Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Authors: Yifan Fang
Yifei Fang
Ruizhe Chen
Haote Xu
Xinghao Ding
Yue Huang
Venue: ICML 2025 poster (2025)
Keywords: Anomaly Detection
Unsupervised Learning
Factor Model
Demeaned Sparse
Primary area: general_machine_learning->unsupervised_and_semisupervised_learning

Abstract: Frequency-domain image anomaly detection methods can substantially enhance anomaly detection performance, however, they still lack an interpretable theoretical framework to guarantee the effectiveness of the detection process. We propose a novel test to detect anomalies in structural image via a Demeaned Fourier transform (DFT) under factor model framework, and we proof its effectiveness. We also briefly give the asymptotic theories of our test, the asymptotic theory explains why the test can detect anomalies at both the image and pixel levels within the theoretical lower bound. Based on our test, we derive a module called Demeaned Fourier Sparse (DFS) that effectively enhances detection performance in unsupervised anomaly detection tasks, which can construct masks in the Fourier domain and utilize a distribution-free sampling method similar to the bootstrap method. The experimental results indicate that this module can accurately and efficiently generate effective masks for reconstruction-based anomaly detection tasks, thereby enhancing the performance of anomaly detection methods and validating the effectiveness of the theoretical framework.

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Yifan Fang Yifei Fang" *3
Ruizhe Chen '14
Haote Xu 12 Xinghao Ding 124 Yue Huang 124
Abstract
2023; Wu et al., 2024). The scarcity and diversity of anoma- lous data present significant challenges collecting suf- ficient quantities for model training. Consequently, unsu- pervised anomaly detection methods, which utilize only nonanomalous samples as the training set, have emerged as a key research focus in this field (Zhang et al., 2024; Ristea et al., 2022; Reiss Hoshen, 2023). Among these methods, reconstruction-baat
crqquency-domaile image anomaly detection methods can substantially enhance anomaly de- tection performance, however, they still lack an interpretable theoretical framework to guarantee the effectiveness of the detection process. We propose novel test to detect anomalies in struc- tural image via a Demeaned Fourier transform (DFT) under factor model framework, and we proof its effectiveness. We also briefly give the asymptotic theories of our test, the asymptotic theory explains why the test can detect anoma- lies at both the image and pixel levels within the theoretical lower bound. Based on test, we derive a module called Demeaned Fourier Sparse (DFS) that effectively enhances detection performance in unsupervised anomaly detection tasks, which can construct masks in the Fourier domain and utilize distribution- free sampling method similar to the bootstrap method. The ex- perimental results indicate that this module can accurately and efficiently generate effective masks for reconstruction irronstrutiion-baeed anomaly detection tasks, thereby enhancing the performance of anomaly detection methods and validating the effectiveness of the theoretical framework.
anomaly detection represents one of the mainstream approaches (Zavrtanik et al., 2022: 2021a; la; Madan et al., 2023). It detects anomalies by measuring differences between samples before and after reconstruc- tion. The underlying assumption is that the model cannot accurately reconstruct anomalous structures not encountered during the training phase (Zavrtanik et al., 2021b).
Recently, Kascenas al. (2023) proposed that anomalies can be viewed as noise added to the nonanomalous distribu- tion, i.e., anomalous structures are considered an 'additive' phenomenon. Anomalies are detected by calculating the residuals between the reconstructed and original images. Furthermore, using residuals for structural change tests is very common in time series analysis, such as the Chow test (Toyoda, 1974), Ljung-Box test (Hassani & Yeganegi. 2020), and Breusch- Pagsa test (Halunga et al., 2017). By the same way, anomalies in images can be considered form of structural change, since the task can be regarded as a regression problem like the aforementioned tests. Fu et al. (2023) used a DFs-based method to detect structural changes factor model, they gave a framework in factor level by capturing information from residuals which con- tain both the time and individual dimension. However, this test is not suitable for image anomaly detection tasks, as anomaly detection in images requires the application of a multi-dimensional DFT across two cross-sectional dimen- sions, which differs from the approach used in time series analysis.
1. Introduction
Anomaly detection techniques, employed to capture anoma- lous structural changes within data that lead deviations from nonanomalous distributions, are widely applied across various domains (Khan & Alkhathami, 2024; Jiang et al.,
"Equal contribution 'Key Laboratory of Multimedia Trusted Perception and Efficient Computing, Ministry of Education of China, Xiamen University, Xiamen, China. "School of Informat- ics, Xiamen University, Xiamen, China. "School of Economics, Remmin University of China, Beijing, China. 'Institute of Ar- tificial Intelligence, Xiamen University, Xiamen, China. Corre- spondence to: Haote Xu <hotxu @ www.xmuu.cucc... Yue Huang <yhuang20100 xmu.edu.cn>>.
In this paper, we follow the method from Fu et al. (2023) and incorporate multi-DFT dual cross-sectional dimension of positional information as primary method, we first demonstrate why anomaly detection can be effectively per- formed at the factor level using by capturing information from residuals. We also derive the test for such issue, as well as asymptotic properties of the statistics under null hypothesis and alternative hypothesis. The null hypothe- sis result let compare with zero spectrum when there
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
is anomaly, the global power and local power of our test show the effectiveness and the aceuracy of our test, re- spectively. Our method reconsider the similarity between structure change and image anomaly detection. Following this motivation, we propose a general optimization module called the Demeaned Fourier Sparse (DFS) module, which specifically designed to enhance reconstruction-base anomaly detection tasks.
2.1. Hypothesis Testing
For an image X E [2# it can be interpreted as se- quence {XAmc: h 1.2..... w 1, 2.2..... W), con- sisting of observed values, where the (h, w)-th element represents the pixel located at the (h, w)-th position. We assume that Xhm is generated according to the following data generating process:
Due to the advantage of residual-based test this module does not require consistent estimation of anomalous factors or prior knowledge of anomaly types to effectively detect anomalous samples. By applying our DFS module, we im- plemented the unsupervised anomaly detection task, the result shows our theoretical properties are robust, also the effectiveness of this module compared the existing litera- ture, which highlighting its strong capabilities in real-world applications.
X Xow Fur Ehals
where F.. is a R x 1 vector of unobserved common factors (To facilitate the expression of formulas and proofs, we assume that the direction of the common factor is consistent with W. practical implementation, À and F are mixed and difficult to distinguish, the factor direction can be arbitrary, this assumption does not affect our theory proof), X is R x 1 vector of factor loadings that may vary across global or local features, R is the factor number, and &hw is the idiosyncratic error. The null hypothesis of no anomaly is given by:
The contributions of this paper are as follows.
Theoretical Contributions. Inspired by the structural changes' detection of factor models, we propose an ap- proach for detecting image anomalies by projecting resid- uals to a multi-fourier space. We derive the asymptotic properties of our test to ensure theoretical completeness, which demonstrates why anomaly can be detected at the factor level by using residuals. We also illustrate how this theory can be applied to reconstruction-based anomaly detection issues.
Ho Anw AAO for all h 1, !! and all w 1, W.
The alternative hypothesis is
HA ARw + Ano for some H and w W,
where Aιo represents factor loading that is not anomalous. W and HH denote the sets of indices (h, w) where Anw * Aso- The alternative hypothesis HA encompasses various types of unknown loadings across different factors.
Practical Contributions. We propose a factor model- based approach for reconstruction based anomaly detec- tion and have developed an effective module called De- meaned Fourier Sparse (DFS). Our method is simple and easy to implement. Our experimental results demonstrate the effectiveness of this approach, which applicable to general iiconstructionn-asedd anomaly detection task.
2.2. Test Statistic
Let = = =ONow - -bo)) F Ehw- Then (1) can be written as
X Nin + + El/ur
The related work of factor models and unsupervised anomaly detection methods presented in Appendix A. Section 2 introduces the hypothesis testing framework and statistical metrics. The asymptotic properties of the pro- posed method are detailed in Section 3. In Section 4, we design the DFS-AD method based on our testing frame- work. Finally, the experimental results presented in Section 5 demonstrate the accuracy of anomaly detection as well as the localization capabilities of DFS-AD.
Let Fw and ÂNo denote the estimation of F., and Ano- respectively. Then, the estimated residual is given by Ênw Xhw AT
Under the null hypothesis Hg. E = Elw for all and w. and (2) represents normal pixel. Consequently, Fu. and Aso are consistent estimators of the true common factors F.. and factor loadings λyu- Under the alternative hypothesis HA, we have: e/1 = F (A - Ano) + Eton- this expres- sion, the first term contains anomalous information. There- fore, our goal is to estimate (2) and subsequently project the anomalous signals in the residuals into the spectral space.
2. Preliminaries
In this section, we propose a test-based approach in the factor model, which can detect anomalies at the factor level through residual analysis. The core concept involves apply- ing the Discrete Fourier Transform (DFT) method to the factors and residuals from the factor model.
Let BX = É It is straightforward to show that Ô is functioned as an annihilator matrix that project X into the estimated residual É The following complex-valued empirical process can then be constructed:
2

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
H
W Eu
loading R,W is positive definite, and the eigen-
C(u,v)=
HW
values of the product of their covariance distinet; (ii)
h=1
H.W The error term { ==1 has weak dependence. (ddt)
H
Fu N
elw and Fw is strong mixing- (iv) the residual terms form Martingale Difference Sequence. Such an assumption is appropriate for anomaly detection, where the objective is to identify irregular, random, and unpredictable anomalies. (v) The higher-order moments of the (-) are finite, this guar- antees the absolute integrability of the Fourier series. Under these assumptions, our goal is to maintain as mild condi- tions as possible. This approach ensures that the theoretical results remain valid under broad range of conditions with- out imposing overly restrictive model specifications, thereby enhancing the applicability of this approach. Furthermore, it supports the analysis of anomaly detection tasks within the deep learning framework. Details of the proofs can be found Appendix
HW
H
W
F in Aba B)(lu,))
HW
Σ
Σ
Ew B)
HW
Ci, v) + C2(u,))
where puu, v) is a demeaned Fourier process, and u, e IR are nuisance parameters such that 2T (H + w represents a Fourier frequency. Here, combining the common factors and the residuals helps prevent the residuals from summing to zero.
3.1. Asymptotic Null Distribution
H
W
pp(u, v) = =6 22
22
um
The asymptotic null distribution shows the properties of our test and its decision-making behavior under non-anomalous conditions. Under Hig, when the factor loadings are non- anomalous, assume that choose the true number of factors.
HW
Intuitively, after we project an image into the Fourier do- main, we can interpret Ci Cu, r) as the "signal" correspond- ing to the anomalous factor loadings and its behavior is approximate to a zero spectral under Hg, under HA it will not. (u, v) as the "noise", contributing to well-defined asymptotic distribution of the Discrete Fourier Transform (DFT) under Ho. The anomalous characteristics of Atoo are reflected through various patterns in the DFT within the frequency domain. Therefore, we can detect unknown types of anomalies by analyzing the deviation of the DFT from the zero spectrum across all frequencies.
Proposition 3.1. Let 5 =- el be a compact subset of R for any constant > O. Under the assumptions mentioned above and Hlg,
V(u - H- H/2
(H, W) - co, where G(u, v) is joint mean-zero joint complex-valued Gaussian process.
Proposition 3.1 demonstrates that under Ho. the term Cs(u, v). which represents the anomalous factor load- ing "noise" in C(u, v) is dominant. For any pact set 5 [-q,9] with q O, V../H VHWČ(u, v) joint mean-zero, joint complex- valued Gaussian process. The covariance kernel K. is equivalent to the product of the asymptotic vari- ance of VHW ΣlI EinI Fw Fu v) and pseudo- covariance induced by the Fourier transform.
In order ensure the detection of unknown types of anoma- lies, is necessary to estimate the order of magnitude that each u. v IR deviates from C(u, v) and in order fa- cilitate the subsequent use of deep learning methods. The following test statistic is constructed:
weakly converges
Đ= = HW Jf b) u) du du.
(4)
The weight function W(-): : R R+ is a function designed to ensure the consistent integrability of Č(u, v), and is a measure used to evaluate the effectiveness of our test.
Based on Proposition 3.1, the asymptotic distribution the test statistic D under Ho is given by the following theorem. Theorem 3.2. When the conditions of Proposition 3.1 hold, under Ho,
3. Asymptotic Theory
D 4, [f1reu, u)] du
This section presents the asymptotic theory underlying our test statistic and explores some of its key proper- ties. To address the practical considerations and require- ments of anomaly detection tasks, we make the following assumptions:(() Common factor: The high order moments of the factor is bounded, and its covariance matrix of factor
(H, W) - 0o, where "S,- means convergence in distri- bution.
Theorem 3.2 presents the asymptotic null distribution the test statistic D. It is important to note that the integration
3

