Unsupervised Anomaly Detection through Mass Repulsing Optimal Transport
Authors: Eduardo Fernandes Montesuma
EL HABAZI Adel
Fred Maurice NGOLE MBOULA
Venue: Accepted by TMLR ()

Abstract: Detecting anomalies in datasets is a longstanding problem in machine learning. In this context, anomalies are defined as a sample that significantly deviates from the remaining data. Meanwhile, Optimal Transport (OT) is a field of mathematics concerned with the transportation, between two probability distribution, at least effort. In classical OT, the optimal transportation strategy of a distribution to itself is the identity, i.e., each sample keeps its mass. In this paper, we tackle anomaly detection by forcing samples to displace its mass, while keeping the least effort objective. We call this new transportation problem Mass Repulsing Optimal Transport (MROT). Naturally, samples lying in low density regions of space will be forced to displace mass very far, incurring a higher transportation cost. In contrast, samples on high density regions are able to send their mass just outside an \emph{exclusion zone}. We use these concepts to design a new anomaly score. Through a series of experiments in existing benchmarks, and fault detection problems, we show that our algorithm improves over existing methods. Our code is publicly available at https://github.com/eddardd/MROT

Published in Transactions on Machine Learning Research (05/2025)
Unsupervised Anomaly Detection through Mass Repulsing Optimal Transport
Eduardo Fernandes Montesuma"
edumfern@@gmailcom com
Université Paris-Saae Paris-Saclay, CEA, List, F-91120 Paloiseau France
Adel el Habazi
adelelladxazi@@mmail.comm
Eoole Centrale de Nantes, Nantes, France
Fred Ngolè Mboula
fred frdd-maunriee.mgdemmmbddeeee
Université Paris-Sor Paris-Sacly,, CEA, List, F-91120 Palaiseau France
Reviewed on OpenReview: https: // openreview. net/ formm?? id=PPG33VVEN
Abstract
Detecting anomalics in datasets is longstanding problem in machine lcarning. In this conlext, anomalies are defined as a sample that significantly deviales from the remaining data. Meanwhile, Optimal Transport (OT) is a field of mathematics concerned with the transportation, between two probability distribution, at least effort. In classical OT, the optimal transportation strategy of distribution itself is the identity, i.e., each sample keeps its mass. In this paper, we tackle anomaly detection by forcing samples to displace its mass, while keeping the least effort objective. We call this new transportation problem Mass Repulsing Optimal Transport (MROT). Naturally, samples lying in low density regions of space will be forced to displace mass very far, incurring a higher transportation cost. In contrast, samples on high density regions are able to send their mass just outside an exclusion zone. We use these concepts to design a new anomaly score. Through a series of experiments in existing benchmarks, and fault detection problems, we show that our algorithm improves existing methods. Our code is publicly available at Q https: // //github.com wotpe//rdd///d//TTT
1 Introduction
An anomaly, or an outlier, is a data point that significantly different from the remaining data Aggarwal 2017), to such an extent that was likely generated by a different mechanism (Hawkins) 1980). From the perspective of machine learning, Anomaly Delection (AD] wants to determine, from set of examples, which ones are likely anomalies, typically through interpretable score. This problem finds applications in many different fields, such as medicine (Salem et al. 2013), cyber-sccurity (Siddiqui et al.) 2019), and system monitoring [[sermann 2006), to name a few. As reviewed in Han et al. (2022), existing techniques for AD are usually divided into unsupervised, semi-supervised and supervised approaches, with an increasing need for labeled data. In this paper, we focus on unsupervised AD which does not need further labeling effort in constituting datasets.
Meanwhile, Optimal Transport (OT) is a field of mathematics concerned with the transportation of masses at least effort Villani et al. (2009). In its modern treatment, one conceptualize transportation problems between probability distributions, which has made an important impact in machine learning rescarch (Monte-
suma et al. 2024a). Hence, OT is an appealing tool, as it can be estimated non- poraaaaaetrriall from samples
from probability distributions. Likewise, the plethora of computational tools for computing OT Peyré &
Cuturi 2020 Flamary et al.] b021 further stresses its usability.
*These authors contributed equally this work.

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In this context, the application of OT for AD is not straightforward, single probability distribution. We present new OT problem ween a distribution and itself, by restricting where a sample can send its mass to. More specifically, we design an exclusion zone, prohibiting samples from koeping its mass, or sending its mass to a small vicinity. Espocially, we assume that anomalies lic in low-density regions space, with only few samples their vicinity. By restricting the transport of mass in the vicinity of samples, anomalics are naturally forced to send their mass the high-density region, which is assumed to be far away from the anomaly samples. Hence, anomalies will have an overall higher transportation effort than normal samples, which find nearby samples outside the exclusion zone. We show conceptual illustration of our method in Figure 1
interested in analyzing
Although OT has been previously used to compare and aggregate signals in the context of AD Alaoui-
Belghiti et al. 2019 2020), to the best of our knowledge ours is the first general purpose OT based algorithm for AD hermhermore, we propose a new [TT problem based on the engineering of the ground-cost, which has
links to OT with repulsive costs (Di Marino et al., 2017]. We benchmark our algorithm comprehensive list. of datasets, including tabular, computer vision, and natural language processing proposed by HHan et. al.)
(2022), besides fault detection (Reinartz et al.
