Set Features for Anomaly Detection
Authors: Niv Cohen
Issar Tzachor
Yedid Hoshen
Venue: Accepted by TMLR ()

Abstract: This paper proposes to use set features for detecting anomalies in samples that consist of unusual combinations of normal elements. Many leading methods discover anomalies by detecting an unusual part of a sample. For example, state-of-the-art segmentation-based approaches, first classify each element of the sample (e.g., image patch) as normal or anomalous and then classify the entire sample as anomalous if it contains anomalous elements. However, such approaches do not extend well to scenarios where the anomalies are expressed by an unusual combination of normal elements. In this paper, we overcome this limitation by proposing set features that model each sample by the distribution of its elements. We compute the anomaly score of each sample using a simple density estimation method, using fixed features. Our approach outperforms the previous state-of-the-art in image-level logical anomaly detection and sequence-level time series anomaly detection.

Published in Transactions on Machine Learning Research (12/2024)
Set Features for Anomaly Detection
Niv Cohen
misc@es.hujj..ec
School of Computer Science and Engineering The Hebrew University of Jerusalem, Isroel
Issar Tzachor
issurizachorr@mailluuu..
School of Computer Science and Engineering The Hebrew University of Jerusalem, Israel
Yedid Hoshen
School of Computer Science and Engineering The Hebrew Universily of Jerusalem, Israel
Reviewed on OpenReview:
Abstract
This paper proposes to use set features for detecting anomalies in samples that consist of unusual combinations of normal clements. Many lcading methods discover anomalies by detecting an unusual part of sample. For example, state-of-the-art segmentation- based approaches, first classify each element of the sample (e.g., image patch) as normal or anomalous and then classify the entire sample as anomalous if it contains anomalous elements. However, such approaches do not extend well to scenarios where the anomalies expressed by unusual combination of normal elements. In this paper, we overcome this limitation by proposing set features that model each sample by the distribution of its elements. We compute the anomaly score of cach sample using a simple density estimation method, using fixed features. Our approach outperforms the previous slate-of-the-arl in image-level logical anomaly detection and sequence-level time series anomaly detection
1 Introduction
Anomaly detection aims to automatically identify samples that exhibit unexpected behavior. In some detection tasks anomalics are quite subtle. For example, let us consider an image of bag containing screws, nuts, and washers []. There are two ways in which sample can be anomalous: (i) one or more of the elements in the sample are anomalous. F.g., a broken screw. (fi) the elements are normal but appear in an anomalous combination. E.g., one of the washers might be replaced with a nut.
In recent years, remarkable progress has been made in detecting samples featuring anomalous elements. Segmentation-bassed methods were able to achieve very strong results on industrial inspection datasets [Bergmann et al.) 2019). Such methods operate in two stages: First, perform anomaly segmentation by detecting which (il any) of the elements of the sample are anomalous, e.g., by density estimation (Cohen & Hoshen 2020 Defard et al., 2021: Roth et al. 2022). Given an anomaly segmentation map, we compute the sample-wise anomaly score as the number of anomalous clements, or the abnormality level of the most anomalous element. the anomaly score exceeds a threshold, the entire sample is denoted as an anomaly.
We denote this paradigm
periction-by-segmentteee
Here, tackle the more challenging case of detecting anomalics consisting of unusual combination of normal elements. For example, consider the case where normal images contain two washers and two nuts, but
Quu code be found htttp:://gishubb.com//t/////t/

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anomalous images may contain one washer and three muts. As cach of the elements (nuts or washers) occur in natural images,
methods will not work. Instead, more holistic understanding of the image is required to apply density estimation techniques. While simple global representations, such as taking the average of the representations of all clements might work in some cases, the result typically too coarse to detect challenging anomalies.
Existing approaches tackle logical anomalies in images by reconstruction-based approaches - e.g., an autoen- coder combining local and global classes (Bergmann et al. 2022 Batzner et al. 2023). These approaches have obtained strong results on some object types, while anomaly detection on other anomaly types remains low. Other approaches provide strong results on all object types but rely on additional supervision (Kim et al. 2024h. For the analogues time-series lask, where there is a global anomaly in the time series, the best-perfon best-performing approach is currently a generalization based approach (Qiu et al., 2021). We aim here to provide strong results on both tasks using a unified density estimation approach.
We propose to detect anomalies consisting of unusual combinations of normal elements using set representations. Our key insight, that we should treat sample as the set of its elements, is driven by the assumption that in many cases the abnormality of sample is more correlated with the distribution of elements than with their ordering. Each sample is therefore modeled as an orderless set of elements. The elements are represented using standard fixed feature embeddings, e.g., a deep representation extracted by a pre-trained neural network or handcrafted features. To describe this set features we model their distribution as a set using a collection of histograms. We compute the histograms for a collection of random projection directions in feature space. The bin occupancies from all the histograms are concatenated together, forming our set representation. Finally, score anomalies using density estimation this set representation. We compare our set descriptor to previous approaches and highlight its connection to the sliced Wasserstein distance (SWD).
Our method, SINBAD (Set INspection Based Aomalies Detection) is evaluated on two diverse tasks. The first task is image-level logical anomaly detection on the MVTec-LOCO datasets. We also evaluate our method on series-level time series anomaly detection. Our method outperforms more complex state-of-thes methods while not using augmentations or training. Note that our method relies on the prior assumption that the elements are normal but their combination is anomalous. In scenarios where the elements themselves are anomalous, it is typically better to perform anomaly detection directly at the element level; E.g., deleecion-byy-seeneeetta
or other methods.
We make the following contribution:
Identifying set representation as key for detecting anomalies consisting of normal elements.
novel bateb method for measuring the distance between samples.
State-of- the-r-- results on logical and time series anomaly detection datasets.
2 Previous work
Image Anomaly Detection. A comprehensive review of anomaly detection can be found in Ruff et al.
2021b. ). Early approaches such as Glodek et al. (2013); Latecki et al. (2007); Eskin et al.
2002
used
handcrafted representations. Deep learning has provided a significant improvement on such
benchmarks
Larsson al.
2016 Ruff et al. 2018 Golan & Eanivy 2018 Hendrycks et al. 2019 Ruff et al.
2019 Perera
& Patel 2019 Salehi al.J [0221 Tack al. (2200). As density estimation methods utilizing pre- prained
deep representation have made significant steps towards the supervised performance such benchmarks
Deecke et al. 2021 Cohen & Avidan 2022 Reiss et al.) 2021 Reiss & Hoshen 2021 Reiss et al. 2022
much rescarch is now dirccted at other challenges (Reiss et al. 2022). Such challenges include detecting
anomalous image parts which are small and fine-grained (Cohen & Hoshen 2020 Li et al. 2021 Defard et al.
2021 Roth et al. 2022 Horwitz & Hoshen 2022). The progress in anomaly detection and segmentation has
2021 Carrera et al. 2016 Jezek et al. 2021 Bonfiglioli et al.) (002). Recently, the MVTec-LOCO dataset Bergmann al. (2022) has put the sihtlight fine-grained anomalies that cannot be identified using single patches, but can only be
been enabled by the introduction of appropriate datasets (Bergmann et al. 2019
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Breakfast Box
Juice Bottole
Pushpins
Screw Bag
Splicing conncectors
Normal data varaition
iba
ibb
Logical Anomalies
Figure 1: In set anomalies, each image element (e-g,, patch) may be normal even when their combination is anomalous. This is challenging the variation in the normal data may be higher than between normal and anomalous elements (e.g., swapping a bolt and washer in the screw bag class).
identified when examining the connection between different (otherwise normal) elements an image. Here, we will focus on detecting such logical anomalies.
