Title: On the SAC-BL Algorithm for Anomaly Detection
Venue: NeurIPS 2025
Accepted: yes
Keywords: SAC-BL algorithm

False positive rate
Abstract: Visual anomaly detection is significant in safety-critical and reliability-sensitive scenarios. Prior studies mainly emphasize the design and training of scoring functions, while little effort has been devoted to constructing decision rules based on these score functions. A recent work  Ma et al. (2025b) highlights this issue and proposes the SAC-BL algorithm to address it. This method consists of a strong anomaly constraint (SAC) network and a betting-like (BL) algorithm serving as the decision rule. The SAC-BL algorithm can control the false discovery rate (FDR). However the performance of SAC-BL algorithm on anomalous examples, or its false positive rate (FPR), has not been thoroughly investigated. This paper provides a deeper analysis of this problem and explores how to theoretically reduce its FPR. First, we show that as the number of testing examples tends to infinity, the SAC-BL algorithm performs well on abnormal data if the scores follow the generalized Gaussian-like distribution family. But such conditions about the number of testing examples and the distribution of scores are overly restrictive for the real-world applications. So, we attempt to decrease the FPR of the SAC-BL algorithm under the condition of finite samples for practical anomaly detection. To this end, we redesign the BL algorithm by incorporating a randomization strategy and propose a novel stochastic BL (SBL) algorithm. The combination of the SAC network and the SBL algorithm yields our method, SAC-SBL. Theoretical results show that the SAC-SBL algorithm can achieve smaller FPR than SAC-BL algorithm while controlling its FDR. Finally, extensive experimental results demonstrate the superiority of our method over SAC-BL algorithm on multiple visual anomaly detection benchmarks.

On the SAC-BL Algorithm for Anomaly Detection

Abstract

Visual anomaly detection is significant in safety-critical and reliability-sensitive scenarios. Prior studies mainly emphasize the design and training of scoring functions, while little effort has been devoted to constructing decision rules based on these score functions. A recent work Ma et al. (2025b) highlights this issue and proposes the SAC-BL algorithm to address it. This method consists of a strong anomaly constraint (SAC) network and a betting-like (BL) algorithm serving as the decision rule. The SAC-BL algorithm can control the false discovery rate (FDR). However the performance of SAC-BL algorithm on anomalous examples, or its false positive rate (FPR), has not been thoroughly investigated. This paper provides a deeper analysis of this problem and explores how to theoretically reduce its FPR. First, we show that as the number of testing examples tends to infinity, the SAC-BL algorithm performs well on abnormal data if the scores follow the generalized Gaussian-like distribution family. But such conditions about the number of testing examples and the distribution of scores are overly restrictive for the real-world applications. So, we attempt to decrease the FPR of the SAC-BL algorithm under the condition of finite samples for practical anomaly detection. To this end, we redesign the BL algorithm by incorporating a randomization strategy and propose a novel stochastic BL (SBL) algorithm. The combination of the SAC network and the SBL algorithm yields our method, SAC-SBL. Theoretical results show that the SAC-SBL algorithm can achieve smaller FPR than SAC-BL algorithm while controlling its FDR. Finally, extensive experimental results demonstrate the superiority of our method over SAC-BL algorithm on multiple visual anomaly detection benchmarks.

1 Introduction

Visual anomaly detection (AD) (Haselmann et al., 2018) is a critical task in computer vision, where the objective is to identify and spatially localize regions in visual data that deviate from normal patterns (Gong et al., 2019; Tien et al., 2023). The AD has wide-ranging applications, including industrial quality control (Bergmann et al., 2019a), medical image analysis (Huang et al., 2024), video surveillance (Lv et al., 2023) and autonomous driving (Huang et al., 2020). Unlike standard supervised learning problems, The AD task is characterized by a scarcity or complete absence of labeled anomalous data, due to the inherent rarity, unpredictability, and diversity of anomalies in real-world scenarios (Bergmann et al., 2019b). Consequently, most existing approaches adopt an

unsupervised learning paradigm, where models are trained solely on normal samples and then identify the anomalies during inference based on the learned distribution information of normality (Wang et al., 2021; Bergmann et al., 2019c; Deng, Li, 2022).