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
is carried out over R. While Proposition 3.1 establishes weak convergence on any compact subset of IR. in practical applications, we use D to ensure that weak convergence holds as (u, v) tends to infinity.
K R. our test statistic δ remains effective for image- level detection. Specifically, our power increases with the growth of HW, which enhances its ability to detect devi- ations from the distribution under Hy, then increases the probability of rejecting the null hypothesis under HA This theorem explains why, as the number of pixels (i.e., the image size) increases, our test is able to detect anomalies at the image level.
3.2. Asymptotic Global Power
The asymptotic global power provides the properties and behaviors of our test under HA. It also offers an explanation of the asymptotic limit of the detection performance given by our test. As the number(size) of pixels(images) being analyzed increases, the detection task transitions from a pixel-level analysis to a more comprehensive image-level analysis.
3.3. Asymptotic Local Power
The asymptotic local power provides an asymptotic lower bound for image anomaly detection. Even when the anomaly is very small(not under the lower bound), the residual-based test remains effective. We now consider the following set- ting for the anomaly factor loadings:
The number of common factors selected under the alterna- tive hypothesis HA, may not be the same as the true value, but our test still has power in most cases. At the image level, pixel-based anomaly detection tasks can be formulated as follows:
H (THH) = Ahw = Ano + THW Gham-
(6)
Here, THW is a scalar that depends on H and W which controls the strength of the anomalies, while Ihw represents the anomaly features for each pixel (h, w) it represents the components that deviate from the normal pixel. In general, Aso is not identifiable in the presence of anomalies, as the anomalies can obseure the true underlying factor loadings. Thus, we assume that H-1 E Ew=1 Ihw 0. It ensures that 9tw does not introduce shift in the overall data, maintaining the balance of the data. In particular, it indicates that the impact of 9hw does not cause significant effects on Avo- This condition rules out the possibility of overestimating the number of common factors, as it en- sures that the anomaly features Ihe cannot be eliminated through projection onto B. Intuitively, this indicates that the anomaly feature 9lo lies in the null space of Xvu-
x= FA +
(5)
Here, F is the matrix of common factors, composed of Fu. and A is the factor loading matrix, composed of Aiw Next, We assume that the number of common factors to be estimated is K.
Proposition 3.3. When K < R for any compact set 5 = 11-9.00 Ø] with an arbitrarily large constant q > 0, as (H, - 0o, sue (C(,,)) v) - C(u, w| = (1), where C(u, v) denotes nonzero spectrum.
The non-zero spectrum provides the convergence of our test under All,, ensuring that the constructed complex-valued empirical process is significantly different to the zero spec- trum. Specifically, this means that the estimator C(u, v) almost surely converges to the true value C(u, v). Intu- itively, the factor model is under-fitted when K < R. In such case, the estimated residuals capture sufficient in- formation about the unexplained common components. As a result, the DFT Ö(u, v) converges to a complex-valued process with non- zero mean value. Otherwise, when it is over-fitted with K3 > R, our test may has less power.
[-q.q] be a compact subset of R. for any constant q > 0. Then under assumptions given above and HA(THW) with THW H -111/2 as (H, W) + Co, 00,
Proposition 3.5. Let
Õlu, (2222-- G(u,u)),
where glu, v) denotes pseudo-covariane between the Fourier series (2m(( +) + +W) and grw B. and G(u, v) is as defined in Proposition 3.1.
In practical applications, the goal is typically to reconstruct the complete image using as little information as possible. This implies estimating R with smaller value of K, thereby promoting information sparsity. In practice, sparsity is often used to describe the relationship between the information uti- lized and the information reconstructed. Therefore, within the context of this analysis, we do not consider the case where K3 > R in the asymptotic analysis.
THw = 1/222 1/2 provides an lower bound for the estimation with K = R. Proposition 3.5 implies that the asymptotic distribution of / /HW ((u, v) under HA(THW) with K R is equivalent valued Gaussian process d(u, v) + G(u, v). When there ex- ists anomaly, which means 6(u, v) A 0 for non-negli subset (u, v). Č(u, e) can capture the anomalous behavior of the factor loadings Apws provided that the subset is in- cluded in 8. To ensure the consistency of the DFT for all u, U. E IR, we have:
non-centralized complex-
Theorem 3.4. When K < R, P(D > CHW) + 1, (H, W) + xc. for any CHw = oHH)).
Theorem 3.4 presents the propertics of the test statistic D under the alternative hypothesis AA 4. indicates that when
4

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Theorem 3.6. Under (T(TTH) with THW H H-1/2 112,, ou, v) and G( are as defined in Proposition 3.5 and Proposition 3.1
Lize
Blamanioeiie Sahetpactoon
D 3 /1 (0(,,)) v) + G(u,,)]l (w,, v) du dv,
Emput Imrsge
Dupat Image X
ar (
as () + oo, where "4 means convergence in distri- bution.
Trmnsfomm
Fourier
P(-)
L Bas
Reconstruction Netwark
DFS Medale
POO
Theorem 3.6 provides the asymptotic distribution of D under the local alternative hypothesis HI ( This indicates that as long as the proportion of anomalies in Anw λιo is not too small, our test still has power. Intu- itively, even if the anomalous characteristics of the factor loadings do not accumulate with the increasing sample size, the spatial information of a single sample still plays a cru- cial role in detecting anomalous changes. Since our test effectively leverages spatial information, it is capable of identifying anomalies individual pixel locations. This indicates that our detection method is more sensitive to smaller anomalies. Furthermore, the theorem establishes the test's ' detection lower bound.
Elemeni-wise Multipiication
Inverse Fourier
Transfonm
Fourier Domain
X Fourier Donain
X
Figure I. Overall framework of method. The framework has two components: (1) DFS module a specific implementation de- rived from the factor model theory, which adaptively learns Fourier space masks to construct the nonanomalous factors and residuals. (2) Reconstruction network aims at reconstructing normal regions. The predicted anomaly map Pred = X X Rea.
of each pixel in X belongs to λιo- Figure illustrates the pipeline of our method:
Assume that each element MA in the mask matrix M fol- lows Bernoulli distribution, i.e., Maw ~ Bernoulli(Phw)) where Pkw is obtained through iterative optimization within the module. Specifically, MAw = (I(), where LA(x) is an indicator function, and A denotes the sample probability Shw, with the condition 8hw < Phvr- This setup ensures that the mask elements are selected based on the probability Phws which is dynamically updated during the optimization process.
4. Methodology
Based on the aforementioned theory, we propose a hypoth- esis test along with its corresponding asymptotic theory. We argue that anomalies can be detected at the factor level through the analysis of residuals. Given the powerful fit- ting capabilities of deep learning networks, they are well- equipped to learn and approximate the factor structure of pixel-level data within the neural network. This allows for effective anomaly detection by leveraging the learned factor representations, facilitating the identification of deviations from the expected patterns in the data.
Sampling Method Firstly, we random sampling a tensor w E # from the [0,1] distribution and then map it to an optimizable variable s via the inverse function of the sigmoid function (.). Then sampling 8kw from (0, 1). In fact, there are no restrictions on the sampling distribution, referring the Bootstrap method, the sampling distribu- tion has restrictions. Theoretically, any commonly used distribution on the interval 10 can be applied.
In this section, we revisit the theoretical foundations of anomaly detection tasks based on reconstruction methods. Reconstruction- based anomaly detection approaches ef- fectively identify anomalies by accurately reconstructing nonanomalous samples during the inference phase, while anomalous samples fail to be reconstructed effectively. We propose the construction of a mask in the Fourier domain, which acts as an auxiliary mechanism for anomaly detec- tion. The mask sparsifies the main information by reducing the number of estimated common factors and manipulates the image through element- wise multiplication. The mask construction follows our proposed progressive theory, and its selection is iteratively optimized using joint loss func- tion. Conceptually, this approach is similar to the bootstrap method commonly employed in non parametric statistical techniques. DFS processes the input image X into X, which also an estimate of X. Subsequently, the output of recon- struction network Xres is used examine whether the À
Decentralization Thus, the probability map ( can be constructed using W, where ((-) is the sigmoid function. The mask elements Mbw = 1 (x can then be approxi- mated in the following optimizable form:
M o s) ~~ (0,1).
s),
Intuitively, as the number of iterations T + oo increases, the sampling of Shw during the iteration process will con- verge to the sample mean. This convergence ensures that the distribution of the sampled values stabilizes, aligning with the expected values. This concept is consistent with the Demeaned Fourier process introduced in Section 2.2, Our goal is to enable the model to optimize the residuals during
5

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Table 1. Anomaly detection/lncalliation results with Img-AUROC/ SES--UURC/PII-AUURREEE AURAOO/C/AAAAA/A/AAAAAA PRO (in %) metric on MyTec-AD. The hest and second best performance are highlighted in bold and underline, respectively.
Curgory
ParchCore
PyranidFlow
SSNF
DRAEM
IFigee
TransPusiot
Ours Base
Ours
corpet
60/ 98.79 18
19649/ / 191.33
/98.11 96.03
96.63/ 95/71 194.22
(9392/ 99:87
97.99 996.27 MM.E7
97.27/ 99.35 191.25
96.77
Rester
Orid
98. 31 195 60
56.17 96.11 192.78
95/95.04
29.90/
$22.80
100,00 10/ 92.30
100,00 99.58 99.51
99.67 99.06
100.00/ 98, 08.00/
Louther
100,00/ 99.03
99 (97.1 195.7
197.70/ 94.83
99.97/ 98.80/98.25
100,00/ /9.67/ 94.57
100,00 99911 196666
100,00/
995668
100,00/ 99,68 99908
Tile
96.11 99.11/9912//// 250
59. 19 995 188.00
99,90/ (94 41 87 83
99.01999 99 (97.10
99.93 (91222 95.35
10IL09 99.65 99.80
99.35 19.77 MM0
100,00/ VM 6M 995
Wixd
99.10/ 688.31
99.50 99.33 88.17
98.81 (8.81/94.79) 91.33
99.61 00.00/069990000 /9148
99.39 (96.64/ 90.72
98.95 /94.3//943///// 99481
100,00 95.18/ / 85,18
00.52 / 19.52 /91.25
PMI
96.29/99.555 (92.58993..00
$1.49 193.58 76,52
98,09/ / /8.62 94.63
97.500 97.45 (9141
/9634/96.35/ / 99.38
28.31 198,03 ( 99792
/1.84/9163 79:71
99.59 / 98,79 /9.66
Transioor
100,00 99.94 Fisl MSS
92
96.71 75.43
(4,42 196.11 11 7M 12
92.55 96 22 (12.19
/4.55/17771 62240
99 56 /177 74.0000 163.22
96.121 93.44 /72.99
19/12 MS. NAM9
Cable
99.57/90.50 /92.00
75.13/ 05:15/91.32 99.32 65.44
96.40 196 94/ 89 21
92, 12:50/ 95.66 153.55
05.35 /17 99.40
08.50 99.08 192.7
90.50 03.63 72.03
00.02 ( 19.62 /91 10
Zipper
99.41/ 99.9999999999999
94.70 (4023 96.00
94.51 / (96.00) 94.33
99.91 / 98.70 96.62
100,00 011.00999999/93333
100.00 000L000/99.00 99.95 996.54
95.75
98.25 99.25 97.55
Objoct
Toothonsdh
99.80/ 98 00000/0//000 /90.91
96 42/ 99.01 186.22
90.52/ 96/ 89 47
99 95. 98 63 (91
100,00/ (96 76/ 92 68
100.00 19809990000 99..9
100.00 995.22/2555 13
99:17 99.88 19.63
Mstal No.
29.02/97..00 97.00 (90.29
77.33 /94.0/9/////7 71.01
99.90/ /97.01/9//// 92.01
99. 10: 99 196.11
(8.85/99.2111 88.68
100.00 /9.66/666/.20 995.3
92.422 99.29 71.22
97.41 99.87 94.97
Hazelout
100.00 0ttp:/99.2//1119 /9.00
82.20 $1.30
96.22/ (1122//99779/87..
99.99 99.42 195.52
99.82 (992299927/999
100,00 99.55 999.84
95.75/ /94/33/799909
96.57/ 10 93.95
Screw
28.01/ (97.99
$3,44786.00 156.11
58,50/98 /9.00/99////// 93.59
95.70/ 98.61 /96.49
92.60 (97.24/9///// 53305
98.63/ 99.17 195.79
69.62/ 97.63 990.44
96.02/ 99.22/ 09
Capanie
2/.22/ (9622) 19644
94/11
780
/50
94
99.40
90.25 197 (87
10. 99..6...88
Bottl
100,00/98.00 /95.80
$5.70 /95.79/
100.00 (95.01/95.11110 186.33
95.59, 99.21 /96.39
96.51 /95.06/
100.00 /9..00
100.00 990.20 84.30
99.92 99,00 92.75
Averag
98.95 08.55/922 /99999977/999
#7.07/ 19500/95.////00 87.04
16.78/ / 95 999
97. (9:99199700 197.05 05 A92.A8
47 /96.99 19.81
2M.42/96.5./00 (98.99
95.04 96, 944
(A.A0/ 99.32 /93.44
the iteration process and use the mask to sparsify the main information.
LSGGMS. The regularization loss Reg || 1I aiming at making the mask more sparse. The Litg is applied in the first half of the training process. In the second half of the training process, the mask is fixed, only the L is applied to continue training the reconstruction network.
Sparsification The obtained mask is applied to the test sam- ple x in the Fourier domain, resulting in a masked image signal. Since the masking in the Fourier domain targets global information, it is equivalent to weakening the main information while increasing the residual, effectively in- troducing a sparsification operation. Such that (K < R) ensures the validity of our test. Thus a binary mask M for suppressing main information is obtained through decentral- ization, which adjusts the common factors in the Fourier space. The real and imaginary parts of the Fourier space after suppression are given by:
5. Experiments
5.1. Experimental Settings
Datasets The experiments were performed on two widely used anomaly detection datasets: (1) MVTec-AD dataset (Bergmann et al., 2019a) serves as a benchmark for AD, which contains 10 object categorics and 5 texture categories, totaling 5,354 colour images. (2) VisA dataset (Zou et al., 2022) large industrial anomaly detection dataset that contains 9,621 normal and 1, 1,200 anomalous samples, cover- ing 12 objects in 3 domains: Complex structure, Multiple instances and Single instance. Division of training and test- ing sets about the datasets can be found in Appendix C.
R(x) R(x) O M,(())= O M,
here, ④ represents element-wiss multiplication. The masked image is then transformed back into the image domain by the inverse Fourier transform. For the residual estimation, this highlights the deviations introduced by anomalics, al- lowing the detection method to more effectively distinguish between normal and anomalous patterns.
Implementation Details The framework were implemented in PyTorch (Paszke et al., 2019) and trained on two datasets with the same settings. Resolution of all input images are set to 256 x 256. The reconstruction network is a vanilla U-Net. Training epochs are set to 800, with batchsize of 2. Adam optimizer are used with an initial learning rate of 10-4, the learning rate decays with factor of 0.2 at the 640 and 720 epoch. The regularization coefficient a is set to 10-6 The mask M binarized at epoch 400 and fixed after binarization. The binarization strategy is to count N epochs before the mask is fixed, when the elements in M,... exceed the threshold skw- where S/ww denotes the element in which illustrated in Equation 7, its count is incremented by 1. When the 400th epoch ends, if the count of the Mb element is greater than the preset sampling threshold multiplied by
Optimization In this section, we use the iterative optimiza- tion process of the loss function to approximate the bootstrap resampling process. This substitution is conceptually simi- lar as it transforms the simulation process into an iterative optimization problem. The training process is constrained by two main losses: the reconstruction loss L and the regularization loss Reg:
L LRec + aL
(9)
where LRec = L2 + LSSIM + LMSGMS includes the three terms, consisting of pixel-wise L2 loss, structural similar- ity loss LSSIM (Wang et al., 2004), and multiscale gradi- ent magnitude similarity (MSGMS) loss (Xue et al., 2013)
6

