Title: Unifying Reconstruction and Density Estimation via Invertible Contraction Mapping in One-Class Classification
Venue: NeurIPS 2025
Accepted: yes
Keywords: Anomaly Detection

One-Class Classification
Abstract: Due to the difficulty in collecting all unexpected abnormal patterns, One-Class Classification (OCC) has become the most popular approach to anomaly detection (AD). Reconstruction-based AD method relies on the discrepancy between inputs and reconstructed results to identify unobserved anomalies. However, recent methods trained only on normal samples may generalize to certain abnormal inputs, leading to well-reconstructed anomalies and degraded performance. To address this, we constrain reconstructions to remain on the normal manifold using a novel AD framework based on contraction mapping. This mapping guarantees that any input converges to a fixed point through iterations of this mapping. Based on this property, training the contraction mapping using only normal data ensures that its fixed point lies within the normal manifold. As a result, abnormal inputs are iteratively transformed toward the normal manifold, increasing the reconstruction error. In addition, the inherent invertibility of contraction mapping enables flow-based density estimation, where a prior distribution learned from the previous reconstruction is used to estimate the input likelihood for anomaly detection, further improving the performance. Using both mechanisms, we propose a bidirectional structure with forward reconstruction and backward density estimation. Extensive experiments on tabular data, natural image, and industrial image data demonstrate the effectiveness of our method.  The code is available at URD.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces an anomaly detection framework for one-class classification that combines reconstruction and density estimation through an invertible contraction mapping. The authors propose a reconstruction process where all inputs are iteratively mapped to a fixed point within the normal data manifold. This makes it so abnormal inputs yield high reconstruction errors. Additionally, the invertibility of the CM enables flow-based density estimation by learning a prior distribution over normal reconstructions and computing input likelihoods via the inverse mapping. By integrating both reconstruction-based and likelihood-based scoring, the method achieves strong performance across tabular and image datasets. Experiments and ablations demonstrate the value of both components in improving detection accuracy.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- The application of contraction mappings to enforce convergence of abnormal samples toward a fixed point on the normal manifold is theoretically grounded. The paper includes clear propositions and quantifies how reconstruction errors grow for anomalies.\n\n- The proposed method merges two dominant OCC paradigms using a single CM-based architecture. This dual use of forward for reconstruction and reverse for flow-based likelihood estimation is efficient and clean.\n\n- The approach outperforms state-of-the-art methods on tabular datasets (+3.78 AP and +0.75 AUROC vs MCM) and improves image-level and pixel-level AUROC on CIFAR-10 and MVTec AD.\n\n- The authors conduct clear component ablations (reconstruction only, density only, full) and show sensitivity to hyperparameters to identify over-contraction issues when the product is too large.\n\nWeaknesses:\n\n- While the paper presents proofs around convergence of contraction mappings (e.g., fixed points within a normal manifold), it assumes the mapping generalises beyond the training data. There is no empirical validation of the fixed-point behavior on hard real-world anomalies.\n\n- Iteratively mapping all inputs, including anomalies, toward a normal manifold may suppress outlier signals rather than enhance them. While this increases reconstruction error superficially, could it not also degrade interpretability or confuse downstream use cases? There should be a discussion on whether the model distinguishes subtle anomalies versus simply \"normalising\" everything.\n\n- The CIFAR-10 experiments use a one-vs-all setting, which favours semantic class separation rather than true anomaly detection and does not reflect real-world conditions with subtle in-class deviations. While MVTec AD is a more appropriate benchmark, relying on a single real-world dataset limits the generality of the conclusions.\n\nQUESTIONS:\nWhile the theoretical analysis shows convergence of inputs toward fixed points within the normal manifold, it would be helpful to include a discussion of how this property behaves on real data, especially when dealing with ambiguous or near-boundary anomalies. Could this contraction mapping project mild anomalies too close to the manifold and reduce separability as a result?