A substantial body of methods have been proposed to tackle the challenge of the AD. These methods can be grouped into four principal categories: knowledge distillation-based methods (Deng, Li, 2022; Tien et al., 2023), synthesizing-based methods (Li et al., 2021; Yan et al., 2021), embedding-based methods (Defard et al., 2020; Roth et al., 2022) and reconstruction-based method (Akcay et al., 2018; Bergmann et al., 2019c). The existing approaches mainly focus on designing the powerful score functions to learn the discriminative information from normal data. However, these methods neglect the deep investigation for the decision rules based on the proposed score functions. A recent work Ma et al. (2025b) points out this issue and tackle it from statistical perspective. Concretely, Ma et al. (2025b) frames the AD task as a hypothesis testing problem and proposes a novel SAC-BL algorithm to address it. The SAC-BL algorithm consists of a strong anomaly constraint (SAC) network and a betting-like (BL) algorithm serving as the decision rule. Theoretically, the SAC-BL algorithm can control false discovery rate (FDR) at prescribed level. However, the performance of the SAC-BL algorithm on anomalous examples, or its false positive rate (FPR), has not been thoroughly analyzed. Factually, only controlling the FDR might result in a poor performance on anomalous data in some worst-case scenario. For example, we consider a trivial decision rule ϕ ( · ) which directly accepts all null hypotheses, equivalent to classify all testing examples as normal data. Obviously, the FDR of decision rule ϕ ( · ) is zero, but its FPR is one. It is therefore imperative to conduct an in-depth study on the FPR of SAC-BL algorithm. Besides, a review (Pang et al., 2021) highlights that existing anomaly detection methods-particularly unsupervised methods-often suffer from high FPR, and how to reduce FPR remains one of the most important yet challenging problems in the field. Then, one natural question arises:

How to theoretically reduce the FPR of SAC-BL algorithm while controlling the FDR at prescribed level?

This paper attempts to address the problems mentioned above.

Based on the analytical framework in Ingster, Suslina (2003); Donoho, Jin (2004); Jin, Ke (2016), we find that as the number of testing examples tends to infinity, the SAC-BL algorithm performs well on anomalous data, provided that the score distribution belongs to the generalized Gaussian-like family. However, such assumptions are often overly restrictive for real-world applications. On the one hand, real-world data distributions are complex and typically unknown, making it difficult to satisfy specific distributional requirements. On the other hand, the number of available testing samples is often limited. For example, in the widely used AD benchmark MVTec (Bergmann et al., 2019b), the testing set for the class 'Pill' contains only 167 images. Hence, it is necessary to enhance the performance of SAC-BL algorithm on the anomalous data under the condition of finite examples for the real-world AD task. To this end, we redesign the BL algorithm by incorporating a randomization strategy and propose a novel stochastic BL (SBL) algorithm. The combination of the SAC network and the SBL algorithm yields our method, SAC-SBL. Theoretical results show that SAC-SBL algorithm can achieve smaller FPR than the SAC-BL algorithm while controlling its FDR at the prespecified level. Finally, we conduct extensive experiments to verify the effectiveness of SAC-SBL algorithm. For example, compared with the SAC-BL algorithm, our method reduces the image-level FPR from 43 . 26% to 29 . 08% while achieving the same TPR for the class 'Pill' in MVTec.

We summarize our main contributions as follows.

We demonstrate that as the number of testing examples tends to infinity, the SAC-BL algorithm achieves a well performance on anomalous testing examples if the distribution of scores belongs to the generalized Gaussian-like distribution family.

To improve the performance of SAC-BL algorithm on anomalous examples in practical AD task, we propose a novel SAC-SBL algorithm which is based on a randomization strategy for the p-values. Theoretically, our proposed method can reduce the FPR of SAC-BL algorithm while controlling its FDR at prescribed level.

Extensive experimental results demonstrate the superiority of our method over SAC-BL algorithm on multiple visual anomaly detection benchmarks.