2021 Montesuma et al. 2024b
Normal samples
Nermal sxmplees
En (6dalt
Anemalion
Eagineered ground- cost
Toasport plan
Tromportation offet
2) between samples of distribution P. That way, samples are forced to send their mass outside a vicinity defined through its nearest neighbors (shaded gray areas). This design leads to transportation plan, such that anomalous samples incur in an higher transportation effort than normal samples. We use these efforts to build an interpretable and generalizable anomaly score.
Figure 1: Mass Repulsive Optimal Transport. Our method engineers the ground-cost c(x, ((()))=)())))
This paper is organized as follows. Section 2 discusses related work in AD and OT Section g discusses our proposed method, called Mass Repulsing Optimal Transport (MROT) Section covers our experiments. Finally, section 3 concludes this paper.
2 Related Work
A Anomaly Detection. Following Han et al. (2022), AD methods can be maïnly divided into categories. First, supervised methods consider AD through the lens of binary classification under class imbalance. Second, semi-supervised methods cither consider partially labeled data, or labeled normal samples, that algorithm can characterize what normal sample is. The third, more challenging category is unsupervised where the training data contains both anomalies and normal samples and labels are not available. This paper considers precisely the last setting. Next, we review ideas in unsupervised AD
AD
The first kind of methods rely on encoder-deoode architectures to detect anomalies. The insight that, by embedding data in a lower dimensional space, anomalics can be detected via the reconstruction error of the auto-encoding function. This is the principle of PPrnccpal Component Analysis (PCA) Shyu et al. (2003) which employs linear encoding and decoding functions, but also of kernelized versions Schölkopf et al. 1997]
Hoffmann (2007), as well as neural nots Vincent et al. (2088) Bengio et al. (2013). which rely on mon-lincar embedding techniques.
The second type strategies are based on the paradigm of 1-class classification. As Schölkopf et al. 1999)
puts, the idca is to define a function that outputs on a small, dense region of the space where normal samples lie, and 1 elsewhere. In this context, [Schölkopf et al. (1997p extends the celebrated Support Vector Machine (SVM) to AD and Liu et. al. (2008) extends Random Forests (RFs) of Breiman 2001
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A third kind of approaches focuses on neighborhoods and clustering, to model the data underlying probability distribution, especially through the density of samples over the space. This is the case ofl-- Nearest Neighbors (K-NV) (Ramaswany et al., 2000), who use distances and nearest neighbors to determine anomalies, Local Outlier Factor (LOF) Breunig et al. (2000), who devised a score that measures the local deviation of a sample
with respect. its neighbors. Finally, Clustering-based LOF (CBLOF) He et al. (2003) proposed an extension of LO basod on the relative sizes of clusters within the data.
Deep Learning-based Anomaly Detection. As Pang et al. 2021 reviews, deep learning is used for AD in mainly 3 ways: for i) learning powerful feature extractors, ii) learning representations of normality, and iii) building an end-to-end anomaly score. Besides iii, i and ii allow for the synergy between the deep and
traditional AD algorithrss. In fact, one can use classical AD strategies over the latent space of a neural net. We explore this possibility in Section 4.1
In addition to this possibility, we give a few examples of how deep learning has contributed to AD For example, Rulf al. (2018) proposed strategy for performing deep one class classification, which echoes the ideas of One Class SVM (OCSVM) Schölkopf et al. (1999)
Furthermore, generative modeling has deeply contributed to AD For instance, generative adversarial nets [Goodfellow et al. 2014) can serve as a modeling step for capturing features of the underlying distri- bution P. For instance, in Schlegl et al. (2019) and Zenati et al. 2018 the authors combine reconstruction errors with the distance of the predictions of the discriminator for building an anomaly score. [Akeay et al. (2018) takes this idea further. This strategy creates a doaeddr--doodd--eeeeede
architecture for the generator network. The model is trained model normal images, by mapping them to the latent space, reconstructing back the image, then mapping again to the latent space. Anomalous images are then detected via the usual reconstruction loss, plus the distance between latent representations and the difference between discrimina- tor's predictions. Another direction consists of using variational inference Dias et al. 2020 or diffusion models [Livernoche et al.] 2024).
As we cover in the next section, our method uses nearest neighbors and OT to model, non-paramettriallly the density of samples over the space. More specifically, we prohibit samples in OT to keep their mass, or sending it over a region of space defined through their NN Differently from Ramaswamy et al. (2000) and [Breunig al. (2000), we do not rely on distances, which might not have meaning in high-dimensions. Rather, we rely on the elfort of transportation, measured through the samples' mass times the ground-cost.
Optimal Transport with Repulsive Costs. In general OT theory (see, e.g., Section 3.1 below), samples are transported based on a ground-cost that measures how expensive it is to move masses between measures. In its original conception by Monge 1781) and Kantorovich (1942), this ground cost is the Euclidean distance c(x), x2) = ||x| - 1- As reviewed in Di Marino al. (2017), may be interesting to consider repulsive costs, i.c. functions that are big when x, and X2 are close to each other and small otherwise. An example of such costs, arising from physics, is the Coulomb interaction c(x1, x2) ([|x; - x1 Still following Di Marino et al. (2017), these kinds of transportation problems proved useful in physics, e.g., for quantum mechanics and N bbby systems.
In this paper, we consider a different kind of repulsive cost, which we call the mass repulsive cost (see, e.g., Scction 3.2 below). Our notion of cost defines an exclusion zone, based on its nearest neighbors, where sending mass is too costly. As a result, our approach captures the local characteristics of the probability distribution being analyzed, especially its density. A special characteristic of our approach is to give a sense of the transportation from distribution to itself.