\n\nLIMITATIONS:\nThe authors briefly acknowledge the limitation to tabular and image data, but the societal impact discussion is overly generic. It would strengthen the paper to include concrete considerations around false positives in safety-critical domains and the risk of over-relying on reconstruction-based signals in real-world deployments.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-06-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes an unsupervised anomaly detection method that unifies reconstruction and density estimation via contraction mappings.\nBy employing Lipschitz-continuous mappings with a Lipschitz constant less than one, any input is guaranteed to converge to a unique fixed point.\nSince this fixed point lies on the normal data manifold, reconstruction errors for anomalous inputs become significantly larger, enabling effective anomaly detection.\nMoreover, the invertibility of the contraction mapping allows for density estimation in the style of flow-based models.\nThe proposed method jointly optimizes both reconstruction and density estimation through a bidirectional architecture,\nachieving competitive or superior performance compared to existing methods on both tabular and image anomaly detection tasks.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n- This paper is well-written and easy to follow.\n- Models that minimize reconstruction error, such as autoencoders, can sometimes reconstruct both normal and anomalous data. The idea of using contraction mappings to address this issue is highly original and important. Additionally, leveraging the invertibility of contraction mappings to integrate with flow-based models is effective and appears to contribute meaningfully to anomaly detection performance.\n\n## Weakness\n- On the other hand, there are several points I did not fully understand or found somewhat unclear; I will list them in the “Questions” section.\n- This is a very minor issue, but in line 183, I believe the expression should more precisely denote the L2 norm.\n\nQUESTIONS:\n- If a contraction mapping is bijective, then its repeated application should also be bijective. However, since the mapping converges to a unique fixed point after repeated applications, it seems to contradict bijectivity. I understand that your method mitigates this issue by limiting the number of iterations. Still, increasing the number of iterations might make the reconstruction error for anomalies even larger. Could you elaborate on this trade-off and how it is handled in your method?\n\n- While your method is based on reconstruction error, it is not encoder–decoder–based like an autoencoder. Instead, it uses a sequence of same-dimensional mappings. In such a case, there is a risk of the network collapsing into the identity mapping. Even with contraction mappings, if the Lipschitz constant is close to 1, the mapping could behave similarly to identity. How is this issue mitigated? My understanding is that the network design in Section 3.1.2 (Equation (2)) and the integration with a flow-based model help regularize the mapping by enforcing alignment with the prior. Is this correct?\n\n- What is the motivation behind jointly optimizing reconstruction error and the flow-based likelihood? If \\phi is trained to minimize reconstruction error, the transformed data distribution should resemble the original input distribution. On the other hand, maximizing the likelihood via a flow-based model pushes the transformed data to match a Gaussian prior. These two goals appear to be in conflict. Is my understanding correct, and if so, how is this trade-off addressed in your method?\n\n- I also have some questions regarding the hyperparameters. For instance, setting $R=2$ or $3$ seems quite shallow compared to typical flow-based models, which often require deep compositions to capture complex distributions. However, from the ablation study, it appears that combining reconstruction and flow-based density estimation yields the best performance. Does this imply that the role of the flow in your method is not to model the data distribution in detail, but rather to act as a complementary scoring mechanism? I would appreciate it if you could clarify the intended role of the flow component.\n\nIf these concerns are resolved, I would be happy to increase both my overall score and confidence.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes to resolve one class classification based unsupervised anomaly detection problem with thecontraction mapping. By iteratively transform samples through the contraction mapping, normal samples stayclose and abnormal samples moves far away, resulting large discrepency. The invertablity property of contractiohmapping enables backward density estimation, and thus present a unifed anomaly detection framework withforward reconstruction and backward density estimation. The method's efficacy is demonstrated throughextensive experiments on a variety of datasets and it shows competitive performance against currentstate-of-the-art methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1.Introducing contract mapping into one class based anomaly detection is interesting2.The performance is promising3. The paper content and illustrations are clear and easy to follow\n\nWeaknesses:\n1. It seems that the main contribution largely comes from the introduced contraction mapping model. The author provides results for single forward reconstruction and backward density estimation in Table 3 and 4.in which the performance boost is observed. However, direct and clear evaluation of performance of contract mapping itself is needed. Just showing the RE an DE is indirect. From my perspective, they are just components of the unified framework.