Structure-aware anomaly detection via semantic prototype reconstruction in hyperbolic space

Abstract

Anomaly detection plays a crucial role in various applications such as industrial defect inspection and safety perception. Existing mainstream approaches typically rely on modeling the distribution of normal samples to build anomaly discrimination models. However, these methods often face challenges in practical scenarios due to ambiguous decision boundaries and limited capability to capture complex semantic and structural variations in defects. To overcome these limitations, we propose a novel industrial defect detection framework based on hyperbolic space. This framework exploits the negative curvature property of hyperbolic geometry to dynamically extract semantic prototypes and embed them into the hyperbolic space, enhancing the model’s ability to represent intricate semantic and structural changes. Furthermore, a semantic prototype-guided attention mechanism is integrated to assist the reconstruction of images, enabling accurate localization of anomalies through reconstruction error. Extensive experiments demonstrate that our method achieves state-of-the-art results on multiple industrial anomaly detection datasets. Notably, on the MVTec AD benchmark, our approach attains 99.8% and 99.1% AUROC scores for image-level and pixel-level tasks, respectively, significantly surpassing current leading methods.

Introduction

Anomaly detection has been widely adopted in industrial manufacturing due to its efficiency and reliability^1–3. Its focus is on identifying small but critical defects within large image datasets. Examples of such defects include scratches, cracks, and missing parts. Anomaly detection addresses diverse defect types with complex features.Slight color changes, shape distortions, and missing components all fall within its scope^4,5. Detecting different anomaly types accurately requires high sensitivity. Robustness to environmental changes and noise remains essential for industrial applications.

Traditional methods typically assume that defects follow stable and manually modelable patterns. Such assumptions limit their effectiveness in handling highly variable and unstructured defect forms found in real-world scenarios⁶. In recent years, deep neural network-based anomaly detection methods have achieved significant progress in industrial vision tasks. Anomaly detection and localization remain particularly challenging in industrial settings due to the scarcity of anomalous samples. Mainstream approaches usually rely on learning the feature distribution of large-scale normal samples during training, and then detect regions that deviate from this distribution during inference^7,8. Some methods generate synthetic anomalies on anomaly-free images by applying random occlusions, noise perturbations, or simulated defects to train models for estimating the decision boundary between normal and abnormal regions^9,10. Others detect anomalies by comparing input features against learned distributions or memory-encoded features^11,12. These approaches generally assume high consistency among training samples and that the feature space can compactly and accurately represent the boundary of normal data distribution. In practical industrial environments, image data often exhibit high feature distribution variability and structural heterogeneity due to sensor noise, imaging angles, illumination changes, and intrinsic part diversity. These factors violate the above assumptions and pose significant challenges for anomaly detection methods.

Recent studies have adopted prototype-based modeling strategies by extracting representative prototype vectors from normal images to characterize the semantic properties of normal regions^13,14. However, most existing prototype-based methods still organize features and prototypes in Euclidean space, which limits their ability to handle industrial anomaly images where anomalous regions are subtle, incomplete, or structurally similar to normal ones. In such cases, the semantic boundary between normal and anomalous features tends to become ambiguous, resulting in reduced discriminative power. To address this issue, we introduce hyperbolic space as a structural prior for prototype-based anomaly detection. As illustrated in Figure 1(a), hyperbolic space is a negatively curved non-Euclidean embedding space that is more effective at representing hierarchical structures and complex semantic relationships in high-dimensional data. By projecting semantic features and prototypes into hyperbolic space, our method can better separate different normal semantic modes while more effectively amplifying subtle deviations from normal prototypes.

Fig. 1.

(a) Comparison of spatial properties between Euclidean and hyperbolic spaces. (b) Existing anomaly detection methods. (c) Our proposed method. (d) Detection performance comparison of multiple methods across different datasets.

In this paper, we propose a hyperbolic space-based framework for industrial defect detection. Unlike existing methods that perform shallow alignment directly on categories, we embed both prototypes and sample features into hyperbolic space. The negative curvature of hyperbolic space allows for a more natural modeling of hierarchical structures and long-tailed distributions. Specifically, we dynamically extract semantic prototypes from input images and structurally model them in hyperbolic space. Additionally, we employ a semantic prototype-guided attention mechanism to assist in reconstructing normal regions of the image, while leveraging reconstruction errors to detect the presence of anomalous regions. We validated our approach on three datasets, and the results demonstrate that our model outperforms the previously best-performing method.