Time series Anomaly detection. A general review on anomaly detection in time series can be found in (Bazquuzz-Garíaa et al., 2021). In this paper,
concerned with anomaly detection of entire sequences, i.e., cases where an entire signal may be abnormal. Traditional approaches for this task include generic anomaly detection approaches such as k nearest neighbors (kNN) based methods, e.g., vanilla kNN Eskin
et al. 2002) and Local Outlier Factor (LOF) Breunig et al. (2000), trre-bases methods Liu et al. (2008) ),
ono-class classification methods Tax & Duin (2004) and SVDD Schölkopf et al. and auto-regrossive mothods that are particular to time series anomaly detection Rousseeuw & Leroy (2005). With the advent of deep Deep one-class classification
learning, the traditional approaches were augmented with deep-learned features:
methods include DeepSVDD Ruff et al. (2018) and DROCC Goyal et al.) (2020). Deep auto-rogressive mothods
include RNN-based prediction and auto-encoding methods Bontemps et al. (2016); Malhotra et al. (2016)
In addition, some deep learning anomaly detection approaches are conceptually different from traditional approaches. These methods use classifiers trained on normal data, assuming they will struggle to generalize
to anomalous data (Bergman & Hoshen 2020 Qiu et al. 2021h.
Discretized Projections. Discretized projections of multivariate data have been used many previous works. Locally sensitive hashing Dasgupta et al. (2011) uses random projection and subsequent binary quantization hash for nigh-dimensional data. It was used to facilitate fast & nearest neighbor search. Random projections transformation is also highly related to the Radon transform Radon (1917). Kolouri ct al. Kolouri et al. (2015) used this reprosentation a building block in their set representation. HBOS Goldstein & Dengel (2012) performs anomaly detection by representing each dimension of multivariate data using a histogram of discretized variables. Rocket and mini-rocket Dempster et al. 2020 2021) represent time scries for classification using the averages of their window projection. LODA Pevný (2016) extends this
work, by first projecting the data using random projection matrix. We differ from LODA in the use of different density estimator and in using sets of multiple elements rather than single sample descriptions.
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3 Set Features for Anomaly Detection
3.1 A Set is More Than the Sum of its Parts
Detecting anomalies in complex samples consisting of collections of elements requires understanding how the different elements of each sample interact with one another. As a motivating example let us consider the screw bag class from the MVTee-LOCO dataset (Fig, D Each normal sample in this class contains two screws (of different lengths), two nuts, and two washers. Anomalies may occur, for example, when an ionalional nut replaces one of the washers. Detecting anomalies such these requires describing all elements within sample together, since each local element on its own could have come from a normal sample.
A typical way to aggregate element descriptor features is by average pooling taking the average of the features describing each element. Yel, this is not always suitable for set anomaly detection. In supervised learning, average pooling often built. into architectures such as ResNet He al. (016) or DeepSets Zaheer jet al. (2017), in order to aggregate local features. Therefore, deep features learned with supervised loss are already trained to be effective for pooling. However, for lower-level feature descriptors this may not be the case. As demonstrated in Fig 2 the average of a set of features is far from a complete description of the set. This is especially true in anomaly detection, where density estimation approaches require more discriminative features than those needed for supervised learning (Reiss et al. 2022). Even when an average pooled set of features works for a supervised task, it might not work for anomaly detection.
Therefore, we choose model a sct by the distribution of its elements in the embodded feature space, ignoring the ordering between them. A naive way of doing so is using a discretized, volumetric representation, similar to 3D voxels for point clouds. Unfortunately, such approaches cannot scale to high dimensions, and more compact representations are required. We choose to represent sets using a collection of 1D histograms. Each histogram represents the density of the elements of the set when projected along a particular direction. We take the bin occupancies of such histograms as our features. We provide an illustration of this idea in Figure 2
In some cases, projecting a set along its original axes may not be discriminative enough. Histograms along the original axes correspond to 1D marginals, and may map distant elements to the same histogram bins (see Fig 2 for an illustration). On the other side, we see at the bottom of the figure that when the set elements are first projected along another direction, the histograms of the two sets are distinct. This suggests set description method: first project each set along shared random direction and then compute a 1D histogram for each set along this direction. We can obtain a more powerful descriptor by repeating this procodure with projections along multiple random directions. We benchmark this approach in section 3.5
3.2 Preliminaries
We are provided a training set S containing set of Ns samples, we denote sample x E S. We assume that all the training samples are normal. We wish to learn model that operates and outputs an anomaly score. We label samples with anomaly scores higher than a predetermined threshold value as anomalies. The unique aspect of our method is that it treats each sample æ as consisting of a set of Ng elements, where we denote each element as e E z. Examples of such elements include patches for images, or temporal windows for time series, We assume the existence of a powerful feature extractor that maps each raw element into an element feature descriptor F(e). We will describe specific implementations of the feature extraction for two important applications: images and time series, in section
new, test sample ž
3.3
Set Features by Histogram of Projections
Motivated by the toy example in section 3.1 we propose to model each set by the histogram of the values of its elements along a collection of directions. We provide an algorithm box Alg summarixing our steps.
Feature extractor. We split each sample to elements e E and extract a feature representation for each (F(e) e E x). We describe the implementation of F in See 4 as differs for the time series and image modalities.
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Distbution
Distbution
Average
Average 2
Y-Hist
Y-Hist
X-Hist
X-Hist
Random Projection-Hist
Random Projection-Hist
Figure 2: Random projection histograms allow us to distinguish between sets where other methods could not. The two sets are similar in their averages and histograms along the original axes, but result in different histograms when projected along a random axis.
Histogram descriptor. While average pooling the features of all elements in the set is obvious set descriptor, it may result in insufficiently informative representations (section 3.1. Instead, we describe the set by computing the set's histogram for each feature dimension and concatenating them.
For each sample, we denote the set of values for the jth dimension of the feature embeddings as sljl [F(e)[] v). Note that each set s[j] consists of N₂ scalar elements coming for each sample. We compute the maximal and minimal values of s(j] for each dimension j, among all the elements from all samples combined (Ns- Næ), and divide the region between them into K bins. Finally, we compute histograms h[j] for each of the ND dimensions, describing the set s[j], and concatenate each histogram of K bins of each histogram to single set descriptor IR. ND-K
Projection. As discussed before, not all projection directions are equally informative for describing the distributions sets. In the general case, it is unknown which directions are the most informative for capturing the difference between normal and anomalous sets. we cannot tell the best projection directions advance, randomly project the features. This minimizes the likelihood of all projections being in catastrophically poor directions, such those illustrated in Fig. 2
In practice, we generate a random projection matrix P E Nox by sampling values for each dimension from the Gaussian distribution N(0, 1). We project the features of cach element of yielding projected features f:
f P-
(1)
We run the histogram descriptor procedure described above on the projected features. The final descriptor h RN-K is the concatenation of the Np histograms.
