Revisiting Energy-based Tabular Anomaly Detection: Energy and Reconstruction are Complementary

Junichiro Niimi

cs.LGAug 14, 2026

Revisiting Energy-based Tabular Anomaly Detection: Energy and Reconstruction are Complementary

Authors: Junichiro Niimi

Organizations: Meijo University, Nagoya Aichi 4688502, Japan

Abstract

Tabular anomaly detection is dominated by classical density-proxy methods (Isolation Forest, OCSVM, LOF), reconstruction-based detectors (Autoencoders, VAEs), and modern non-parametric scorers (COPOD, ECOD, Deep SVDD), all of which approximate the inlier distribution only indirectly; explicit energy-based models are largely absent. Motivated by the recent revival of EBMs in deep learning (e.g., Energy-Based Transformers, JEPA), we revisit the classical Deep Boltzmann Machine (DBM) for this task and hypothesize that its mean-field energy combines more effectively with a reconstruction-based score than same-lineage pairs do. We evaluate a two-hidden-layer DBM on two tabular benchmarks spanning distinct domains (UCI Bank Marketing and NSL-KDD) against eight classical and modern baselines across twenty random seeds. The DBM mean-field energy matches the strongest baseline (the Autoencoder) on Bank Marketing and statistically beats it on NSL-KDD, while significantly outperforming the remaining seven on both datasets. When fused with the Autoencoder via rank fusion, the DBM energy yields a statistically significant improvement on both datasets (AUROC=+0.014, p<0.01 on Bank Marketing; +0.002, p<0.001 on NSL-KDD); every non-DBM-derived base model instead fails to improve or significantly degrades the AE-paired ensemble. Our position is that classical EBMs, exemplified by the DBM, deserve a place in the tabular anomaly detection toolbox as a non-redundant complementary view to the reconstruction-based scores that dominate current practice.

Figures & tables

Structure	$[H_{1},H_{2},\cdots]$	AUROC	PR-AUC	F1 (best)	$\Delta F$ -sum	epoch
Moderate	$[48,24]$	$\mathbf{0.737}$	$\mathbf{0.309}$	$\mathbf{0.370}$	$\mathbf{0.676}$	$32$
Single-RBM	$[48]$	$0.732$	$0.304$	$0.369$	$0.656$	$55$
Strong	$[32,16]$	$0.677$	$0.238$	$0.305$	$0.613$	$\mathbf{3}$
Expansion	$[96,128]$	$0.663$	$0.236$	$0.307$	$0.631$	$33$
Three-layer	$[48,24,12]$	$0.608$	$0.170$	$0.248$	$0.571$	$\mathbf{3}$

Table 1: Single-seed ablation of five DBM hidden-layer configurations on Bank Marketing. epoch = last epoch with val reconstruction-BCE improvement (max 60); a value of

3

signals immediate fine-tune termination.

\Delta F

-sum is the AUROC of

\sum_{c}|\Delta F_{c}|

, the column-wise intervention aggregate used in [ 29 ] . Bold marks the selected configuration. Full discussion in Appendix A .

Method	AUROC	PR-AUC	F1 (best)
DBM $F_{\text{before}}$	$0.713\pm 0.009$	$0.275\pm 0.009$	$0.339\pm 0.008$
AutoEncoder	$\mathbf{0.731\pm 0.004}^{\phantom{***}}$	$\mathbf{0.275\pm 0.006}^{\phantom{***}}$	$\mathbf{0.350\pm 0.004}^{\phantom{***}}$
VAE	$0.685\pm 0.003^{*\phantom{}}$	$0.252\pm 0.007^{\phantom{***}}$	$0.301\pm 0.006^{***}$
Isolation Forest	$0.617\pm 0.004^{***}$	$0.164\pm 0.003^{***}$	$0.252\pm 0.003^{***}$
One-Class SVM	$0.607\pm 0.001^{***}$	$0.178\pm 0.001^{***}$	$0.265\pm 0.001^{***}$
LOF	$0.659\pm 0.001^{***}$	$0.205\pm 0.001^{***}$	$0.276\pm 0.001^{***}$

Table 2: Single-method performance on Bank Marketing (

n=20

seeds). Mean

\pm

SE; significance markers from paired two-sided

t

-tests vs. DBM

F_{\text{before}}

{}^{*}p<0.05

{}^{**}p<0.01

{}^{***}p<0.001

. Best per column in bold.

Method	AUROC	PR-AUC	F1 (best)
DBM $F_{\text{before}}$	$\mathbf{0.995\pm 0.000}$	$\mathbf{0.981\pm 0.001}$	$\mathbf{0.943\pm 0.003}$
AutoEncoder	$0.993\pm 0.000^{***}$	$0.960\pm 0.001^{***}$	$0.933\pm 0.002^{*\phantom{}}$
VAE	$0.984\pm 0.001^{***}$	$0.881\pm 0.004^{***}$	$0.870\pm 0.004^{***}$
Isolation Forest	$0.964\pm 0.002^{***}$	$0.702\pm 0.021^{***}$	$0.765\pm 0.011^{***}$
One-Class SVM	$0.983\pm 0.000^{***}$	$0.960\pm 0.000^{***}$	$0.921\pm 0.000^{***}$
LOF	$0.767\pm 0.005^{***}$	$0.266\pm 0.001^{***}$	$0.450\pm 0.005^{***}$

Table 3: Single-method performance on NSL-KDD DoS subset (

n=20

seeds). Conventions as in Table 2 .