Our contributions are summarized as follows:

• We propose a dynamic anomaly detection framework based on hyperbolic space. This framework uses the structure-aware ability of hyperbolic geometry to better capture local geometric features.

• We introduce an adaptive prototype-aware approach that dynamically extracts semantic prototypes from input images. This mitigates semantic distribution shifts between training and testing phases and improves detection consistency across different categories and scenarios.

• Our method achieves state-of-the-art image-level and pixel-level AUROC performance on three benchmark datasets by integrating hyperbolic space to enhance geometric discriminability and employing adaptive prototype learning to improve robustness across diverse industrial environments.

Related work

Anomaly detection

Anomaly detection aims to identify unknown or inconsistent samples that deviate from the distribution of normal data^15–17. It plays an important role in a wide range of critical applications, including industrial surface defect inspection, medical image analysis, and intelligent video surveillance^18,19. In industrial visual inspection, anomalies usually manifest as surface defects on products, such as scratches, cracks, stains, and missing components^20,21. Among existing methods, reconstruction-based methods and prototype-/memory-based methods have become two important research directions in industrial anomaly detection in recent years^22–24.

Reconstruction-based methods detect anomalies by learning to recover normal samples and measuring the discrepancy between the input image and its reconstructed counterpart. Typical models, such as Autoencoders (AEs), Generative Adversarial Networks (GANs), and Variational Autoencoders (VAEs)^25–28, are trained only on normal data under the assumption that anomalous regions cannot be faithfully reconstructed. To improve robustness, some studies introduce sparse constraints, feature compression, or masking and inpainting strategies to encourage the model to better capture normal semantics and highlight anomalous regions^21,29,30. Although these methods have achieved promising results, they may still over-reconstruct anomalous regions when defects are subtle, structurally incomplete, or semantically similar to normal patterns.

Prototype-/memory-based methods represent normal data using prototype vectors, memory banks, or representative feature dictionaries^15–17. During inference, anomaly scores are computed based on the discrepancy between input features and the stored normal representations. Compared with single-center distribution modeling, these methods are better suited to capturing multiple normal modes and handling intra-class variations^18,19. Recent studies have further improved compactness and robustness by aggregating region-level semantic prototypes from normal samples^22–24. However, most existing methods still organize these representations in Euclidean space and usually treat them as relatively static anchors, which limits their ability to model hierarchical semantic relations and subtle structural deviations in complex industrial images. These limitations motivate us to combine dynamic semantic prototypes with hyperbolic structural modeling for anomaly detection.

Hyperbolic space

Hyperbolic space is a non-Euclidean manifold with constant negative curvature, where the volume grows exponentially with radius. This geometric property gives hyperbolic space a strong advantage in modeling data with hierarchical or tree-like structures. Such structure-aware representations are particularly relevant for human-centered perception tasks, where relational and topological dependencies play a key role^31,32. Compared to Euclidean space of the same dimension, hyperbolic space has higher representational capacity. It can represent complex structured relationships more compactly and accurately using the same number of embedding parameters^33,34. For this reason, hyperbolic space has been widely used in natural language processing, especially in tasks such as hierarchical semantic modeling, knowledge graph embedding, and word meaning evolution, where it has shown clear performance gains^35–37. In computer vision, hyperbolic embeddings have also achieved notable progress. Researchers have applied hyperbolic geometry to face recognition, image classification, action recognition, few-shot learning, and object segmentation. For example, in few-shot learning, hyperbolic space can amplify small geometric differences between samples, improving class separability³⁸. Recent studies have shown that hyperbolic geometry can effectively model hierarchical semantic relations and structured feature dependencies, improving representation capacity in complex visual learning tasks³⁹. In addition, hyperbolic geometry has been incorporated into various deep neural architectures, enabling more effective modeling of hierarchical and structured representations in non-Euclidean spaces⁴⁰.

Although research on hyperbolic space in anomaly detection is still in its early stages, existing studies have preliminarily demonstrated its unique modeling potential. Traditional anomaly detection methods typically rely on Euclidean space for statistical computations, similarity measurements, or feature clustering. However, these approaches often struggle to effectively model complex fine-grained structures and multi-scale semantic information in high-dimensional images. In contrast, hyperbolic space naturally adapts to nested structures and geometric hierarchies, enabling it to capture structural shifts between normal and anomalous samples more effectively in the feature space. By amplifying subtle local differences, hyperbolic embeddings show particular promise in detecting micro-defects in industrial images.