2 Background

Different from previous literature, Ma et al. (2025b) studies the AD problem from the perspective of multiple hypothesis testing, and propose the SAC-BL algorithm to tackle it. We first introduce the hypothesis testing framework introduced by Ma et al. (2025b) for the AD task. To avoid confusion, we use the same mathematical notations as Ma et al. (2025b) in our paper. Denote by X the feature space of the normal examples, and X follows the underlying distribution D . In most cases, D is unknown. Given a testing set T test = { X test 1 , X test 2 , X test 3 , · · · , X test n } 2 , Ma et al. (2025b) frames the AD task as the following multiple hypothesis testing problem:

where H i ;0 and H i ;1 are called null hypothesis and alternative/non-null hypothesis, respectively. In the context of anomaly detection, if H i, 0 is rejected, we declare that X test i is anomalous.

In statistics, the decision to accept or reject the null hypothesis is determined by the concept of p-value . Its general definition is presented as follows.

Definition 2.1. ( P-value (Casella, Berger, 2002) ) Given a sample ˜ X 3 . A statistic p ( ˜ X ) is called p-value corresponding to the null hypothesis H 0 , if p ( ˜ X ) satisfies

for every 0 ≤ t ≤ 1 .

A small p-value usually provides strong evidence against the null hypothesis. If the cumulative distribution function F ( · ) of testing statistic T for null hypothesis is known, then the corresponding p-value can be defined as

where ˆ T is the observation of T . It is easy to demonstrate that p ( ˆ T ) satisfies the condition in Eq. (2). It is noteworthy that the p-value has clear statistical interpretation. For example, suppose the p-value of a anomalous testing example X test i is 0 . 01 . This means that for any coming testing example X test j , the probability that X test j is more anomalous than X test i is 0.01. In other words, it is extremely difficult to find a more anomalous example than X test i . Hence, we are highly confident that X test i is abnormal.

It is known that the probability of type-I error should be controlled at the prescribed significant level in single hypothesis testing. Similarly, in multiple hypothesis testing, the false discovery rate (FDR), as the generalization of probability of type-I error, should be controlled. The statistical advantages of FDR have been detailedly discussed in (Benjamini, Hochberg, 1995; Benjamini, Yekutieli, 2001).

Given the null hypotheses { H 1;0 , H 2;0 , · · · , H n ;0 } , let R be the set of indices of the rejected null hypotheses. Similarly, denote by H 0 and H 1 the set of indices for the true null hypotheses and false null hypotheses for { H 1;0 , H 2;0 , · · · , H n ;0 } , respectively. Besides, let n 0 = |H 0 | be the number of true null hypotheses. In statistics, if one null hypothesis is rejected, it is said to make a discovery. FDR is the expected proportion of false discoveries among the rejected hypotheses.

Definition 2.2 ( FDR(Benjamini, Hochberg, 1995) ) . Dnote by V the number of true null hypotheses rejected for the hypotheses H 1;0 , H 2;0 , · · · , H n ;0 . Additionally, let U be the number of rejected hypotheses. The false discovery proportion (FDP) is defined as:

The expectation of FDP is defined as the FDR, namely

We given a new insight for controlling the FDR. Following Ma et al. (2025b), the normal data is set to be positive. The FDP is closely related to the TPR and FPR. Using the notations of confusion matrixm, the FDP is expressed as

Note that P = |H 0 | and N = |H 1 | . For a given testing set, P and N are fixed. It is well-known that there is a tradeoff between the detection performance of normal and abnormal examples for a trained score functions. Therefore, we cannot only consider the true positive rate (TPR) or false positive rate (TPR) when designing the AD algorithm. Factually, an ideal AD algorithm should achieve low FPR while maintaining a high TPR, which leads to a small FDP based on Eq. (3). Thus, controlling FDR tends to achieve a well tradeoff between the detection performance of both normal and abnormal examples if R̸ = ∅ .

To control the FDR for the AD task, Ma et al. (2025b) proposes a new SAC-BL algorithm, which consists of a SAC network serving as the score function and a BL algorithm as the decision rule. The SAC network is composed of two parts: a reconstrunction network (R-net) and a discriminative network (D-net). The core of SAC network is to apply strong constraints for the training process of discriminator by hard pairs generated from reconstruction network, which can improve the discriminative capability for weak anomalies. The details about SAC network can be found in Section 4 of Ma et al. (2025b). The BL algorithm is defined as follows.

