We propose SYRAN, an unsupervised anomaly detection method based on symbolic regression. Instead of encoding normal patterns in an opaque, high-dimensional model, our method learns an ensemble of human-readable equations that describe symbolic invariants: functions that are approximately constant on normal data. Deviations from these invariants yield anomaly scores, so that the detection logic is interpretable by construction, rather than via post-hoc explanation. Experimental results demonstrate that SYRAN is highly interpretable, providing equations that correspond to known scientific or medical relationships, and maintains strong anomaly detection performance comparable to that of state-of-the-art methods.
Figures & tables
Figure 1: Box plots and critical difference diagrams comparing the AUC-ROC performance of the SYRAN ensemble (a,b) and its best-performing ensemble member (a,c) against baseline anomaly detection methods.
Figure 2: Symbolic invariants learned by SYRAN on three ADBench datasets. Significant deviations of function values from 1 are indicative of anomalous behavior.
Figure 3: Mean AUC-ROC performance across benchmark datasets for SYRAN and its best ensemble member across different hyperparameterizations compared to LOF as its strongest competitor. The default configuration is highlighted in gray.
Complexity weight γ
Best Equation
AUC-ROC ( % )
0.001
(x31.0500)x6⋅(−0.0526)⋅2.2257
99.19
0.01
x2cos(x4−1.6023)
97.60
0.10
x2+∣1−x2∣
97.73
0.50
x2
97.73
Table 1: Effect of the complexity weight on symbolic expressions for the breastw dataset [ 32 ] . Here, xi represent the i -th feature of the dataset ( x2 corresponds to xcs in Section 5.2 ). We state the best ensemble member equation for each γ .
Anomaly detection is often applied to data stored in relational databases, yet most existing methods require flattening multiple tables into a single feature matrix. This flattening can obscure entity identity, schema structure, and multi-hop dependencies, limiting the detection of anomalies that depend on relational context rather than isolated feature values. Beyond preserving relational structure, relational anomaly detection raises an additional challenge: how to incorporate symbolic behavioral evidence into learned relational representations. To address these challenges, we study relational anomaly detection, where the goal is to identify anomalous entities or events in a multi-table database. We propose RAD, a rule-augmented relational anomaly detector that combines heterogeneous graph representation learning with refined symbolic rule signals. RAD derives candidate rules from random-forest paths over flattened summaries of the entities or events being scored, refines them into compact interpretable predicates, injects the resulting rule features into the graph model, and learns anomaly scores using reconstruction-based and pairwise-ranking supervision. To evaluate this setting, we introduce a relational anomaly detection benchmark spanning three settings: LANL cybersecurity event detection and two unexpected user-churn anomaly tasks derived from Amazon and H&M relational databases. Experiments show that RAD improves anomaly ranking over flattened tabular detectors and relational baselines under natural class imbalance, achieving the best average rank on AUROC and AUPRC across the benchmark. Ablations show that direct rule injection and ranking-based supervision are key contributors to performance, while edge reconstruction is not uniformly beneficial. Our code and data are available at: https://github.com/noahd15/RAD_RelationalAnomalyDetection.
Unsupervised anomaly detection scores each point of an unlabelled, contaminated sample in a single pass, and increasingly must also explain why a point is flagged. Yet the dominant detectors give a score with no account of which features drive it, and explanations are bolted on post-hoc with SHAP or LIME, which re-query the detector thousands of times per point and only approximate it. We introduce WAND, an unsupervised tabular anomaly detector that is explainable by design. WAND organises its computation around directions on the unit sphere, scoring each point by how far its projection escapes a sub-Gaussian extreme-value baseline. The originality of our approach is that the witness directions that flag a point, being vectors in feature space, are its explanation, a per-feature attribution obtained at no cost over scoring and, since the score is differentiable, recoverable by gradients. Scoring is linear in the sample size, and a probe-efficiency bound guarantees every anomaly a witness, hence an explanation. Across 47 ADBench datasets WAND attains the best mean Friedman rank at ROC-AUC parity with 16 unsupervised baselines, so the gain is interpretability at no accuracy cost; its native explanations are more accurate and faithful than post-hoc SHAP/LIME and ECOD at a fraction of the query cost. WAND is thus a practical, interpretable solution for explainable anomaly detection.
Lamine Diop
EPITA Research Laboratory, Le Kremlin-Bicˆetre FR-94276, France
Time-series anomaly detection trades off predictive accuracy, computational efficiency, and interpretability. We use a large language model not as the detector but as the author of one: an autonomous research loop in which the model repeatedly edits a single short NumPy program under a leakage-free objective, keeping the best-scoring detector it finds. The loop discovers two compact detectors, one for univariate and one for multivariate series, that describe short windows by their local spectral features and compare them with the training-region distribution through a covariance-aware distance. On the TSB-AD benchmark these detectors lead the field across metrics, ahead of the strongest classical, deep, and foundation-model baselines including Time-RCD, yet they train no network and use no GPU, and the multivariate detector is faster than every similarly performing baseline. LLM-driven program search is thus a practical route to accurate, efficient, and transparent detectors.
David Berghaus
Lamarr Institute Germany · Fraunhofer IAIS Germany