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Table 2. Anomaly detecction/ncalivattonn results with Img-AUROC/ PiX-UUOOCA PRO (in %) metric on VisA. The best and second best performance are highlighted in bold and underline, respectively.
Categety
PrachCrre
PyramidFlow
SSNF
DRAFM
IFiNNa
TransFusics
Oun-Riee
Ours
PCB1
27.70 99922 /95.08
$5.19 15.19/98002 98002 / 72.85
91.12/99.AU0 /M1/999..9
78.42/55.30// 25.64
95.29 /90.00//////0 778.3
99.96 99, 14/ 96.41
97 .7.06/// / 29945 86.89
97.46 99.42 /93004
Complec
PCR22
31.31/11 197.0000 11.80/991..0 62
94.21 (97442 89:03
92.58 /98.21 58430
R9.13 10.13/7704/2477
16.64/95,50
744.1
09.06 988.8 99.27
98. 08.00/ 98.19
77.25
09.22 (9..20 7333
PCRS
97. 14. /9545/ 199.21
81.33 98 91.90
58.24 198.15//500
80 28.53 (91.50
94.89 94.89/97888 /97.22 /01.51
08 88 197
02. 96.37
96. 30 98.77
98,09 999.1/ 24.34
PCB4
99.52/99 00.22/97.0///0 /87.19
96.22/ 94.63 81,44
94.52 /96.39 82.55
98,11 96,93 88144
95.80 (9.50/98.///// 991.98
99.75 /98.07 992.71
99.15 /9.0/4/299990 79.13
99.25 (94,59 500.7
Macammi
14.41/99.12//000
$2.50 98.72 993.0
95.10 /99.58///////// / 196112
88.52 02.52/99..00 99.7979977777
29.68 990.66 /22.98
99,89 (90 / 99.67
92. 02.46/999-0 096.//99922...3
96.94 /99778 99.52
Moliple
Macaroni
70.22/ 01/2/9977//990
73.01 y 19403 88572
63.24/97.05 191.00
77.03 99.69 (98.54
80.43 (99,9/ /98.78
98.43 (99 98.99
72.27 / /9337/9333311
90.08 /99.74/ 99.91
Capsiles
77.11 01111/99.0//77.0 /87.22
$7.99) 96.12 19:60
/2.41/999500 192994
73.81 99800 (92:25
90.65 90.65/199.4 /9944) 994.58
98.70 (99.77 999.37
95.23 (9.50)9999995
95.92 /99.87 99.19
Candle
98.69 /9877/877//// /99,42
$4.08 / 187 237.22/33 755.4
50.50 /9..0/96.011/ /910//9/1330 /91.3
91,04 01.04/95.22 /8.24/992///8.52
92.95 /97.17 /57.94
98. 20 (M.66)
1100//220 992.2
85.89 08.899/93.78.. 75.95
88.44 /95.54/ 59 20.20
Castew
97.56/ 99.11/99///9911
95.40/95993// / 189331
57.54/96.20 99082
90,20 11178/3/3/45114 (45.24
90.72 10072/99.10 (99.19/9/////// /9118
92.82/ 195.54 94.66
90.72/ 99500 88892
96.04 /98800 57.36
Single
Chewlaggum
99.22/95.02
/91.03
91.33/94991
82.003
/96.78 19013
88.23 09,23/97.04/70.3
98,96 198.53 /5:45
99.98 195.36 91,44
96.32 99.55/ 99.2//99..//99.22 79.52
97.25 /98.29 50.10
Fryum
94.09 /93.41 /88773
91.84/94012
/ 187.78
/92.429
78.61
96.20/9143 86.04
92.65 196.6) / 55405
98866 MS.M8 / $6.58
98.36 90.09955000 20 81.94
97.84 /94.71 5M.93
Pipe fryum
99.69 998.99 997,70
76.8% 96.66 19.12
95.13 /98.2/ 1222//922 92.70
94.72/ /9/1//9.1//00 190.32
97.28 99.999 29 /93.74
99.90 (95 99.62
98.84/ 99.5999.555 54) 98:25
98.68 /99.41 82
Average
93.39 99..9/ /92.90
$6.52/ 95:46 185.56
09.24/97.82//88 89.25
N0.13 90.13/92.57/4426
94.62 / 998.39 11999/9940 59901
98.69 /M8.///8.44./4 94.67
94,88 86.13
96.34 / /98.199991.9.55
straightforward as it allows us to translate the theoretical framework into an optimization problem suitable for train- ing machine learning models. We evaluate our method using three metries. Among them, the image-level area under the receiver operator curve (Img-AUROC) reports the perfor- mance of image-level anomaly detection. The pixel-level AUROC (Pix-AUROC) and per-region-oveellap (PRO) report the performance and the accuracy of pixel-level anomaly localization, respectively. NPS:Note that statistical signifi- cance testing of the reported metrics is necessary, since the main contribution of this paper lies in the statistical theory. Moreover, because the method must be validated on practi- cal detection problems, more comprehensive measures are required to assess overall performance. In particular, when verifying the asymptotic global theory (i.e., for the entire image), contributions from individual pixel p-values may be obscured (p-values are poorly adapted to imbalanced data, as they typically fail to account for anomaly sparsity). Therefore, we opt to use evaluation metrics commonly em- ployed in the anomaly detection literature as our assessment criteria.)
96
Peformanct
Img. AUROC on My Tec AD Pb--UUROC AD
Img. AUROC on VisA
AURDC on Visk PRO on VisA
24
PRO on MvTec AD
0.3
Figure 2. Average performance of anomaly detection and localiza- tion under different ulg.
the number of iterations during the statistical period, it is assigned to 1, otherwise 0, the preset sampling threshold is 0.5. All experiments were conducted on an NVIDIA RTX 3090 GPU, Details about the hyperparameter selection study can be found in Appendix D.
5.2. Benchmarking
Compared Methods We compare our method with feature- based methods PatchCore (Roth et al., 2022), Pyramid- Flow (Lei et al., 2023), SSNF (Chiu & Lai, 2023), and Reconstruction-baased methods DRAEM (Zavrtanik et al., 2021a), IFgNet (Chen et al., 2024), TransFusion (Fučka et al., 2024), in order to illustrate the effectiveness of DFS module, we added an extra setting for our method, that is, removing DFS from our framework (Noted as Ours-Base), only retaining the reconstruction network.
Table presents a comprehensive summary of the anomaly detection and anomaly localization results on the MVTec- AD dataset. The average scores across all categories demon- strate that our method is competitive to most of the com- pared featured-based based and reconstruction-based approaches. Table 2 provides the corresponding results the VisA dataset, which are also competitive.
Both the MVTec-AD and VisA datasets feature complex structures and a diverse set of anomalies, highlighting the robustness of our approach and validating the theoretical foundations underpinning our method. To further assess the efficacy of the proposed DFS module, we report the results of the baseline model (Ours-Base) in Table I and
Evaluation Metrics We use the area under the receiver op- erating characteristic (ROC) curve at the testing stage as a substitute for the power of the test statistic 0 to evalu- ate the proposed method. This substitution is conceptually
7

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Input
GT
PatchCore
PyramidFlow
SSNF
DRAEM
IFgNet
TransFusion
Ours-Base
Ours
My Tee-AA
EP
FP
VisA
a
Figure 3. Qualitative comparison for anomaly localization on two datasets. From left to right: the input images, ground-truth, and the anomaly maps produced by compared methods.
Table The improvements achieved by our method over Ours-Base are 3.65%. 1.73%, and 9.14% for MVTec-AD, and 1.46%, 0.56%, and 5.02% for VisA, across the Img- AUROC, PiX-A and PRO metrics, respectively. The PRO metric, being particularly sensitive to subtle anomalies, further substantiates the effectiveness of the asymptotic local power in the asymptotic theory we proposed.
Input
GT
X
X
Pred
To further validate our proposed asymptotic global power, we compared our method against baseline approaches on varying image sizes using the MVTec-AD dataset. The sults in Table 3 demonstrate that, compared to the baselines, our method maintains high efficacy and exhibits greater stability as image size increases, thereby confirming the validity of the proposed asymptotic global theory.
Figure 4. Visualization results of intermediate step in our method.
Table 4 only contains the computational cost of the layer used for computing features in Patchcore, the core subset of PatchCore and kNN search not reported.)
In general, (eature-based methods are typically fine-tuned using pretrained models, leading relatively stable per- formance. In contrast, reconstruction-ben based methods often require data augmentation during the training process, and the external images introduced for augmentation tend to lack sufficient diversity, resulting in inconsistent performance across different datasets. Guided by the theory we have proposed, our method constructs residual using the model itself detect anomalies without the need for additional data, enabling precise anomaly detection. Furthermore, shown in Table 4, our approach significantly reduces both model parameters and floating point operations compared to (eeehods.((S..)))) NPS:Note that the back- bone PatchCore WideResNet50 (Zagoruyko, 2016), the coreset subsampling phase of PatchCore uses 1% feature vectors which are then indexed and stored in GPU memory enable fast search for nearest neighbors during infer- ence. Therefore, the mechanism of PatchCore makes its Params and FLOPs varies with the amount of data, therefore
5.3. Hyperparameter Selection Study
Section 4 mentions that our resampling method is inspired by the Bootstrap method nonparametric statistics, which does not assume specific distribution for the signal. To evaluate the influence of the sampling function, conduct experiments with different sampling functions, different means of mask sampling threshold as well as the number of sampling epochs N to assess their impact on performance. The detailed results are provided in Appendix D.2. The results of sampling functions show that resampling using uniform and normal distributions yields nearly identical per- formance. The results of the mean mask sampling threshold (Ht)) are shown in Figure 2. As μıg changes from 0.1 to 0.9, the three metrics remain nearly identical trends across both datasets. This indicates that the DFS module is largely un- affected by the expected value of the sampling distribution. Additionally, results for varying the number of sampling epochs N indicate that performance remains stable N in-
other reconstruction-basee methods,
8

---

Some qualitative comparison results are presented in Figure Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Table Anomaly detecction/locllizttin results Img-AUROC/ Pix-AUROC/ PRO (in %) on MvTec-AD different resolutions.
Resolution
256
512x512
x024 1024
Category
Ours-Base
Ours
Ours-Base
Ours
Ours- Base
Ours
carpet
97.27 /99.35 29,,25
99.00 99.45 99.77
77.89/ / 86.57 67.97
98.27 / 998 /6 /96.73
62.08 / 76.48 46.55
94.78 / 95.02 190.1
grid
99.67 / /99.08 / /93.12
100.00 / 98.90 94,91
97.33 / 97.49/ 79.79
99.67 999.29 /66.86
96.74/ 99.22 61.34
100.00 99.40 99.699
leather
100.00 99.29 19668
100.00 / 99.68/ /99.08
100.00 / 999445 / /93822
100.00/ 99.68 999.35
98.06 / 99.60/ /8.556
99.18 97.31 11/ 19.16
tile
99.35 / 997.77 / /90779
100.00 / 98.69 195.13
96.83 / 99.67 886.95
100,00/ / 98.58/ / 19.116
93.65 / 88.88 77.62
98,70/ 96,57 92.84
wood
100.00/ 98.18/ / 85.18
99.82/ 97.52 91.25
63.86/ 84.21 66.599
99.91 / 93.93 /88.99
98.16. 88,26/ 69.69
97,54/ / 90,31 84.33
pill
91.84 /91.63 / 79.71
99.59 / 98.79 99.664 7..4
82.38 / 74.72/ 558.93
99.78 / 98.55 / /96999
77.93 / 58.43/ 40.51
96,15 / 94,91 / 77.84
transistor
96.12 / 93.45 / 772.4
99.42 95.32 / 84,98
87.46/ / 83.69 / 14.611
97.96 / /87.09 / 76.114
60,38 / /2.66/ 29.21
95,38 / 75.71 / '74.58
cable
90.50 / /9388 / 772.03
99.03 99.99/9/99 10
71.44/ / 85.75 /50.73
97.34 / 95 /2/ 88,47
52,44 / 80,73 /44.61
94.04 87.00 / 76.11
zipper
95.75 /99.32 99.97
98.35 99.25/ 19.955 .6
97.14 / 99,09 81.13
95:46 / 98.58 95.99
97.30 / 99.80 87.50
94.38 / 99725 93.71
toothbrush
100.00 19..22 75.13
99.17 99.05 /93.62
92.78 89,78 60.22
100.00 / 99.70/ / 94,06
81,94/ 82.32 /3.00
98.89 99.11 86.00
metal but
92.42/ 99.29 77.29
97.41 96.87 /84.97
90.47 / 99.62 63.76
97.07 /9..6/ 88.44
88.51 / 92.53 52.80
98.68 85.01 / 79.100
hazelnut
95.75 / /94.73 79.44
98.57 19. 19/10/ 93.95
80.00 / 69.99/ 53.06
99.07 19.112 / 196.39
68.25 / 60.73 38.33
98.79 / 99.91 / 195.45
screw
69.62 997.63 / /90.44
96.02 99.20 96.69
51,98 / 96.26 187.43
96.74 97.82/ 993.77
63.05 / /95.91 85.01
96.18/ /9.32/ 89.82
capsule
97.25 / /97.7// 87.67
94, 10 95.25 888888 18
87.16/ 92.80 774.78
97.85 / 88, 38.76/ /8871
87,91 /91,44/ 77.31
96.22. / 67.43/ 77.997
bottle
100.00/ 98.30/ 84.29
99.92 98.05 92.75
100.00 / 97.44/ / 67.35
98.73 / 95.98 /88.37
98.41 / 99.55 / /42.78
98.89/ / /9.06/ 85.55
average
95.04 /96.9/ 14.10
84.10
98.69 99.32 93.24
85.11 / 89.97 69.24
98.52 / 96.13 /91.69
81.65 / 82.83 58.79
97.19/ 99149/ 86.82
creases from 5 to 35. To optimize computational efficiency, we select μ 0.5 and = 5 for our experiments.
and provide a theoretical demonstration of its effectiveness. The asymptotic theory elucidates why this test can detect anomalies both the image and pixel levels within the theoretical lower bound. Then we diseuss the relationship between the test and practical problems, we propose DFS as an optimized reconstruction anomaly detection module and develop the DFS-AD framework. Furthermore, we in- troduce a Bootstrap- resampling method to optimize our framework. We quantitatively and qualitatively demon- strate the effectiveness of our method in reconstruction- based anomaly detection tasks.
5.4. Qualitative Results
3. The proposed DFS-AD demonstrates visually accurate localization performance. Additionally, the intermediate steps in our method are visualized in Figure 4. The anoma- lous image X is initially masked in the Fourier domain using DFS module. The resulting image X is treated as the nonanomalous factor and subsequently reconstructed by the reconstruction network to generate the correspond- ing nonanomalous sample X Ree- This approach effectively removes the abnormal residual, enabling precise anomaly detection and localization. Further qualitative results are provided in Appendix E.
Acknovledgements
The authors are grateful to the Area Chairs and the anony- mous reviewers for their constructive comments. This work was supported in part by the National Natural Science Foun- dation of China under Grant 82172073, Grant U19B2031, Grant 61971369, Grant 52105126 Grant 82272071, and Grant 62271430; and in part by the Dreams Foundation of Jianghuai Advance Technology Center; and in part by the Open Fund of the National Key Laboratory of Infrared Detection Technologies.
Table 4. Computational cost evaluation of comparison methods. Number of model parameters (Params) and doating-point opera- tions (FLOPs) are reported.
PreeCCum
PynmmiFFlw
SSNE
DRAFM
IFgNet
TranFussum
Oun
Panau (M)
2486
22.64
204.67
97.42
69.58
54.53
28.37
FLOPs (G)
9.24
479.20
243.77
198.66
135.23
148.38
38,20
Impact Statement
6. Conclusion
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
In this work, we propose novel method for detecting im- age anomalies using demeaned Fourier transform (DFT)