Definition 2.3 ( BL algorithm (Ma et al., 2025b) ) . Given the p-values p 1 , p 2 , · · · , p n corresponding to the null hypotheses H 1 , 0 , H 2 , 0 , · · · , H n, 0 , let p ( i ) be the i -th order statistics from the smallest to the largest. For a pre-specified level α ∈ (0 , 1) , define

where γ, δ are two positive real numbers and satisfy γ + δ ≤ 1 . Then, the null hypothesis H ( i ) , 0 is rejected if i ≤ i ∗ BL .

In statistics, α is usually set to 0 . 05 . Eq. (4) indicates that the BL algorithm rejects the null hypothesis H i, 0 if p γ i ≤ αδ n i ∗ BL and |R| = i ∗ BL . In practice, for a testing example X test i , if the distribution of testing statistic (or scores) is known, the p-value p i corresponding to null hypothesis H i ;0 can be presented as p i = 1 -F ( s ( X i )) where F ( · ) is the cumulative distribution function of the scores and s ( · ) is the score function. If the distribution information is unknown, the computation of the p-value relies on a calibrated set consisting of normal examples. Specifically, given a calibrated set T cal = { X cal 1 , X cal 2 , . . . , X cal k } , p i is presented as

3 Asymptotic FPR of SAC-BL algorithm

False positive rate (FPR) is a significant evaluation criterion for AD task, which can be expressed as

Although Ma et al. (2025b) has proved that the SAC-BL algorithm can control FDR. However, relatively little is known about the theoretical properties of FPR of SAC-BL algorithm. In this section, we aim to analyze the asymptotic performance of its FPR. Our analytical framework follows Donoho, Jin (2004); Jin, Ke (2016)

In the vast majority of theoretical literature on multiple hypothesis testing (Benjamini, Hochberg, 1995; Benjamini, Yekutieli, 2001; Storey, 2002; Blanchard, Roquain, 2008), the p-values are assumed to be available, equivalently, the distribution of test statistic for each null hypothesis is known. In this

situation, the p-value can be represented as p i = Ψ( ˜ T i ) 4 where ˜ T i is the observation of test statistic T i corresponding to hypothesis H i and Ψ( · ) is the survival function of test statistic. For analytical simplicity, we suppose that the observation ˜ T 1 , ˜ T 2 , · · · , ˜ T n are continuous random variables. We first define a function class as follows.

Based on the function class F , we have the following definition.

Definition 3.1 (Generalized Gaussian-like Distribution Family) . A random variable X is said to follow the generalized Gaussian-like distribution with the location µ and the degree τ , denote X ∼ G ( µ, τ ) , if the survival function Ψ( · ) of X satisfies Ψ( t -µ ) ∈ F .

It is easy to verify that Gaussian distribution belongs to this distribution family. Similar to Donoho, Jin (2004); Ingster, Suslina (2003); Jin, Ke (2016), our analytical framework relies on the above generalized Gaussian-like distribution family. Specifically, we assume that the test statistic T i 5 corresponding to the hypothesis H i satisfies:

Following Donoho, Jin (2004), we set µ = ( τr log n ) 1 /τ . Besides, we focus on the sparse region in which the number of true null hypothesis is larger than that of the true alternative hypothesis (Donoho, Jin, 2006). In this case, n 1 = |H 1 | = n 1 -β where β < r < 1 (note that n 1 n → 0 ). Based on the framework above, we derive the asymptotic FPR of SAC-BL algorithm.

Theorem 3.2. Suppose that the test statistic T i corresponding to H i satisfies the condition in Eq. (6) for i ∈ [ n ] . Then, for the p-values p 1 = Ψ( ˜ T 1 ) , p 2 = Ψ( ˜ T 3 ) , · · · , p n = Ψ( ˜ T n ) , the FPR of SAC-BL algorithm converges to zero in probability, namely

The proof of Theorem 3.2 is presented in Appendix B.1. Based on the Theorem 3.2, we have the following theoretical result.

Theorem 3.3. Suppose that the conditions in Theorem 3.2 hold. Then, we have

The proof of Theorem 3.3 is presented in Appendix B.2.