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Block
56 X6
Block 2
Block 28 28
14
C0O
Level 1
Lsvel 2
Level
b) c-
2-P
2-2 2+2
t-
246
Figure 3: For both image and time series samples, extract set elements at different granularity. For images (left), the sets of elements are extracted from different ResNet levels. For time scries (right), we take pyramids of windows at different strides around each time step.
3.4 Anomaly Scoring
We perform density estimation on the set descriptors, expecting unusual test samples to have unusual descriptors, far from those of the normal train set. We define the anomaly score as the Mahalanobis distance, the negative log-likelihood in feature space. We denote the mean and covariance the histogram projection features of the normal data as a and Σ:
a(h) (h (h-1)²² (h-1))²
3.5 Connection to Previous Set Descriptors and the Wasserstein Distance
Classical set descriptors. Many prior methods have been used to describe sets of image elements, among them Bag-oo-Features Csurka et al. (2004), VLAD Jégou et al. (2010). and Fisher- Vectors Sáncheez et al. (2013). Those methods begin with a preliminary clustering stage (K-means or Gaussian Mixture Model). They then describe the set. using the zeroth, first, or second moments each cluster. The comparison Appendix shows that our method outperforms clustering- based methods in describing our feature sets.
Wasserstein distance. Our method is closcly related to the Wasserstein distance, which measures the minimal distance required to transport the probability mass from one distribution to the other. As computing the Wasscrstein distance for high-dimensional data such as ours is computationally demanding, the Sliced Wasserstein Distance (SWD) Bonneel et al. (2015), was proposed as an alternative. The SWD, between two sets, I and y, has a particularly simple form:
SWD((a,)) 9) = || hp
(3)
where hpa: hpy are the random projections histogram of sets x and y, that we defined Sec 3.3
As the histogram projections have a high correlation between them, it necessary to decorrelate them. This is done here using a Gaussian model. We refer to Appendix G for an explanation of why Gaussian model is appropriate here. The Mahalanobis distance used here therefore performs better than the simple SWD; distance. While this weakens the connection to the Wasserstein distance, this was crucial for most time-scries datasets (see Table 12]. In practice, we opted to use ANN with the Mahalanobis distance rather than simply computing the Mahalanobis distance to as it worked slightly better (see Appendix
We note that the Sliced Wasscrstein distance can be calculated directly, without using the suggested histogram binning. However, our histogram- based approach is necessary for our density estimation method. Namely, compute &NN using the Mahalanobis distance, we require feature representation for each point, rather than

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relying just pairwise distance function between samples. We compare approach directly with the precise and binned Sliced Wasserstein distance in Appendix
Algorithm Set-based Anomaly Detection with Histogram Projections
1: Input:
Training set & with Ns normal samples, feature extractor F, feature dimension ND, number of projections Np, number of histogram bins K
2: Output: An anomaly detection score
Generate random projection matrix P e NoXNN
for each sample x 8 do do 6: Extract features F(e)
for each element
7: Project features: P- F-
8: end for
9: for each projection dimension j = 1 to Np do
10: Build and histogram for each projected dimension, each with bins
11:
For each sample æ we populate the histogram ha of projected features in the j'th dimension Each histogram is populated with Ng values, the number of
with the projected features fli] elements extracted from each sample
12: end for
n 13: Concatenate histograms for each sample: h2 = - [Np]]
&. resulting dimension is Np
14: end for
15: Compute mcan i and covariance E of hy (the set hs &h x E S))
16: for test sample 전 do
17: Compute ha as above
18: Calculate anomaly score: a(hs) (h2 10 (hs -1)
19: end for
4 Application to Image and Time Series Anomaly Detection
4.1 Images as Sets
Images can be seen as consisting of a set of clements at diffcrent levels of granularity. This ranges from pixels to small patches, low-level elements such lines corners, up to high-level elements such as objects. For anomaly detection, we typically do not know in advance the correct level of granularity for separating between normal and anomalous samples (Heckler et al. 2023). The correct level may depend the anomalies we will encounter, which are unknown during training. Instead, we use multiple levels of gramularity, describing image patches of different sizes, and combine their scores.
In practice, we use representations from intermediate blocks of a pre-trained ResNet (He et al. 2016). As a ResNet network simultancously embeds many local patches of each image, we pass the image samples through the network encoder and extract our representations from the intermediate activations at the end of different ResNet blocks (sce Fig 3) We define cach spatial location in the activation map an element. Note that different blocks have different resolutions, they yield different numbers of elements per layer. We run our set methods with the elements at the end of each residual block used and combine the results in an ensemble as detailed in the appendix (App C.1
4.2 Time Series as Sets
Time series data can be viewed as set of temporal windows. Similarly to images, it is generally not known in advance which temporal scale is relevant for detecting anomalies. Le., what is the duration of windows which includes the semantic phenomenon. Inspired by Rocket Dempster et al. (2020), we define the basic clements of a time series as collection of temporal window pyramids. Each pyramid contains windows. All the windows in pyramid are centered at the same time step, each containing 7 samples (Fig The first level window includes elements with stride 1, the second level window includes elements with stride
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Table Anomaly detection on MVTec-LOCO. ROC-AUC (%). See Tab for the full table.
f-AnoGAN
MNAD
ST
SPADE
PCore
GCAD
SINBAD
Breakfast box
69.4
59.9
68.9
81.8
77.7
87.0
97.7 + 0.2
Juice bottle
82.4
70.5
82.9
91.9
83.7
100.0
97.1 + 0.1
Pushpins
59.1
51.7
59.5
60.5
62.2
97.5
88.9
Logical
Screw bag
49.7
60.8
55.5
46.8
55.3
56.0
81.1 = 0.7
Splicing connectors
68.8
57.6
65.4
73.8
63.3
89.7
91.5 0.1
Avg. Logical
65.9
60.1
66.4
71.0
69.0
86.0
91.2 1 0.8
Breakfast box
50.7
60.2
68.4
74.7
74.8
80.9
85.9 0.7
Juice bottle
77.8
84.1
99.3
84.9
86.7
98.9
91.7 + 0.5
Structural
Pushpins
76.7
90.3
77.6
74.9
78.9
3.7
Screw bag
46.1
56.8
87.0
59.8
86.6
70.5
92.4
- 1.1
Splicing connectors
63.8
73.2
96.8
57.1
68.7
78.3
78.3
0.3
Avg. Structural
62.7
70.2
88.3
66.9
78.9
80.7
85.5
0.7
Avg. Total
64.3
65.1
77.4
68.9
74.0
83.4
88.3 + 0.7
Table 2: Anomaly detection MVTec-LOCO. ROC-AUC (%). See Tab
0 for the full table
SINBAD
ElficientAD (reported) EfficientAD (reproduced)
PUAD
SINBAD ElficientAD
Logical
91.2 - 0.8
86.8
85.9 0.4
92.0
92.7 0.6
Structural
85.5 = 0.7
94.7
93.8 0.4
94.1
95.8 + 0.5
All
88.3 = 0.7
90.8
89.8 = 0.5
93.1
94.2 + 0.6
2, etc. Such window pyramid is computed for each time step in the series. The entire series is represented as set of pyramids of its elements. Implementation details for both modalities are described in Sec C.2
5 Results
5.1 Logical Anomaly Detection Results
Logical Anomalies Datasct. We use the recently published MVTec-LOCO dataset Bergmann et al. (2022) to evaluate our method's ability to detect anomalies caused by unusual configurations of normal elements, This dataset features five different classes: breakfast boz, fuice bottle, pushpins, screw bag and splicing connector (sce Fig M Each class includes: (i) a training set of normal samples (~ 350 samples). (ii) validation sel, containing a smaller set of normal samples (~ 60 samples). (iii) a test set,, containing normal samples, structural anomalies, and logical anomalies (~ 100 each).