Optimal Transport-based Anomaly Detection. Previous works Bellhitt et al., 2019 2020
have considered OT for AD These works proposed a distance-based detection mechanism, in which isolated samples are considered anomalies. OT contributes to this setting, by defining a rich metric between samples. Especially, these works considered
AD in time series data, and OT is used to compute distances between those time series in the frequency domain, under Chebyshev ground cost. In comparison with these methods, ours is notably general purposc, that is, we do not assume data, to be time scries. Instead of using Chebyshev ground cost, we model the AD problem with repulsive cost.
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3 Proposed Method
3.1 Optimal Transport
OT is a field of mathematics, concerned with the displacement of mass between a source measure, and a target measure, at least effort. In the following, we cover the principles of ofT in continuous and discrete settings. We refer readers to Peyré Cuturi (2020) for computational exposition of the main concepts, and Montesuma al. (2024a) for applications in machine learning. In the following, we are particularly interested in the formulation by KKantorvvich 1942), which is defined as,
Definition 3.1. (Kantorouich Formulation) Let P and Q be 2 probability distributions e: X' X + R be ground-cost, measuring the effort of transporting wnils of mass from to Let T(P,Q) by E P(X x X): Sx r(x, B)dz = (BB and Sx Y(A, v)dy = P(A)} be the set of transportation plans, whose marginals are P and Q. The optimal transportation problem is wrilden as,
set X. Let
x" = OT(P,))= = arg inf SET(P.Q) J c(z, (I,,))d(Z,,))
(1)
Equation T defines the transportation problem as an infinite dimensional linear program on the variable called transport plan. In our case, instead of having access to a closed-form P, one has samples (xf (x each x; (P) P with probability p:- In such cases, may approximated with an empirical measure,
J=
P(x) = E p, - x P)), Σ P = =11p 2 1,
(2)
Plugging back equation into equation
1 leads to finite lincar program,
9 arg min (y, C) My c(x) (P)
I(p.q) fr: Σ vig 9ú. Σ Yas P. and Ya 2 0}
(3)
(ET(p.q)
Ca
where the optimization variables the coefficients Yi of the transport plan. Problem 3 is finite lin- ear program, hence the solution is sparse matrix with at most n + m - non-zero elements (Peyré & Cuturi 2020). Solving it through the celebrated Simplex algorithm (Dantzig 1983), which has com- putational complexity On log n) and storage
plexity Onz²) (i.e., storing each 7ij). We show example of empirical OT in Figure
A faster alternative was introduced by Cuturi (2013), who shown that adding an entropic regu- larization equation 3 leads to a problem that can be solved through Sinkhorn's algorithm (Sinkhorn] 1967). From continuous perspective, this is equiv- alent to penalizing Lullbeck-Leibler (KL) diver- gence between and the trivial coupling P &&Q Q=
Sumor indes
Figure 2: Optimal transport plan y between samples of P (blue) and Q (orange). On the left, we connect samples (i,j) for which 7/ 0. On the right, we show the entries of 7*
P**)Q(x). This regularization term is related to the entropy of as discussed in (Peyré & Cuturi 2020
Chapter 4), hence this problem is called entropic OT Next, we define the entropic OT problem,
Definition 3.2. (Entropic Optimal Transport) Under the same conditions of Definition 3. let e 2 be an entropic penally. The entropic OT problem is given by,
2t OT. -ary inf f (E((P..) Jxxx c(x, +cKL(T
(4)
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dy (x, 1)
where KL()(E) Sxxx log meosures y and (Peyré & Cuturi
dr (2, v) + Sxxx((E(xx,)) (r(x,))) is the KL divergence between
de
2020, Eq. 4.10).
We can obtain equivalent discrete formulation by plugging back equation 2 into which leads to,
9e = arg min (y. O)p-cH(1 = Σ To Cij yCT(p.q)
+
Yiy (log 76 - 1),
(5)
==
where H1)) denotes the entropy of the transportation plan. Since equation 5 relies on the Sinkhorn (1667) algorithm rather than linear programming, it has (Ln)) computational complexity, where L is the number of iterations. In general, the KL and entropic terms in equations and have smoothing effect on the transportation plan "e- As result, 9. has more non-zero elements than 7.
In the next section, we explore new transportation problem with single probability distribution. This problem is understood the transportation of P to itself, when the samples form P are forced to send their mass outside of their immediate neighborhood.
3.2 Mass Repulsing Optimal Transport
In this section, we propose new OT problem called [ROT This problem is inspired by the Kantorovich formulation, described section 3.1 However, instead of considering two different probability distributions
P, Q, it considers the transport P to itself. Due to the properties of if we consider OT y OT(P, P), supported in the set {(ar,)) E X), that is, each point keeps its own mass (Santambrogio 2015). This motivates our new problem, in which we force points to repell its mass. Henceforth, we assume that comes from metric (X, i.e., c(x, d(z, v), p [1,+ For pairs E X, y E X, and
L E [0, +00),
E(x,y) ((x,y) y
(6)
otherwise,
where (x) denotes the vicinity x (e.g., k-nearest neighbors, or an p- centered at r). the next remark, we give some geometric intuition behind the cost engincering, as well as some motivation for the choice of L. More generally, in the next section we give theoretically motivated choice for L.