Methods

Overall architecture

To improve the accuracy and generalization of industrial anomaly detection, we propose a dynamic anomaly detection framework based on hyperbolic space, as shown in Fig. 2. Our method exploits the hierarchical nature of hyperbolic space as a structural prior for organizing semantic features and prototypes. Such a design allows the model to better capture heterogeneous normal semantic patterns and to amplify subtle deviations that are otherwise difficult to separate in Euclidean space. Specifically, image features are first mapped into hyperbolic space to provide a geometry better suited to modeling hierarchical semantic structure. Based on the resulting hyperbolic representations, the INP Extractor dynamically derives semantic prototypes from normal patterns. These prototypes are then used by the SP-Guided Decoder to guide the reconstruction of normal regions, thereby improving the separation between normal and anomalous responses in the reconstruction space. Finally, reconstruction errors are used to localize anomalous regions. In the following sections, we provide a formal description and detailed analysis of each core module.

Fig. 2.

The overall architecture of our method. The input image first passes through an encoder to extract Euclidean features. These features are then mapped into hyperbolic space to obtain hierarchical hyperbolic features. A semantic prototype extractor and an MLP process these features. The decoder, guided by semantic prototypes, generates reconstructed features. These reconstructed features are compared with the original features. The model is optimized using Dynamic Focal Loss and Coherence Loss. Finally, the network outputs an anomaly heatmap.

Hyperbolic space mapping

A commonly used isometric model for computation in hyperbolic space is the Poincaré model. To facilitate intuitive analysis of hyperbolic geometry, we adopt the n-dimensional Poincaré ball model to represent hyperbolic space with constant negative curvature. Specifically, a hyperbolic space with curvature c<0 can be defined as , where the Poincaré ball is given by: , and the Riemannian metric tensor is given by: . where denotes the Euclidean metric, and is the conformal scaling factor.

Given two vectors , the inner product in the tangent space is defined as:

1

where denotes the Euclidean inner product. To handle vector operations in hyperbolic geometry, the algebraic structure of gyrovector space is introduced. In the Poincaré ball model, point addition is defined using Möbius addition. The Möbius addition of two points is given by:

2

To map from Euclidean space to hyperbolic space under the Poincaré model, we introduce the exponential map, which projects Euclidean vectors from the tangent space onto the manifold. Assuming the reference point is the origin , the mapping function is defined as:

3

Near the boundary of the Poincaré ball model, the rate of distance growth in hyperbolic geometry is significantly faster than that in Euclidean space. This phenomenon can be quantitatively described using a specific mathematical formulation. The hyperbolic distance between two points is defined as:

4

It is evident that when either or , that is, as either point approaches the boundary of the Poincaré ball, the norm . Consequently, the hyperbolic distance , which indicates an exponential expansion of distance near the boundary.

Adaptive prototype-aware

Prototype-aware module is built upon query-based cross-attention, in which learnable prototypes are dynamically updated from hyperbolic feature maps and subsequently injected into the decoder as structural guidance for normal-region reconstruction. This design enables the model to capture diverse normal appearance variations and hierarchical semantic structures that cannot be adequately represented by a single global prototype. Assume there are M semantic prototypes and a set of learnable initial prototype vectors , which are used to guide the aggregation of normal semantic features. Here, M denotes the number of prototypes, and C is the feature dimension. The initial values are randomly initialized and updated during training to learn representations of normal semantics. Specifically, given the pre-trained dual-branch features , where W and H denote the width and height of the input image. We use the pre-trained dual-branch features as the key-value pairs, and T as the query. Attention weights are calculated to linearly aggregate global semantic information from , thus generating representative regional information. Denote: where is the learnable projection matrix for the query, and are the projection matrices for the key and value, respectively. , . Next, the aggregated attention result is added to the initial prototype T via residual connection to obtain the updated prototype vector T', which is further refined through a feedforward network to produce the final semantic prototype P, expressed as:

5

The M semantic prototypes act as a compact dictionary of recurring normal semantic modes. The query-based cross-attention implements a soft assignment over spatial features, aggregating the normal feature field into M region-level summaries and thus covering heterogeneous normal appearances beyond a single global prototype. Next, we use multiple normal semantic prototypes to reduce noise interference and effectively suppress reconstruction of abnormal regions, guiding the reconstruction of normal areas. Specifically, we use the semantic prototypes in dual-space as the key-value pairs, and linearly combine the semantic prototypes as query inputs to the decoder. Let the output of the first layer l-1 be , then the output of the l-th layer is: where are learnable parameters, and P is the normalized prototype matrix input. Attention scores are computed by . Then, the final output is calculated as:

6

Loss optimization

Coherence loss

To ensure that the extracted semantic prototypes can stably represent normal semantics, we introduce a Coherence Loss . This loss minimizes the distance between each normal feature and its nearest semantic prototype. Finally, the average distance over all features yields the Coherence Loss. It is defined as:

7

where denotes the Euclidean distance.

