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Latent Anomaly Detection Through Density Matrices

Joseph Gallego-Mejia, Oscar Bustos-Brinez, Fabio A. González

TL;DR

The paper addresses anomaly detection by learning a density score from a density matrix $rho$ built from adaptive Fourier features that approximate a Gaussian kernel $k_gamma$. It introduces two end-to-end variants, ADDM (shallow) and LADDM (which adds an autoencoder to learn a latent representation $z$ and latent features $o=[z, l_Euc, l_cos]$). Density estimates are computed as $f_gamma_hat(x) approx M_gamma^{-1} phi_aff(x)^T rho phi_aff(x)$ with a spectral decomposition $rho approx V^T Lambda V$ to enable scalable evaluation, and anomalies are detected by a threshold $tau$ derived from the target anomaly proportion $r ext{%}$. On twenty datasets, both ADDM and LADDM offer competitive or superior performance against fourteen baselines, with ADDM excelling in low-dimensional settings and LADDM gaining advantages in high-dimensional ones.

Abstract

This paper introduces a novel anomaly detection framework that combines the robust statistical principles of density-estimation-based anomaly detection methods with the representation-learning capabilities of deep learning models. The method originated from this framework is presented in two different versions: a shallow approach employing a density-estimation model based on adaptive Fourier features and density matrices, and a deep approach that integrates an autoencoder to learn a low-dimensional representation of the data. By estimating the density of new samples, both methods are able to find normality scores. The methods can be seamlessly integrated into an end-to-end architecture and optimized using gradient-based optimization techniques. To evaluate their performance, extensive experiments were conducted on various benchmark datasets. The results demonstrate that both versions of the method can achieve comparable or superior performance when compared to other state-of-the-art methods. Notably, the shallow approach performs better on datasets with fewer dimensions, while the autoencoder-based approach shows improved performance on datasets with higher dimensions.

Latent Anomaly Detection Through Density Matrices

TL;DR

The paper addresses anomaly detection by learning a density score from a density matrix built from adaptive Fourier features that approximate a Gaussian kernel . It introduces two end-to-end variants, ADDM (shallow) and LADDM (which adds an autoencoder to learn a latent representation and latent features ). Density estimates are computed as with a spectral decomposition to enable scalable evaluation, and anomalies are detected by a threshold derived from the target anomaly proportion . On twenty datasets, both ADDM and LADDM offer competitive or superior performance against fourteen baselines, with ADDM excelling in low-dimensional settings and LADDM gaining advantages in high-dimensional ones.

Abstract

This paper introduces a novel anomaly detection framework that combines the robust statistical principles of density-estimation-based anomaly detection methods with the representation-learning capabilities of deep learning models. The method originated from this framework is presented in two different versions: a shallow approach employing a density-estimation model based on adaptive Fourier features and density matrices, and a deep approach that integrates an autoencoder to learn a low-dimensional representation of the data. By estimating the density of new samples, both methods are able to find normality scores. The methods can be seamlessly integrated into an end-to-end architecture and optimized using gradient-based optimization techniques. To evaluate their performance, extensive experiments were conducted on various benchmark datasets. The results demonstrate that both versions of the method can achieve comparable or superior performance when compared to other state-of-the-art methods. Notably, the shallow approach performs better on datasets with fewer dimensions, while the autoencoder-based approach shows improved performance on datasets with higher dimensions.
Paper Structure (17 sections, 14 equations, 2 figures, 7 tables, 3 algorithms)

This paper contains 17 sections, 14 equations, 2 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Baseline Anomaly Detection Methods
  4. Anomaly Detection Through Density Matrices: ADDM and LADDM
  5. Autoencoder
  6. Fourier Features
  7. Quantum Measurement-based Estimation
  8. Anomaly Detection Step
  9. Complexity Analysis
  10. Experimental Evaluation
  11. Experimental Setup
  12. Baseline Methods
  13. Datasets
  14. Metrics
  15. Results and Discussion
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Structure of the proposed method, in its deep version (LADDM). Step 1, autoencoder representation learning. Step 2, training of the model using both the reconstruction error and the maximum likelihood estimation of the density matrix $\rho$. Step 3, estimation of the density of a new sample using the learn density matrix $\rho=V^T\Lambda V$. Step 4, anomaly detector using the proportion of the anomalies. The shallow version (ADDM) presents the same structure, except for the first Step.
  • Figure 2: Results for Friedman-Nemenyi test over the metrics. Our methods (LADDM and AD-DMDKE) are the most different from others.