Localized anomaly detection with a calorimeter normalizing flow

Raunav Mendiratta; Jason Weitz; Javier Duarte

physics.ins-detOct 30, 2026

Localized anomaly detection with a calorimeter normalizing flow

Authors: Raunav Mendiratta, Jason Weitz, Javier Duarte

Organizations: Westview High School, San Diego, USA · Department of Physics, University of California San Diego, La Jolla, USA

Abstract

Normalizing flows trained as fast calorimeter simulators also give the likelihood of each shower, which can serve as an anomaly score. Because this likelihood sums over every voxel, a signal confined to a few voxels can be lost in the fluctuations of the rest of the shower. For example, in a public benchmark dataset in which a long-lived neutral particle decays to two photons, the CaloFlow likelihood used in previous work selects less than 5% of the signal at a background efficiency of $10^{-3}$ when the particle has the largest simulated mass and decays at or just behind the front face of the calorimeter. In this work, we define a regional anomaly score by summing the squared latent coordinates of the same flow within small detector regions around the point where the particle enters the calorimeter, with the threshold set using background showers alone. At the same background efficiency, this regional anomaly score selects more than 95% of the signal. The per-coordinate likelihood terms, summed over the same regions, perform equally well, but the squared latent coordinates have a known distribution for background showers and clearly illustrate when a shower departs from it. Therefore, normalizing flows may be sensitive to new physics, even if the global likelihood is not.

Figures & tables

Figure 1: Energy deposited in the three calorimeter layers (rows) by individual showers. The columns show a particle with $m_{X}=$5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ decaying upstream at $d_{\mathrm{dec}}=$0.33\text{\,}\mathrm{m}$$ , a particle of the same mass decaying at the calorimeter front face at $d_{\mathrm{dec}}=$1.00\text{\,}\mathrm{m}$$ , and a background photon. The upstream decay produces two separate cores, while the front-face decay produces a single core that is hard to tell from that of the photon. The axes give the voxel indices in the two transverse directions, labeled $\varphi$ and $\eta$ as in Ref. [ 39 ] . All panels use the same logarithmic energy scale; white voxels contain no deposited energy. — Figure 1: Energy deposited in the three calorimeter layers (rows) by individual showers. The columns show a particle with $m_{X}=$ 5\text{,}\mathrm{G}\mathrm{e}\mathrm{V} $decaying upstream at $d_{\mathrm{dec}}=$0.33\text{\,}\mathrm{m}$ , a particle of the same mass decaying at the calorimeter front face at $d_{\mathrm{dec}}=$ 1.00\text{,}\mathrm{m}$$ , and a background photon. The upstream decay produces two separate cores, while the front-face decay produces a single core that is hard to tell from that of the photon. The axes give the voxel indices in the two transverse directions, labeled $\varphi$ and $\eta$ as in Ref. [ 39 ] . All panels use the same logarithmic energy scale; white voxels contain no deposited energy.

Figure 2: Concentric square regions in (a) layer 0, (b) layer 1, and (c) layer 2. The impact position is at the origin of each panel. Together with the three complete layers, these regions form the

14

regions of the regional score. Dashed lines show the region boundaries, and darker shading denotes regions closer to the impact position. All panels cover the same

480\text{\,}\mathrm{m}\mathrm{m}$\times$480\text{\,}\mathrm{m}\mathrm{m}

transverse area.

term per coordinate	summed over	$\varepsilon_{S}$ , $1.00\text{\,}\mathrm{m}$	$\varepsilon_{S}$ , $1.04\text{\,}\mathrm{m}$
$\tfrac{1}{2}z_{a}^{2}$	all coordinates (global latent norm)	$0.42\pm 0.10$	$0.46\pm 0.15$
	three layers	$0.50\pm 0.09$	$0.65\pm 0.13$
	regions (regional score $M$ )	$0.964\pm 0.005$	$0.972\pm 0.002$
$-\ell_{a}$	all coordinates	$0.002\pm 0.004$	$0.006\pm 0.007$
	regions	$0.86\pm 0.07$	$0.87\pm 0.08$
$\tfrac{1}{2}z_{a}^{2}-\ell_{a}$	all coordinates (image likelihood)	$0.006\pm 0.009$	$0.05\pm 0.02$