Hyperbolic alignment loss

Considering that distance definitions in hyperbolic space differ significantly from Euclidean space, we cannot directly apply conventional Euclidean distance functions. Here, we adopt a hyperbolic distance metric as a substitute. For two embedded vectors in hyperbolic space with curvature c, the distance is defined as:

8

where, denotes Möbius addition, and represents the Euclidean norm. Based on the above hyperbolic distance, the Hyperbolic Alignment Loss is defined as:

9

where and denote the hyperbolic embeddings of encoder and decoder features, respectively.

Dynamic focal loss

To guide the model to focus more on challenging regions, we introduce a dynamic weighting mechanism based on region-wise difficulty. Specifically, let and be the encoder and decoder feature maps at layer l, respectively. Let denote the residual similarity in a local region, and the average similarity across the region. The Dynamic Focal Loss is defined as:

10

here, γ> 0 is a temperature hyperparameter, denotes dynamic weighting. fl(· ) denotes the spatial flattening operation that converts feature maps into vector representations for similarity computation. denotes a dynamic feature modulation function implemented as element-wise scaling of the decoder features , guided by the region-wise difficulty factor , which emphasizes hard regions during optimization. denotes the inner product computation.

The overall loss function of the proposed model is defined as:

11

The coefficients , , and are balancing parameters for different loss terms.

Experiments

Datasets

MVTec-AD dataset⁴¹ comprises 15 categories, including 5 object types and 10 texture materials. Each category provides a collection of defect-free (normal) images along with multiple types of real-world anomalous images, accompanied by pixel-level annotations of defect regions, enabling its use in segmentation tasks. Specifically, the dataset contains 3,629 normal images for training, and 1,982 anomalous images together with 498 normal images for testing. The dataset exhibits significant variability in defect types and substantial appearance differences across categories. Moreover, the anomalous regions vary in size and shape, often exhibiting complex structures. These characteristics make MVTec-AD particularly suitable for evaluating the robustness and generalization ability of models across diverse real-world scenarios.

VisA dataset⁴² is designed for industrial visual inspection tasks and covers 12 representative categories, including electronic components, tool parts, and natural objects. The dataset consists of high-quality images, comprising 8,659 normal images for training, 962 additional normal images, and 1,200 anomalous images for testing. Region-level anomaly masks are provided, enabling its use in classification, detection, and segmentation tasks. VisA offers images that closely resemble real-world industrial environments, characterized by complex background textures, subtle anomalies, multi-scale structures, and high intra-class variability, thus presenting new challenges for model robustness and generalization.

Real-IAD dataset⁴³ is collected from real-world industrial production lines and comprises six categories of industrial components. The data were acquired from multiple production batches and diverse industrial equipment, encompassing both normal samples and various types of anomalies. A subset of the images is annotated with segmentation masks. Real-IAD preserves the background context and noise inherent to in-situ environments. The dataset includes 30 distinct objects, with 36,645 normal images for training, 63,256 additional normal images, and 51,329 anomalous images for testing. In contrast to synthetic or standardized datasets, Real-IAD exhibits significantly higher real-world complexity, including variations in lighting, occlusion, background clutter, and ambiguity in anomaly definitions.

Experimental setup

All experiments were conducted on a single NVIDIA GeForce RTX 4090 GPU. The software environment was based on Python 3.9 with CUDA 11.2 configured to support GPU acceleration. All images were resized to a uniform resolution of 256×256 pixels. During training, models were optimized with a batch size of 8 for a total of 200 epochs. We employed the StableAdamW optimizer with an initial learning rate of 0.001, using a stepwise decay schedule for dynamic adjustment. The datasets were partitioned into training, validation, and test sets with a ratio of 8:1:1 to ensure scientific rigor and stable model evaluation.

In anomaly detection research, evaluation metrics must account for both anomaly identification and localization performance. We adopted the Area Under the Receiver Operating Characteristic Curve (AUROC), Average Precision (AP), and F1-score-max(F1 max) as metrics for anomaly detection accuracy. For anomaly localization, we employed the Area Under the Per-Region Overlap Curve (AUPRO) to assess the model’s performance at the region level.In all experiments, the balancing coefficients , , and are empirically set to 0.2, 0.4, and 0.4, respectively.