\n\n2. Sensitivity of data distribution. Since the contraction mapping iteratively transforms data toward a fixed point derived by training the model on normal data, what if the normal data distribution itself is chaotic? Or if the normal data distribution is ring-shaped or U-shaped? Can the fixed point locate in an appropriate position? Will the reconstruction discrepancy identification work well in such cases?\n\nAs contraction mapping is an introduced mathematical tool, further justification for its use in anomaly detection should be supplemented.\n\nQUESTIONS:\n1. Are there any experiments to directly evaluate the effect of using contract mapping instead of indirect evaluation?\n\n2. Can you further discuss the affects of different kinds of normal data distribution on CM results? It's interesting to see in what circumstances dose CM work well and in which case it may not well distinguish abnormal samples.\n\nLIMITATIONS:\nNo Limitations", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a new method for unsupervised anomaly detection that combines a forward contraction mapping with density estimation via a reversed normalizing flow. The contraction mapping is designed to map any input data point toward a fixed point on the normal data manifold. Specifically in this paper, the mapping $f$ is chosen as a 1-Lipschitz neural network, regularized with spectral normalization and using the LipSwish activation function. The network is trained to minimize the reconstruction error $||f(x) -x||_2$ for arbitrary input \n$x$. Then, leveraging the invertibility of $f$, the method then uses the inverse map of $f$ to transform a Gaussian  prior distribution into a real data distribution, allowing for estimating the likelihood of the input. Overall, the approach jointly minimizes the reconstruction error in the first stage and the negative log-likelihood in the second stage. The method is evaluated on multiple anomaly detection benchmarks and demonstrates competitive performance relative to existing anomaly detection methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. While the individual components of the proposed approach (i.e., the reconstruction map and the use of normalizing flows) draw heavily from prior work, the primary contribution lies in their combination. The experimental results provide reasonable evidence that combining the reconstruction map with normalizing flows leads to improved anomaly detection performance compared to using either component alone.\n2. I appreciate that the authors included a detailed sensitivity analysis of the hyperparameters $\\lambda$ and $\\alpha$, which  illustrate the trade-off between reconstruction error and the MLE loss.\n\nWeaknesses:\n1. The term \"well-constructed anomalies\" was not defined or explained in the introduction, which makes it difficult to grasp the motivation behind the proposed method. It appears to refer to reconstructed anomalous inputs that fall outside the normal data manifold, and I am not even sure if I understand this correctly. Clarifying this early on would make the paper more accessible, especially for\n    readers less familiar with reconstruction-based anomaly detection.\n2. The theoretical guarantees provided for the reconstruction map appear shallow. In particular, Proposition 1 does not rigorously prove that the fixed point $x^*$ will be close to the original input $x$. The paper does not show that, for any chosen mapping $f$ and optimization algorithm, the reconstruction loss $||f(x) - x||_2$ can be made arbitrarily small. \n3. The choice of the activation function $\\psi$ of the neural network employed appears important to me, yet it is omitted from the main text. Only in the appendix do the authors mention that $\\psi$ is the  LipSwish activation function.  Given that ReLU is also 1-Lipschitz and commonly used, why do the author choose LipSwish over ReLU? Does LipSwish bring empirical benefit, possibly related to the extra learnable parameter $\\beta$?\n4.   The paper lacks a theoretical justification for why the integration of reconstruction mapping and normalizing flows should be more effective than using either component independently.\n\nQUESTIONS:\nSee Weaknesses\n\nLIMITATIONS:\nSee Weaknesses", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-02"}