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
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Stock, J. H. and Watson, M. Forecasting in dynamic factor models subject to structural instability. The Methodology and Practice of Econometries. A Festschrift in Honour of
David F. Hendry, 173:205, 2009.
Toyoda, T. Use of the chow test under heteroscedasticity. Econometrica: Journal of the Econometric Society, pp. 601-608, 1974.

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
A. Related Work
Unsupervised Anomaly Detection Unsupervised Anomaly Detection (AD) can be categorized into reconstruction--assl AD methods, flow-based AD methods, and feature-based AD methods. Reconstruction-batt by comparing the differences between reconstructed images and the original images. To enhance the reconstruction accuracy of the network, Bergmann et al. (2019b) utilized the L2 norm and the Structure Similarity Index Measure (SSIM) as the loss function. Furthermore, AnoGAN (Schlegl et al., 2017) constructs latent space representing the distribution of normal images and employs an optimization- based approach to search for the closest embedding of the input image within this space, then the embedding is decoded to obtain the reconstructed image. In addition, DRAEM (Zavrtanik et al., 2021a) enhances the network's 's reconstruction capability through the use of synthetic anomalies. Moreover, this method trains a discriminator sub-network to detect anomalies. Considering the potential presence of background noise in real-world scenarios, IFgNet (Chen et al., 2024) adopts a multi-task framework that enables the network to simultancously detect anomalies and extract foreground objects, which can eliminate interference effectively. Recently, to address the issues of overgeneralization and loss of detail commonly seen in reconstruction models, ? proposed method based on a transparency-ban based diffusion process that improves the detection accuracy significantly. Flow-based methods utilize pre-trained models to extract features from nonanomalous samples and model their probability distributions, attributing anomalies. A notable example is PyramidFlow (Lei et al., 2023), which employs
AD methods identify anomalies
to the detection of out-of--istrrbbun pyramid- traee normalizing flows and volume normalization to facilitate high-resolution anomaly localization. Similarly, Chiu & Lai (2023) introduced a self-supprvised normalizing flowdbased model that integrates synthetic anomalies into the flow-based framework. This approach conditionally optimizes the model by maximizing the likelihood of nonanomalous features while minimizing the likelihood of synthetic anomaly features, allowing the model to capture the distribution of nonanomalous features more accurately. Feature-based anomaly detection methods detect anomalies by comparing the differences in features between normal and abnormal images. PatchCore (Roth et al., 2022) extracts a core subset of patch-level features from training images using a pre-trained network. Anomaly scores are calculated by measuring the distance between the features of the test images and the core subset. OCGAN (Perera et al., 2019) proposes to add Gaussian noise to the input, while ARNet (Ye et al., 2020) uses some transformations to erase some important attributes in the images. However, they do not provide interpretabilityy or theoretical underpinning for the method.
Factor Model Factor models are commonly used for analyzing structured data (Bai & Perron, 1998), such as in finance, economics (Forni et al., 2000), and weather forecasting. Most studies assume that factor loadings are time- inme-invrriant (Bai, 2003), capturing the relationship between observed data and unobserved common factors. In recent years, research in such field has increasingly focused on testing and estimating factor models with structural changes (Stock & Watson, 2009; Breitung& & Eickmeier, 2011; Cheng et al., 2016). However, such tests and estimations often specify on a certain type of structural change, like abrupt structural changes or smooth structural changes. Fu et al. (2023) proposed tests for structural changes in time-varying factor loadings scenarios, considering both abrupt and smooth structural changes. In the context of common image anomaly detection tasks, the data can be considered structured (Bergmann et al., 2021), as these tasks usually assume that anomalies occur according to the practical scenario. Compared to economic data, this is reflected on dimensionality, structural changes in panel data can be mapped to anomalies in images. For example, a structural change corresponds to anomaly in certain pixel positions within the image, while in panel data it reflects the break in both cross-sectional level and time-series level.
Discrete Fourier Transform The Discrete Fourier Transform (DFT) is commonly used in image processing, with the goal of transforming image information into the spectral domain for further analysis (Rao et al., 2011). The coefficients in the frequency domain contain global information about the image (Lin et al., 2019). Signal decomposition methods can be employed to decouple the image's features in the frequency domain. Li et al. (2023) proposed an embedded Fourier enhancement method to improve low-light images, aiming to enhance illumination information at the feature level using Fourier methods. Abdulaal et al. (2021) introduced an asynchronous multivariate time series anomaly detection and localization method for asynchronous multivariate time series data. Based on the theory of factor models, we reconsider the theoretical construction of reconstruction based anomaly detection by projecting the image features into the Fourier domain, representing the image information as a combination of signal and noise. In this way, we propose simple and effective mask-based anomaly detection method.
12

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
B. Proofs
B.1. Proof Proposition 3.1
Let (()) be W W diagonal matrix, where the m-th diagonal element in P(u) is sp(),, with P(m) (2ml wp))
πI Em= ( w/ representing demeaned Fourier process. Let ((v) be an H x H diagonal matrix, where the n-th diagonal element in ((v) is (p(),, with (P(n) 22m t+) W Σn c (w) representing demeaned Fourier process. 1. an x vector with each entry being H- The matrix An contains the set of non-anomalous factor loadings. Buw is matrix containing the K largest eigenvalues of the matrix H - and A = H- Ao) B) is rotation matrix.
Based on the definition C(u, v),, under the null hypothesis Ho,
TTD(u)
FFT W
Ō(u, v)
1
W
D(UFF
FT
F W
FT)(U) W
(T FT W
Ag (
<V(v)
W
= Ca, v) + C,(
Similar to Sec.2, we decompose the complex-valued empirical process at the matrix level into the form of "signal" and "noise". Under the null hypothesis Hlo, (2) represents a non- con-anomalous factor model. Based on this, we consider C₂ (u, v) and proceed to decompose follows:
()(u,))= (T)(
(T FTF A AT)(),
W
W
W
it represents the error between the estimated common factor E and the true factors F. The difference between the common factor and the residuals is represented as F FA-! This operation is a standard factor model error decomposition, we have
F ((u) FA-1)
(T)( FT (F - FFA
Ca(u,))
Ag (()a
W
W
W
FT (F (F-FA) (T - FA)
A A-1AT
W
W
1Čaa(u, v) - (u, w)]A-1 Ag (v))a,
the asymptotically dominant term of Ca v) is Cas(u, v). Next, we focus on Cas(u, v). Under the null hypothesis Ho- we have the expression: F- FA loadings and the true factors, and is modeled through the residuals and the matrix Ag, then have
ree F. FA F HW + HW + EA,FT F FA
Fl represents the error term between the factor
HW
P f FA)
(Co(lu,)A--a0 v(v)a
A 4-14/000 (TT(())
W
(()) (E + FAT)E
FT EA F
B (1)a
W
HW
W HW
-Cabba (u, v) + Cass(u, +) A-1 1 #(v)a.
By comparing the orders of the decomposition terms, we identify the asymptotically dominant term: AJ ((v)a according to the definition of A. Now, we proceed to
Casolu, BB A. (v)a== consider (s(u,v).
(, & TTaleAA HW
H
(T FT (v)) C,(u,v) FT (TD(U)O
W
W
= Coa(u,, Oso(u, v).
13

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
obtain the asymptotic dominant term Cba (u, v). By the way, we combine Cass (u, u) B) A Φ(v)a and Cua(u, and have
Cba(u, v)
( B
u))
1
Σ
Σ
Fu ke By(u,))
HW
H1
W
H W
E
- A
A Fw Bu(u,t) +
Σ Σ
A Fw Bu( u),
HW
HW
1= m=
where B limm, B, the asymptotic leading term of Caa (u, o) A Φ(v)a is given by: HW Σm=1E A Fwe Belu, v). We can therefore conclude that
H Σ A
HWWC(u,)
A
Fu B)B op(().
sup M
HWW
represents the difference between WWW Clu, v) and the second term, which converges at a rate op(1) in the large sample limit, consistent with the weak convergence theory of the complex-valued empirical process. Let Ao = plpppllip....p H.W os A, and define C( A VHW Ei - F.. BeBpppu r)} it is straightforward to show that VHW Σ Ew=n Be o) is asymptotically tight in (()), where (()) denotes the space of complex-valued continuous functions on 5. For any u1, U2 E 6, we apply the Lagrange mean value theorem to obtain:
H
W
H1 W
@pp(u,)) v)
(E, () - -2)) + @up(u, v) (E, ()(o)) - 02)
Fa Be [f(u), v1) - c0u2, 02)] = ΣV
ΣΣN
Beno
Ou
Ov.
1=11 mm
b= mmml
where (E, 1) is a point on the intermediate path of (u1, 11) and (u2, 12), then
W
E
F (un 01)
Fu Benw 02)
HWW
VHW
H
W
(Dolu..) du
(E, (E:n)(u - U2) + Oz((,, v)
Σ
E
(E, - 12)
HW
dv
By the assumption that the residual terms form Martingale Difference Sequence and applying the Cauchy-Schwarz inequality, derive the following:
H
W
Σ
E [I Berw < max (E1(F=11)) max (E (E (Elleavll'') max [B]]2
HW
A=
where the order of last term is O(1). Thus,
W Σ
2
F
VHW Σ
Fu ( HW
Σ
F. Behnn 02)
S [C[((u,, (uz, u2)11
therefore, /HW D' W Fú, (u,, v) is stochastically equicontinuous (Newey, 1991). It ensures the uniform behavior of the complex-valuee empirical process over the space 6, as the sample sizes H and W increase. the process does not exhibit large fluctuations at different points, thus providing essential control conditions for proving the weak convergence and tightness of the process. Next, let E(u,
FFFFEeud ((u, v), where d is arbitrary nonrandom vector VHW IH1 Ew,, F.. Be u) VHW ΣH Σw E(u,v). We assume that Ele(u, w]] = 0. For some r 2 2, applying the Cauchy-Schware Schwarz inequality we also have the following bound:
with ||d]| = 1. Then, we have: d
Elie(u, r)] < Be p7] Fild Fwip(l, o)) < co.
14

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
This
allows us handle the higher-order moments and ensure their existence. H Var VHW h==1 Ew= E(u, u)
Let H(u,v)
0 for each fixed
Then, according to Central Limit Theorem, we have:
W ((u,v) S N(O, 1).
VHW v /H(u, v)
h=1 mm
Based the Cramer-Won Wold device,
1 H W
G(u, u) VHW
Ag F Ben v) S N(., 5.),
w=1w= w=1
for each fixed u, u, where > limæ,w W- +00 Var (VHW Σ 11
W Ewe1
Ag F BEhw) [(1- ei2Tur dr) (1 e cl dr)
where 7 follows the uniform distribution U10, 1). Given that is totally bounded, we have: G(u, v) = G(u.v) g. where G(u, 8) is a joint mean-zero, zero, complex-valued Gaussian process. Therefore, we conclude that: vHWC(u, = (u,,v) in as (H,W) - under HHg.
B.2. Proof of Theorem 3.2
Given Proposition 3.1 under Ho
δ b= /1 G(, w)| W(u, v)dudo + op(1);
where (1) denotes small term that converges to zero in probability. To establish weak convergence, we need to prove that ò converges in distribution D as (H, W) oc, there exists compact subset å E R large enough such that [s [[G(u, w)ll W (w,, v)dudv < T- for any positive value e, 8 denotes the complement of and we have:
= // G(u, w)| G(u,,)]]
W(u, v) du du S1. [(u,,)) v)) w)| W(u, v) du du,
and similarly for D:
D /1. IG Wl, du I1. [G(u,,)) W(u, v) du de.
Thus, we need to show: Do ISo ((u, wilf W(u, v) du du converges in distribution to Dg
ISs [[((u, )I Ww, u) du de; and Dø- = SSs- (o(u..>)i W(u, v) du du converges in distribution to Dje
JS [[G(u, u) W(u, v) du do.
We start by analyzing the part over 5:
Ds = 1/ W(u, v) du dv.
From Proposition 3.1, we know that G(u, v) converges to G(u, v) in distribution as (H, W) + oo, i.c G(u,o) $ Glu, v). To show that Ds converges to Ds, we need to ensure that the integrals of G(u, v) and G(u, v) converge, and that the integrands remain uniformly integrable. For any fixed u, v, we assume G(u, v) and G(u, v) are integrable. By using Jensen's inequality and Fubini's theorem, we can bound the expectation For some 2:
W
E G(u, w)ll" < Alolll" E
F Bebw
(sup max (p(u, u))² < C. u,veō
HW
15