Budiddoan
Engigeentl well L exPisdil
Engineeuel coel sppuet
Reprćkiv ooel
1.0
08
To
To
TD
-1
-1
-1
-1
044
Figure 3: Comparison between different ground costs for (0,0), 0). From left to right: Squared euclidean 5 for two choices of L, and repulsive cost.
cost c(x, vo) = || xoll2, Eugineered cost. (c.f., equation
Remark 3.1. (Geometric intuition) Assume x= 2, fized. Our cost engineering strategy replaces the base ground cost y(ro,y), in the neighborhood No ((xo) with value L. Assume, for argumentation purposes, that L + +oo. This choice means that it is infinitely costly for each point to keep its own mass. As result, samples are encouraged to send their mass lo the immediate oulside of their neighborhood thuss we call this idea mass repulsive OT
X
To ground some intuition, we show, in Figure 3 a comparison using the squared Euclidean distance as the ground-cost, i.e., c(x, y) I| v113- Alongside the base cost, we show engineered cost, and the repulsive
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cost (1 + c(z, v)) Here, we define the neighborhood of Zo through the p-closed ball centered at Iu. i.e., M(xo)= = {y : || vll2 S p). Likewise, we use p= and, I. = (10) 2). This particular choice leads to nice theoretical properties for the continuous MROT problem (see equation
7
below).
Due to the particulor choices we made, the supremum takes its value on the frontier (N(=))== R2 I1ro - = = in which case c(zg, 2) p², independently of zo. This choice for I. effectively faltens the ground cost around the vicinity of zo. As we show in the next section, this is essentiol for ensuring the existence of the continuous [ROT] problem. In practice, it is beneficial to set L to a large, finâte constant to ensure points transported outside N(xo). Note that, not harm the eristence of the díscrete MROT solution. We show an ezample of this choice by setting L = s==pxexl1==-- 2-
remark in the nezt section, this does
Our principle of mass repulsion is different from repulsive costs Di Marino et al. 2017]: 7,, which are designed to model the interaction between particles in multimarginal OT Pass 2015). Indeed, we design a transportation problem from a probability distribution to itself. Hence, while repulsive costs incentive transportation towards distant points in space, our mass repulsing cost induces transportation just outside an exclusion zone. Henceforth, we focus on p = for the Euclidean distance. The ground-cost we propose essentially defines an exclusion zone around points x, where those points are discouraged from sending their mass.
Our main hypothesis for anomaly delection is that anomalous points lie in low density regions of P. On the one hand, If these points are forced to send its mass outside its vicinity, it will be forced to send it to high density regions of P othorwise, mass conservation in OT would not hold. On the other hand, if the exclusion zone smaller than high densily reyions of P, points on these regions will be sent close-by. As a result, anomalous points will have a higher transportation cost (c.f., equations 9 and 10 below) than normal points. We thus consider the following OT problem,
/ c(x ((l,,))d((z + +KL( P), *ET(P,P)
MROT.(P)
inf
which, like the continuous entropic OT problem in definition approximated empirically through equation 2
3.2 also admits a discrete version when is
2e = MROT.(P)= min
(r,
(8)
>ET(p.p)
Considering again Figure 3 we focus on comparing the commonly used squared Euclidean cost, our en- gineered cost, and a repulsive cost associated with the Coulomb interaction (Di Marino et al. 2017 Tn traditional OT sending mass to distant regions of space is costly. As result, points are encouraged to keep their own mass as close as possible. Consequently, OT from P to itself is the trivial plan = Id.
For our engineered cost, we use N(x) = (y ||x yl S p), p= 1. Both our engineered cost ē and the repulsive cost c(x,,o)) assign high costs to points near Zg. In particular, within the neighborhood M(xo), our engineered cost ē(ι, xo) reaches its maximum. Hence, there is an important difference between MROT and [T with repulsive costs. Our engineered cost promotes transportation to the immediate region outside N, whereas the repulsive cost drives samples as far as possible from Zg. As discussed previously, this feature of the ground-cost allows us to build an anomaly score.
3.3 Theoretical Analysis
In this section, we provide a proof for the existence of MRRTT plans. While at least one solution of equation 7 exists, its uniqueness is not, in general, guaranteed. In this section, we denote the set of measures over set X' by P(X)) and the power-set of X' by IP))
From classical OT theory, the Kanlorovich problem (c.f., equation 1 has a solution under mild conditions on the ambient space X and the ground-cost c, see (Santambrogio (2015). Indeed, X is required to be Polish (i.c., complete, separable metric space) and e must be lower semi-contimnous. For simplicity, we state our results on d-dimensional Euclidean spaces, which are the setting for our experiments. We start by re-stating (Santambrogio 2015. Theorem 1.5),
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Theorem 3.1. (Eristence of Optimal Transport plans) Let X be Polish space, P, and Q P(X), and c: e: X' x X' + [0, +o] be lower semi-conlinuone Then equation 1 admils a solution.
Since we are dealing with d dimmensional Euclidean spaces, the condition on the ambient space is already satisfied. It remains the question of the lower semi-continuity of our engineered cost. In the next thcorem, we prove that, for the squared Euclidean cost, a solution for MROT exists,
Theorem 3.2. Let c(z, y) = || y1ž and ((x)) {y R 1| ull S pt. problem (equalion 7 admils solution.