The anomalies in each class are divided into structural anomalics and logical anomalies. Structural anomalics feature local defects, somewhat similar to previous datasets such as Bergmann et al. (2019) Conversely, logical anomalies may violate 'logical' conditions expected from the normal data. As one example, an anomaly may include a different number of objects than the numbers expected from normal sample (while all the featured object types exist in the normal class; see [i Other types logical anomalics in the dataset may include cases where distant parts of an image must correlate with one another. For instance, within the normal data, the color of one object may corrclate with the length of another object. Those corrclations may break in an anomalous sample.
Baselines. We compare to baseline methods used by the paper which presented the MVTec-LOCO dataset Bergmann et al. 2022): Variational Model (VM) Steger (2001). MNAD, f-AnoGAN Schlogl et al. (0017)
AE / VAE. Studen! Teacher (ST), SPADE, PatchCore (PCore) Roth el al. (022)). We also compare to GCAD Bergmann et al. (0022) a reconstruction- based method, based on both local and global deep ResNet features, which was explicitly designed for logical anomaly detection; Efficient.AD Batzner et al. (0223)
reconstruction- based method with a loss aimed preventing an autoencoder from reconstructing well
anomalous unseen images. We also report the results by PUAD Sugawara & Imamural 2024 ensemble
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Table Anomaly detection on the UEA datasets, average ROC-AUC (%) over all classes.
See 7 for the complete table with error bounds.
OCSVM
IF
RNN
FD
DSVDD
DAG
GOAD
DROCC
NeuTral
Ours
EPSY
61.1
67.7
80.4
82.6
57.6
72.2
76.7
85.8
92.6
98.1
NAT
86.0
85.4
89.5
91.5
88.6
78.9
87.1
87.2
91.5
96.1
SAD
95.3
88.2
81.5
93.1
86.0
80.9
94.7
85.8
98.9
97.8
CT
97.4
94.3
96.3
79.0
95.7
89.8
97.7
95.3
99.3
99.7
RS
70.0
69.3
84.7
65.4
77.4
51.0
79.9
80.0
86.5
92.3
Avg.
82.0
81.0
86.5
82.3
81.1
74.6
87.2
86.8
94.4
96.8
method combining Batzner et al. (2023) and Rippel ct al. 2021)) h. Finally, we report SINBAD+ EEficiinttA a simple average of EfficientAD's per-sample results and ours. As the last set baselines does not always report per-class accuracies, we report them in a different table.
Metric. Following the standard metric in image-level anomaly detection we use the ROC-AUC metric.
Results. We report per-class results image-level detection of logical anomalies and structural anomalies in Tab Interestingly, we find complementary strengths between our approach and GCAD, a reconstruction- based approach by [Bergmann et al. (2022). Although GCAD performed better on specific classes (e.g., pushpins), our approach provides better results on average. Notably, our approach provides non-trivial anomaly detection capabilities on the screw bag class, while baseline approaches are close to the random baseline performance (which is 50% ROC-AUC). EfficientAD Batzner et al. (2023), focuses on structural anomalies and achieves impressive results on them, but underperforms on logical anomalies (see Tab 2
As our method is comparatively strong specific classes (e.g., Screw bag, Logical, 81.1% compared to 56.0% of GCAD and 55.5% of EfficientDD), it serves as a valuable component in ensemble methods. For example, a simple combination of our method with EfficientAD Batzner el al. (2023 (SINBAD+ SUNNBA++EJicienAADD outperforms all other methods and ensembles (See table 2
Our approach also improves upon dettettion-y--segmenttttt some classes. This is somewhat surprising, as one may assume that would perform well in these cases. One possible reason for that is the high variability of the normal data in some of the classes (e.g., breakfast boz, screw bag, [i This high variability may induce false positive detections for baseline approaches. While different methods provide complementary strengths, on average, our method provides state-of-the-art results in logical anomaly detection. See also the discussion at Sec
methods in detecting structural anomalies in approaches
5.2 Time Series anomaly Detection Results
Time series dataset. We evaluate on the five datasets used in NeurTraL-AD Qiu et al. 2021 RacketSports (RS). Accelerometer and gyroscope recordings of players playing different racket sports. Each sport is designated as a class. Epilepsy (EPSY). Accelerometer recording of healthy actors simulating four activity classes, e.g. an epileptic shock. Naval air training and operating procedures standartization (NAT). Positions of sensors mounted on body parts of a person performing activities. There are six different activity classes the dataset. Character trajectories (CT). Velocity trajectorics of pen on tablet. There are 20 characters in this dataset. Spoken Arabic Digils (SAD). MFCC features ten Arabic digits spoken by 88 speakers.
Baselines. We compare the results of several baseline methods reported by Qiu et al. 20221) The methods cover the following paradigms: One-class classification: One-class SVM (OC-SVM), and its deep versions, DeDDSVDD (*DSVDD*) Rulf et al. (2018), DROCC Goyal et. al. 20200). Tree-based delectors: Isolation Forest. (IF) Liu et al. 2008 Density estimation: LOF, a version of nearest neighbor anomaly detection Breunig ct al. (2000) DAGMM ("DAG") Zong al. (2018) density estimation an auto-cncoder latent space.
Auto-regressive methods RNN and LSTM-ED "ED" deep neural network-based version of auto-regressive
prediction models Malhotra al. 2016)) Transformation prediction GOAD Bergman & Hoshen 2020 and

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NeuTrrL-AAD Qiu et al. (2021) are based on transformation prediction, and are adaptations of RotNet-based approaches such as GFOM Golan & El-Yaniv (2018).
Metric. Following Qiu et al. 2021). we use the series-level ROC-AUC metric.
Results. Our results are presented in Tab. 3 We can observe that different. baseline approaches are effective for different datasets. ANN-based LOF is highly effective for SAD which is a large dataset but achieves worse results for EPSY. Auto-regressive approaches achieve strong results on CT. Transformation- prediction approaches, GOAD and NeuTraL achieve the best performance of all the baselines. The learned transformations of NeuTrraL achieved better results than the random transformations of GOAD. Our method achieves the best overall results both on average and individually on all datasets apart from SAD, where it is comparable but a little lower than Neu Tral. We note that unlike NeuTraaL, our method is far simpler, does not use deep neural networks, and is very fast to train and evaluate. It also has fewer hyperparameters.