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
This inequality ensures that G(u,, u) is uniformly integrable over d. With the dominated convergence theorem, we can then show that:
/f,1/ "/ ((u,e) du du S S/. [IG(u, v)ll" (u, v) du dv.
By the dominated convergence theorem, can establish that: Do S Do. Next, consider the part over 8%
D&= 1/ [((,, w) W(u, v) du.
We have that: J aau w|l w(,, u) du dd we know: F [So. C(u, o|| W(u, u) du du)
E (JIs= [IG(u, W(u, v) dudu). By uniform integrability, II E [ b)]l W(u, v) du dv. Therefore, Dge is also uniformly integrable, and by applying the dominated convergence theorem again, we can conclude that: Dz S Dzc. By combining the results from Dj and Dæe, we get:
D Ds Doo 4 Ds + D3- D. Thus, Đ converges in distribution to D, ic. S D.
B.3. Proof of Proposition 3.3
Consider the singular value decomposition (SVD) of A: A USV where U and V are unitary matrices containing the left- and right-singular vectors of A, respectively. S is an H x W diagonal matrix with singular values sgw.1,... SHW,L along the diagonal, arranged in descending order, where I, = ran Here, we assume that the first R singular values exert the most significant influence on the features, thereby preserving the maximum amount of information.
L
8 SHw,R UUVRRURVLL +
Σ R+11
SHW,U.VV ( + -
Then we have
X FA FF((-)) +
This implies that the factor loading matrix effectively represented in a R-dimensional space, where the asymptotic leading terms of A can be captured by the first R singular values. Given H-'W-'(A AF (T under HA,
X FAT + E. We can decompose Õ(u, v.) like what we do in Proposition 3.1:
(T)( W
FFT
C(u,v)=
X
W
FT (())
PBT
FT W
FA
(0)a +
I-
W
W
W
= Calu, v) + v).
Similarly, when K we obtain Cu,u u) the asymptotic leading term, we can decompose Č₂ (u, v) as:
vu(u,v) =Caa (FA) ((u)
FA(FA] T
FA FAT4(u) Φ(v)a
W
-aaa v) + Cas(u,
By comparing the order of each decomposed term, we found that Cas(u,, v) serves the asymptotic leading term. As
have mentioned A A (R) + A (-R), we have
A (0))
AT
A(-R) (- (-RO
A
B +
B
HW
HW
A + A6-
16

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Then
(FA) T ((U)
FA(FA)
Cas(u, =
FA Φ(U)a
W
W
(FA (TTA)T((C ((n)
T
FA, (FA))
FA D(U)a
W
W
(FAs) ( (())
[(FA.) + (FAb)
FA. FAI
FA D(v)a
W
W
(FAc) T (T)())
(FAA)) ( FA (FAb)
FA, + (PA))FAA
FA FATT(c)a
W
CCaba (u, v) (u.o( Cu.
By comparing the order of each decomposed term, the Casa (u, 2) is the asymptotic leading term of Cab(u, v). The same operation can be applied:
F FA
F (A)FFA Φ(u)a
Cabo =B
HW2
FT FR() A F (R) A F BEW
(R)A FAT
BBw
H3W3
HW2
By spilting A AR)) (--R), we have Cata Cutaau, (u,v) v) Cabon v) Canablu, (u, v). Obviously Caban (u, v) is the asymptotic
leading term of Caba (u, v).
i (F)) (R)
FT (T 1
Cabaa(U,1)
(u, v) BB
HW2
F (V (I)
F ((( FTF
H2W3
F FV
F FV (
x B
HW
HW2
Catma (u, v) x (B
(())(())) F FV(R)))))) ((()))((((TTT + o1).
HW
Each decomposition term in matrix Cabaa (u, v) has nonzero order. thus Eabaa (u, v) + C(u, v) is non-zero spectrum.
B.4. Proof of Theorem 3.4
When K R. u(u,u) is nonzero spectrum. In the majority of cases, C(u,v) is non-zero and meaningful, thus ensuring that C(u, u) сиW holds true in the limit. As (H, W) co, D HW JJ& [IC(u, w)ll3 (u, v) du dv. and
CHW - 0 Therefore, for the majority of samples, D will inevitably be greater than a small cHw This leads to the probability P(D > CHw)
B.5. Proof Proposition 3.5
Let gf is a set of Sows Then under (A(THW),et =E THW Ihw, and X
FAJ +
et, then
Ö(u,v)= = FT
FFT 1-
F TT
FFT
F FAJ TAja
1-
e'o(v)a
W
W
W
W
= v', v)(
17

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
We follow the procedure outlined in Proposition 3.1 and decompose Co (u, v).
(()) (F- FA-1)
(T F FT (F- FA-1)
o)),,))
Ag (v)a
W
W
W
F FA)
(T FA)
A ATA @(v)a
W
W
= =ICaluu, v) Caslu, w) ((v)a.
By comparing the order of each term, we find that Cao(u, 0) is the asymptotic leading term of C² (u, v).
ele' F
FA el F
A F
F. -- F-FA=
Biw
HW
HW
HW
Thus,
FT (T)(( FA), A-1A0 Φ(v)a
(())
W
(T)( et (TT + FFA))
+ FT(() et AO F
B A-1A-11- ((v)a
W
HW
W
HW
Caaa (una(u, (u, + Casu(u, BHW (u))a
By the same way, the Caiss (u, v) is the asymptotic leading term of Co (u, v). We have
Casalu, v) B a(v)a (f AF) T (u)) Ao
Ag Ag
A (()) Ao
Ag Ao
11 ((v)a+ +
-1 AA (
HW
H
HW
H
Cd abba (u, v) Õ" (u, v),
and (asso(u,)) ) is the asymptotic leading term. Next, consider C%(u,)). v).
FFT
I
E'4(U)a
W
((T)E)) ((U)a
FT FTelo(U)a
W = Üba v) Cu.lu,
W
W
By comparing the order, obtain E% (u. u) is the asymptotic leading term of Cj, tu, v),, then decompose C% (u, v).
Öha (u, v) (F AF) ( (())
ATF F 4 (0 (
W
W
Cbaa (u, v) + Chas (u, v),
and the asymptotic leading term is C% (u,,) v), then we combine the asymptotic terms Chas v) and - (u, v) and obtain
Clat(u,, u) - Caoss (u, v) A FT (u))et(()))))
(T)T An
AJ Ao
A AJ D(u)a
W
HW
H
W Σ
A Fa Bylu,v) Σ
THW HW
A F., F Dew B)(uu,)).
HW
We have
A W
H W
sup 6,
VHWČ(u, VHW
A] F Bu(u,
Σ
Σ
Ag F Sh Bill, v)
= =0p(1).
HW
hm w=1
Am w=1
18

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Let G(u,, v) v E E Fo (u((, (e v) and δ(u, v) #W Σx= E Ag F FT 9ow Biplu, v). As have mentioned in Proposition 3.1 that G(u, v) G(u, v),, Such that
H
v(u,v)=
Σ
Ag Fw F.. 9hw Byp(u,))
HW
1=1
H
W
H w
Ag (F
4F Bp(l,,
Σ
Ag Bp(u, v).
HW
HW
h=1w=1
h= w=1
We
have
H- Eh=1 Ew= 9hve
straightforward
to
show
that
E
H Ew= Ag FF
2r) Bip(u, 0. and E
1 Hw
Σ h== E Ag (F w/ g Bep(u, w)|
HW
0 OH OH-IW-I). Thus,
W (F Dym -0,(HH-1/2--122)
1
ΣE
sup
M
HW
follows that (u, v) + o(u, v). where v) denotes pseudo-covariance between the Fourier series 2ms 4 it and B. B. So we have vHWC(u,e) = o(u, + G(u,e) v) under HA (THW)) with Tow H 11////-11 1/2
B.6. Proof of Theorem 3.6
Under (THW))::HH Ahu No + THW By Proposition 3.5, we have
VHWCCu, + o(u, v) + G(u,
By the continuous mapping theorem, we have
SS. [ " v) du de 4 Sf. (i((,,,) + G(u, w)ll v) du dv.
By similar arguments in the Proof of Theorem 3.2, it follows
D 4 // (l + G(u, ol, (, du
C. The Division of Datasets
MVTec-AD Dataset: MVTec-AD dataset (Bergmann et al., 2019a) is collected for real-world industrial production scenarios unsupervised anomaly detection. There are 10 object types: Bottle, Cable, Capsule, Hazelnut, Metal Nut, Pill, Screw, Toothbrush, Transistor, Zipper. And 5 texture types: Carpet, Grid, Leather, Tile, Wood. The dataset has totaling 5,354 images with pixel-level annotations. Among them, the training set contains 3,629 anomaly- images, and the testing set contains 1,725 images with both normal and anomaly samples.
VisA Dataset: VisA dataset (Zou et al., 2022) is a large dataset collected for industrial anomaly detection, covering 12 objects domains: Complex structure, Multiple instances and Single instance. Among them, Complex structure domain has objects: PCB1, PCB2, PCB3, PCB4. Multiple instances domain has 4 objects: Macaroni 1, Macaroni 2, Capsules, Candles. Single instance domain has 4 objects: Cashew, Chewing gum, Fryum, Pipe fryum. The dataset has 10,821 images with pixel-level annotations. Among them, the training set contains 8,721 anomaly-free images, and the testing set contains 2,100 images with both normal and anomaly samples.
D. Hyperparameter Selection Study Results
D.I. Hyperparameter Selection Settings
Equation 7 defined the decentralization strategy. This section reports the results of different sampling functions mask sampling threshold s, different mean of mask sampling threshold s, as well as the number of sampling epochs N. We first
19

