Energy-Based Models for Anomaly Detection: A Manifold Diffusion Recovery Approach

We present a new method of training energy-based models (EBMs) for anomaly detection that leverages low-dimensional structures within data. The proposed algorithm, Manifold Projection-Diffusion Recovery (MPDR), first perturbs a data point along a low-dimensional manifold that approximates the training dataset. Then, EBM is trained to maximize the probability of recovering the original data. The training involves the generation of negative samples via MCMC, as in conventional EBM training, but from a different distribution concentrated near the manifold. The resulting near-manifold negative samples are highly informative, reflecting relevant modes of variation in data. An energy function of MPDR effectively learns accurate boundaries of the training data distribution and excels at detecting out-of-distribution samples. Experimental results show that MPDR exhibits strong performance across various anomaly detection tasks involving diverse data types, such as images, vectors, and acoustic signals.

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Peer review

Reviewer Tcts5/10 · confidence 1/52023-07-03

Summary

The authors introduce a novel algorithm, Manifold Projection-Diffusion Recovery (MPDR), for training energy-based models (EBMs) that improve the performance of anomaly detection tasks. These tasks are highly relevant in real-world applications like industrial surface inspection, machine fault detection, and particle physics. Unlike conventional EBM training methods, MPDR harnesses low-dimensional structures in the data to generate more informative negative samples. It works by initially perturbing a data point along a manifold approximating the training dataset and then trains the EBM to maximize the probability of recovering the original data. A significant aspect of this new method is its use of Manifold Projection-Diffusion (MPD), which replaces Gaussian noise with perturbations reflecting the data's low-dimensional structure. This approach provides more meaningful insights into variations within the data. The authors show that MPDR provides consistent density parameter estimation under standard assumptions, is compatible with any energy function, and can work with multiple autoencoders - an advantage over existing algorithms. Moreover, it demonstrates good results even with lightweight autoencoders, making it computationally efficient. The paper also presents practical strategies for deploying MPDR, such as two-stage sampling, energy function design, and ensemble techniques using multiple autoencoders. Through experimental testing on various data types, including images, representation vectors, and acoustic signals, the authors demonstrate that MPDR significantly improves unsupervised anomaly detection tasks, outperforming other deep generative models.

Strengths

**Originality:** The paper is highly original in its formulation and approach to training energy-based models (EBMs) for anomaly detection. The development of the Manifold Projection-Diffusion Recovery (MPDR) represents a creative combination of existing methodologies, such as the use of autoencoders for efficient training and manifold projections for capturing low-dimensional data structures. The method of perturbing data points along a low-dimensional manifold that approximates the training dataset and then training the EBM to maximize the recovery of the original data is a fundamentally new approach. **Quality:** The quality of the paper is commendable. The authors present a clear problem statement, propose a novel solution, and provide empirical evidence supporting their claims. They also delve into the theoretical backing of the proposed method, offering a consistent density parameter estimation under standard assumptions. The paper includes detailed experimental results, highlighting the strength of MPDR across diverse anomaly detection tasks and data types. **Clarity:** The exposition in the paper is clear and well-organized. The authors have done an excellent job explaining complex concepts and methodologies, which makes the paper accessible even to readers who may not be experts in the field. The use of illustrative figures and well-explained algorithms further enhances the clarity of the paper. **Significance:** The significance of this work is substantial, given the wide range of practical applications of anomaly detection. By providing a more efficient and effective way to train EBMs, MPDR could significantly improve performance in areas like industrial surface inspection, machine fault detection, and particle physics. Furthermore, the ability of MPDR to perform effectively with lightweight autoencoders means it can be used in scenarios where computational resources are limited, making it relevant to a broader audience.

Weaknesses

While the paper presents a compelling new approach, here are a few areas that could be addressed or clarified further: **Theoretical Analysis:** While the authors provided a theoretical justification for consistent density parameter estimation under standard assumptions, it would be beneficial to include more analysis of the proposed method's convergence properties. Understanding how the choice of manifold affects the convergence and stability of learning would also be useful. **Comparison with Other Methods:** The paper could benefit from a more comprehensive comparison with other state-of-the-art methods for anomaly detection. Not only should this include direct quantitative comparisons on common datasets, but also qualitative discussions about when and why one might prefer the proposed method over others. **Parameter Sensitivity:** It is not clear how sensitive the results are to the choice of parameters within the MPDR framework. It would be beneficial for practical applications to know how much tuning is needed to achieve optimal performance and how robust the method is to variations in these parameters. **Real-World Applications:** While the experiment demonstrates the effectiveness of the proposed algorithm in various data types like images, vectors, and acoustic signals, applying the model to real-world datasets and providing case studies can strengthen the paper. This will help readers understand its practical implications better. **Computational Complexity:** The paper mentions that MPDR performs well with lightweight autoencoders, which indicates computational efficiency. However, a more explicit discussion or analysis of the computational complexity of the proposed method, including both training time and inference time, would provide valuable information to practitioners considering this method.