Table 1: Signal efficiency at

\varepsilon_{B}=10^{-3}

for the two front-face samples (

m_{X}=

5\text{,}\mathrm{G}\mathrm{e}\mathrm{V}

, $d_{\mathrm{dec}}=$1.00\text{\,}\mathrm{m}

and

1.04\text{\,}\mathrm{m}

). For each latent coordinate

a

, the term

\tfrac{1}{2}z_{a}^{2}

, its log-Jacobian contribution

-\ell_{a}

(Appendix A.5 ), or the per-coordinate likelihood term

\tfrac{1}{2}z_{a}^{2}-\ell_{a}

is summed over all

504

coordinates, over the three complete layers, or over the

14

regions. In the last two cases the largest standardized sum defines the score. Only the sums over regions select nearly all of the signal.

Figure 3: Background rejection

1/\varepsilon_{B}

as a function of the signal efficiency for the front-face samples at (a)

d_{\mathrm{dec}}=

1.00\text{,}\mathrm{m}$$ and (b)

1.04\text{\,}\mathrm{m}

. Each score is thresholded at the

\varepsilon_{B}

quantile of its conformal

p

-values on the test photons. The dashed line marks

\varepsilon_{B}=10^{-3}

, the working point of Table 1 , and the dotted line a random selection,

\varepsilon_{S}=\varepsilon_{B}

. The regional score and the regional likelihood run close to

\varepsilon_{S}=1

along the right edge of each panel. The likelihood score was evaluated only at the seven marked background efficiencies, the lowest of which lies on the upper edge of each panel. Bands show the standard deviation over the five trained flows.

Figure 4: Squared latent coordinate $z_{a}^{2}$ of each voxel for (a) test photons in the energy bin of the front-face samples and (b) signal showers with $m_{X}=$5\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ decaying at $d_{\mathrm{dec}}=$1.00\text{\,}\mathrm{m}$$ . The left block shows the mean over showers and over three of the five trained flows, and the right block shows, for one flow, the shower with the median regional score in each sample. Columns show the three layers, drawn to scale over the same area as in Fig. 2 , and the dashed square marks the innermost layer-1 region. For photons the mean is close to one almost everywhere and a single shower fluctuates about it, while for the signal the excess is concentrated at the impact position, on average and in a single shower. — Figure 4: Squared latent coordinate $z_{a}^{2}$ of each voxel for (a) test photons in the energy bin of the front-face samples and (b) signal showers with $m_{X}=$ 5\text{,}\mathrm{G}\mathrm{e}\mathrm{V} $decaying at $d_{\mathrm{dec}}=$1.00\text{\,}\mathrm{m}$ . The left block shows the mean over showers and over three of the five trained flows, and the right block shows, for one flow, the shower with the median regional score in each sample. Columns show the three layers, drawn to scale over the same area as in Fig. 2 , and the dashed square marks the innermost layer-1 region. For photons the mean is close to one almost everywhere and a single shower fluctuates about it, while for the signal the excess is concentrated at the impact position, on average and in a single shower.

Figure 5: Fraction of test photons and of signal showers with

m_{X}=

5\text{,}\mathrm{G}\mathrm{e}\mathrm{V}

and $d_{\mathrm{dec}}=$1.00\text{\,}\mathrm{m}

selected as a function of the threshold on

-\log_{10}\hat{p}

, for (a) the global latent norm and (b) the regional score. The dashed line marks

\hat{p}=10^{-3}

. For calibrated

p

-values the fraction of photons selected is close to

\hat{p}

, so the photon curve follows the diagonal, and a score separates the signal when its signal curve stays high where the photon curve is low. Panel (c) gives the fraction of signal showers at

1.00\text{\,}\mathrm{m}

and

1.04\text{\,}\mathrm{m}

for which each region attains the maximum; “inner” and “second” denote the innermost region of a layer and the one surrounding it, and “other” collects the remaining regions. Bands and error bars show the standard deviation over the five trained flows.

