into the signal region.
- 2.2 Background estimation: ANODE can accompany any background-estimation technique or provide one directly, but direct estimation imposes stringent density-accuracy requirements while reducing the need for decorrelation.
3 The ANODE Method
ANODE is an unsupervised, model-agnostic anomaly-search method that estimates data and background densities in discriminating features, using sidebands to interpolate the background into a localized signal region. Its likelihood-ratio selection supports signal-sensitive classification and several data-driven background-estimation strategies, while accommodating correlations between features and the localization variable.
- Signal and sideband regions: ANODE defines a signal region around a scanned localization feature m0 and uses the remaining values as sidebands.The signal region is m0 ± δ; the sideband excludes that interval.
- Density-ratio construction: Under the ideal mixture model, R(x|m) is the optimal test statistic, while in the absence of signal it equals 1.Signal-sensitive regions therefore correspond to deviations from the predicted background behavior.
- Density-ratio construction: The method estimates pdata(x|m) in the signal region and pbackground(x|m) from sidebands, then interpolates the background density into the signal region.Classification uses their likelihood ratio R(x|m).
- Neural density estimation: ANODE uses normalizing flows as a proof-of-concept density estimator, with masked autoregressive flows providing efficient neural density estimation.Conditional dependence on m is implemented by supplying m to the flow functions.
- Background estimation: The learned background density enables direct background estimation through density integration or importance sampling after applying an R(x|m) threshold.A separate sideband fit can also interpolate the selected background into the signal region.
- Comparison with CWoLa hunting: Unlike CWoLa hunting, ANODE does not require x and m to be independent, provided signal-region features are smoothly predictable from sidebands.This permits use of features strongly correlated with m.
4 Details of the Sample
The study illustrates ANODE with a simulated large-radius dijet resonance search using the LHC Olympics 2020 research and development dataset. It defines a signal region around the resonance and evaluates the method using high-level jet features, with density estimation requiring feature transformations and a fiducial restriction.
- Sample construction: The benchmark uses simulated large-radius dijet events from the LHC Olympics 2020 challenge research and development dataset, including a hypothetical W′ boson signal.The signal has mW′ = 3.5 TeV and arises from W′ decays into X and Y bosons with masses 500 GeV and 100 GeV.
- Sample construction: Events are selected with at least one large-radius jet having pT > 1.3 TeV after detector simulation and anti-kt clustering with R = 1.0.The X and Y decay products are captured by a single large-radius jet because of their large Lorentz boost.
- Feature definition: The resonant feature is the invariant mass of the leading two jets, mJJ, while the discriminating features are four-dimensional jet observables.The leading jets are ordered by jet mass, with mJ1 < mJ2.
- Feature definition: ANODE is demonstrated using high-level features, including jet masses and τ21 observables that distinguish two-prong signal jets from mostly one-prong background jets.The lighter jet mass peaks near mY, the jet-mass difference near mX − mY = 400 GeV, and τ21 is lower for the signal jets.
- Sample construction: The injected sample contains 1000 signal events, is split equally into training and testing samples, and defines the signal region as mJJ ∈ [3.3, 3.7] TeV.Each sample contains approximately 500,000 background and 500 signal events; the signal region retains approximately 60,000 background and 400 signal events.
- Baseline result: The baseline signal-to-background ratio in the signal region is S/B = 0.6%.This baseline provides the reference for assessing enhancement from ANODE.
- Density-estimation setup: Density estimation uses linearly scaled features followed by a logit transformation, with a fiducial requirement 0.05 < x < 0.95.The fiducial restriction retains 95% of the signal and 72% of the background in the signal region.
5 Results
ANODE learns density-based anomaly scores that concentrate background near R(x|m)=1 while isolating signal overdensities, enhancing bump-hunt sensitivity and enabling data-driven background estimates. Its performance is comparable to CWoLa in the nominal dataset, remains effective with correlated SR–SB differences, and predicts background efficiencies accurately away from extreme tails.
- Sensitivity: ANODE concentrates background near R(x|m)=1 while signal events populate a higher-R tail in relatively background-sparse regions.This pattern identifies localized overdensities through the learned density ratio.
- Sensitivity: Approximately 17% background spread around R=1 reflects SR and sideband density accuracy, interpolation quality, dataset size, network flexibility, and training.The reported width is attributed to multiple estimation and training sources.
- Sensitivity: ANODE achieves an AUC of 0.82, with performance comparable to CWoLa hunting across ROC and SIC evaluations.CWoLa performs somewhat better at higher signal efficiencies and much better at lower signal efficiencies.
- Background Estimation: For background efficiencies down to approximately 10^-3, both estimation methods are accurate within about 25%, while direct integration has about 10% bias.Both methods diverge from truth in the extreme tails of the R distribution.
- Correlated Features: With shifted correlated features, ANODE retains significant signal enhancement with maximum significance improvement near 4, whereas CWoLa becomes essentially random.The shifted dataset makes SR–SB differences larger than SR background–signal differences.
- Correlated Features: Direct density estimation gives background-prediction performance comparable to the unshifted dataset under the shifted-feature test.This supports ANODE's potential use as a complete anomaly-detection approach in that setting.
6 Conclusions
ANODE is presented as a model-independent anomaly-search method based on neural density estimation. It directly learns background and data densities, supports background-efficiency estimation, and is positioned as a proof of concept with broader future applications.
- ANODE directly learns the background and data probability densities in a signal region.Their ratio serves as a classifier for anomaly detection.
- The learned background density can estimate background efficiency after applying a classifier threshold.
- ANODE is robust against data correlations that can break other model-agnostic sideband methods such as CWoLa.
- The results are presented as a proof of concept, with possible applications beyond high energy physics and with alternative density-estimation algorithms.The paper also identifies higher-dimensional feature spaces and newer neural density estimators as future directions.
A Comments on optimality
The paper distinguishes likelihood-ratio optimality for the learned data distribution from optimality for any particular BSM model. ANODE is asymptotically optimal only for rejecting the data as background-only under its stated hypotheses.
- The Neyman-Pearson lemma gives likelihood-ratio tests maximum power for simple hypotheses at fixed level.It does not provide a uniformly most powerful classifier for composite alternatives or profiled nuisance parameters.
- ANODE compares a sideband-estimated background density with the learned density in a signal-sensitive region.
- ANODE is asymptotically optimal for rejecting the learned data distribution as background-only, but not for rejecting any particular BSM model.If the learned background and data densities are equal, the rejection power equals the test level.