Questions

1. **Theoretical Analysis:** Could the authors provide a more rigorous analysis of the convergence properties of their proposed method? Specifically, how does the choice of manifold affect the stability and speed of learning in MPDR? 2. **Comparison with Other Methods:** It would be beneficial to see a wider comparison with other state-of-the-art anomaly detection methods. Could the authors elaborate on why one might choose MPDR over other established methods in specific scenarios? 3. **Parameter Sensitivity:** How sensitive is the MPDR algorithm to the initial choice of parameters? Is there a recommended procedure for parameter tuning, or guidelines that could assist users in achieving optimal performance? 4. **Real-World Applications:** Could the authors possibly demonstrate the application of their model on real-world datasets or provide case studies? This could help showcase the practical implications of the proposed method. 5. **Computational Complexity:** The paper mentions that MPDR can work well even with lightweight autoencoders, but could you please clarify further on its computational complexity, training time, and inference time? Would there be any scalability issues when applying this method to larger datasets? Looking forward to the authors' response to these points.

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

1: Your assessment is an educated guess. The submission is not in your area or the submission was difficult to understand. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

2 fair

Contribution

2 fair

Limitations

From the information provided, it does not appear that the authors have explicitly discussed the limitations and potential negative societal impacts of their work. Therefore, here are some suggestions for addressing these points: **Limitations:** 1. **Robustness to Noise:** How well does the MPDR framework handle noise in the data? Real-world data often contain a significant amount of noise, which may distort the underlying manifold structure that MPDR relies on. 2. **Scalability:** The scalability of the model hasn't been addressed. Can the method be applied efficiently to very large datasets? What would be the computational requirements in such cases? 3. **Multimodality:** How effectively can MPDR handle multimodal or highly dimensional distributions? This is a common challenge in many real-world anomaly detection tasks. **Potential Negative Societal Impacts:** While this study primarily focuses on improving anomaly detection methods, which generally have positive implications (e.g., defect detection in manufacturing, early detection of diseases, etc.), any technology has the potential to be misused. 1. **Privacy Concerns:** Anomaly detection tools can potentially be used to identify outliers or anomalies in personal behavior or characteristics, leading to potential privacy concerns if misused, especially in contexts like surveillance or social profiling. 2. **False Positives/Negatives:** In critical applications, false positives or negatives can have serious repercussions. For instance, in health care, a false positive might cause unnecessary worry or treatment, whereas a false negative could delay necessary intervention. It would be beneficial to see the authors address these potential issues and discuss how they might be mitigated.

Reviewer Pi2D5/10 · confidence 3/52023-07-05

Summary

Paper proposes MPDR, a novel method of using auto-encoders for training EBM. Some practical techniques are introduced. Extensive numerical experiments are done.

Strengths

Numerical experiments cover a large scope of benchmarks. And it shows superiority on most benchmarks.

Weaknesses

No theoretical guarantee is provided. And it doesn't show dominant superiority on some datasets.

Questions

Can we have some theoretical analysis on when proposed method has significant advantage, when not? Or is the proposed method better in general and some loss are just random?

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

3 good

Contribution

3 good

Limitations

No negative societal impact is seen.

Reviewer gSqM5/10 · confidence 5/52023-07-05

Summary

This paper introduces an energy-based model based on the manifold of low-dimensional data. To train the EBMs, this paper takes the idea of maximum recovery likelihood and adds a layer of autoencoder to approximate the low-dimensional manifold representing the data. This introduces perturbation along the low-dim manifold representing the dataset. Simulation results are provided to show the performance.

Strengths

- It is easy to follow. - Sufficient literature review.

Weaknesses

- The novelty of this paper is moderate at best. It only adds a deterministic encoder/decoder to the original problem of likelihood recovery. - The accuracy of the manifold approximation is as good as the autoencoder approximation of the dataset. - The generation of the negative samples concentrated neat the manifold could potentially introduce biases. - lack of comparison to some recent advancements in the topic such as: * Anomaly Detection in Networks via Score-Based Generative Models by Gavrilev et. al, 2023. * Enhancing Unsupervised Anomaly Detection with Score-Guided Network by Huang et. al, 2022.