Then the continuous MROT
Proof. Let Zo E R and p > 0 be given. Our proof relies on the dccomposition of Rd in 3 regions: i) ||tt - =ll2 > Ps ii) | - yll2 < P, and iii) || - ul12 = For i) and ii), ē is a continuous function of its inputs. Indeed, for | - ylla > p. 0(2.y) = |00 u1l. Likewise, for ||20 - yll2 < p, we have 0(z, v) = p². Now, we need to prove that ē is continuous on || - ylle - p. We sub-divide this into 2 cases. First, assume we have a sequence (yn): yn - y, Yn & Nzz). Since Yn (xxo), we have 0(zo. yn) = 1| ye122 From the contimuity of the squared Fuclidean distance, ((xo, - ||:00 ul = p². Second, assume we have sequence (v'.), v/h + y, s/1 E (zoo) Then, ē(zo, V/n) 0² C(zo. y) = p². Hence, ė is continuous.
]
The principles used in the previous theorem can be generalized to other costs and neighborhoods, as long as: i) the base cost c(x,y) is continuous, ii) the neighborhood (xx) is compact, and iii) the supremum ( 2) is attained in the boundary of M(x), Vx E X. The boltom line of our analysis is that, in general, one needs to carefully engineer the ground cost, through the choice of the base ground cost e, and the neighborhood function N.
Although existence may hold, the fact that the engineered cost introduces a flat region on the neighborhood M(x) of makes it difficult to establish the uniqueness of the MROT plan. We recall from (Figalli & Glaudo, 2021 Section 2.6) that, in classical OT the existence of a unique transport plan is related to the existence of a Monge map between P and Q. From (Figalli & Glaudo 2021 2.7.1), this means that we need 3 conditions: i) x + C(z, y) is differentiable, ii) y + V=ū(I, y) is injective, and iii) for p > O, and B (x E X ||a||| S p): Vē(x, y)1 < C, for every æ E Bp. As it turns out, neither of these conditions holds for our engineered cost, so the MROT plan is likely not unique.
Now, let us focus on the discrete case, which is of practical interest to us. Note that (p,, is non-empty "e PiPy E T(p, p). Furthermore, this set is compact, and the objective in equation g is continuous. Then, by Weierstrass theorem (Santambrogio 2015 Box 1.1), there exists a minimizer to the discrete setting. Here, note that we did not have to impose continuity or lower semi-continuity on the neighborhood function N or the ground cost ē. As in classical OT the solution to this problem is not unique, since the objective function is not strictly convex. Following this line, adding the entropic penalty (i.e., e >> 0 in equation 8 makes the objective strictly convex (Peyré & Cuturi 2020, Section 4.1), which guarantees the uniqueness of the OT plan, and hence, the uniqueness of the MROT plan.
3.4 Building and Generalizing an Anomaly Score
Through MROT we want to build a score for the samples based on how anomalous they are. Assuming that anomalies lie in a low density region of space, our MROT problem forces those samples to send their mass to distant parts of the feature space. In contrast, normal samples can send their mass to the immediate neighborhood outside the exclusion zone. As a result, we can sort out normal from anomalous samples using the transportation effort,
E (2x(x,)) = / (((,,))dd(((l))
T(x)= )>((-)))
(9)
where 7(ylz) corresponds the conditional probability, calculated through the joint 7(z v), given x. For empirical measures, this quantity can be calculated follows,
4 = T(x (P)
do (x .2) ),)))
(10)
Ps
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where p; is the importance of the i-th sample (e.g, Ps = =n for uniform importance). Interestingly, equation 10 is similar to the barycentric map, widely used in domain adaptation Courty et al. (2016). T has 2 shortcomings as an anomaly score. First, it is hardly interpretable, as its range depends on the choice of ground-cost c. Second, it is only defined in the support of P. We offer a solution to both of these problems.
Concerning interpretability, we propose to transform using the Cumulative Distribution Function (CDF) of its values. Let PT be the probability distribution associated with T(x). The [CDF] is simply (T Pr((-oo,t). Naturally, since PT is not available, it may be approximated from samples {t:]==1> obtained through equation 10. We do so through, Kernel Density Estimation (KDE)
to and, FT(() = [ Pr())
(T( = no
(11)
where @ is a kernel function (e.g., the Gaussian kernel xxxx) = exp( -a² /21), and a is the bandwidth, controlling the smoothness of Pr and determined through Scott's rule (Sott 1979p. Equation 11 is an approximation for the density of transportation efforts [t,),-1. The CDF is appealing, as it a monotonic function over transportation effort E Ik, and it takes values on [0, I], both of which are desirable for an anomaly score.
The question of how to extrapolate the anomaly score for use the CDF values as anomaly scores, we need to recalculate t = (xx) for a new sample x ~ P, which is challenging, as 7 is only defined on the support of P. A naive approach would be to append x to the set (rr")) and solve a MROT problem again. Naturally, this is not feasible, solving an OT problem for each new sample is compulationally expensive.
samples remains. For example, even if we
In this paper, we present the more efficient idea of modeling the relationship x + (x(()x) from the samples we receive, that is, we create labeled data set {x: (P)) 44)=s where 4 = (x)())) Our anomaly score comes, then, through a function (xx (P) fit to the labeled dataset through regression. The fitting s can be done with standard regression tools, such as Ordinary Least Squares (OLS) (Support Vector 4 Regression (SVR) [Smola& & Schölkopf 2004), near- est neighbors or gradient boosting (Fiedmmnn 2002). 6
Algorithm 1: Mass Repulsive Optimal Trans- port.