Compare with state-of-the-arts

The anomaly detection results on the MVTec AD dataset are summarized in Table 1. Our method achieved the highest image-level AUROC (I-AUROC) scores in 11 out of the 15 categories, outperforming all competing methods. Notably, in challenging categories such as Cable, Metal nut, Tile, and Wood, Ours attained detection accuracies of 99.8% or higher, significantly surpassing other approaches. For instance, on the Tile category, Ours achieved an I-AUROC of 99.2%, outperforming the next best method, ADdiffusion (97.0%), by 2.2 percentage points. These results demonstrate the superior robustness of our method in handling complex anomalies.

Table 1.

Performance Comparison of Different Methods. The first and second numbers denote I-AUROC (image-level detection score) and P-AUROC (pixel-level localization score), respectively. The penultimate “Mean” row corresponds to the average performance over texture categories, while the final “Mean” row reports the overall average across all categories.

Category	RIAD44	CutPaste45	InTra46	PNI47	SimpleNet48	LGFDR49	GLAD50	DiffusionAD51	AdvRAD52	ADdiffusion53	Ours
Object Categories
Bottle	99.9/98.4	98.3/97.6	100/97.1	100/98.87	100/98.0	100/99.2	100/98.9	–	99.6/–	99.8/99.4	99.8/99.5
Cable	81.9/84.2	80.6/90.0	70.3/91.0	99.7/99.1	99.9/97.6	99.3/98.9	99.9/98.1	99.6/98.1	79.2/–	100/99.2	100/99.8
Capsule	88.4/92.8	96.2/97.4	86.5/97.7	99.7/99.3	97.7/98.9	97.4/99.2	99.5/98.5	98.9/98.3	90.5/–	99.7/98.8	99.5/99.6
Hazelnut	83.3/96.1	97.3/97.3	95.7/98.3	100/99.3	100/97.9	100/99.2	100/99.5	100/99.7	97.3/–	99.8/99.8	99.8/99.9
Metal nut	88.5/92.5	99.3/93.1	96.9/93.3	100/99.29	100/98.8	99.9/97.9	100/98.8	100/99.7	91.0/–	100/99.8	100/99.8
Pill	83.8/95.7	92.4/95.7	90.2/98.3	96.8/99.0	99.0/98.6	99.2/99.0	98.1/97.9	99.0/99.3	94.4/–	98.0/99.8	99.0/99.8
Screw	84.5/98.8	86.3/96.7	95.7/99.5	99.5/99.6	98.2/99.3	96.3/99.6	96.9/99.1	99.4/99.0	99.3/–	96.8/97.0	99.7/99.6
Toothbrush	100/98.9	98.3/98.1	100/98.9	99.7/99.0	99.7/98.5	100/99.1	100/99.4	100/98.8	100/–	100/99.2	100/99.0
Transistor	90.9/87.7	95.5/93.0	95.8/96.1	100/98.0	100/97.6	100/98.1	98.3/96.2	99.8/92.9	90.6/–	100/99.3	100/99.5
Zipper	98.1/97.8	99.4/99.3	99.4/99.2	99.8/99.4	99.9/98.9	100/98.6	98.5/97.9	100/99.2	99.7/–	99.9/99.4	99.9/99.4
Mean	89.9/97.8	94.3/95.8	93.0/96.9	99.5/99.1	99.6/98.1	99.2/98.8	99.3/98.6	99.7/98.7	93.9/–	99.2/99.1	99.8/99.6
Texture Categories
Carpet	84.2/96.3	93.1/98.3	98.8/99.2	100/99.40	99.7/98.2	98.6/98.7	99.0/98.5	99.3/99.0	–	73.8/–	99.7/99.4
Grid	99.6/98.8	99.9/97.5	100/98.8	98.4/99.7	99.7/98.8	98.8/98.8	100/99.6	100/99.7	–	100/–	100/99.7
Leather	100/99.4	100/99.5	100/99.5	100/99.5	100/99.2	100/99.6	100/99.8	100/99.8	–	100/–	100/99.9
Tile	98.7/89.1	93.4/90.5	98.2/94.4	100/98.4	99.8/97.0	100/98.4	100/98.7	99.9/99.7	–	95.4/–	100/99.2
Wood	93.0/85.8	98.6/95.5	97.5/88.7	99.5/97.0	100/94.5	99.9/97.3	99.4/98.4	99.9/96.7	–	98.3/–	100/96.7
Mean	95.1/93.5	97.0/96.3	98.9/96.1	99.5/98.7	99.8/97.5	99.5/98.6	99.3/98.5	99.7/98.6	–	–	99.9/98.9
Overall Mean	91.6/94.1	95.2/95.9	95.1/96.6	99.6/99.1	99.6/98.4	99.6/99.1	99.3/98.6	99.8/98.5	–	–	99.8/99.1