Questions

- How can we mitigate these biases to ensure the trained energy-based model generalizes well to out-of-distribution samples? - The encoder in this algorithm is assumed to be deterministic how does the deterministic assumption impact the performance and the ability to capture the full range of variations in the data? - Could authors please elaborate more on the conditions under which the consistency of the estimation by maximizing $\log p(x|z^~)$ holds?- How does the use of a latent chain and latent space improve the sampling process compared to the visible chain? What are the advantages and disadvantages of using a latent LMC? - The paper mentions that the perturbation design, including the encoder-decoder pair (fe, fd) and noise magnitude $\sigma$, significantly impacts the algorithm's performance. How can we effectively select the optimal perturbation design for different datasets and anomaly detection tasks? - The paper mentions that the autoencoder (fe, fd) and the noise magnitude $\sigma$ should be independent of $\theta$ and remain fixed during training. How does the fixed nature of the autoencoder and noise magnitude impact the algorithm's adaptability to different datasets and anomaly types? - Can you provide more details on how the simultaneous use of multiple perturbations enhances the algorithm's performance? Are there any potential trade-offs or challenges associated with this approach? - One of the main potential issues is the memory overhead in manifold ensembles. When utilizing multiple autoencoder manifolds in MPD, there is a memory overhead associated with processing multiple groups separately. How can this memory overhead be managed effectively to ensure efficient training while utilizing multiple autoencoders? - Can you provide more insights into how the choice of Dz affects the algorithm's ability to detect anomalies? How can we determine the optimal combination of autoencoders with varying Dz for different types of data? Specifically, in high-dimensional data such as images, how does MPDR framework deal with curse of dimensionality and maintain good performance with relatively smaller autoencoders?

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

5: You are absolutely certain about your assessment. You are very familiar with the related work and checked the math/other details carefully.

Soundness

2 fair

Presentation

3 good

Contribution

2 fair

Limitations

Please check weaknesses and questions.

Reviewer dsqA6/10 · confidence 4/52023-07-07

Summary

This paper introduces an EBM-based model for anomaly detection in the latent manifold space. The proposed model first trains an autoencoder that maps a data point $x$ into the low-dimensional $z$, and then two-stage sampling strategy is developed to generate the original data via the LMC algorithm. Several notable designs are developed to complete the proposed model, such as manifold projection diffusion, two-state sampling, and perturbation ensemble. Experiments on images, vectors and acoustic signals show the strong performance of the method.

Strengths

(1) The idea that generation of negative samples from the manifold space sound good. This makes sure the starting points highly reflecting information, resulting in more discriminative generation. (2) Several practical strategies such as two-stage sampling and resembling multiple autoencoder are developed to benefit the anomaly detection performance. (3) Extensive empirical results and ablation studies show the efficiency of the model.

Weaknesses

(1) lack of the results of widely-used anomaly detection dataset MVTec-AD (2) Lack of well-known anomaly detection baselines, such as UniAD [1] and DRAEM [2] [1] Zhiyuan You et.al. A unified model for multi-class anomaly detection. [2] Vitjan Zavrtanik et.al. Draem-a discriminatively trained reconstruction embedding for surface anomaly detection.

Questions

1. The method artificially inject Gaussian noises into the latent space. What is the correlations between the simulated abnormal data and the real abnormal data in the test dataset?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

yes.

Reviewer ekma7/10 · confidence 4/52023-07-28

Summary

This paper proposes a novel anomaly detection algorithm utlizing energy-based models (EBMs). The proposed method, Manifold Projection-Diffusion Recovery (MPDR), is based on recovery likelihood, a framework for learning energy functions by denoising data from artifically injected Gaussian noise. MPDR uses deterministic encoder and decoder to first project the data onto a latent space, add Gaussian noise in the latent space, and finally decode the noisy latent representation onto the data manifold. This would capture more relevant modes of variation in the data than Gaussian perturbations on the original data. The recovery likelihood of MPDR is derived and is shown to result in consistent estimation of the energy function. Negative samples are generated via Langevin Monte Carlo in the latent space. The paper also introduces additional variations, including different types of energy functions and the use of ensembles to generate diverse negative samples. Experiments demomstrate the effectiveness of MPDR for out-of-distribution detection on MNIST and CIFAR-100 datasets, as well as anomaly detection for acoustic signals.

Strengths

Anomaly detection is a long standing problem in machine learning with a rich literature, and this paper proposes a novel step in the development of new algorithms. The idea of using recovery likelihood as well as projecting data onto a low-dimensional latent space are not novel, but this paper combines them to produce an original idea. The relevant background and motivation are presented with sufficient clarity, and the proposed method seems logically sound. The toy example presented in Figure 2 empirically verifies that MPDR captures more relevant modes of variation in the data.