Theorem 3.2 indicates that as the number of testing samples tends to infinity, the SAC-BL algorithm performs well on anomalous examples, provided that the distribution of the scores belongs to the generalized Gaussian-like distribution family. However, such conditions are overly restrictive in real-world applications, as the distribution of real-world data is often complex and typically unknown. Besides, for the widely used AD benchmark MVTech (Bergmann et al., 2019b), the testing set for the class 'Pill' only contains 167 images and the SAC-BL attains the image-level FPR of merely 43% on it (see Table 1). Hence, we need to explore how to improve the performance of SAC-BL algorithm on the anomalous data under the condition of finite examples for the practical AD task.

4 Decreasing the FPR of SAC-BL algorithm

In this section, we explore how to reduce the FPR of SAC-BL algorithm while controlling its FDR at the prescribed level. Our main focus is to redesign the BL algorithm.

Note that FPR = FP FP + TN = |R c ∪H 1 | |H 1 | . where R c is the complement of R . For a given testing set, FP + TN is equal to the number of true anomalous examples in testing set. Therefore, we can reduce the FPR by reducing FP, equivalent to increasing the number of rejecting null hypotheses |R| . Recall the BL algorithm: i ∗ BL := max { i ∈ [1 : n ] : p γ ( i ) ≤ δαi n } . The null hypothesis H i, 0 is rejected if and only if p γ ≤ δα i ∗ . Denote

( i ) n BL

Note that ξ BL only take the value in C . For a given p-value p i satisfying p γ i > δα n ξ BL , our idea is to develop a strategy that changes the value of p i to ˜ p i such that ˜ p γ i = ξ BL without violating the core conditions of controlling FDR for the SAC-BL algorithm. Then we could increase the number of rejecting null hypotheses and further reduce the FPR of SAC-BL algorithm.

Denote by Λ the random variable uniformly distributed on (0 , 1) independent of p-values and we define

Moreover, for each i ∈ [ n ] := { 1 , 2 , 3 , · · · , n } , we denote

If there are multiple indices that satisfy the argmax, we take the largest index. Then, we define the stochastic BL (SBL) algorithm as follows.

Definition 4.1. ( Stochastic BL Algorithm ) Given the random variable Λ uniformly distributed on (0 , 1) , p-values p 1 , p 2 , · · · , p n corresponding to the null hypotheses H 1 , 0 , H 2 , 0 , · · · , H n, 0 and the prespecified level α ∈ (0 , 1) . For each i ∈ [ n ] , define

and

where γ, δ are two positive real numbers and satisfy γ + δ ≤ 1 . Then, the null hypothesis H i ;0 is rejected if

The SBL algorithm in Definition 4.1 is mainly used to derive our theoretical results. Additionally, we have another simple version of SBL algorithm.

Definition 4.2. ( Stochastic BL algorithm: version 2 ) Given the p-values p 1 , p 2 , · · · , p n corresponding to the null hypotheses H 1 , 0 , H 2 , 0 , · · · , H n, 0 , let p ( i ) be the i -th order statistics from the smallest to the largest. Denote by Λ the random variable uniformly distributed on (0 , 1) . For a pre-specified level α ∈ (0 , 1) , define

where γ, δ are two positive real number and satisfies γ + δ ≤ 1 . Then, the null hypothesis H ( i ) , 0 is rejected if i ≤ i ∗ SBL .

The Following theory reveals the relation between the SBL algorithms in Definition 4.1 and the one in Definition 4.2.

Theorem 4.3. The SBL algorithm in Definition 4.1 is equivalent to the one in Definition 4.2.

The proof of Theorem 4.3 is presented in Appendix B.3. Obviously, the SBL algorithm in Definition 4.2 has less computation steps than the one in Definition 4.1, and thus is more suitable for the practical

Algorithm 1: Practical SAC-SBL Algorithm

Input: Training set T tra , calibrated set T cal = { X cal 1 , X cal 2 , . . . , X cal k } testing set T test = { X test 1 , X test 2 , . . . , X test n } , prescribed level α ∈ (0 , 1) , generator of synthetic anomalous example G ( · ) .