1 function mrot (P))) e))
7he MROT (P):
4 + (6.((///
(P) (P));
.x² ;x]
P + (DE((
+ Regresssion (x} Pr (t))::)
5
return v;;
In general, one can expect the relationship between
(P))) to be non-linear, hence, it is generally necessary to use a non-linear regression model. We show a summary of our strategy in Al- gorithm
(P) and (x(T(x(())
x;
4 Experiments
We divide our experiments in 3 parts. Section 4.1 shows results on AdBench (Han et al. 2022 shows our experiments in fault detection on the Tennessoe Eastman Process (Montesuma ct al. 2024b; Reinartz et al. 2021). Finally, Section 4.3 explores the robustness of our methods to various paper-paraaeters and design choices.
Section 4.2
In the following, we consider adaptive neighborhoods Nx(x) which take the k-nearest neighborhood of x, and set C₁ max:=1,.... Cie, where j E (N)(x(P).. Before going through our experiments, we show toy example, going through all the steps in our algorithm. See Figure 3 for a comparison.
An introductory example. Before comparing our method with prior art, we give an introductory example that illustrates how we create anomaly scores out of samples. In this example, we sample normal examples
x; ~ NN 0,2512), and anomalous samples y, ~ 3[[[3, -3),03000011 where I, a 2 2 identity matrix. The dataset for this toy example consists of the concatenation (P) (x),"" Na, Ymm, where 500
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and 25, which means that roughly 5% of the total number of samples anomalies. These samples are
shown in Figure 5
We compare our engineered cost with regularized Coulomb interaction,
Ciy (1+ |xc xj (P) 11)) This cost is shown alongside proposed engineered cost in Figure Note that, in Figures (a c) the lower right corner of the malrices corresponds to anomalies.
Nexaed
From Figure 5 OT with the Euclidean cost results in a trivial OT solution, because the diagonal entrics Ci are zero. Hence, Y" I, achieves transportation cost. This not the case for our engineered cost, and the regularized Coulomb interaction, shown in Figures (c) and (d)- To emphasize this idea, we show in Tigure/ōthe transportation plans acquired by MROT and the regularized Coulomb cost. For the Coulomb cost, the anomalies send and receive less mass than normal samples. This is somewhat expected, as the anomalous samples are clustered together in tight region of IR?, resulting a high cost. As a result, these samples are encouraged send their mass elsewhere. This phenomenon does not happen with engineered cost, which encourages samples to send their mass just outside to their exclusion zone.
0
-2
2
Figure 4: Toy example.
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100
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100
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200
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20
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10
+10
400
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100
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(a) Euclidean cost.
(b) Engineered cost.
(c) Coulomb cost.
Figure 5: In (a), (b) and (c), we show the ground-cost Cip between samples Figure using the Euclidean distance, our engineered ground-cost (equation and the regularized Coulomb interaction cost.
Hence, there an important distinction between our approach and using repulsive costs in OT Considering our last remark about the regularized Coulomb interaction, samples send their mass to distant regions of space, thus reducing Czy. As a result, anomalies will incur in a smaller transportation cost than normal samples. Although counterintuilive, one still construct detection rule out of this idea.
In Figure 7 present the anomaly scores derived from our MROT strategy and the regularized Coulomb cost. These scores are computed by first determining the transportation effort (c.f., equation 10 for each sample _! !!(") then estimating the density of these efforts, and finally transforming the density into [0, 1] score via its CDF as outlined in section 3.4 As shown in Figure 7 the [MRT approach assigns higher anomaly scores to anomalous samples compared normal ones, simplifying the process of setting threshold for anomaly detection. In contrast, the regularized Coulomb cost exhibits the opposite behavior. Indeed, anomalous samples send their mass to distant parts of space (i.e., to normal samples), which, due the nature of the Coulomb cost, lead to a smaller transportation effort. Nevertheless, as discussed previously, it is still possible to establish a detection rule, albeit being counterintuitive.
As we discussed throughout section 3.4 the scores in Figure 7 are only defined in the support of P. We then explore the regression of this score through regression, which be done with variety of standard

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Neni
100
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(b) OT with the regularized Coulomb cost.
Figure 6: In (a), we show the MROT t transportation plan (left side), the matching (center) and the transport plan (right side). Likewise, in (b) we show the transportation plan and strategy for the regularized Coulomb cost. As shown in (b) left side and right side, the anomalics (lower right corner) sends and receives much less mass than other normal samples.
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(a) MROT anomaly score.
(b) regularized Coulomb cost anomaly score.
Figure 7: Anomaly score comparison between the MROT (ours, a) and OT with repulsive costs (b). Scores are computed by calculating the transportation effort (Eq. 10) for each sample a;." estimating their density, and normalizing to the [0, 1] range via the CDF (c.f., section 3.4). Overall, our strategy assigns higher scores (close to 1.0) to anomalous samples, while assigning smaller scores to normal samples.
regression algorithms. In Figure show the results for eXtreme Gradient Boosting (XGBoost) Chen & Guestrin 2016) and [VR Smoola & Schölkopf 2004). which both define the anomaly score over the whole ambient space R As an important remark, the relationship between x; and its score likely non-linear.
Indeed, this idea is evidenced in Figure 8 As result, one needs non-linear regression algorithm.
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XGBaee
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(a) MROT regressed score.