Weaknesses

The related works section cite most of the relevant previous works, but are not covered in sufficient detail, perhaps owing to the pade limitations. It is important to cover previous work exhaustively to demonstrate where the proposed method stands in relation.

Questions

The experiments cover two image datasets and acoustic signals, which are relatively high dimensional data. It would be interesting to demonstrate the performance of MPDR on tabular data [1] (both low and high-dimensional) and explore the relevant modes which arise from applying MPDR to such datasets. [1] Han, S., Hu, X., Huang, H., Jiang, M. and Zhao, Y., 2022. Adbench: Anomaly detection benchmark. *Advances in Neural Information Processing Systems*, *35*, pp.32142-32159.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

3 good

Contribution

2 fair

Limitations

The paper covers the limitations of MPDR in sufficient detail, namely the sensitivity to the autoencoders used for latent space projection, questions around application to high-dimensional text or image data, and the fact that MPDR is not optimized to generate samples from simple distributions.

Reviewer ekma2023-08-10

Review updated

My major concerns have been addressed, and I am happy to see the good performance on ADBench. I have updated my recommendation for acceptance.

Authorsrebuttal2023-08-13

Additional response

Due to space limitations, our previous response was unable to address all the questions. Therefore, we would like to offer additional responses to the unanswered questions here. >How does the fixed nature of the autoencoder and noise magnitude impact the algorithm's adaptability to different datasets? Please note that the autoencoders are trained on the training data. Although the autoencoders remain 'fixed,' they adapt to the specific dataset and approximate the manifold structure of each dataset. Additionally, the fixed noise magnitude doesn't imply the application of the same perturbation across different datasets. Since distinct autoencoders are used for various experiments, a Gaussian perturbation in the latent space corresponds to a distinct operation in the input space. > How does the simultaneous use of multiple perturbations enhance the algorithm’s performance? Any potential trade-offs? One hypothesis is that ensembling over multiple perturbations increases the MCMC’s coverage of a high-dimensional input space. We empirically observe that the gain from ensembling usually diminishes as the number of components increases. Therefore, there is usually a trade-off between the marginal improvement in performance and the marginal computational cost. > how the choice of Dz affects the algorithm's ability to detect anomalies? $D_z$ is a hyperparameter of the proposed algorithm that has to be tuned, while the performance is generally robust across a wide range of $D_z$. Please see Table 9 in Appendix for the sensitivity analysis with varying $D_z$ value. > Solution for memory overhead of using multiple autoencoders? Multiple possible solutions may present. We may distill or quantize the autoencoders. Instead, we may design the autoencoders so that a significant portion of parameters are shared across the autoencoders. We believe this is an exciting direction for future work. We thank you again for your deep and thorough review. Please let us know if you have additional questions. Best regards, Authors.

Reviewer dsqA2023-08-16

Thanks for the experiments compared with UniAD and DRAEM. However, the experiment results are lower than UniAD. The reasons and results analyses should be clarified.

Authorsrebuttal2023-08-19

Regarding MVTect-AD Experiment

Dear Reviewer dsqA, Thanks for taking the time to read our response. We would love to provide further discussions on the MVTec-AD Experiment. **MPDR outperforms UniAD in certain classes of MVTec-AD.** Even though UniAD's mean AUROC is higher (with a very small gap), UniAD does not dominate MPDR. In the detection task, MPDR achieves higher or equal scores in 8 out of 15 classes, and in the localization task, MPDR wins in 5 out of 15 classes. This result suggests that MPDR captures the patterns in data that UniAD neglects. **MPDR and UniAD use different approaches to prevent the reconstruction of anomalies.** UniAD is an algorithm that aims to prevent the reconstruction of anomalies. MPDR, especially MPDR-R, which uses an autoencoder's reconstruction error as energy, shares the same goal, as the energy should be large for anomalies. The difference is that UniAD prevents anomaly reconstruction through a novel neural network design, while MPDR addresses the problem with novel recovery likelihood learning. Besides, **MPDR has the advantage of being more widely applicable than UniAD.** As a general learning algorithm for energy-based models, MPDR is compatible with a wide range of network architectures and can be applied to diverse data types. MPDR has been tested on 2D data, tabular data, images, audio signals, and feature vectors from a pre-trained network. However, UniAD is specialized for anomaly detection using feature vectors and has only been tested for image data. Best regards, Authors.

Reviewer Tcts2023-08-18

I acknowledge I have read the rebuttal.

Reviewer Pi2D2023-08-18

Thanks for the reply. I will keep my original assessment.

Reviewer gSqM2023-08-18

I thank the authors for their response and the extra experiments they provided. I have completely read the authors' rebuttal and other reviews and thus I increase my score.

Program Chairsdecision2023-09-21

Decision

Accept (poster)

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