Interpretable Multi-Hypersphere Deep Anomaly Detection for Open-set Supervised Anomaly Detection

Zhiji Yang; Fangyong Wang; Yue Li; Xianli Pan; Jianhua Zhao

cs.LGSep 19, 2026

Interpretable Multi-Hypersphere Deep Anomaly Detection for Open-set Supervised Anomaly Detection

Authors: Zhiji Yang, Fangyong Wang, Yue Li, Xianli Pan, Jianhua Zhao

Organizations: School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, China · Beijing National Center for Applied Mathematics, Capital Normal University, Beijing, 100048, China

Abstract

Multi-class open-set anomaly detection requires a model to characterize the normal acceptance domain formed by multiple heterogeneous subdistributions using only class-labeled samples from known normal classes, and to identify previously unseen anomalies at test time. Existing single-hypersphere methods cannot explicitly represent class-specific locations and acceptance ranges, while current multi-hypersphere or multi-class approaches do not fully integrate inter-class boundary constraints, learnable acceptance ranges, and interpretable decisions. To address these limitations, we propose Interpretable Multi-Hypersphere Deep Anomaly Detection (IMHD-AD). IMHD-AD constructs an independent hypersphere for each known normal class in a shared feature space. With target-inside and non-target-outside constraints, IMHD-AD embeds the class-specific hypersphere centers and radii directly into the final network layer and jointly optimizes them with the shared representation. The minimum signed boundary score across hyperspheres simultaneously determines open-set acceptance or rejection and provides a faithful geometric explanation of each decision. On MNIST, Fashion-MNIST, and CIFAR-10, IMHD-AD achieves the highest AUC in 28 of 30 open-set comparisons. A two-dimensional synthetic study further shows that model architecture must balance the compactness of known normal classes against the separability of unknown anomalies.

Figures & tables

图 1: Architecture of the proposed interpretable multi-hypersphere deep anomaly-detection model.

Input	Training set $\mathcal{D}$ ; number of classes $K$ ; hyperparameters $\nu$ , $\mu$ , and $\lambda$ ; learning rates $\eta_{\mathrm{p}}$ and $\eta_{\mathrm{d}}$ ; and number of epochs $T$ .
Output	Network parameters $\theta$ and $\{\mathbf{w}_{k},b_{k}\}_{k=1}^{K}$ ; hypersphere parameters $\{C_{k},R_{k}\}_{k=1}^{K}$ .
1	Initialize the network parameters and Lagrange multipliers $\alpha_{k}$ and $\beta_{k}$ .
2	for epoch $=1$ to $T$ do
3	Sample a mini-batch and compute $z_{i}=\phi(x_{i};\theta)$ .
4	Compute $g_{k}(x_{i})=\mathbf{w}_{k}^{\top}z_{i}+b_{k}$ .
5	Compute $s_{ik}=g_{k}(x_{i})+\\|z_{i}\\|_{2}^{2}$ .

表 1: Training procedure for IMHD-AD.

Dataset	Classes	Number of samples	Image size
MNIST	10	60,000 (train) + 10,000 (test)	$28\times 28$ (grayscale)
Fashion-MNIST	10	60,000 (train) + 10,000 (test)	$28\times 28$ (grayscale)
CIFAR-10	10	50,000 (train) + 10,000 (test)	$32\times 32$ (RGB)

表 2: Summary of the public benchmark datasets.

Dataset	Conv. layers	Conv. channels	Embedding dim.	Dropout
MNIST	2	6, 16	64	0.2
Fashion-MNIST	2	8, 24	96	0.3
CIFAR-10	3	32, 64, 128	256	0.5

表 3: Network configurations for the public image datasets.

Category	Parameter	Setting
Training	Random seed	42
	Optimizer	Adam
	Initial learning rate	$3.0\times 10^{-4}$
	Weight decay	$0.5\times 10^{-6}$
	Maximum epochs	200
	Batch size	200

表 4: Experimental hyperparameters for IMHD-AD.

Dataset	Known-class combination	MC-SVDD	DeepSVDD	DMSVDD	DeepMAD	IMHD-AD
MNIST	(0, 2)	91.23	86.14	97.64	94.07	96.77
	(0, 6)	94.96	96.52	88.75	96.56	96.99
	(0, 7)	93.71	87.68	97.15	96.85	95.40
	(0, 9)	91.83	91.85	87.96	91.61	97.25
	(4, 9)	92.85	95.23	83.89	83.70	97.50
CIFAR-10	(0, 2)	50.19	47.32	71.63	57.12	90.28

表 5: AUC comparison on MNIST, CIFAR-10, and Fashion-MNIST with two known normal classes and eight unseen anomaly classes.

Dataset	Known-class combination	MC-SVDD	DeepSVDD	DMSVDD	DeepMAD	IMHD-AD
MNIST	(2, 6, 9)	78.38	78.66	81.32	96.13	97.45
	(3, 6, 7)	73.60	76.76	78.03	96.60	96.68
	(0, 8, 9)	89.04	87.55	92.09	92.14	96.51
	(0, 6, 8)	89.43	88.09	86.44	87.29	93.92
	(0, 5, 8)	85.23	84.46	89.22	94.82	97.61
CIFAR-10	(2, 6, 9)	51.02	46.68	69.48	78.92	88.02

表 6: AUC comparison on MNIST, CIFAR-10, and Fashion-MNIST with three known normal classes and seven unseen anomaly classes.

Panel of Figure 8 — 图 2: Feature-space visualizations of the synthetic data under different network complexities.

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Anomaly Detection Interpretable Anomaly Detection Hyperspherical Representation Learning

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Anomaly detection is a crucial machine-learning task with wide-ranging applications. Deep Support Vector Data Description (Deep SVDD) is a prominent deep one-class method, but it is vulnerable to hypersphere collapse, often relies on heuristic choices for hypersphere parameters, and provides limited interpretability. To address these issues, we propose Interpretable Maximum Margin Deep Anomaly Detection (IMD-AD), which leverages a small set of labeled anomalies and a maximum margin objective to stabilize training and improve discrimination. It is inherently resilient to hypersphere collapse. Furthermore, we prove an equivalence between hypersphere parameters and the network's final-layer weights, which allows the center and radius to be learned end-to-end as part of the model and yields intrinsic interpretability and visualizable outputs. We further develop an efficient training algorithm that jointly optimizes representation, margin, and final-layer parameters. Extensive experiments and ablation studies on image and tabular benchmarks demonstrate that IMD-AD empirically improves detection performance over several state-of-the-art baselines while providing interpretable decision diagnostics.

Abstract

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