5.3 Implementation Details
We provide here the main implementation details for our image anomaly delection application. Further implementation details for the image application can be found in App Implementation details for the time scries application can be found in the supplementary material in App
ResNet levels. We use the representations from the 3rd and 4th blocks of WideResNee50x (resulting in sets size 7 7 and 14 x 14 elements, respectively). We also use all the raw pixels in the image as additional set (resized to 224 x 224 elements). The total anomaly score is the average of the anomaly scores obtained for the set of 3rd ResNet block features, the set of 4th ResNet block features, and the set of raw pixels. The average anomaly score is weighted by the following factors (1, 0,11) respectively (see App D for our robustness to the choice of weighting factor).
Multiple crops for image anomaly detection. Describing the entire image as a single set might sometimes lose discriminative power when the anomalies are localized. To mitigate this issue, we can treat only part of an image as our entire set. To do so, crop the image to smaller images by factor of c, and compare the elements laken from each crop. We compute an anomaly score for each crop scale and for each center location. We then average over the anomaly scores of the different crop center locations for the same crop scale c. Finally, for each ResNet level, we average the anomaly scores over the diffcrent crop scales. We use crop scales of (1.0, 0.5, 0.33). The different center locations are laken with a stride of 0.25 of the entire image. We observe that combining different scales offers only a marginal advantage (see Tab(4), while baseline methods require significantly more forward passes through the feature extractors Roth et al. (2022)
Runtime. A simple implementation of our method can run in real time (> 20 images per second) without multiple crops. Using multiple crops can be simply parallelized on multiple GPUs.
5.4 Ablations
We present ablations for the image logical AD methods. For further ablations of the histogram parameters and for the time series application, see appendix F
Using individual ResNet levels. In Tab 4 we report the results when different components of our multi-level ResNet ensemble are removed. We report the results using only the representation from the third or fourth ResNet block (*Only 3" /OOnly 4"). We also report the results using both ResNet blocks but without the raw-pixels level (*No Pixels").
No multiple crops ablation. We also report results without the multiple crops ensemble (described in Sec. 5.3). We foed only the entire image for the set extraction stage (*Only full"). As expected, using multiple receptive fields is beneficial for classes where small components are important to determine abnormality.
Ablating our histogram density-estimation method. In Tab we ablate different aspects of our histogram set descriptors. Sim. Aug. We show a simple averaging Lee et al. (2018) of the set features (sce also Fig. 2 ablating our entire set-features approach. This yields a significantly worse performance. No We replace the random histograms with similar
proj. We ablate our use of random projections 3.3)
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Table Ablation for logical image AD. Logical Anomalies. ROC-AUC (%).
Only 3
Only
No pixels
Only full
Ours
Breakfast box
95.9
95.7
96.8
97.2
97.7
Juice bottle
93.0
97.0
95.8
97.0
97.1
Pushpins
79.2
67.0
74.0
89.9
88.9
Screw bag
79.8
70.4
76.6
76.2
81.1
Splicing connectors
84.7
85.6
86.1
90.7
91.5
Average
86.5
83.1
85.9
90.2
91.2
Table 5: MVTec-LOCO ablation (no raw-pixels). Logical Anomalics. ROC-AUC (%).
Sim.
Avg.
No Proj.
No Whit.
Ours
Breakfast box
84.6
91.7
95.9
97.0
Juice bottle
98.0
97.3
97.5
96.2
Pushpins
63.5
69.3
73.4
73.7
Screw bag
65.0
68.2
72.5
77.5
Splicing connectors
87.4
84.5
87.9
85.9
Average
79.7
82.2
85.5
86.1
histograms using the raw given features. No whil. We ablate our Gaussian modeling of the set features. The whitening is not essential for the image modality, as it is for the time-series data (see Tab[12). We compare these variants of our mothod using the 3rd and 4th ResNet blocks, as the raw pixels level adds significant variance overshadowing the difference between some of the allernatives. While ablation may give stronger results in specific cases, our set approach together with the random projections and whitening generally outperforms.
Ablating the number of bins and the number of projections. While generally would like to have as many random projections as possible; and large number of bins per histogram, long we have enough statistics to estimate the occupancy in each of them; We find that in practice the values we choose are large enough. We show in the App. II that while significantly lower values in these parameters degrade our performance, the benefit from using larger values saturates.
6 Discussion
Complementary strength density estimation and reconstruction based approaches for logical anomaly detection. Our method and GCAD Bergmann et al. (2022), a reconstruction- based approach, exhibit complementary strengths. Our method is most suited to detect anomalies resulting from the distribution of featured objects in each image. E.g., object replacements, additional or missing objects, components indicating logical inconsistency with the rest of the image. The generative modeling by GCAD gives stronger results when the positions of the objects are anomalous. E.g,, one object containing another when it should not, or vice versa, as in the Pushpins class. The intuition here that our approach treats the patches as an unordered set, and might not capture exact spatial relations between the objects. Therefore, it may be a natural direction try and use both approaches together. A practical way take advantage of both approaches would be an ensemble we suggest in Sec Ultimately, future research is likely to lead to the development of better approaches, combining the strengths of both methods.
Rclation to previous random projection methods. Our method is related to several previous mothods. HBOS Goldstein & Dengel 2012) and LODA Pevný (016)) also used similar projection features for anomaly detection. Yet, these methods perform Histogram-basee based density, ignoring the dependency across projections. As they can only be applicd to single element, they do not achieve competitive performance for time series AD. Rocket and mini-rocket Dempster al. (2020) 2021) also average projection features across selected windows from given sample but do not apply to image data.
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Additional discussion points can be found in App
7 Limitations
Detecting structural anomalies. Our approach aims to detect specific, yet important, types of anomalies
image-level logical anomalies and the analogues time-series sequence-level anomalies. It is not particularly effective for detecting local structural anomalics, such scratches or dents in images of objects Bergmann et al. 2019 Zou et al. (2022), object-level anomalies Reiss et al. (2021), or local time-series anomalies Blázquez- García et al. [2021]. We evaluate our method on appropriate datasets. Currently, only one dataset evaluates logical anomalies Bergmann al. (2022). Yct, this very comprehensive datasot contains 5 different. sub-tasks, where each sub-task features numerous different types of anomalies. An anomaly detection method wishes to discover (Reiss al., method with
must. rely on some assumptions regarding the nature of the anomalies
2023 Therefore, when the type of anomalies is unknown, methods tailored to different types of anomalies (Reiss et al. 2022 Roth et al. 2022 Batzner et al. 2023
recommend combining
Element-level anomaly detection. Our method focuses on sequence-level time series and image-level anomaly detection. some applications, & user may also want a segmentation map the most anomalous elements of each sample. We note that for logical anomalies, this often not well defined. E.g., when we have an image with 3 nuts as opposed to the normal 2, each of them may be considered anomalous. To provide clement-level information, our method can be combined with current segmentation approaches by incorporating the knowledge of global anomaly. E.g.; removing false positive segmentations if an image is normal. Directly applying our set features for anomaly segmentation is left for future research.
Class-specific performance. In some classes we do not perform as well compared to bascline approaches. A better understanding of the cases where our method fails would be beneficial for deploying it practice.
8 Conclusion
We presented a method for detecting anomalies caused by unusual combinations of normal elements. We introduce set features dodicated capturing such phenomena and demonstratc their applicability for images and time series. Extensive experiments established the strong performance of our method. As with any anomaly detection method, our approach is biased to detect some abnormality modes rather than others. Using few anomaly detection methods together may allow enjoying their complimentary benefits, and is advised in many practical cases.