1 Utilize G ( · ) to construct the new training set T mix based on the training set T tra .

2 Train the SAC network on T mix , and then obtain the score function s ( · ) for images or pixels.

3 Calculate the empirical p-value corresponding to X test i :

4 Draw a sample Λ from the uniform distribution on (0 , 1) .

5 Determine the index i ∗ BL :

Output: Declare that X test ( i ) is anomalous if i ≤ i ∗ SBL .

applications. The version of SBL algorithm in Definition 4.1 is used to establish the connection between the SBL algorithm and BL algorithm for the theoretical analysis.

In the context of anomaly detection, the underlying distribution information of normal data is usually unknown. Therefore, we can use the method in Eq. (5) to compute the p-values in practice. Combining the SAC network proposed by Ma et al. (2025b) and the SBL algorithm in Definition 4.2 yields our method, SAC-SBL. Its detailed steps are presented in Algorithm 1.

Now we present our core theoretical results to demonstrate the superiority of SAC-SBL algorithm over the SAC-BL algorithm in terms of FPR while controlling the FDR at the prescribed level.

Theorem 4.4. Given the random variable Λ uniformly distributed on (0 , 1) , the p-values p 1 , p 2 , · · · , p n corresponding to the null hypotheses H 1 , 0 , H 2 , 0 , · · · , H n, 0 in Eq. (8) and the prespecified level α ∈ (0 , 1) . The following conclusions hold:

the FDR of SAC-SBL algorithm satisfies

For the index sets R SAC -BL and R SAC -SBL , we have

If there exists a p-value p i which satisfies P ( p γ i < δαi ∗ BL n ) > 0 , then we have

The proof of Theorem 4.4 is presented in Appendix B.4. Theorem 4.4 suggests that the SAC-SBL algorithm does not reject fewer null hypotheses than the SAC-BL algorithm almost surely while controlling the FDR at the prescribed level. In other words, SAC-BL algorithm tends to classify more testing examples as anomalous data. Based on Theorem 4.4, we can easily obtain the following theoretical results.

Theorem 4.5. Suppose the conditions in Theorem 4.4 hold, then the FPR of the SAC-BL algorithm and that of SAC-SBL algorithm satisfy

Moreover, if there exists a p-value p i which satisfies P ( p γ i < δαi ∗ BL n ) > 0 , then we have

The proof of theorem 4.5 is presented in Appendix B.5. Theorem 4.5 indicates that the FPR of the SAC-BL algorithm is not smaller than that of SAC-SBL algorithm almost surely. Besides, the condition 'there exists a p-value p i which satisfies P ( p γ i < δαi ∗ BL n ) > 0 ' is nearly always satisfied in practice, since it only excludes the trivial situation where all hypotheses are rejected. Therefore, our proposed method can achieve a smaller FPR over the SAC-BL algorithm.

5 Experiment

In this section, we perform extensive comparison experiments to verify the effectiveness of our SAC-SBL method.

5.1 Experimental Settings

Baseline. We compare the decision-making performance of our proposed SAC-SBL method with the SAC-BL method (Ma et al., 2025b), using the default settings for SAC-BL.

Datasets. Experiments are conducted on three widely used anomaly detection datasets. The MVTec dataset (Bergmann et al., 2019b) consists of 5354 images across 15 object and texture categories. The VisA dataset (Zou et al., 2022) is an industrial anomaly dataset comprising 10821 images from 12 objects across 3 domains. The BTAD dataset (Mishra et al., 2021) includes 2830 images of 3 industrial products, showcasing body and surface defects.

Evaluation Metrics. We use the true positive rate (TPR), false positive rate (FPR) and f1-score (F1) to evaluate the effectiveness of the proposed method.

Implementation Details. In SAC-SBL, we repeat the stochastic perturbation 100 times and take the meaning value. All experiments are conducted on a workstation with eight NVIDIA GeForce GTX 3090 GPUs and two 2.2GHZ Intel CPUs.