As shown in Table 2, our method achieved state-of-the-art performance across three representative industrial anomaly detection datasets, demonstrating its effectiveness and robustness. On the MVTec-AD dataset, we obtained 99.9/99.4 for image-level tasks and achieved 73.2/69.7/95.1 for pixel-level tasks, consistently outperforming existing approaches. In particular,our AUPRO score exceeds that of DiAD by 4.4 percentage points. On the VisA dataset, our method achieved 99.2/96.8 at the image level, significantly outperforming Dinomaly (98.9/96.2), and reached 52.8/55.2/94.4 at the pixel level, demonstrating competitive anomaly localization performance. On the more challenging Real-IAD dataset, our method achieved 88.1/82.5 (image-level) and 47.7/50.3/95.0 (pixel-level), surpassing all previous methods in every metric. Notably, we improved the image-level F1_max by 2.3 points and AUPRO by 1.1 points over the second-best approach. In addition to detection accuracy, we also analyze the computational efficiency of different methods. Our method maintains strong detection performance while requiring relatively low computational cost, with only 38.4M parameters and 9.6G FLOPs. Compared with the best-performing baseline, Dinomaly (148.0M parameters and 104.7G FLOPs), our method demonstrates a clear advantage in computational efficiency. These results collectively demonstrate that our method not only delivers superior detection accuracy but also exhibits notable advantages in precise anomaly localization. Representative samples of anomaly localization results are visualized in Figs. 3, 4, and 5.

Table 2.

Performance Comparison Across Different Datasets. Image-level(I-AP/I-F1 max), Pixel-level(P-AP/P-F1 max/AUPRO).

Method	MVTec-AD		VisA		Real-IAD		Params	FLOPs
	Image-level	Pixel-level	Image-level	Pixel-level	Image-level	Pixel-level	(M)	(G)
RD4AD24	96.5/95.2	48.6/53.8/91.1	92.4/89.6	38.0/42.6/91.8	79.0/73.9	25.0/32.7/89.6	80.6	28.4
UniAD54	98.8/96.2	43.4/49.5/90.7	90.8/85.8	33.7/39.0/85.5	80.9/74.3	21.1/29.2/86.7	24.5	3.6
SimpleNet	98.4/95.8	45.9/49.7/86.5	87.0/81.8	34.7/37.8/81.4	53.4/61.5	2.8/6.5/39.0	72.8	16.1
DesTSeg55	95.5/91.6	54.3/50.9/64.8	89.0/85.2	39.6/43.4/67.4	79.2/73.2	37.9/41.7/40.6	35.2	122.7
DiAD56	99.0/96.5	52.6/55.5/90.7	88.3/85.1	26.1/33.0/75.2	66.4/69.9	2.9/7.1/58.1	1331.3	451.5
MambaAD4	99.6/97.8	56.3/59.2/93.1	94.5/89.4	39.4/44.0/91.0	84.6/77.0	33.0/38.7/90.5	25.7	8.3
Dinomaly57	99.8/99.0	69.3/69.2/94.8	98.9/96.2	53.2/55.7/94.5	86.8/80.2	42.8/47.1/93.9	148.0	104.7
Ours	99.9/99.4	73.2/69.7/95.1	99.2/96.8	52.8/55.2/94.4	88.1/82.5	47.7/50.3/95.0	38.4	9.6

Fig. 3.

Visualization of each category in MVTec AD dataset.

Fig. 4.

Visualization of each category in VisA dataset.

Fig. 5.

Visualization of each category in Real-AD dataset.