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
conduct experiments with standard uniform (U(0, 1)) and standard normal distribution (N( 1)) sampling functions with the same setting of pl, on MvTec-AD dataset, where (4s changes from 0.1 to 0.9 increasing by 0.1 a time, and the number of sampling epochs N are both set to 5 by default. Then we further conduct experiments for different M4 on VisA dataset, as well as different N on both MvTec-AD and VisA dataset where sampling function is set to uniform distribution by default. When exploring one of the two hyperparameters, the other is set to the default, the settings are as follows: the A4s changes from 0.1 to 0.9 increasing by 0.1 at a time, and the number of sampling epochs N changes from 5 to 35, increasing by 5 at a time, the default 24 is 0.5, and the default N is 5. Meanwhile, we conducted parameter selection experiments for different regularization coefficients a
D.2. Detailed Results
Table 5, Detailed anomaly detection/localizztion results with Img-AUROC/ / PiX-AUROC/ / PRO (in %) metric of different μa, on MvTec-AD under sampling function with Uniform Distribution.
Camprey
GLI
102
13.5
0L7
carpet
91099/ (9.59 197.49 26
99.16
(99.51 99119
99.43 197.61
95.11 19.5 99
90.00
/9945299552 96077
99.00 99.50 197.45
V5.01/ 19.43 (97 335
99.35 FSRCEO 90L00
95.11 /9.11/99917/ 195.75
100.0/95.07// 95.60
100,00 M2 195.67
100000: 59.07) 1N
100,00 (9.15 96624
100,00
21 22
95.91 (9.55 99000
Lexher
100/0/999557/
79
M0:05 (8.51 197 78
103/00/ 99 55) 998 TM
100000 9990099/966 99 15
100.00/
999 68 199.0
100/00/ 59 07 M9 11
100 000
95.67 99.11
100 00 199 68 199.77 17
100/00 999 09 190 24
1ile
100/0//99.39/
1000000 68 195.15
100000/ 198 94.44
100000 998.2/ 99.35
1000000
195.13
100000 198
100.000 (99.99 94.86
100.000 96. 195.78
99.82/ 98, /94.44
100.00/
99172
39
91.71
91.57
59.56 /92.35
PIB
99.15/99.911 /96.98 56
99.99
18.91/99900/ 00000/997.5 997.4
99.51/96.91
99.59 19..9/198.991999 97.64
00 00.22/9673/99333
98.47/ 18.50 51
99.61 (10.0/99..444 197
54.84/ 19.20 /97.11
98/6//94
43
99.25 (9129/91773/ 15
27/5575555 $4.55
99/67/
41
(55.3355333333 54.
58.92/96.100 86-33
95.35/
20..066..55 (95 79
(55/67 20
07/ (95077 M5:63
Canle
45.80/96 96. 1144/881222 23
91.90 07.01/97667/ 88.65
96 18.16/1983111 190.36
45.52/ 96 501 (91
94101 /98.6/996211111 91 10
900 54 142.00
09.80/
00.60 985 (92.99
Zipper
98.95 19.28 196.02 013
19.50:150 0000/99920 /99:29/ 50.30
08.99/19335/95
/99..5
95.35
08:11/99022
197.0
95.35 10, 14 14.197..3 23
95.19 /93.57 196.1
50.17/ 08.33 / 795.89
Tootbooh
100.0//98.23/ 59.54
99.44/99.51/99 91.85
100.00 (95.0 94.45
(99/17//9/05/ 93.02
100:00/59.33
(0500
00/ 99.21) 194.6
100.00 23 594.02
100:00 596 34/ 194
Maail Not
97,12/ /9/ 99. /94.99
97.25 196.57 85.07
07.50//66/2/88 53.73
97.12/
31
(7.41/95.87 $4.97
97.56/997.00 86.12
96.63/ 12.12 10
96.33 195822 $5.50
54.62 98.50 $4.79
Haodho:
99,93 (98 99 09
05.30 002 /99/07 99.71
59 59 299/9/999999999
96.93/ 05 199774
05.57//99.0// 95.95
98.82/99300 /9377 33.73
98.93 99 11 11/ 79
99. 1114//555 59.009 22.22
71 199.9992 99.64
19436
99-74
91.67/
47
Capsulc
96.17 01.77/1..00 99.14 14444/11.00 96
99.23 195.79 56.96
94.59 96.00/ $5.00
94.15/ 18.12/ 20.04
04.10/95:25 /9525 18.18
94.20
99:53 $8.24
93.82/ 99 28 (00 A00
94.38 /91.63 90L78
95.97 90.77 99.73
Buol:
99.54 03 192.7
09.00 /97.87/ 92:11
99.68 97.8779777770 / 99.15
99.66 97, 69 191
99.92 /58055 92. 75
99.92/99
997 07 99277
100.00 00/ 997 97.88/ / 92.40
100.00 997 X2| /92.20
100x00: 97. 78 91 1.00
Avssaee
97.91 /97 097.666690 (1.66/99.07 67
07:30/ /11::/5555//00 92.35
19.15/99819 99.883
96.15/ 19
95 69 /5..3/ 99.24
18.50/100
98
59 29//3/244
55 13 93.00
Table 6. Detailed anomaly
results with Img-AUROC/ / PiX-AUROC / PRO (in %) metric of different Jtc on MvTec-AD under sampling function with Normal Distribution.
Cangrey 0.1
0.2
0.3
0.I.
0.3
D.G
0.7
0.8
D.9.
carpei
09.53 197.13
/99:44 96.92
95:47:
99.19 195.63
98.45/
99L43/ 99699
19944/999499/
S6L5O
56.70/ (999999943
99.00
94.82/
99662
89.04
/98.5/ 94.41
72.79/ 05.55/
Gnd
192/0/99904/ (95.56
99.522 /99L11 19.2. 19.19
99.55 99.14 VeUAs
192.00/99800 95.98
100L000
19316
99.30 99.35 96.75
99.06/ V1. (57) 113
95.23 29833/115722 15.
98.58 92165 /90.54
Lexha
100,00 /99.64/ 98.94
100,00 99 /99:20
190668 99.22
100000 /99.68 $9.19
100,00 99.68 199.18
100/00 / 59.68 190 20
100,00 99.65 24.24
100.00 99.66 /99:22
59.95 19.52 99900
100.00
100/00
191.77
Wood
/92/0/99.355 91.15
1000000 97. (91.3
99.56 96.15 99014
99:14) 96 76/ 99
09.50/96.67991 91.10
102000/ 96.54 91. 26
100000
95. 70/ 91.45
1000000
96 21 /91.42
96.25/ 98.594455 /90000
95.36/ 08 99.99
0w..2/99676//// 96-49
19.0 19125
99.65/ 96. 75 99
19.15/100 99.18/98.68. 92397
92.98 90..6 /9335
09.09 51 30
92.20
99.36
97.49 98 191.11
Taassoo
19.36/15.59// 84.96
09.33/9531
/9.17/195.28/ 65.16
99.12/9537/ $5.80
19656/960221 85.74
19.62/95555 $6.35
Calle
98.65 (98 54 /06.191111 17
94 16.24/95.35/00 50.59
98.71/98 00. / 90.33
97.75
19. 19 15
/95.2/ 10399999099 50.79
98.67 (98 37/ 99093
$7.85/
59, 51 191 .44
20/ /55.80 92 20
99 10.12//0 98.73 / 993399
Zipper
45.69 19.32 12..2299923 24
95.27 29..77/99////99 97.57
95.61 29..5/2//00 99124
05.77 90. 17/
95. 96 (90.10 95.35
9% 983 85 99-95
41.79/ 98 /99 m9
91.65 /96.00 94 45:45
92. 19 18 / $9.00
Tcothrush
192/0/999.11
100.00/ 9915
100000. 190.00 94444
99.00 94.97
1000000 195.19
99.34 05.13
99/71 19.26 20
(9928) 15.36
100000190226 18.16
Metdl Nat
96668 46..46//50 983.19 19
16.73 /27.2/7.//55 85.01
(7.02/966.1//5 $4.26
97.56/
10.500 23
97.31 156.9911111100 65.79
95.85 $5.05
96.87/ 23 154 10
96.73 /55335333 55 54.54
94.18 46 / $3.76
Haodivr
98.61/ 90 /99 25
94.05 00-01//99/1/99 99.95
96 00.45/1900 91.44/9944/944 94.15
96.57/ 00. 14.77
10 19 291.99
00.11/ 99 10/99 S4
99.71 /99 190 54 00
99. 11 / /94.0
Soew
$7.00 /98 31..111/22.9
93.11 /55.66/999 9ALTO
95, 00.661988877 /9533
92.91 20 19672
19.77 199.39 977
94.20 04:20/9933 993.779995777 97.20
93.87/
19677
23/ 23//9223 66.6
Dil 00.10/99920 (96.5)
Capuric
96.13/ 67 /86 72
17.25/ 195.13 87.82
96 19. 19999955000 95.44/ (51.5
93.1%/ 96 90 125
M1 70 1O 301
01.82/ W 22 185 413
M1.24 (57.14 5M mm
96. 16- 17.17. / $7.59
01.511111900 /93.91
99.92 0W.92/99333 /98.29/ 98.68
103000197.99 90.31
99.51/ 18, 992.6
99.02 (98005) 92.066
59.92 192.2222222222 19..3
99.91/
99 75 02.222100 10
99.02/99333
$2.02
19.92 97.55 /9200
Awerge
47.35 996 65
/1/ 97
96 16 18 /91.09
95.17/ 96 (91.05
94 00 95.14 14
09/ 905 15 997.93
91.70/ VT /90.67
96.99 /18.0//80000 90.61
95 07/ 927 (91.89
20

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
18
97-
PerfomancisS
06
PerformancsSS
Ing- AUROC-U FIR.AUOOC U
Imn- -UR--UN
AUROCU N
05
PRO U
05
PRO N
94
95-
93
92-
012
0.2
0.3
0.5
03.8
0.9
0.2
0.3
0.1
0.5
0.5
0.9
(a) Uniform Distribution
(b) Normal Distribution
Figure 5. Average performance of anomaly detection and localization of the mask sampling function with uniform distribution and normal distribution as the μt, changes, U represents uniform distribution, N represents normal distribution.
Table 7, Detailed anomaly detecttion/localization results with Ing-AUROC/ / PiX-AUROC/ PRO (in %) metric on VisA under different pe.
Casgery
Air 6.1
Jiu 0.2
pi 0.4
0.5
Air
PCBI
98.03 /9///999///0 /91.76
08.32:
99.38/99.77
07.00/
99.37
46
58.34
99.778
95.15 99.60 /95.74
97.32 (9.357
97.51/99.55/92..
PCB2
99.43/99.61 (66774
99.38
98.59 96,52
09.53 198.00 88.49
99.17/99.79
/ 89.74
99
701
58.00
58.00/50
591.00
99.01 99900///000 91.27
99.16/99.01
/91.54
98.50/ 98.59 99:97
PCBS
98.78/99.22 /94.35
05.42:
99.21 194.56
08.25
99.13 94.55
98.65
29.23 99.95
96.90
14 95
55.00
99.29 99 95.04
95.94 19.94/199:23. 194.75
90.41 99.22 94.36
PCB4
99.53/96.23 84.96
99.36/
93.21 77.31
09.251
94.45/14 79.99
(97.57 (85.5)
09.72:
Maxemai
96.50 99.73/99.////3 59.55
96.75
19.72.
07.55
198
95.30 1966/9999000 997.08
96600
196.99
94..5 99.59) 96.27
Macuoni2
0944//99.8/99.0
88.22:
99.57 98.91
99.85
19.74/ 99.99
91.44
19.79
19.15
90.00
19.04/ 99.90
93.76 /99.2299924
(9..)/999.//// 199.14
84.51 /99.//99999900/0 199.11
$3,47/ 99.31/ 99.04
Capoales
96.22 00/.0/99.0//00 /9.00
05.50 00.37 03.3090.86 07.88
06.35) 09.82 90.95
97.13/
89.55 00.46
95.02/
06.00
$4.80 99.87 198.55
93.50 /96.9/9955511 99.99
92.72/199.7// /99.93
92.57 (9.577/9..84///9 /9.448
Cadi:
94.57/94.58 577.5
93:49: 94.73/ 87.08
89.55 95.99 87.16
55.71
89.06/95.74998.55
88.22/955.0 85.68
Cadex
83399
98.00: 98.93/ 85.69
07.06/ 98.73
96.04/
94.62
97.58/99
99.28/99.21 (90.10
97.95/99.JJ 89.39
Chevingram
07.54/07.82//6.77
97.70/ 76.34
07.36/ 98.00 78.20
92.403 98.1 78.20
99.28 98 80.10
97.52 /98.5/ $1.98
97.78/98.46 84.03
97.20/99.4//3330
96.02/ 99 96.33/ 92.81
Fryum
07.96/09.34/55..0 /8.991
90.04 94.70 68.54
99.65 64.266 18
06.99 92. 94 663
91 84 96. /888688
06. 19. 68 $8.56
96.12/ 76/ $9.30
97.18 /9..48/96442/44 /89 0n
90.04 96.81 99.65
Pge frysm
97.06/99.31
91
97.63/ 99
96
99.96
96 88.
A8
96 68/ 99 19988
98.30
18
79575
97.95 999.99999...00
16
95.80
97.00/
96.85 97.89 99.58
06.63/ 09800/ 89.96
96.65 99.199
96.34/
96.29 99.399 0
96.05
192.28
/98,68
94,35 99.71/
Table& 8. Detailed anomaly detecttion/locaiization results with Img- AUg-AUROC/ / PiX-AUROC/ / PRO (in %) metric on MvTec-AD under different N.
Category
10
25
30
earpet
/9.00/99.45// 96.77
97.71 /99.49/ 19.18
99.16 19.6/99.00// /
91.93/ 99.20 99600
87.96 98 99,14
98.39 /9944/9//////// 11/43/997 /97.05
98.84 999,11 977200
Grid
10000/98.000 99.00/999009999
98.91 /95.61
100.00/ 98.93/ US.11
98
/5.50
99.75 98.99 /95
/92//998/0 95.37
Leather
100.00 /99.08
100.00 / 99.64 98.96
100.00 99.67 99.110
100.00/ 99.67 / 199.1
100.00/ / 99.67 / 99.100
100.00 /99.66/9666// 99.05
100.00 99.67 99.10
Tile
100,00 /95.13
100/000 98, 12 93, 29
100.00/ 90 94.29
100,000 98 80/ 195.74
100.00/ / 97. 79 92,57
100,00 95. /9.72
100.00 98 78 95.21
Wood
/9.82/97.52/9 91.25
89.61
99.47 /97.20//////
/ 94.68 $7.71
99.01 /9.99
99.47 /95.88 90.35
96.27/
Pill
99.59/98879/ / 99.664
99.37 /9.88/ 97.48
99.73 99.799 997558
99. / 98, 79 97.55
99.36/ 99.77/
99:65 / 99,779 / 99.66
99.54 99, 76 97.59
Transistor
99.42/95.52/ 84.98
99.25 99.30/ 84.61
99.58 19/.18/ 84.63
99.25 / 95.33 84.70
99.50/ 95.39/ 85.35
99.42/ 95.27 /84.77
99.25 795.04 84.99
Cable
99.03 09.03/98.62/91 91.10
98.54/ 98.28 90.23
98.58 98.38 38// 90.85
99.06 / 98.47 90.97
98.29 98.59/ 99..9/991..0 /9.889
98.43 / 99.445 /91.35
98.76 9984433 90.89
Zipper
98.35 93.35/99.25/999 /97.56
95.67 29.23/111 97.46
98.53 19999999///// 997558
95.98 / 999 97.54
98.71 /9.31///////// /97.57
95.69 / 99,221 99.552
97. 99 24 97.39
Toothbrush
/9.17/99.05// 93.62
100.00 99.21 194.96
100.00/ 94.92
100.00/ 99.15 94.95
100.00 /99.21 95.06
100,00 99.19 /94.80
100.00 99.. 19.188 94.95
Motal Not
97..8/96.87//// /94.977844.9 84.97
97.02/ 82/688//60 85.11
97.50/ 01:00/97.00 99.001 86.09
97.36 / 96.73 $4.57
97.07 97.00/ 84,98
97.17 117/995 /97.7877/8 /85.32
97.36 96. 85.25
Hazelnur
98.57 / /5.1//99.1///// 993995
99.29 99.82 19.26
98.36 / 98.99/9///// 93.57
97. (9.89/ / 99. 194 10
98.64 98.98/98888 93.43
98.64 / 99.99 /93.50
98.68 /9..01 93.98
Screw
96.02. 99.23/99//// 99.69
99.28
93.65 199.15 /6.443
99, 96.55
94,16/
/96.52
94,55 99.157 /91.1/9996666
93.54 96.04
Capsule
94.10 / 96.29/66/// 188.8
94.34 96.84 84/ /88.30 30
94.23 95. 22 87.41
93. 03.58/ 96.48 58.88 19
94.18 96.76/ 88.65
94.30 /90.996673/ / 07
94,30 96. 89/ 88.35
Bottle
99.92. / 98.05/ 92.75
99.84 19.98.00 06/ 92.774
99.84 98.0988800.0 92.877
99.76/ / 98.08 /292.7 192.78
100.00 / 98.99..00 92,72
99.84 / 99.06 / /9..81
99.84/ 19.003 92.66
Avenge
98.69/ /8.39/99.///// 93.24
198, 18 193044
98.60/ 93.25
97. 98. 04
97.86/ 24 93.14
98.35 98.22/ /93.31
98. / 93.25
21