5.2 Experimental Results

We evaluate our proposed method against the SAC-BL algorithm on three datasets, demonstrating its superior performance and generalization. The image-level and pixel-level results on MVTec are presented in Table 1, and the results on VisA are shown in Table 2. because of the space limitation, the results in BTAD are reported in Appendix A. From the Tables 1 and 2, we can see: 1) Our mehtod achieves the comparable performance in terms of TPR. For example, in Table 1, our method have the same image-level TPR as the SAC-BL algorithm in 13 classe among 15 classes. On average, the image-level and pixel-level TPRs of two methods are also comparable. 2) Our method yields lower FPR and higher F1-scores across all classes, regardless of image-level or pixel-level evaluation. As shown in Table 1, compared to the SAC-BL algorithm, our method reduce the image-level FPR from 21 . 74% to 15 . 22% , and improve the image-level F1-score from 84 . 44% to 87 . 5% , the direct improvements of 6 . 52% and 3 . 06% , respectively. 3) In those classes where the SAC-BL algorithm struggles to identify anomalous examples, our method can considerably reduce the FPR. For instance, For classes 'Carpet' and 'Pill', our method achieve the improvements of 11 . 24% and 14 . 14% in terms of image-level FPR. Overall, these experimental results demonstrate the superiority of our method over the SAC-BL algorithm.

6 Conclusion

In this paper, we focus on investigating the SAC-BL algorithm in terms of FPR. First, we demonstrate that the FPR of SAC-BL algorithm converges to zero in probability based on the generalized Gaussianlike distribution family. Then, we propose a novel SAC-SBL algorithm which can reduce the FPR of SAC-BL algorithm while controlling the FDR at the prescribed level. Finally, we conduct extensive experiments to verify the effectiveness of our proposed method.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant 624B2106, the Key R & D Program of Hubei Province under Grant 2024BAB038, the National Key R & D Program of China under Grant 2023YFC3604702, the Fundamental Research Funds for the Central Universities under Grant 2042025kf0045.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors provide a deeper analysis of this problem and explores how to theoretically reduce its False Positive rate. First, the authors show that as the number of testing examples tends to infinity, the SAC-BL algorithm performs well on abnormal data if the scores follow the generalized Gaussian-like distribution family.\n\nFirst, the authors show that as the number of testing examples tends to infinity, the SAC-BL algorithm performs well on abnormal data if the scores follow the generalized Gaussian-like distribution family. But such conditions about the number of testing examples and the distribution of scores are overly restrictive for the real-world applications. So, the authors attempt to decrease the FPR of the SAC-BL algorithm under the condition of finite samples for practical anomaly detection. To this end, the authors redesign the BL algorithm by incorporating a randomization strategy and propose a novel stochastic BL (SBL) algorithm.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nPaper is well formatted, using LaTex formatting\n\nReferences are Up to date, perhaps more could be added\n\nBackground materials are up to date and well presented\n\n\n\nWeaknesses:\n\nAlgorithmic content could be clearer, see:\n\nhttps://shantoroy.com/latex/how-to-write-algorithm-in-latex/\n\nFor examples of algos\n\nIn the background, literature cited could focus more on limitations\n\nEquations not of authors should be cited\n\nI think the mathematics could be explained more in the text.\n\nExperimental analysis is short, such as Table 1 being discussed, \n\nMore graphical results should be presented.\n\nFuture work could be discussed in conclusion or before\n\nQUESTIONS:\nAlgorithmic content could be clearer, see:\n\nhttps://shantoroy.com/latex/how-to-write-algorithm-in-latex/\n\nFor examples of algos, is that possible?\n\nIn the background, literature cited could focus more on limitations, is that possible?\n\nEquations not of authors should be cited when needed?\n\nI think the mathematics could be explained more in the text.\n\nExperimental analysis is short, such as Table 1 being discussed, \n\nMore graphical results should be presented.\n\nFuture work could be discussed in conclusion or before\n\nLIMITATIONS:\nMore limitations could be expanded upon", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the theoretical properties of the SAC-BL algorithm for visual anomaly detection, with a particular focus on its underexplored false positive rate (FPR). Specifically, the authors first demonstrate that the FPR of SAC-BL can be low under certain distributional assumptions. However, these assumptions may be too restrictive for real-world applications. To address this, the authors propose a new algorithm, SAC-SBL, which introduces stochasticity into the decision rule to reduce the FPR while maintaining the false discovery rate (FDR) control. Theoretical analysis shows that SAC-SBL achieves a lower FPR than SAC-BL almost surely. Experiments on MVTec, VisA, and BTAD datasets confirm consistent improvements in FPR and F1-score without compromising TPR. Overall, this paper is well-motivated, theoretically rigorous, and empirically validated, offering a novel perspective on the design of decision rules in anomaly detection.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths \n- Despite being theory-heavy, this paper is well-structured and readable. Motivations, algorithms, and results are presented clearly.\n- This paper proposes a novel SAC-SBL algorithm, and provides solid theoretical results showing that the proposed method can decrease the FPR control while maintaining FDR, compared the SAC-BL algorithm.\n- The proposed method is simple to implement and integrates naturally into existing SAC-BL frameworks. \n- This paper validates the proposed method across multiple challenging datasets and at both the image and pixel levels, showing consistent performance gains in terms of reduced FPR and improved F1-score, with comparable or improved TPR.\n\n## Weaknesses\nI have not identify the major weaknesses, but I have some recommendations:\n- It would be better if the title of the paper included the background of the research problem. For example, it could be revised to “On the SAC-BL Algorithm for Visual Anomaly Detection.”\n- The authors should provide more detailed experimental setup.\n- I am confused of the results of line 725 in Appendix, please give a more clear interpretations.\n\nQUESTIONS:\nSee weaknesses\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-06-26"}