Figure 6: Signal efficiency at $\varepsilon_{B}=10^{-3}$ as a function of the mass $m_{X}$ and the decay position $d_{\mathrm{dec}}$ for (a) the regional score, (b) the global latent norm, and (c) the likelihood score $-\log p_{\mathrm{CF}}$ , the baseline of Ref. [ 39 ] . Each entry is the mean over the five trained flows. Panels (a) and (b) come from a separate evaluation of the flows compared to the one in Table 1 , and at the front-face points, the two agree within the spread over the flows. The color scale is logarithmic and saturates at $10^{-3}$ , the background efficiency, so the darkest entries are not separated from photons. The calorimeter front face is at $d_{\mathrm{dec}}=$1.00\text{\,}\mathrm{m}$$ . — Figure 6: Signal efficiency at $\varepsilon_{B}=10^{-3}$ as a function of the mass $m_{X}$ and the decay position $d_{\mathrm{dec}}$ for (a) the regional score, (b) the global latent norm, and (c) the likelihood score $-\log p_{\mathrm{CF}}$ , the baseline of Ref. [ 39 ] . Each entry is the mean over the five trained flows. Panels (a) and (b) come from a separate evaluation of the flows compared to the one in Table 1 , and at the front-face points, the two agree within the spread over the flows. The color scale is logarithmic and saturates at $10^{-3}$ , the background efficiency, so the darkest entries are not separated from photons. The calorimeter front face is at $d_{\mathrm{dec}}=$ 1.00\text{,}\mathrm{m}$$ .

Anomaly Detection

Anomaly detection finds rare or unexpected patterns in data, from defective products and system faults to fraud and security threats.

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Conformal calibration and look-elsewhere effect in anomaly detection for new-physics searches

Jack Y. Araz, Michael Spannowsky

Department of Physics and Astronomy, University College London, London, WC1E 6B, UK · Department of Engineering, City St. George’s, University of London, London, EC1V 0HB, UK · Institute for Theoretical Physics, Campus Süd, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany+1

Post-Hoc Calibration Conformal Prediction Anomaly Detection

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Machine-learned anomaly detection is reshaping searches for new physics, but it has outrun the statistics used to interpret it. A raw anomaly score has no calibrated meaning, a model that scans many regions inflates the look-elsewhere effect, and the asymptotic significances the field relies on are blind to the background mismodelling that anomaly detectors are especially prone to. We propose a calibration layer, built on conformal prediction, that turns any anomaly score into a defensible significance with distribution-free, finite-sample guarantees. Conformal prediction converts scores into valid local p-values, weighted and Mondrian variants repair the sideband-to-signal-region exchangeability failures that resonant searches suffer, and a Gross-Vitells step carries the result through to a look-elsewhere-aware global significance. The layer does two things at once. It exposes miscalibration that the standard pipeline cannot see, and it corrects it without retraining the detector. On public LHC Olympics data, a classifier develops a substructure-mass correlation that makes sideband-calibrated background p-values anti-conservative. Taken at face value, this manufactures a

\sim 46σ

excess from background sculpting alone, which the label-free weighted correction removes, restoring an honest null. When run as a blind wide-mass bump hunt, the standard asymptotic and unweighted procedures fabricate

\gtrsim10σ

excesses and

\approx5σ

excesses even in signal-free windows, while the conformal layer raises no false alarms and its global false-positive rate is verified on background-only pseudoexperiments. The result is an auditable, detector-agnostic path from an uncalibrated score to a trials-factor-aware significance, ready to be folded into experimental anomaly searches.

Abstract

Anomaly Detection

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