9 Acknowledgment
This research was partially supported by the Israeli Science Foundation, the Israeli Council for Higher Education (VATAT), the Hebrew University Data Science grants (CIDR), the Malvina and Solomon Pollack scholarship, and a grant from KLA. Niv Cohen was partially supported by the Israeli data science scholarship for outstanding postdoctoral fellows (VATAT). We thank Paul Bergmann for kindly sharing numerical results for many of the methods compared on the MVToc-LOCC dataset.
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Contents
Introduction
1
Previous work
2
3
Set Features for Anomaly Detection
4
A Set is More Than the Sum of its Parts
4
Preliminarics
4
3.3
Set Features by Histogram of Projections
Anomaly Scoring
6
3.5
Connection Previous Set Descriptors and the Wasserstein Distance
Application to Image and Time Series Anomaly Detection
Images as Sets
2
Time Series as Sets
Results
Logical Anomaly Detection Results
5.2
Time Series anomaly Detection Results
5.3
Implementation Details
10
4
Ablations
10
6
Discussion
11
Limitations
12
Conclusion
12
Acknowledgment
12
A Full Results Tables
19
B Set descriptor comparison
19
C Implementation Details
20
Image Anomaly Detection
20
C.2 Time Series Anomaly Detection
21
D Logical Anomaly Detection Robustncss
21
E Further Image Anomaly Detection Ablation
21
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F Time Series Anomaly Detection Ablations
21
G Using the Central Limit Theorem for Set Anomaly Detection
23
Further Discussion
23
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A
Full Results Tables
The full table for the image logical anomaly detection experiment can be found in Tab The full table for the time series anomaly detection experiments including error bounds for our method and baselines that reported them can be found in Tab 7 The difference between the methods is significantly larger than the standard error.
Table 6: Anomaly delection on the MVTec-LOCO dataset. ROC-AUC (%).
VM
AE
VAE
f-AG
MNAD
Logical Anom.
Breakfast box
70.3
58.0
47.3
69.4
59.9
Juice bottle
59.7
67.9
61.3
82.4
70.5
Pushpins
42.5
62.0
54.3
59.1
51.7
Screw bag
45.3
46.8
47.0
49.7
60.8
Splicing connectors
64.9
56.2
59.4
68.8
57.6
Avg. Logical
56.5
58.2
53.8
65.9
60.1
Structural Ano.
BBeekffast box
70.1
47.7
38.3
50.7
60.2
Juice bottle
69.4
62.6
57.3
77.8
84.1
Pushpins
65.8
66.4
75.1
74.9
76.7
Screw bag
37.7
41.5
49.0
56.8
Splicing connectors
51.6
64.8
54.6
63.8
73.2
Avg. Structural
58.9
56.6
54.8
70.2
Avg. Total
57.7
57.4
54.3
64.3
65.1
ST
SPADE
PCore
GCAD
SINBAD
Breakfast box
68.9
81.8
77.7
87.0
97.7 + 0.2
Structural Ano. Logical Anom.
Juice bottle
82.9
91.9
83.7
100.0
97.1 = 0.1
Pushpins
59.5
60.5
62.2
97.5
88,9 = 4.1
Screw bag
55.5
46.8
55.3
56.0
81.1 1 0.7
Splicing connectors
65.4
73.8
63.3
89.7
91.5 1 0.1
Avg. Logical
66.4
71.0
69.0
86.0
91.2 1 0.8
Breakfast box
68.4
74.7
74.8
80.9
85.9 1 0.7
Juice bottle
99.3
84.9
86.7
91.7 - 0.5
Pushpins
90.3
58.1
77.6
74.9
78.9 + 3.7
Screw bag
87.0
59.8
86.6
70.5
92.4 + 1.1
Splicing connectors
96.8
57.1
68.7
78.3
78.3 - 0.3
Avg. Structural
88.3
66.9
78.9
80.7
85.2
- 0.7
Avg. Total
77.4
68.9
74.0
83.4
88.3
1 0.7
B Set descriptor comparison
Clustering- set descriptors. We compare our histogram-based approach to the VLAD and Bag-of- Features approaches. We report our results in Tab We use the number of mcans 100 cluster, but this result. persists when varied the number of clusters. We do not report the results on Fisher-Vectee Vectors the underlying GMM model (unlike K-means) requires unfeasible computational resources with set dimensions. Taken together, it soems that the underlying clustering assumption does not fit the sets we wish to describe as well our set descriptors.
KNN versus distance to the mean. We found that using the Gaussian model only to whiten the data and taking the distance the nearest neighbors (and not the center) worked better for the MVToc-LOCO dataset (see Tab(3). The nearest neighbors density estimation algorithm better models the density distribution when the Gaussian assumption is not accurate description of the data.
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Table UEA datasets, average ROC-AUC (%) over all classes including error bounds
OCSVM
IF
T.OF
RNN
LSTM-ED
EPSY
61.1
67.7
56.1
80.4 + 1.8
82.6 + 1.7
NAT
86
85.4
89.2
89.5 + 0.4
91.5 + 0.3
SAD
95.3
88.2
98.3
81.5 + 0.4
93.1 + 0.5
CT
97.4
94.3
97.8
96.3 + 0.2
79.0 = 1.1
RS
70
69.3
57.4
84.7 + 0.7
65.4 - 2.1
Avg.
82.0
81.0
79.8
86.5
82.3
DeepSVDD
DAGMM
GOAD
DROCC
NeuTraL
Ours
EPSY
37.6 + 0.7
72.2 + 1.6
76.7 + 0.4
85.8 + 2.1
92.6 + 1.7
98.1 + 0.3
NAT
88.6 + 0.8
78.9 + 3.2
87.1 + 1.1
87.2 + 1.4
94.5 + 0.8
96.1 + 0.1
SAD
86.0 0.1
80.9 1.2
94.7 + 0.1
85.8 + 0.8
98.9 = 0.1
97.8 + 0.1
CT
95.7 + 0.5
89.8 + 0.7
97.7 + 0.1
95.3 + 0.3
99.3 0.1
99.7 + 0.1
RS
77.4 0.7
51.0 4.2
79.9 + 0.6
80.0 + 1.0
86.5 + 0.6
92.3 + 0.3
Avg.
81.1
74.6
87.2
86.8
94.4
96.8
Table 8: MVTec-LOCO ablation: using no raw-pixels level. ROC-AUC (%).
Mahalanobis (dist. 14)
BoF
VLAD
Ours
Breakfa.
93.6
84.7
87.9
97.6
Juice bo.
91.6
93.8
97.5
97.0
Pushpins
79.9
78.2
79.1
88.6
Screw b.
68.2
69.9
64.1
81.7
Splicing,
78.2
Average
82.3
82.3
83.7
91.2
C Implementation Details
Histograms.
We use the cumulative histograms as our set features for both data modalities (of S.c.3.3) 3.3
C.1 Image Anomaly Detection
Parameters. For the image experiments, we use histograms of K bins and 1000 projections. For the raw-pixels layer, used projection dimension of 10 and no whitening due to the low number of channels. To avoid high variance between runs, we did 32 different repetitions for the raw-pixel scoring and used the median score. We use k = for the ANN density estimation.