comments: SUMMARY:
This paper presents a theoretical analysis for the SAC-BL algorithm, focusing on its false positive rate (FPR), which has not been well-analyzed before. The authors propose SAC-SBL, a stochastic variant of SAC-BL, and demonstrate that it achieves a lower FPR while maintaining its false discovery rate (FDR) at under prescribed level.  Theoretical results are backed by experiments on various anomaly detection benchmarks.

STRENGTHS AND WEAKNESSES:
Strengths 
1, this paper provides a rigorous theoretical analysis for the asymptotic FPR of SAC-BL algorithm.
2, The proposed method is a well-motivated extension of SAC-BL, the motivation and contributions are clear.
3, The presentation is clear and the proofs in Appendix are sound. 
4, Experimental results across three benchmarks show the improvements in terms of FPR and F1-score, adding practical credibility to the theory.

Weaknesses
1, The title lacks clarity about the application domain. Readers unfamiliar with SAC-BL may not recognize this as a paper on visual anomaly detection.
2, Apart from the Gaussian distribution, are there any other distributions that belong to the generalized Gaussian-like distribution family?
3, There exist some typos in line 198 of submission.  
4, Do the results in Theorem 4.5 depend on the distributional assumptions in Definition 3.1?

QUESTIONS:
the calibrated set plays a key role in estimating empirical p-values. How does it influence the experimental results?

LIMITATIONS:
N/A
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-06-29

comments: SUMMARY:
This paper improved the (Ma et al., 2025) by theoretically analyzing the FPR and proposing a stochastic version of the BL method. This paper discussed the decision rule of AD methods following (Ma et al., 2025), which is an important problem when applying AD to the real world. The author presented a theoretical guarantee and empirically showed the advantages over the original SAC-BL methods.

STRENGTHS AND WEAKNESSES:
Strengths
- This work focuses on FPR, which is more practical.
- The proposed stochastic method is simple and effective.
- This paper discussed an important problem in UAD, but it is not ready for publication.

Weakness
- The content in total is an incremental work of (Ma et al., 2025) so far. Compared with (Ma et al., 2025), only an extended FPR analysis and a stochastic variant of BL are not enough for NIPS acceptance.
- It is not intuitive to see how the SBL can improve the FPR.
- It seems that the BL method relys on a calibrated set while how this set is constructed is unclear in the main text.
- No investigation on the related works. The authors should also compare the results with recent SOTA.
- see more comments in the question.

QUESTIONS:
- How to generalize the decision rule to any AD algorithm and any AD task? Since the p value is only calculated from the anomaly score, it should not be limited to the vision AD task and the reconstruction based SAC method. 
- How to obtain the calibrated set? Is it a subset of trianing set or it requires the labeled anomaly?
- If the calibrated set is a subset, how to select the subset?
- how the hyper-parameters, such as alpha and the synthetic strategies affect the performance?

LIMITATIONS:
The author does not discuss the potential negative societal impact.
TITLE: Official Review
CLARITY: 1
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-01