Ablation studies

Overall ablation

As shown in Table 3, we conducted a systematic ablation study on the VisA dataset to validate the effectiveness of each proposed component. including the hyperbolic representation module which implicitly reflects the effect of the Hyperbolic Alignment Loss. The baseline model, with all modules removed, achieved only 87.5 and 93.3 in image-level and pixel-level AUROC, respectively, indicating significantly degraded performance. Upon introducing the hyperbolic space module (Hyp), the image-level AUROC improved to 92.4, demonstrating the positive impact of hyperbolic representation on anomaly detection. With the further addition of semantic prototypes (SP), image-level performance increased to 98.2 and pixel-level AUROC reached 96.1, suggesting that SP effectively enhances the model’s generalization capability. Introducing the consistency loss and Dynamic Focal Loss led to further performance gains; notably, the incorporation of dynamic focus boosted pixel-level AUROC to 98.9. When all components were combined, the model achieved the highest AUROC scores of 99.8 (image-level) and 99.1 (pixel-level), confirming the synergistic benefits of each module in accurately detecting and localizing anomalous regions.

Table 3.

Ablation study on ViSA datasets.

Hyperbolic	SPs				I-AUROC	P-AUROC
×	×	×	×	×	87.5	93.3
\checkmark	×	×	×	×	92.4	95.5
\checkmark	\checkmark	×	×	×	97.4	95.6
\checkmark	\checkmark	\checkmark	×	×	98.2	96.1
\checkmark	\checkmark	\checkmark	\checkmark	×	99.4	98.9
\checkmark	\checkmark	\checkmark	\checkmark	\checkmark	99.8	99.1

The impact of coherence loss

To further evaluate the impact of the Coherence Loss on anomaly localization performance, we conducted a visual analysis across multiple categories of the VisA dataset. As illustrated in Fig. 6, the first row shows the input anomalous images, the second row presents the pixel-level distance maps generated without , and the third row depicts the maps obtained with applied. It is evident that, without the consistency loss, the generated heatmaps often exhibit strong background responses and blurry anomaly regions. This issue is particularly pronounced in categories such as cable and metal nut, where anomalous regions are difficult to distinguish accurately. In contrast, with the introduction of , the response in anomalous regions is significantly enhanced, with sharper localization boundaries that closely align with ground-truth anomalies. These results demonstrate that substantially improves the discriminative and focusing ability of the model for anomaly localization. We further visualize the effect of the coherence loss by plotting the kernel density estimation (KDE) of pixel-level anomaly scores on the cable. As shown in Fig.7, introducing reduces the overlap between the score distributions of normal and anomalous pixels, indicating improved separability and more robust anomaly scoring.

Fig. 6.

Visualization of the impact of Coherence Loss . (a) Input anomalous image. (b) Without . (c) With .

Fig. 7.

Visualization of the impact of the coherence loss .

Influence of the number of SPs

As shown in Table 4, we conducted an ablation study on the number of semantic prototypes M. The experimental results indicate that the model performance stabilizes when M> 4, while excessively large M leads to mild degradation. This is because M determines the semantic granularity of normal modeling. A small M may compress different normal regions into coarse prototypes, while a large M can over-fragment features and introduce noisier assignments, making the model slightly less robust. In this study, we set M = 6 to balance model effectiveness and robustness.

Table 4.

Performance Metrics with Different M Values.

M	1	2	4	6	8	12	50
I-AUROC	96.2	97.5	98.7	98.9	99.1	99.1	98.6
I-AP	98.2	98.6	98.9	99.2	99.2	99.2	99.1
P-AUROC	97.3	97.8	98.7	98.9	98.6	98.6	98.1
P-AP	47.2	49.3	51.9	52.8	52.9	52.7	51.3

Effect of curvature parameter c

As shown in Table 5, the model achieved peak performance at c = 0.01, attaining 99.8% I-AUROC and 99.1% P-AUROC, which significantly outperforms the Euclidean baseline (c = 0) that yielded 99.1% I-AUROC and 97.4% P-AUROC. Notably, the improvement in P-AUROC is particularly substantial, with hyperbolic space providing an 1.7 percentage point gain over Euclidean space. The results indicate that a moderate hyperbolic curvature (c = 0.01–0.1) maintains consistently high performance, with I-AUROC exceeding 98% and P-AUROC remaining above 97%. However, when curvature becomes excessively large (c \ge 0.3), performance begins to degrade markedly, underscoring the critical importance of proper curvature selection. We further evaluate the effect of curvature by comparing the anomaly score distributions in Euclidean and hyperbolic spaces. As shown in Fig. 8, which depicts the pixel-level anomaly score distributions for the metal nut subcategory, we observe that hyperbolic space achieves better separation between normal and anomalous pixels.The false positive rate (FPR) and false negative rate (FNR) in hyperbolic space are reduced to 2.3% and 9.8%, respectively, compared to 4.1% and 11.3% in Euclidean space.