(b) regularized Coulomb cost regressed score.
Figure 8: Regression of the anomaly score for MROT (ours, a) and OT
with repulsive costs (b). While our with repulsive costs does the inverse.
method attributes higher scores to anomalous regions of the space, OT
4.1 Comparison AdBench
AdBench (Han al 2022 is benchmark in AD with 57 different kinds of datasets, grouped into 47, 5 and 5 real-world, vision and Natural Language Processing (NLP) datasets. In our experiments, we focus real-world and natural-language datasets. We compare, in total, 17 methods, grouped into classical,
diffusion-based and our proposed methods based on OT For classic methods, we consider Isolation Forest
IsoF)
[Lu et al. 2008), OCSVM (Schölkopf et al. 1999 NN LOF Breunig et al. 2000 CBLOF He
et al. 2003), Empirical Cumulative Distribution Functions (FCOD) [Li
al. [2022), Copula- Based OD
[COPOD] (Li al.) (2020), Lightweight On-line Detector of Anomalies (LODA) (Pevný 2016), Feature
Bagging (Lazarevic & Kumar 2005). and [Histogram-based Outlier score (HBOS) (Goldstein & Dengel
2012
For diffusion based, consider the 4 variants of Diffusion Time Estimation (DTE) Livernoche
ct al.
2024
We also consider OT with repulsive costs Di Marino et al. (2017), and MROT (ours).
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ALCPR
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(a) Real-world.
NL.P
Figure AdBench result summary. AUC-ROC and AUC-PR per dataset available in the appendix. We compare, in total, 17 algorithms over 47 real-world datasets (a) and NLP datasets (b). For real-world datasets, our MROT has saate-of-theeart performance, especially when compared with recently proposed flow-based models. For higher dinensional data, our method
We show our summarized results in Figurcs g (a) and (b), for real-world and NLP datascts, respectively. First, we note that MROT and OT with repulsive costs have superior performance with respect other methods real-world datasets. This result highlights the usefulness of OT theory in the analysis of probability distributions. Furthermore, on average, our MROT has better performance than OT (77.36% versus 75.97% ROC-AUC), proving the effectiveness of our cost engineering strategy. We refer readers to our appendix for detailed results per dataset.
Here, we highlight two limitations of our method. The first limitation of using MROT comes from the scal- ability of OT) As linear program, it has at least O(n?) storage, and On log n) computational complexity, where is the number of samples. In experiments, limited the number of samples to 20, 000, by
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down-sampling larger datasets. As reported in Figure g (a), and our detailed results given in the Appendix, this process does not. affect performance. Furthermore, from Figure (b), see that method struggles in high dimensional AD such as those in the NLPP datasets of Han et al. (0222). This limitation stems from the use of OT in high-dimensions [Montesuma et al. 2024a Section 8.1).
4.2 Tennessee Eastman Process
In this section, we focus in comparing anomaly detection methods for fault detection in control systems. In this context, as defined by [sermann (0006). a fault is an anomaly in at least one of the system's 's variables.
To that end, we use the Tennessee Eastmann (TE) process benchmark (Downs Vogel 1993 especially the simulations of Reinartz al. (2021). pre-processod by Montesuma
(2024b).
This benchmark is composed by simulations of large-scale chemical plant, from which collection of 34 physical and chemical quantities measured over time. There are total of 29 states for this plant: I healthy state, and 28 faulty states. Furthermore, the plant can operate under 6 different modes of operation, depending on the specified production requirements. We refer readers to the aforementioned references, as well as our appendix, for detailed descriptions of the variables, faults and modes of operation.
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(d) Mode 4.
(e) Mode 5.
(f) Mode 6.
Figure 10: t-SNE embeddings of the Tennessee Eastman process data per mode. In blue, show the normal samples, whereas we show anomalous samples in shades of reds, corresponding to the actual fault category they correspond. While most anomalous samples do cluster in a region outside the non-faulty cluster, some faulty samples do not.
From the original simulations Reinartz et al. (0021). we consider 100 simulations for the normal state. These simulations last for 600 hours, with time-step of hour, and concern sensors mcasuring different physical and chemical properties. On top of these 100 simulations, we take faulty simulation for each kind of fault, leading to total of 128 simulations. Based this set of simulations, extract windows of 20
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hours from the original signals. Each window is then considered a sample for anomaly detection. On each window, compute the mean, and standard deviation of each variable, leading to vectors μ, E TR We use the concatenation of these vectors as features for anomaly detection. These steps lead to datasets with 3840 samples, 840 (21.875%) of them being anomalous, one for cach mode of operation.
MROT
MROT
MROT
ROT
ROT
ROT
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LOF
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Algorishon
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IcNN
K-NN
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DTEC
DTTC
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ECOD
(OPOD)
COPOD
COPOD
AUC-ROC
AUC-ROC
AUC-ROC
(d) Mode 4.
(e) Mode 5.
(f) Mode 6.
Figure 11: Aggregated anomaly detection on the Tennessee Eastman data per mode of operation. First, MROT outperforms OT with repulsive costs over all modes. Second, MROT and DTE-C have state-of-the- art performance, superior previously proposed methods.
We start our analysis by embedding the obtained features in IR² through 1-Stochastic Neighbor Embeddings (t-SNE) Van der Maaten & Hinton 2008), which is shown in Figure 10 We scatter the embeddings of each mode's data, showing that most faulty samples cluster outside the normal data cluster. However, some faulty samples are close to normal ones. This phenomenon is expected, as the effect of faults evolves over time. As a result, windows taken early stages of simulation resemble those of normal samples.