---

Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
Table Detailed anomaly detecttion/lccalization results with Img- AUg-AUROC/ PiX-AUROC/ PRO (in %) metric on VisA under different N.
Category
=
N =
N = 15
N = 20
N = 25
N = 30
N 35
PCB1
97.46
93.04
/99,49 93.29
98.11 09.44// 93.42
97.
99.12
97.70 992.76
97.55 193.01
16.19/ 99.19/
POH2
99.22/ / 198 87.33
99.11 / 98.27 83.12
99.22/ / 98 76
98 16.86// 9W 42/ 84479
99.45/9..4 / 18444.0
99.58 / 58.57 86.14
99.01 5%% 10 82.24
PCRS
/99//9999//99. 95.34
98885
95.41
99.01 99.00/ 99.44
99.06 99.23 95.15
99.17
08.900 /99.27 /95.99
98.82 /99.21 195.35
PCB4
99.28 996.5 80.87
99.21 99.96 80.08
10..12/93.85 78.62
99.17/94.11/79 79:26
99.25/93.78/ 77.00
99.41/94.22 79.06
99.13 94,00 79.36
Macaroni
96.94 /9.78 98.82
97.04 /9.76 98.64
96.90/ 99.77 98.71
97.17/99.72/ 99.20
96.63/ 98.66
97.35/ 98.55
96,46 99.79 99.49
Macaroni
90.08 /99.74 98.81
$7.98 / 99.73 98.90
89.76 99,72 98.84
89.78 99.7//99.2///5. 99.78
87.63 37.63//9.722 98.82
89.59 / 99.79/99///// 98.93
89.59 99.75 98.91
Capsules
95.92 /9.887 19.19
95.77 / 99.86 98.58
95.95 99.85 99..3
95.98 99.88/9.8/// 99.38
96.88 19.9999888 98.44
95.73/ /9.8999988. 98.440
96.45 99.87 98.37
Candle
$8.44 /5.84 899220
$7.06 / 95.56 89,20
87.40 95.68 88.93
86.90/ (5..0/99548/890
86.89 01.99995.33 88.55
$8.06 95.58 18.118
87.30 95.16 / 88.38
Cashew
96.04/ / 98.80 87,36
95.64/ / 98.81 88.46
96.70 99. 89.77
98.84/ 99.84/99.40 99.16/996//// 88.28
98 99.1999333 38.24
94.82/ 99.08/ 88.19
94.34 99.06/ 88.07
Chewinggum
97.28 998.29
98.56 98.39
98.74/ 98.25
97. 77.777
98.34 98.41
97.70
Fryum
97.84 / 94,71 88.93
98.98 94,87 89.08
99.56 94.83 89:45
97.92 95 95.2955.333 89.71
98.88 94.92 89.25
99.36 94.98/ 89.35
98.74 94.81 (1/8//1/ 89.26
Pipe fryum
98.68 19.44 95.882
98.46 95.49
98.34 99.36 95.32
99.99944 95.71
98.34 99.31 /95.37
Average
96.34 /9.15
96.18 98.18 90.91
96.57 / 98. 16 91.31
96.34/ 98. 16 90.93
96.25 / 98. 18.13//8888 90.68
96.39 19.999991 /9,09
96.17 16.17/ 99.11 90.63
Table 10. Detailed anomaly detection/localizztion results with Img-AUROC/ PiX-AUROC/ PRO (in %) metric on MvTec-AD under different a.
Category
a =
2
le
0=
1e
carpet
81,99
65.77 04.7//9..1////.
79.21 /96.9966///
86.39 86.39
95.95/ 98.68 /93.79
99.00/ 99:45 96.77
97.15/99.111 / /96008
47.99/999 96.91
Grid
00000/99.31/93.00
100.00
193.39
08.50/96..1 81.33
59.63
100.00 99.90 94.91
100.00/ 96.25
99.42/99.
Leather
100,0099..0 198.12
100.00 / 99.68 /98.55 19
100.00 99.69 99.11
100.00/ 99.59 199.15
100.00 99 09..68//55 99.08
100.00 99.532 99.55/997.9
100.00 /99.62/9//2/ 98.85
Tile
96.07 /97.7//976///// 92.53
98.70 97.83 19282
91.23 (7.88 /7.866
100.00 97.82 99.00
100.00 98.69 /95.13
99.13 97.68 /93.36
99.12 16.96//61 99,80
Wood
100.0/95.51 88.55
99.52
97.46 07.46/91.577 /91.57/ 82.31
98.77 95.42 190.09
99.82 97.52/ 91.25
100,00 95.61 89.36
100.00 /94.72/ 86.85
Pill
/4.76/90.26/ 2 81.69
94.33 88.49 /79.56
98.94 08.94/98.68 96.55
98.15 98,59 17.554
99.59/ 98.79 (88.9/997.5 99.66
99:07 / /97.69 /92.43
97.27 /9.55/ 83.11
Tomnistor
/96.1/ /3.25
99.71 99691 /83775
98.88 19..06/5/ 85.53
99.12 95. 85.02
99.32/ 84.98
99.87 /96.996666 84.57
99.87 /17./95443 43,54/3/ 83.58
Cable
98.16/99.39/ 899.1 89.14
98.20 98.43 $8.79
99.53 /98.07 89.13
95.91/ 98.02 19.50
99.03 99.62 19.10
98.84 / /9..24 86.98
98.63 19.8 18 85.29
Zipper
94.01
96.32 99.31 93.77
95.04 / /98.60 95.96
98.48/ / 98.86 19.881
98.35 99.25 97.56
97.11 / 99,31/ /97.20
97.64 / 19..33 97.19
Toethbrush
100.0/99..00
99.44 99..3
19.72, /9.332
100,000 19.17 /95.01
19.05 93.62
100000 99.18 94.67
100.00 /99
Motal Nut
95.65//96.55 74.36
95.89 96.40/ 75.66
95.99 03.99/94.. /94.4/ 83.17
93:26 94.62 $3.35
97.41 / 84.97
96.14 / 96.38 38.33///50 75.24
93.21 / 19:26/ 69.19
Hazelnur
V7. 77.54//8..41// / 88.22
97.43 998 /82.99
97.64 98. 19, 88.04
98 82/ 99 26 23
98.57 90.100 93.95
95.64 / /95.54 /86-00
97.86 922.97 $1.64
Screw
91.10/ 98.99/ 95.57
93.20 /98.97 955.58
96.23
97.80
9999999333
$6.94 /9.57 93.09
77.06 97.79
Capsulc
95.60 09.69/ 18.122 90.58
94.81 18,1888.88. 90.62
94.58 99.84//8/4/ 88.57
94.26 / 96.41 87.64
94,10/ 96.25/ 88.18
96.89 / 98.61 93.25
97.05 19.856 93.39
Bottle
99.92. 98.9988.66 / 88.78
100.00 198. 69 /
100.00/ 98, 93.22
100,000 97, 97.64/ /91.75
99.93 98.05/ 92,75
100,00 98.66/ 993448
100.00 /98 63 92.82
Avenugo
97..100/// 89.27
95.45 99.04 788.74
19 99.59/688888 / 89.01
97.92/ 97.78 92.42
98.69 99.39983222 93.24
97.79 97.98 /91.36
V7. 97 89.32
22

