stat.MLSep 29, 2026

Post-Anomaly Detection Inference for Deep SVDD

Authors: Cao Le Cong Thanh, Dang Quang Vinh, Vo Nguyen Le Duy

Organizations: University of Information Technology, Ho Chi Minh City, Vietnam · Vietnam National University, Ho Chi Minh City, Vietnam

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Abstract

Deep Support Vector Data Description (Deep SVDD) has become a prominent framework for unsupervised anomaly detection by learning latent representations that compactly characterize normal data around a center. Despite its empirical success, anomaly decisions produced by Deep SVDD are typically made solely based on anomaly scores without rigorous statistical guarantees, thereby limiting their reliability in safety-critical and high-stakes applications where false positives must be strictly controlled. In this paper, we propose PADI (Post-Anomaly Detection Inference), a novel framework that equips a trained and frozen Deep SVDD detector with statistically valid inference by leveraging the Selective Inference framework. Specifically, PADI performs inference conditional on the event that a test instance is identified as anomalous by Deep SVDD, thereby enabling rigorous statistical assessment of anomaly decisions. Based on this formulation, we derive valid selective p-values that quantify the statistical significance of the detected anomaly. Using these p-values, we theoretically establish control of the false positive rate (FPR) at a user-specified significance level αα (e.g., α=0.05α=0.05). Furthermore, we extend the proposed framework to Deep Semi-Supervised Anomaly Detection (Deep SAD), providing a principled approach for statistically reliable inference in semi-supervised anomaly detection settings. Extensive experiments on both synthetic and real-world benchmark datasets robustly support the theoretical findings. The results demonstrate that PADI consistently achieves proper FPR control while attaining superior true positive rates compared with existing approaches.

Figures & tables

Figure 1: Overview of the proposed PADI method. A trained and frozen Deep SVDD model first maps test instances into the latent space and selects anomalies whose distance-based anomaly scores exceed a fixed threshold. For each selected anomaly, PADI performs SI conditional on the anomaly-selection event induced by the Deep SVDD detector. Using independent normal reference samples, PADI computes a selective pp -value that quantifies the statistical significance of the detected anomaly while accounting for the selection event. The right panel illustrates how PADI differs from conventional inference: unlike naive pp -values that ignore the anomaly-selection process, PADI provides statistically valid inference for detected anomalies.
Figure 2: Geometric illustration of the truncation region Z\mathcal{Z} on the 1D line parametrized by zz . The final region is constructed through four steps: (1) Zregion\mathcal{Z}_{\mathrm{region}} (blue) identifies the intervals where the line intersects the affine regions P1,…,PK\mathcal{P}_{1},\dots,\mathcal{P}_{K} . (2) Zscore\mathcal{Z}_{\mathrm{score}} (orange) defines the intervals where the quadratic anomaly score exceeds the threshold τ\tau , computed independently per region. (3) Zsign\mathcal{Z}_{\mathrm{sign}} (green) enforces the global sign-pattern constraint required to linearize the ℓ1\ell_{1} -norm discrepancy. (4) The final truncation region Z\mathcal{Z} (red) is the intersection of the sign constraint with the union of region-specific valid scores. The observed test sample zobsz_{\mathrm{obs}} lies within this final valid set.
Figure 3: GPU-based parallelization of PADI for convolutional Deep SVDD encoders. Before the line-search computation, the frozen CNN parameters, the input data X,A,BX,A,B , and the current feasible interval are copied to GPU memory. The forward propagation is performed by custom Numba-CUDA kernels for Conv2D , BatchNorm , SILeakyReLU , SIMaxPool , and the final fully connected layer. The Conv2D , BatchNorm , and fully connected kernels propagate the forward values and affine coefficients X,A,BX,A,B . In addition, the SILeakyReLU and SIMaxPool kernels compute local interval constraints induced by the LeakyReLU branches and max-pooling indices. These local constraints are combined on the GPU to update the feasible interval. The final transformed data X(L),A(L),B(L)X^{(L)},A^{(L)},B^{(L)} and the final interval are then returned to the host.
Figure 4: Results on synthetic data.
Figure 5: Runtime comparison between the proposed Numba-CUDA implementation and the PyTorch-based implementation for CNN-based PADI.
Figure 6: Results on real tabular datasets. Left: empirical FPR of PADI, OC, and Naive. Right: empirical TPR of the statistically valid methods. PADI achieves stronger TPR than OC while preserving FPR control.
Figure 7: Results on real image data from the MVTec AD dataset. Left: empirical FPR of PADI, OC, and Naive across five classes. Right: empirical TPR of the statistically valid methods. PADI maintains FPR control while achieving higher TPR than OC.
Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Results on synthetic data for the extension to Deep SAD.
Figure 9: Results on real tabular datasets for the extension to Deep SAD. Left: empirical FPR of PADI, OC, and Naive. Right: empirical TPR of the statistically valid methods. PADI consistently achieves stronger TPR than OC while preserving FPR control.
Figure 10: Results on real image data from the MVTec AD dataset for the extension to Deep SAD. Left: empirical FPR of PADI, OC, and Naive across five classes. Right: empirical TPR of the statistically valid methods. PADI empirically maintains FPR close to the significance level while achieving higher TPR than OC.
Figure 11: Empirical FPR of PADI under non-Gaussian data distributions.
Figure 12: Effect of the reference set size on the empirical FPR and TPR.
Figure 13: Effect of the data dimension on the empirical FPR.
Figure 14: Additional evaluation on synthetic data for all competing methods.
 4 blocks5 blocks6 blocks7 blocks
Phase I0.010810.011020.011510.01142
Phase II9.4651512.7130024.6065345.99283
Phase III0.248530.165480.195140.15108
Appendix
Table 1: Runtime breakdown (in seconds) of the three phases in Algorithm 1 using the proposed Numba-CUDA implementation.
StageRuntime (s)
Deep SVDD training615.34
Deep SVDD inference0.9007
PADI9.3698
Appendix
Table 2: Runtime Breakdown: Training, Inference, and the Proposed PADI Method

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