Figure 1: AUROC of

S_{\alpha}=\alpha F^{\mathrm{std}}_{\text{DBM}}+(1{-}\alpha)F^{\mathrm{std}}_{\text{AE}}

on Bank Marketing ( left ) and NSL-KDD ( right ); shaded band

=\pm 1

SE, star marks the peak, dashed red line shows the parameter-free rank-fusion reference. On both datasets the peak sits above the AE-only (

\alpha{=}0

) and DBM-only (

\alpha{=}1

) endpoints.

Score	AUROC	$\Delta$ vs. AE ( $p$ )	$\Delta$ vs. DBM ( $p$ )
AutoEncoder	$0.7306\pm 0.0040$	—	$+0.017\,(0.10)$
DBM $F_{\text{before}}$	$0.7134\pm 0.0091$	$-0.017\,(0.10)$	—
Convex (LOSO-tuned)	$0.7430\pm 0.0048$	$+0.012\,(0.003)^{**}$	$+0.030\,({<}10^{-3})^{***}$
Rank fusion	$\mathbf{0.7450\pm 0.0051}$	$+0.014\,(0.008)^{**}$	$+0.032\,({<}10^{-3})^{***}$

Table 4: Hybrid scoring on Bank Marketing (

n=20

seeds). Mean

\pm

SE of AUROC;

p

-values from paired two-sided

t

-tests against AE alone and DBM alone. Best in bold.

Score	AUROC	$\Delta$ vs. AE ( $p$ )	$\Delta$ vs. DBM ( $p$ )
AutoEncoder	$0.9933\pm 0.0001$	—	$-0.0016\,(2{\times}10^{-5})$
DBM $F_{\text{before}}$	$0.9949\pm 0.0003$	$+0.0016\,(2{\times}10^{-5})$	—
Convex (LOSO-tuned)	$0.9949\pm 0.0003$	$+0.0018\,({<}10^{-5})^{***}$	$+0.0000\,(10^{-4})^{***}$
Rank fusion	$\mathbf{0.9953\pm 0.0002}$	$+0.0021\,({<}10^{-8})^{***}$	$+0.0004\,(0.003)^{**}$

Table 5: Hybrid scoring on NSL-KDD DoS subset (

n=20

seeds). Conventions as in Table 4 .

	Bank Marketing				NSL-KDD (DoS)
Partner of AE	Solo	Ens.	$\Delta$	$p$	Solo	Ens.	$\Delta$	$p$
(AE alone)	$0.731$	$0.731$	—	—	$0.993$	$0.993$	—	—
DBM $F_{\text{before}}$	$0.713$	$\mathbf{0.745}$	$\mathbf{+0.014}$	$\mathbf{0.008^{**}}$	$0.995$	$\mathbf{0.995}$	$\mathbf{+0.002}$	$\mathbf{6{\times}10^{-9\,***}}$
DBM-recon.	$0.714$	$0.735$	$+0.004$	$0.09$	$0.995$	$0.995$	$+0.002$	$2{\times}10^{-12\,***}$
VAE	$0.685$	$0.717$	$-0.013$	${<}10^{-6\,***}$	$0.984$	$0.990$	$-0.003$	$8{\times}10^{-12\,***}$
LOF	$0.673$	$0.716$	$-0.015$	${<}10^{-6\,***}$	$0.767$	$0.957$	$-0.036$	$1{\times}10^{-14\,***}$

Table 6: AE-paired rank-fusion ensembles on Bank Marketing and NSL-KDD (

n=20

). Solo = partner AUROC; Ens. = ensemble with AE;

\Delta,p

= paired test of ensemble vs. AE alone. DBM-recon. = BCE between input and DBM mean-field reconstruction. Only the two DBM-derived partners significantly improve AE; every non-DBM partner significantly degrades it.

Energy	AUROC	PR-AUC	F1 (best)
$F_{E}$ (used)	$\mathbf{0.713\pm 0.009}$	$\mathbf{0.275\pm 0.009}$	$\mathbf{0.339\pm 0.008}$
$F_{\text{proper}}$	$0.711\pm 0.009$	$0.271\pm 0.010$	$0.335\pm 0.009$
$\Delta\ (p)$	$-0.002\ (0.055)$	$-0.004\ (0.009)^{**}$	$-0.004\ (0.009)^{**}$

Table 7: Mean-field energy

F_{E}=E(\mu)

vs. proper variational free energy

F_{\text{proper}}=E(\mu)-H(\mu)

on Bank Marketing. Mean

\pm

SE across

n=20

seeds;

p

from paired two-sided

t

-tests.