Preprocessing. Before feeding each image sample to the pre-trained network we resize to 224 x 224 and normalize according the standard ImageNet mean and variance.
Considering that classes in this dataset are provided in different aspect ratios, and that similar objects may look different when resized to square, found it beneficial to pad each image with empty pixels. The padded images have 1 aspect ratio, and resizing them would not change the aspect ratio of the featured objects.
Software. For the whitening of image features we use the ShrunkCovuriance function from the scikit-learn library Pedregosa et al. (2011) with its default parameters. For KNN density estimation we use the faiss library Johnson et al. (2019).
Computational resources. The experiments
on single RTX2080-GT GPU.
20

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Published in Transactions on Machine Learning Research (12/2024)
Table 9: Robustness to the choice of A. Average ROC-AUC (%) on logical anomalies classes.
0.2 0.1 (Ours)
0.05
0.02
90.2
91.2
91.4
90.7
Table 10: Ablation for values of r, the number of random projections. Average ROC-AUC MVTec-LOCO, logical anomalies. K =5, 5, = 0.6 (%).
2000
1000
500
200
100
Avg. Logical 91.2
91.2
90.6
89.6
86.1
C.2 Time Series Anomaly Detection
UEA Experiments. We used each time series as an individual training sample. We chose kernel size of - 9, r = 100 projection, K = 20 quantiles, and a maximal number of pyramid levels of L = 10, cach using consecutive strides. The results varied only slightly within a reasonable range of the hyperparameters. E.g. using 5, 10, 15 levels yielded an average ROCAUC 97, 96.8, 96.8 across the five UEA datasets.
Padding. Prior window extraction, the scries x is first right and left zero-padded by to form a padded series x'. The first window wi is defined as the first observations in padded series S' i.e. wi +2, We further define windows at higher scales W*, which include observations sampled with stride At scale e, the original series is right and left zero-padded by T to form padded series S
Spoken Arabic Digils processing We follow the processing of the dataset as done by Qiu et al. Qiu et al. (2021). In private communications the authors explained that only use sequences of lengths between 20 and time steps were selected. The other time series were dropped.
Computational resources. The experiments were run on a modest number of CPUs on computing cluster. The baseline methods were run single RTX2080-GT GPU.
D Logical Anomaly Detection Robustness
We check the robustness of our results to the parameter À the weighting between the raw-pixels level anomaly score lo the anomaly score derived from the ResNet features 5.3 As can be seen in Tab our results are robust to the choice of X.
E Further Image Anomaly Detection Ablation
We include here the ablation tables for the number of random projections and number of bins, for logical anomaly detection (Tab 10
F Time Series Anomaly Detection Ablations
Number of projections. Using a high output dimension for projection matrix P increases the expressively but also increases the computation cost. We investigate the effect of the number of projections on the final accuracy of our method. The results are provided in Fig. We can observe that although a small number of
Table 11: Ablation for values K, the number of bins per histogram. Average ROC-AUC on MVTec-LOCO, logical anomalies. r == 1000, a 0.6 (%).
K
20
10
3
2
Avg. Logical
91.1
91.3
91.2
91.2
90.8
90.2
21

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Published in Transactions on Machine Learning Research (12/2024)
96
96
ROCAUC/9)
94-
ROCAUCINS
84
90
82-
R5
R5
86
EPSY
NA
NA
125
150
Number of prolectionss
Number bins
Figure 4: Ablation of accuracy vs. the number of projoctions (left) and the number of bins (right) for different time series datasets.
Table 12: An ablation of distance calculation methods for different time series datasets. ROC-AUC (%)..
EPSY
RS
NA
CT
SAD
Quantized SWD (No whitening)
62.1
70.9
93.6
98.5
78.8
SWD
90.7
84.8
91.9
99.3
88.4
With Whitening
98.1
92.3
96.1
99.7
97.8
projections hurts performance, even a moderate number of projections is sufficient. We found 100 projections to a good Iradeoff between performance and runtime.
Number of bins. We compute the accuracy of our method function of the number of bins per projection. Our results (Fig. 4 show that beyond very small number of bins, a larger number of bins does not help. We found 20 bins to be sufficient in all our experiments.
Effect of Gaussian density estimation. We ablate the use of Gaussian modeling in Tab. 12 (Quantized SWD). We can see that our approach achioves far better results, attesting to the importance of modcling the correlation between projections.
Comparison to the Sliced Wasserstein distance. In Sec we highlight the connection between our approach and the Sliced Wasserstein distance. An empirical comparison between the approaches can be found in Tab. 12] Our results show that computing the SWD without histogram binning be much more accurate than with binning (Quantized SWD). However, the binning is necessary for our whitening technique (Sec 3.4,, which significantly outperforms standard SWD. We also note that increasing the number of bins (making the quantization finer) does not improve the accuracy of our full approach.
Comparing projection sampling methods. We compare three different projection selection procedures: (i) Gaussian: sampling the weights in P from a random Normal Gaussian distribution (ii) Using an identity projection matrix: P= T_ (iii) PCA: selecting P from the eigenvectors of the matrix containing all (raw) features of all training windows. PCA selects the projections with maximum variation but is computationally expensive. The results are presented in Tab. 13 We find that the identity projection matrix under-perfornet the other approaches (as it provides no variable mixing). Surprisingly, we do not see a large difference between PCA and random projections.
Table 13: An ablation of projection sampling methods for different time series datasets. ROC-AUC (%).
EPSY
RS
NA
CT
SAD
ld.
97.1
90.2
91.8
98.2
78.3
PCA
98.2
91.6
95.8
99.7
96.7
Rand
98.1
92.3
96.1
99.7
97.8
22

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Published in Transactions on Machine Learning Research (12/2024)
Table 14: An ablation of time-series number of pyramid levels. ROC-AUC (%), 9.
5
10 (Ours)
12
15
Avg. time-scrics 96.7
96.9
96.8
96.8
96.7
Table 15: An ablation of time-series window size. ROC-AUC (%), 7 == 10.
(Ours)
Avg. time-scries 96.8
96.8
96.8
96.8 96.6
Effect number of pyramid levels and window size. We ablate the two hyperparameters of the time-series feature extraction: the number of pyramid windows used L, and the number of samples per window (see Sec 4.2 We find that in both cases the results not sensitive to the chosen parameters
(Tab 14 15
G Using the Central Limit Theorem for Set Anomaly Detection
We model the features of cach window f as normal set IID obscrvations coming from probability distribution function p(f). The distribution function is not assumed to be Gaussian. Using Gaussian density estimator trained on the features of elements observed during training is unlikely to be effective for element-level anomaly detection (due to the non-Gaussian p(f)).
An alternative formulation to the one presented in section 3 is that each feature f is multiplied by projection matrix P. and then each dimension is discretized and mapped to a one-hot vector according to its relevant histogram bin. This formulation therefore maps the representation of each element to a sparse binary vector. The mean of this one-hot vector representation of elements in the set recovers the normalized histogram descriptor precisely (therefore this formulation is equivalent to the one in section 3 As the histogram is a mcan of the onc-hot representations of elements, it has superior statistical properties. In particular, the Central Limit Theorem states that under common conditions the sample mean follows the Gaussian distribution. While typically in anomaly detection only a single sample is presented at a time, the situation is different when trcating samples as sets. Although the clements are often not IID, given multitude of elements, an IID approximation may still be applicable. This explains the high effectiveness of Gaussian density estimation in our formulation.
