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Why the Counterintuitive Phenomenon of Likelihood Rarely Appears in Tabular Anomaly Detection with Deep Generative Models?

Donghwan Kim, Junghun Phee, Hyunsoo Yoon

TL;DR

This work investigates whether the counterintuitive likelihood phenomenon observed in image-domain anomaly detection extends to tabular data. By defining a domain-agnostic criterion and conducting large-scale experiments on 47 tabular datasets and 10 CV/NLP embeddings, it shows that the phenomenon is rare in tabular domains and that simple likelihood testing with normalizing flows (NF-SLT) often outperforms baselines. The authors provide theoretical and empirical analyses linking dimensionality and feature correlation to the robustness of likelihood-based detection, including an intrinsic-dimension perspective and entropy-driven considerations. The findings support the practical viability of flow-based likelihood methods for tabular anomaly detection and highlight the importance of data structure in model reliability.

Abstract

Deep generative models with tractable and analytically computable likelihoods, exemplified by normalizing flows, offer an effective basis for anomaly detection through likelihood-based scoring. We demonstrate that, unlike in the image domain where deep generative models frequently assign higher likelihoods to anomalous data, such counterintuitive behavior occurs far less often in tabular settings. We first introduce a domain-agnostic formulation that enables consistent detection and evaluation of the counterintuitive phenomenon, addressing the absence of precise definition. Through extensive experiments on 47 tabular datasets and 10 CV/NLP embedding datasets in ADBench, benchmarked against 13 baseline models, we demonstrate that the phenomenon, as defined, is consistently rare in general tabular data. We further investigate this phenomenon from both theoretical and empirical perspectives, focusing on the roles of data dimensionality and difference in feature correlation. Our results suggest that likelihood-only detection with normalizing flows offers a practical and reliable approach for anomaly detection in tabular domains.

Why the Counterintuitive Phenomenon of Likelihood Rarely Appears in Tabular Anomaly Detection with Deep Generative Models?

TL;DR

This work investigates whether the counterintuitive likelihood phenomenon observed in image-domain anomaly detection extends to tabular data. By defining a domain-agnostic criterion and conducting large-scale experiments on 47 tabular datasets and 10 CV/NLP embeddings, it shows that the phenomenon is rare in tabular domains and that simple likelihood testing with normalizing flows (NF-SLT) often outperforms baselines. The authors provide theoretical and empirical analyses linking dimensionality and feature correlation to the robustness of likelihood-based detection, including an intrinsic-dimension perspective and entropy-driven considerations. The findings support the practical viability of flow-based likelihood methods for tabular anomaly detection and highlight the importance of data structure in model reliability.

Abstract

Deep generative models with tractable and analytically computable likelihoods, exemplified by normalizing flows, offer an effective basis for anomaly detection through likelihood-based scoring. We demonstrate that, unlike in the image domain where deep generative models frequently assign higher likelihoods to anomalous data, such counterintuitive behavior occurs far less often in tabular settings. We first introduce a domain-agnostic formulation that enables consistent detection and evaluation of the counterintuitive phenomenon, addressing the absence of precise definition. Through extensive experiments on 47 tabular datasets and 10 CV/NLP embedding datasets in ADBench, benchmarked against 13 baseline models, we demonstrate that the phenomenon, as defined, is consistently rare in general tabular data. We further investigate this phenomenon from both theoretical and empirical perspectives, focusing on the roles of data dimensionality and difference in feature correlation. Our results suggest that likelihood-only detection with normalizing flows offers a practical and reliable approach for anomaly detection in tabular domains.
Paper Structure (24 sections, 14 theorems, 36 equations, 11 figures, 26 tables)

This paper contains 24 sections, 14 theorems, 36 equations, 11 figures, 26 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Normalizing Flow
  4. Counterintuitive Phenomenon of Likelihood
  5. Definition of Counterintuitive Phenomenon
  6. Experiment
  7. Why Is The Simple Likelihood Test Successful in Tabular Data?
  8. High Dimension Perspective
  9. Feature Correlation Perspective
  10. Conclusion
  11. Definition of OOD Detection and Anomaly Detection
  12. Rigorous Formulation of Definition \ref{['def:counterintuitive']}
  13. Additional Description from Section \ref{['high_dim_perspective']}
  14. Euclidean Norm Gets More Concentrated as Dimension Increases
  15. Euclidean Norm Is Almost Identical in High-dimensional Space
  16. ...and 9 more sections

Key Result

Theorem 5.4

Let $P=\prod^d_{i=1}p_i(x_i)$ and $Q=\prod^d_{i=1}q_i(x_i)$ be independent $d$-dimensional continuous probability density models in $\mathbb{R}^d$ with same conditions as Lemma lemma:kldivergence_approx. Let $P_{\theta}$ be a well-trained density estimation model approximates $P$ (i.e., $p_{\theta}(

Figures (11)

  • Figure 1: (Left, Center) : Estimation of ID according to changes in $\rho$. The x-axis represents the value of $\rho$ and the y-axis represents dimension. The horizontal line represents the ambient dimension. (Right) : Log-scale scatter plot for ID estimate and ambient dimension using the TwoNN method of real dataset and synthetic dataset sampled by $\mathcal{N}(0,I_d)$. The estimated ID for the image dataset are from Table 1 in pope2021the. The distance recorded in the legend represents the average value of the distance between each point and the green line indicating when the ID estimate and ambient dimension are the same. Each blue/red point corresponds to one ADBench/image dataset.
  • Figure 2: Performance of NF-SLT implemented by NICE across different dimensions. The y-axis represents AUROC, and the x-axis indicates the dimensionality of the data. The titles of each subfigure specify distributions' parameters. Subplots 1 and 2 have a larger In/Out of Distribution difference than Subplots 3 and 4, and the speed of performance decay as the dimension increases becomes faster as the distribution difference becomes smaller.
  • Figure 3: Performance of NF-SLT implemented by RealNVP across different dimensions. The y-axis represents AUROC, and the x-axis indicates the dimensionality of the data. The titles of each subfigure specify distributions' parameters.
  • Figure 4: Histogram of log-likelihood values for the 1st subfigure in Figure \ref{['fig:dimsionality_test']}
  • Figure 5: Histogram of log-likelihood values for the 2nd subfigure in Figure \ref{['fig:dimsionality_test']}
  • ...and 6 more figures

Theorems & Definitions (25)

  • Definition 3.3: Occurrence of Counterintuitive Phenomenon, Informal
  • Theorem 5.4: Impact of Dimensionality on Likelihood Gap
  • Corollary 5.6: Dimensionality and AUROC Upper Bound
  • Definition 2.1: Occurrence of Counterintuitive Phenomenon
  • Proposition 3.1
  • proof
  • Theorem 3.2: klartag2007centralfleury2007stabilityguedon2014concentration
  • Proposition 3.3
  • proof
  • Theorem 3.4: guedon2011interpolating
  • ...and 15 more