	Bank Marketing			NSL-KDD (DoS)
Method	$n_{\text{fit}}$	Train (s)	Score (ms/1k)	$n_{\text{fit}}$	Train (s)	Score (ms/1k)
DBM $F_{\text{before}}$	$35{,}930$	$175.8$	$1.56$	$60{,}609$	$468.1$	$2.07$
AutoEncoder	$35{,}930$	$16.7$	$4.42$	$60{,}609$	$28.4$	$4.52$
VAE	$35{,}930$	$10.2$	$3.33$	$60{,}609$	$17.8$	$3.40$
Deep SVDD	$35{,}930$	$2.36$	$\mathbf{0.13}$	$60{,}609$	$4.20$	$\mathbf{0.26}$
One-Class SVM	$8{,}000$	$0.25$	$56.3$	$8{,}000$	$0.40$	$54.6$

Table 8: Wall-clock cost of every method, measured on one machine (Apple M2 Max, CPU only). Train = fit from scratch, including the DBM’s pretraining and fine-tuning (

233{+}60

249{+}100

epochs under the early stopping of Section 3.1 ); Score = scoring all

45{,}211

74{,}826

samples, per

1{,}000

n_{\text{fit}}

is the training-set size each method receives. Means over 3 seeds (BM) and 1 seed (NSL-KDD).

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Model	$\Delta H$	pos.	AUROC $F_{E}$	AUROC $F_{\text{proper}}$	$\Delta$ AUROC	$p$
Bank Marketing	DBM-2L	$+0.433\pm 0.022$	$20/20$	$0.7134\pm 0.0091$	$0.7111\pm 0.0093$	$+0.0024$	$0.055$
Bank Marketing	RBM	$+0.420\pm 0.040$	$20/20$	$0.7209\pm 0.0030$	$0.7218\pm 0.0019$	$-0.0009$	$0.66$
NSL-KDD (DoS)	DBM-2L	$+1.138\pm 0.143$	$20/20$	$0.9949\pm 0.0003$	$0.9946\pm 0.0003$	$+0.0003$	$6\times 10^{-4\,***}$
NSL-KDD (DoS)	RBM	$+1.268\pm 0.111$	$20/20$	$0.9952\pm 0.0003$	$0.9949\pm 0.0002$	$+0.0003$	$6\times 10^{-5\,***}$

Appendix

Table 9: Scope of the entropy-term comparison across architectures and datasets (

n=20

seeds each).

\Delta H=H_{\text{anomaly}}-H_{\text{inlier}}

(nats), mean

\pm

SE; “pos.” = seeds with

\Delta H>0

\Delta\text{AUROC}=\text{AUROC}(F_{E})-\text{AUROC}(F_{\text{proper}})

, with

p

from a paired two-sided

t

-test. The Bank Marketing / DBM-2L row is the one reported in Table 7 .

Figure 2: Pairwise interaction effect $\bar{I}_{c,c^{\prime}}$ across $n=20$ seeds. Red: pairs whose non-additive coupling contributes more to the anomaly score than to the inlier score; blue: the reverse. Bonferroni-significant cells ( $p<5.5{\times}10^{-4}$ ) account for $31/91$ pairs.

$c$	$c^{\prime}$	$\bar{I}_{c,c^{\prime}}$	SE	paired $t$	frac. pos.
job	education	$-0.711$	$0.043$	$-16.5$	$0.00$
housing	month	$-0.664$	$0.053$	$-12.6$	$0.00$
month	day_of_week	$+0.611$	$0.048$	$+12.6$	$1.00$
contact	month	$+0.370$	$0.026$	$+14.4$	$1.00$
job	age	$-0.254$	$0.011$	$-23.8$	$0.00$

Appendix

Table 10: Top five column pairs by

|\bar{I}_{c,c^{\prime}}|

. “frac. pos.” = fraction of the 20 seeds for which the per-seed statistic is positive (

0.00

1.00

= full sign stability).

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Faculty of Electrical and Computer Engineering Technion – Israel Institute of Technology Haifa, Israel

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Open paper AskPDF arXiv

While deep features have transformed anomaly detection in images and video, their impact on tabular data has been less substantial, partly due to the limited availability of strong deep representations. Recently, prior-data fitted networks (PFNs) have emerged as a promising source of such representations for tabular data. In this work, we investigate how PFN representations can be adapted and leveraged for anomaly detection. The question is harder than it looks. No anomalies are available before deploy- ment, so model parameters cannot be tuned with supervision, and the reference set that defines normal behavior may itself contain the very anomalies it is supposed to reveal. We begin our study using frozen TabPFN features. Scoring each sam- ple by its distance to its nearest neighbors in feature space already gives strong results. We identify which layers to use and a feature-extraction procedure suited to the task. Next, to further improve performance, we use the reference set to fine- tune the model, so that the resulting features better separate normal samples from anomalies. On the ADBench benchmark, our fine-tuning free approach (ZEN) reaches a higher mean AUROC than every baseline, and our fine-tuned method (FOCUS) improves on it further. Our approach also generalizes across PFN models.

Abstract

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