H Further Discussion
Incorporating deep features for time series data. Our method outperforms the state-of-the-art in time series anomaly detection without using deep neural networks. While this is interesting and surprising result, we believe that deep features will be incorporated into similar approaches in the future. One direction for doing this is replacing the window projection features with a suitable deep representation, while keeping the set descriptors and Gaussian modeling steps unchanged.
Fine-tuning deep features for anomaly detection. Following recent works in anomaly detection and anomaly segmentation, we used fixed pro-trained fcatures as the backbone of our method. Although some methods fine-tune deep features for anomaly detection based on the normal-only training set, we keep them constant. Doing allows interpretable examination of the relative strength of our novel scoring function with respect to prior works that use fixed features. Yet, we expect that fine-tuning such features could lead to further gains in the future.
23


Reviews:
[{"review_id": "V5rqSfyFk2", "reviewer": "Reviewer_aQQj", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n- The paper introduces a novel approach to anomaly detection that focuses on logical anomalies, which is an important aspect often overlooked in existing methods.\n- The use of projections and quantization simplifies the feature representation, making the method computationally efficient.\n\n## Weaknesses\n\n- There is uncertainty about whether simplifying the feature representations by using histograms actually improves performance, given that Sliced Wasserstein (SW) distance can already efficiently operate on discrete distributions (sums of Diracs) without requiring quantization into histograms.\n- The presentation in Section 3.5 seems misleading as it suggests the necessity for quantization for SW when, in reality, SW could be directly applied to raw distributions of features.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper aims to develop a novel method for detecting anomalies in images or time series data, specifically targeting what the authors term \"_logical_ anomalies.\" These anomalies arise from unusual combinations of items rather than from the appearance of an unusual item itself. The proposed method simplifies bags of local features for computational efficiency by projecting them into 1D space and then quantizing them to obtain histograms.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "## Critical Adjustments\n\n1. To strengthen the paper, a quantitative comparison between the proposed method and SW distance without quantization should be conducted. This would help validate the claim that histograms indeed offer an advantage.\n2. The authors should clarify their statement about quantization in Section 3.5 to avoid any misleading information for readers.\n\n## Non-Critical Adjustment\n\nCorrect all typos in the text, such as \"article amsmath algorithm algorithmic\" on page 6.", "broader_impact_concerns": "-", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "6sTsu9V4YP", "reviewer": "Reviewer_p7s6", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The main strength of the paper is the quality of the empirical results obtained, in particular on the logical anomalies in the MVTec-LOCO benchmark. The ablations are also helpful. The discussion in Appendix H helps motivate why it is particularly reasonable to use a Gaussian density estimation model in the context of random-projection-based methods.\n\nThe main weakness is that the method's description is vague and confusing. In particular, the paper claims that there are two main distinctions from the closely related prior work by Pevny, LODA, using a different density estimator and \"using sets of multiple elements rather than single sample descriptions\". While there is some further, helpful discussion of the difference in the density estimator in Appendix G/Table 13, I don't fully understand this \"sets of elements\" description; the paragraph \"Histogram descriptor\" in Section 3.3 is especially confusing. Section 3.2 says each example is described by a set of elements (patches or windows), but then in 3.3 there's a set of values for each dimension. What do the dimensions range over -- does this correspond to the elements of section 3? But when you refer to it as a set, it sounds like you mean that it is unordered -- if so, how does this indexing make sense? Are the projections simply computed on some raw representation of the data (interpreting patches as flattened into vectors of pixels, for example)?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work proposes a variant of a random projection method for anomaly detection; the method applies these projections to features extracted from pretrained deep neural networks in images and simple window pyramids in time series, and (effectively) uses a model of the projections as Gaussian for density estimation. It obtains strong performance on a benchmark for \"logical\" anomalies that had largely eluded previous efforts. The paper also exhibits a number of ablations evaluating the contribution of various components and hyperparameter settings to these results.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Following the above, my main request is a precise and complete description of the method, preferably using complete pseudocode or mathematical definitions of the components, in particular the histograms.", "broader_impact_concerns": "There is no broader impact statement, but I am not particularly concerned about this for this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "qZ7w72wOl2", "reviewer": "Reviewer_HGet", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**: \n\n1. The set representation for anomaly detection is novel to me.\n2. Evaluating the method on the time series task is important.\n\n\n**Weaknesses**:\n\n1. It seems that the proposed method is only applicable to a very specific case of anomaly detection, where the combinations of elements are unusual. The authors should at least discuss the applicability of the proposed method in other scenarios.\n2. There are too many notations that are not properly defined. For example, when it comes to a set, we usually use curly brackets. In Section 3.2, it first use $x1, x2… x_{N_S} $ and then $[e_1, e_2..e_{N_E}]$ to define sets. In addition, $ e_1, e_2..e_{N_E}$ are not formally defined at all. Therefore, it is hard for me to understand the formulation from the very beginning. Due to a similar reason, it is also hard for me to understand the histogram descriptor in Section 3.3 (e.g., $s[j]$, $f[j]$).\n3. The Mahalanobis distance is adopted when computing the anomaly score in Section 3.4. The implicit assumption here is the histogram projection follows a Gaussian distribution, which does not necessarily hold in practice. The authors should at least have some discussions on it. \n4. The proposed method is only evaluated on two datasets. Other benchmarks (e.g.,VisA), should also be considered.\n\nOverall, I don't think this work is ready to be published in a top journal such as TMLR.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "**Summary**: This paper proposes a novel method for anomaly detection using set features, where anomalies are identified based on unusual combinations of normal elements. Specifically, the authors introduce a technique that models each sample as a distribution of its elements and uses random projection histograms for feature representation, followed by density estimation for anomaly scoring. The proposed method is evaluated on two benchmark datasets against several baseline algorithms.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "As in Weaknesses, in order to meet the criteria of a top-tier venue such as TMLR, the authors need to\n\n1. make major revisions to section 3 to improve the presentation of the methodology\n2. revise section 5 to make a more comprehensive evaluation", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "kv0PaUjm8U", "reviewer": "Reviewer_UZyq", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. Anomaly detection is a critical task warranting thorough investigation.\n2. Modeling sample distributions shows promise and effectiveness in anomaly detection.\n\n\nWeaknesses:\n1. The Introduction should clarify existing solutions to this issue and compare them with the proposed approach.\n2. In the experiments, it is important for all baselines to report standard deviations to assess their robustness. Additionally, comparing with recent baselines used in time series anomaly detection would strengthen the evaluation.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses anomaly detection where anomalies are characterized by unusual combinations of normal elements. The authors propose set features that model each sample based on the distribution of its elements. Experiments across several datasets demonstrate the effectiveness of the proposed model.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see the weaknesses.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The authors propose an approach to improve anomaly detection when the anomaly is the result of anomalous combinations of features, and not of single anomalous features. Experiments that include ablation studies validate the claims. The proposed method could be of interest for those studying anomalies of sets of objects and other logical anomalies.
Decision: Accept (poster)