Table 5.

Performance Metrics Comparison Under Different Curvature Values.

c	0	0.01	0.05	0.1	0.3	0.5
I-AUROC	99.1	99.8	99.2	98.4	97.8	95.3
P-AUROC	97.4	99.1	99.1	97.6	95.4	95.4

Fig. 8.

Comparison between Euclidean and hyperbolic representations in anomaly scoring and localization. The left and middle panels show the anomaly score distributions in Euclidean and hyperbolic spaces, respectively, while the right panel presents representative anomaly localization comparisons. The vertical dashed line indicates the optimal threshold based on the F1 score. The area to the left of the threshold corresponds to false negatives (FNR), while the area to the right corresponds to false positives (FPR).

Discussion

Our experimental results across multiple datasets consistently demonstrate the effectiveness of modeling semantic and structural information in hyperbolic space for anomaly detection. By leveraging the negative curvature of hyperbolic geometry, our method captures subtle semantic differences and hierarchical relationships more effectively than conventional Euclidean approaches. This leads to substantial improvements in both image-level and pixel-level AUROC scores, especially in complex cases with fine-grained or ambiguous anomalies. The adaptive semantic prototype mechanism enhances the model’s robustness against domain shifts and category variance, while the coherence and Dynamic Focal Losses further boost feature discrimination and localization precision. These results underscore the synergistic benefit of jointly modeling semantic structure and geometry, offering new insights into designing high-capacity representations for industrial visual inspection.

Despite its strong performance, the proposed method also presents certain limitations that open avenues for future exploration. First, the increased computational complexity introduced by hyperbolic embeddings and attention-guided reconstruction may hinder deployment in real-time or resource-constrained environments. Second, the sensitivity of the model to the curvature parameter c indicates the need for more adaptive or learnable curvature designs. In practice, a smaller curvature is typically a stable choice when a dataset exhibits modest semantic variability and relatively homogeneous normal patterns, whereas a moderately larger curvature can better separate richer part-to-whole structures and long-tail variations across more diverse categories. Furthermore, while dynamic prototype learning alleviates semantic drift to some extent, its effectiveness remains influenced by training data distribution and prototype size.Another challenge arises when detecting extremely low-contrast defects in reconstruction-based anomaly detection methods. When anomalous regions share similar local appearance with normal patterns, reconstruction models may partially restore these regions, which reduces anomaly discrimination capability. Future work may consider integrating lightweight geometric modules, multi-modal cues (e.g., text, audio), and self-supervised pretraining to further improve model generalizability and interpretability. Additionally, extending hyperbolic representation learning to other domains such as video anomaly detection or temporal event modeling may broaden its application scope.

Conclusions

In this work, we propose a novel structure-aware anomaly detection framework that leverages the geometric properties of hyperbolic space to enhance both the semantic representation and localization of anomalies in industrial inspection tasks. By dynamically extracting semantic prototypes and embedding them into a negatively curved space, our model effectively captures hierarchical semantic variations and local structural differences. Extensive experiments on three challenging benchmarks demonstrate that our method consistently achieves state-of-the-art performance at both image and pixel levels. Incorporating hyperbolic geometry into deep anomaly detection provides a promising direction for improving detection accuracy, especially in complex, real-world scenarios. Future work will explore lightweight hyperbolic modules, adaptive curvature learning, and multi-modal extensions to further enhance the scalability and applicability of the proposed framework.

Author contributions

Conceptualization, F.Y. and L.J.; methodology, F.Y.; software, F.Y.; validation, F.Y. and G.Y.; formal analysis, F.Y.; investigation, G.Y.; resources, L.J.; data curation, F.Y.; writing–original draft preparation, F.Y.; writing–review and editing, L.J.; visualization, G.Y.; supervision, L.J.; project administration, L.J.; funding acquisition, L.J. All authors have read and agreed to the published version of the manuscript.

Data availability

The data from the current study are available from the corresponding author upon reasonable request. The proposed method was evaluated on three publicly available surface defect detection datasets that are widely used in the field of object detection: MVTec-AD (https://www.mvtec.com/company/research/datasets/mvtec-ad), VisA (https://amazon-visual-anomaly.s3.us-west-2.amazonaws.com/VisA_20220922.tar) And Real-IAD (https://huggingface.co/datasets/Real-IAD/Real-IAD).

Declarations

Competing interests

The authors declare no competing interests.