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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
E. More Qualitative Results
This section reports more qualitative Results on the MvTec-AD and VisA dataset. Figure depicted results from different categories of the compared methods. Figure 7 depicted intermediate step and anomaly localization results under different sampling function and different Пa. Due the different sparsity of different channel masks, some will show color or brightness reduction.
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Figure 6. Qualitative comparison for anomaly localization on two datasets. From left to right: the input images, ground-truth, and the anomaly maps produced hy all compared methods.
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Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate
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Figure 7. Intermediate step and anomaly localization results under different sampling function and different n..
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Reviews:
[{"review_id": "QoIWG0aPnb", "reviewer": "Reviewer_LTm8", "summary": "The paper \"Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate\" proposes a novel approach for unsupervised anomaly detection in structural images using a factor model framework combined with Discrete Fourier Transform (DFT). The authors introduce a test to detect anomalies by analyzing weighted residuals in the Fourier domain, comparing them to a zero spectrum under a null hypothesis of no anomaly. They develop the Demeaned Fourier Sparse (DFS) module, which constructs masks in the Fourier domain to enhance reconstruction-based anomaly detection. The method leverages residuals to identify anomalies without requiring prior knowledge of anomaly types, offering both theoretical rigor (via asymptotic properties) and practical applicability. Experimental results on datasets like MvTec-AD demonstrate competitive performance in anomaly detection and localization, with the approach being computationally efficient compared to some existing methods.\n\n## Update after rebuttal\nAfter reviewing the authors’ rebuttal, I appreciate the effort they’ve put into addressing the feedback. Their responses have largely alleviated my concerns, and despite some differing perspectives, I will maintain my original score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer LTm8,\n\nFirstly, we express our gratitude for your thorough review and insightful comments on our paper. Your recognition of the rigor and contribution of our work is greatly appreciated. We also greatly appreciate your rigorous and professional feedback. Your suggestions are highly meaningful, especially for a theory-driven paper like ours. We have taken your feedback as an opportunity to further refine our manuscript. We address your concerns below:\n>W1:” more test on variable image sizes”\n\nWe agree with the viewpoint that the asymptotic global theory should be validated on more image sizes, as you mentioned in your comment: \"it could have emphasized how its theoretical edge translates into tangible benefits.\" To address this, we have applied our method to larger-scale scenarios, with experimental results on MvTec-AD dataset presented for 512×512 and 1024×1024 image sizes. Due to space and time limitations, we only provide the average results of metrics.\n|Resolution|Ours|Ours-Base|\n|:---:|:---:|:---:|\n|256 $\\times$ 256|98.69/98.32/93.24|95.04/96.59/84.10|\n|512 $\\times$ 512|98.52/96.13/91.69|85.11/89.97/69.24|\n|1024 $\\times$ 1024|97.19/91.49/86.82|81.65/82.83/58.79|\n\nresults show that our method continues to maintain a high degree of effectiveness and shows greater stability compared to the baseline method.\nIf possible, we will include relevant comparative results in future versions.\n>W2:” Robustness isn’t fully substantiated.”\n\nWe agree that statistical significance tests for the reported metrics are necessary, as the main contribution of this paper is statistical theory. Since we apply our method to practical detection problems, more comprehensive metrics are needed to assess overall performance. Specifically, in the validation of the asymptotic global theory (for the entire image), the p-value contribution of a single pixel might be overlooked (p-values are less adaptive to imbalanced data, as they often fail to account for the sparsity of anomalies). Therefore, we prefer to use metrics commonly employed in the anomaly detection field as our evaluation criteria. We provide partial p-values for our method on MvTec-AD dataset.\n| |Ours| Ours-Base|\n|:---:|:---:|:---:|\n|P-value| 0.0276 | 0.0792 |\n\nAs can be seen, after incorporating our method, The p-value is below the significance level of 0.05, indicating that we can reject the null hypothesis (nonanomalous) with a high probability.\nIf possible, we will include corresponding statistical significance tests for both our method and the comparative methods in future versions.\n\n>W3:” As mentioned in the Claims and Evidence section”\n\nPlease see our response to W1.\n\n>W4:”There are some minor issues, which I mentioned in the previous sections”\n\nPlease see our response to W1 and W2.\n\nWe sincerely hope that our clarifications above have increased your confidence in our work. We will be happy to clarify further if needed. We thank you again for sharing your valuable feedback on our work.", "claims_and_evidence": "- **Strengths:**\n    - The paper’s theoretical backbone is a standout feature. Section 3 delivers a rigorous statistical framework through asymptotic properties, such as Theorems 3.2, 3.4, and 3.6, which mathematically define how residuals behave under normal and anomalous conditions. The derived detection rate of $H^{−1/2}W^{−1/2}$ offers a concrete lower bound for spotting subtle anomalies, marking a meaningful contribution to the field.\n    - Experimental evidence supports its claims effectively in several cases. The results show the method can outperform established approaches like PatchCore and DRAEM in specific scenarios, suggesting practical potential.\n- **Weaknesses:**\n    - The connection between theory and practice isn’t fully realized. The asymptotic properties assume large image dimensions (H, W → ∞), but typical dataset images (e.g., 256x256 in MvTec-AD) don’t meet this scale. The paper doesn’t bridge this divide with empirical validation, such as testing on varying image sizes. To strengthen its case, it could have emphasized how its theoretical edge translates into tangible benefits, making the evidence more compelling against competitors.", "methods_and_evaluation_criteria": "- **Strengths**:\n    - The methodology shines with its originality, blending a factor model with DFT and introducing the DFS module. This module projects residuals into the Fourier domain and iteratively optimizes masks (e.g., via Bernoulli sampling and sigmoid mapping) to isolate anomalies efficiently. A key advantage is its self-contained residual generation—detecting anomalies using the model itself without external data structures like PatchCore’s memory bank or DRAEM’s dual-network setup. This lean approach enhances its appeal for unsupervised settings.\n    - Efficiency is a clear strength, as evidenced in Table 3. The proposed method significantly undercuts heavier models like SSNF (294.67M parameters, 102.23G FLOPs) and PyramidFlow (162.20M parameters, 81.13G FLOPs), positioning it as a viable option for resource-constrained environments, such as industrial applications with limited compute power.\n- **Weaknesses**:\n    - Robustness isn’t fully substantiated. The absence of statistical significance tests (e.g., p-values or confidence intervals) for the reported metrics means readers can’t confidently assess whether the performance edge is meaningful or due to chance.", "theoretical_claims": "The asymptotic theory is a robust intellectual achievement. Section 3’s derivations (e.g., Proposition 3.1, Theorem 3.6) establish how residuals behave under null and alternative hypotheses, providing a statistical lens for anomaly detection. This clarity sets a high bar for theoretical rigor in the domain.", "experimental_designs_or_analyses": "- Strengths:\n    - The experimental design is well-structured, testing the method on established datasets (MvTec-AD and VisA) against strong baselines like PatchCore and DRAEM. This ensures a fair comparison within the anomaly detection community’s standard benchmarks.\n    - Hyperparameter exploration in Appendix E adds depth. By varying sampling functions and epochs, it demonstrates the method’s robustness and adaptability, giving readers insight into its operational flexibility.\n    - Qualitative results (Figures 6, 7) are visually persuasive, effectively showcasing the method’s ability to pinpoint anomaly locations in images, which aligns with its localization claims.\n- Weaknesses:\n    - As mentioned in the **Claims and Evidence** section, testing on variable image sizes, if possible, would benefit the paper by demonstrating that the asymptotic property holds in practice.", "relation_to_broader_scientific_literature": "- The paper builds thoughtfully on prior work, extending factor models (Fu et al., 2023) and reconstruction-based detection (Zavrtanik et al., 2021) into a new image-focused framework, blending time-series inspiration with visual analysis.\n- Its Fourier-based approach is a fresh take, diverging from the CNN-heavy norm and offering a frequency-domain perspective that could inspire further exploration.", "essential_references_not_discussed": "Appendix A effectively addresses relevant references, providing a comprehensive overview of related work that ties the method to the current research landscape.", "other_strengths_and_weaknesses": "- Strengths:\n   - The paper is clear and well-written.\n   - The method offers a theoretically robust framework, with rigorous asymptotic properties and a novel Fourier-based approach enhancing anomaly detection.\n- Weaknesses:\n   - There are some minor issues, which I mentioned in the previous sections.", "other_comments_or_suggestions": "The paper struggles to clearly showcase its comparative advantages—such as greater efficiency, competitive performance, and a self-contained residual construction that detects anomalies without the need for additional data—over other methods, and it would benefit from more prominently integrating these strengths into the abstract or introduction to convince readers of its practical value.", "questions_for_authors": "N/A. Please check other sections.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "u02b0SzYCv", "reviewer": "Reviewer_a5G4", "summary": "This paper proposes a reconstruction-based method for anomaly detection, by using the construction of a mask in the Fourier domain to sparsify the information by reducing the number of estimated common factors of the input images. Then a U-Net is used to reconstruct the images, using the reconstruction error as the anomaly score. Experiments are conducted on MVTec AD and VisA datasets.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer a5G4:\n\nWe appreciate your review of our paper and the insightful comments you provided. We are glad to hear that you find the method we proposed to be both simple and effective. We have addressed the issues you raised as follows:\n\n>W1:” Lacking important reviews in the literature”\n\nAs mentioned in your comments, we will include a related literature review in the subsequent version. We greatly appreciate your thorough feedback. Unlike the papers mentioned, which operate at the feature level, our method performs a demeaned Fourier transform in the frequency domain and provides a complete theoretical framework and proof, which is not mentioned in previous literature. (We were motivated by the application cases and abstract it into a theoretical model, based on this, we further give the Asymptotic theory of our model. ) \nRegarding the comment \"However, this paper still focuses on one-model-one-task,\" since our contribution is mainly focus on the theory filed, our initial approach was to validate it on benchmark problems. However, your comment has been very insightful. Based on your suggestion, we have attempted to apply a multi-task, one-model approach on MvTec-AD dataset：\n||Ours|Ours-Base|\n|:---:|:---:|:---:|\n|Average|93.3/94.6/85.9|58.0/70.8/39.3|\n\nCompared to the baseline method, our approach still shows a significant improvement in the multi-class task.\nWe will refine the complete theoretical framework and methodology in future work. Thank you for your constructive suggestions.\n\n>W2:” The theoretical part and the method design part are very disconnected”\n\nWe thank the reviewer for raising this question, as it will help others better understand our work. (Maybe some of our expressions in writing let you not clearly see our writing logic, but in fact, the theoretical derivation of the previous sections and the application of the later sections are closely related, you could see when apply the DFT method, our theory clearly explains why the applied method works). As we mentioned in Section 4, the decentralization operation corresponds to the construction of the complex-valued empirical process in our theoretical section. It simultaneously constructs the test statistic, which helps us derive the asymptotic lower bound. Based on our asymptotic theory, in order to establish the quantitative relationship ($K<R$) between the true common factors and the estimated values, we designed the sparsification operation. This operation weakens the main information (non-anomalous parts) in the original image, causing the residuals (anomalous parts) to occupy a larger proportion, making anomalies easier to detect. The experimental results based on the above operations validate our asymptotic theories.\n\n>W3:” The paper is very incremental”\n\nThank you for raising this question, In terms of theory, our contribution is not simply an extension of existing one-dimensional theories to two-dimensional space. We also provide a complete asymptotic theory along with the corresponding proofs. This method and its applications really make sense: first, it requires the development of new hypothesis tests, complex-valued empirical processes, test statistics, and upper and lower bounds for the asymptotic theory. Second, unlike the one-dimensional time series theory, the theory at the two-dimensional image level requires spatial relationships to serve as the central element of the theory. Finally, the proof section involves new derivations and experimental validation of the constructed asymptotic theory, with validation metrics and methods that differ from those in the one-dimensional time series scenario. We are committed to constructing a complete theoretical framework, not merely extending the methods and theories.\n\n>W4:”it seems that only DFS Module was added on the U-Net. What if the DFS Module is added on the other architectures, such as UniAD?”\n\nWe thank the reviewer for raising this question. Since our main contribution is the theoretical part, it is essential to validate the effectiveness of the theory and its corresponding asymptotic properties using a simple model, this is very common in the Monte Carlo simulation section of such theoretical articles, and our main aim is similar. This allows us to eliminate the additional effects introduced by complex models. Therefore, we opted for the relatively simple U-Net architecture. As pointed out in your comment, adding the DFS module to the UniAD architecture should indeed be effective. However, the UniAD framework operates at the feature token level, which conflicts with our frequency domain operations. This results in our module not being directly applicable to the UniAD framework. We appreciate your valuable suggestion, which will be of great help for our future work.\n\nWe sincerely hope that our clarifications above have increased your confidence in our work. We will be happy to clarify further if needed. We thank you again for sharing your valuable feedback on our work.", "claims_and_evidence": "Not always.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "[a] OCGAN: One-class Novelty Detection Using GANs with Constrained Latent Representations. CVPR 2019.\n[b] Attribute Restoration Framework for Anomaly Detection. TMM 2020.\n[c] A Unified Model for Multi-class Anomaly Detection. NeruIPS 2022.", "other_strengths_and_weaknesses": "The proposed method is simple yet effective to improve the reconstruction-based anomaly detection. However,\n1. Lacking important reviews in the literature. At the very beginning, OCGAN [a] proposed to add Gaussian noise to the input, while ARNet \n [b] used some transformations to erase some important attributes in the images, which have a very similar idea to this paper. Then, some MAE-based methods used masks to remove part of the inputs, transferring the reconstruction task to image inpainting. Finally, UniAD [c] illustrated that traditional AE failed in AD because of the identity shortcut of the model, by simply using the transformer-based architecture, the model's generalizability can significantly be improved, and can even be used for all tasks with only one model. However, this paper still focuses on one-model-one-task.\n2. The theoretical part and the method design part are very disconnected.\n3. The paper is very incremental. In terms of theory, the author's contribution is only to simply extend the existing one-dimensional theory to two-dimensional space.\n4. Also, in terms of experimental design, it seems that only DFS Module was added on the U-Net. What if the DFS Module is added on the other architectures, such as UniAD? \n\n[a] OCGAN: One-class Novelty Detection Using GANs with Constrained Latent Representations. CVPR 2019.\n[b] Attribute Restoration Framework for Anomaly Detection. TMM 2020.\n[c] A Unified Model for Multi-class Anomaly Detection. NeruIPS 2022.", "other_comments_or_suggestions": "See strengths and weaknesses.", "questions_for_authors": "See strengths and weaknesses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1YcYDuvpww", "reviewer": "Reviewer_2ntV", "summary": "This paper proposes a novel test for detecting anomalies in structural images using a discrete Fourier transform (DFT) under a factor model framework, which enables interpretable and effective reconstruction-based anomaly detection.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer 2ntV:\n\nFirstly, we express our gratitude for your thorough review and insightful comments on our paper. Your recognition of the novelty and contribution of our work is greatly appreciated. We also greatly appreciate your rigorous and professional feedback. Your suggestions are highly meaningful, especially for a theory-driven paper like ours. We have taken your feedback as an opportunity to further refine our manuscript. We address your concerns below:\n\n>Q1:” On page 3, line 154, the question arises as to why the first statistic is weakly convergent and, after integration, becomes convergent in distribution”\n\nThank you for raising this question. Due to space limitations, we did not provide a detailed explanation. In our initial assumption, the Fourier transform is not performed over the $ \\mathbb{R} $, but rather over a finite interval. However, our integration is carried out over the entire real domain, which corresponds to the weak and strong convergence relationships in functional analysis. This setting is also valid for the weight function $W$.\n\n>Q2:” The regularization coefficient alpha has not been discussed”\n\nWe have added experiments regarding the regularization coefficient $\\alpha$ and provide some of the results here:\n|  $\\alpha$  |   Ours     |\n| :---: | :----------------------------------------: |\n| 1e-2  | 95.50 / 97.10 / 89.27 |\n| 1e-3  | 95.45 / 97.04 / 88.74 |\n| 1e-4  | 96.19 / 96.58 / 89.01 |\n| 1e-5  | 97.92 / 97.78 / 92.42 |\n| 1e-6  | 98.69 / 98.32 / 93.24 |\n| 1e-7  | 97.79 / 97.98 / 91.36 |\n| 1e-8  | 97.03 / 96.97 / 89.32 |\n\n>W1:” on page 3, in the section on constructing the complex-valued empirical process, it is not explained why the common factors and residuals are combined together”and” this is actually used in the later asymptotic theory section. Another issue is that the title of the paper mentions residual estimation, yet the theoretical section also incorporates common factors”\n\nThe purpose of constructing the complex-valued empirical process by combining the common factors and the residuals is to unify the estimated quantities within the test statistics. Additionally, combining the common factors and the residuals helps prevent the residuals from summing to zero. In the process of estimating the residuals, the common factor $F$ is also involved, but $F$ differs across theoretical framework assumptions (as we mentioned in Section 3.1, maybe is not based on the factor structure). However, $F$ can be expressed in the form of residuals in all cases. Therefore, the estimation of the residuals is a central component of the complete theory, with $F$ serving as an intermediate variable.\n\n>W2:” the manuscript does not include experiments examining the effects of different sparsity levels on model performance”\n\nThank you for raising this question. In our theoretical section, our asymptotic theory suggests that as long as the estimated values are smaller than the true values, the anomaly detection task will be effective. In the experimental section, we constrain the sparsity using the regularization coefficient. Since we are applying this in an unsupervised setting, the level of sparsity is learned by the model itself. We believe that the sparsification operation enhances the effectiveness of the anomaly detection task, but the degree of sparsity cannot provide a precise bound. This is because the sparsity degree may vary for different types of unknown anomalies. Nevertheless, this raises an insightful question, and we will continue to explore this in our future research.\n\n>W3:” Writing suggestion”\n\nThank you for carefully pointing out our small mistake. We will make the correction in the subsequent version. Additionally, we will provide the pseudocode of our method to make it more intuitive and clearer for the readers.\n\n>Q3:” Whether the module constructed by this method potentially be applicable to a broader range of anomaly detection problems”\n\nWe have validated our method on multi-task tasks, and the results are as follows. Theoretically, our method remains effective in more anomaly detection methods. Here, we attempt to apply our method to the multi-class task on MvTec-AD dataset：\n||Ours|Ours-Base|\n|:---:|:---:|:---:|\n|Average|93.3/94.6/85.9|58/70.8/39.3|\n\nCompared to the baseline method, our approach still shows a significant improvement in the multi-class task.\nWe will follow up on this and conduct corresponding tests in the subsequent versions.\n\nWe sincerely hope that our clarifications above have increased your confidence in our work. We will be happy to clarify further if needed. We thank you again for sharing your valuable feedback on our work.", "claims_and_evidence": "To my understanding, the claims are articulated clearly enough to be comprehensible.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria make sense for the anomaly detection problem at hand.", "theoretical_claims": "Yes. Most of the theoretical proofs are logically sound. However, some aspects are controversial. On page 3, line 154, the question arises as to why the first statistic is weakly convergent and, after integration, becomes convergent in distribution.", "experimental_designs_or_analyses": "Yes. Most of the experimental design and analysis are reasonable. The regularization coefficient alpha has not been discussed. Therefore, the results of incorporating additional values for alpha should be considered.", "relation_to_broader_scientific_literature": "This paper adopts a factor model commonly used in time series analysis in economics. Building upon existing methods for detecting structural changes in time series data, this paper proposed a test which incorporates a multi-DFT dual cross-sectional dimension of positional information for image anomaly detection which also provides the relevant asymptotic theory.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strength: \n- This paper provides a relatively complete theoretical framework and derivations, particularly matching the properties of the asymptotic theory with the practical issues. The proofs are based on a well-established factor model framework, and the related theory is fairly comprehensive.\n- The module proposed based on the test is simple and quick to implement, with validation provided regarding its resource consumption.\n- The experimental results validate the effectiveness and consistency of the theory.\n\nWeakness:\n- As I mentioned above, the experimental design could be further improved.\n- This paper requires more detailed explanations. For example, on page 3, in the section on constructing the complex-valued empirical process, it is not explained why the common factors and residuals are combined together. However, this is actually used in the later asymptotic theory section. Another issue is that the title of the paper mentions residual estimation, yet the theoretical section also incorporates common factors. The authors have many ideas to convey, but the details are not sufficiently thorough.\n- The sparsification method proposed in the manuscript appears to offer several advantages; however, these benefits need to be supported by more detailed experimental analyses. Currently, the manuscript does not include experiments examining the effects of different sparsity levels on model performance. Adding such experiments would provide valuable insights into how varying degrees of sparsity influence performance outcomes. This additional analysis would significantly strengthen the manuscript and validate the proposed method's effectiveness.\n- The description of the method in the paper is not very intuitive. It would be helpful to summarize the method into a structured algorithm, presented in pseudocode format.", "other_comments_or_suggestions": "Some minor problems:\n- The notation in Equation 9 seems unusual since both loss terms are labeled as L_reg. \n- There are similar typos in the paper. Please review and correct them accordingly.\n- Whether the module constructed by this method potentially be applicable to a broader range of anomaly detection problems, including non-reconstructive anomaly detection methods?", "questions_for_authors": "See comments above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Although this paper did not receive unanimous support from the reviewers before and after the rebuttal, with unchanged ratings of 4, 4, and 2, the area chair believes that the two reviewers who rated it positively provided useful comments on the paper's contributions.

From **Reviewer LTm8**:
> The method offers a theoretically robust framework, with rigorous asymptotic properties and a novel Fourier-based approach enhancing anomaly detection.

From **Reviewer 2ntV**:
> This paper provides a relatively complete theoretical framework and derivations, particularly matching the properties of the asymptotic theory with the practical issues.

Several concerns were raised, such as (1) the connection between theory and practice, (2) the experimental design, (3) the effects of different sparsity levels, and (4)  applicability to a broader range of anomaly detection problems. The authors addressed these concerns by incorporating experiments involving "more tests on variable image sizes," "statistical significance tests," " the regularization coefficient," and "validation on the multi-task MvTec-AD dataset." These efforts were acknowledged by the reviewers. 

After considering all assessments, the area chair recommends that we accept the paper.
Decision: Accept (poster)