Next, we benchmark the AD performance of algorithms, and their capability to generalize to un- that within the same mode of operation. In this experiment, ber of anomalous samples per fault category to (5, 15, ,0).. This results in percentage of (4.45% 8.53%, 12.28%, 15.73%, 18.92%, 21.87%} of anomalous samples. In Figure [] we report our aggre- gated results over all percentage of anomalies. We refer readers to our appendix for results per percentage.
downsample the mum-
The main idca of this methodology is evaluating how different algorithms perform, under a variable per- centage of faults. Our results are shown in Figure 10 (b). Among the tested methods, MROT has better performance and remains stable throughout the range of percentage of anomalies.
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4.3 Ablations
Hyper-parameter robustness. Here, analyze the robustness of our method with respect to the entropic regularization penalty 6, and the number of nearest neighbors k in Mr. In our experiments, we evaluated our method on the values e E {0, 1001110011,100_01111111100000000 ) where = 0 implies the use of exact OT that is, linear programming. For MROT we use & E (5, 10, 20, ... ..... Note that while is related to transportation plan, k is linked to the ground-cost. We summarize our results Figure T2] where we present box-plots over the AUC-ROC scores on each dataset in the AdBench benchmark, for each combination of prpee-parameters.
From Figure 12 note that both MROT is robust to the choice of entropic regularization and number of nearest neighbors. As general guideline, it is better to limit the number of nearest neighbors, anomalous examples are likely rare. As consequence, using high value for may lead to the inclusion of normal points in the neighborhood of anomalous ones.
1P
120
100C-300C
13 0.01 01 14
(a) MROT.
(b) ROT.
Figure 12: Hyper-parameter sensitivity. In (a), we show the performance of MROT for variable number of nearest neighbors k, and entropic regularization Overall, our method is robust to the choices of these typer-parameters, but using a lower is generally better. In contrast, we show in (b) the performance of OT based AD with the regularized Coulomb cost, for which using a higher entropic penalty e improves performance.
Chapter 4), We give an additional reasoning on the choice of c. From the classical OT Peyré & Cuturi 2020
using 6 > leads to a nonsparse plan. This means that there are more indices (i,j), for which Vij > 0. Like- wise, if we let E > +oc, then converges to the trivial coupling 7ij = P.Pj. Assuming uniform importance, i.e., P; n-1, and plugging these results back into equation 10, have 4; n- Σÿ= (1, " (P))), that is, for large entropic penalty, end up simply computing the average cost of each sample. The implication of this analysis is as follows: the MROT plan captures some structure in the relationship between anomalous and normal points, which contributes to a more accurate anomaly score.
(P)
Robustness to regression algorithm. As we mentioned in our toy example, it is necessary to use nonlin- ear regression model for generalizing the anomaly score, as the relationship between x!" and (t,),) is likely nonlinear. We ablate on the choice of regression algorithm, exploring the use of gradient boosting (Fried- Smola & Schölkopf 2004) and k-nearest neighbors regression. We show a summary of our
man 2002) SVR
Note that, general, the performance boosting and [SVR] is similar and stable among different choices for in N& in our engineered cost. The performance of k-nerrsst neighbors regression is
results in Figure 13
sub-optimal, especially due to overfitting.
In addition to the pipeline shown in Algorithm 1 it is possible to add model selection step, e.g., through k-fold cross-validation. The idea is to partition the training dataset (x!") (T)))) into a training and validation set. Then, among pool of possible regression models, the one that has the minimum error in the validation set.
5 Conclusion
In this paper, we introduced a novel, general purpose anomaly detection algorithm based on optimal transport theory. Our method works under the assumption that the local neighborhood of anomalous samples is likely
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XGBoast
Support Vector Regression
k-Nearcst Neighbors
100
80
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60
40
20
10.0
20.0
40.0
10.0
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40.0
10.0
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Number of neighbors
Figure 13: Ablation of MROT performance with respect regression model choice. Overall, gradient boost- ing and [VR have similar performance, whereas nenearest neighbors has sub-optimal performance due to overfitting.
more irregular than that of normal samples. Based on this idea, we engineer the ground-cost in optimal trans- port, to encourage samples to send their mass just outside an exclusion zone, defined through its k --nnerest neighbors. While this idea bears some similarity to optimal transport with repulsive costs (Di Marino et al. 2017), the defined ground-cost anomaly delectors. We thoroughly experiment on the AdBench (Han et al. 2022) benchmark, and the Tennessee Eastman process (Downs & Vogel 1993 Reinartz al. 2021 Montesuma et al., 2024b). showing that our method outperforms previously proposed methods in real-world benchmarks.
repulsive, and, we show in our experiments, it leads better
Limitations. Our methods faces 2 limitations from OT computation. First, we need to store a n X m matrix, which incurs a (nn²) storage complexity. Furthermore, computing Te" has O(n log computational complexity. For that reason, the AdBench experiments, we had to downsample larger datasets to n 20, 000 samples. Second, OT notoriously hard estimate in higher dimensions. As a result, our method struggles with high-dimensional features, such those in the NLP datasets of the AdBench benchmark.
In the literature, there are different ways of tackling these limitations, but we leave those for future works. For instance, workaround to the first limitations is to focus on mini-batee OT Fatras et al. 2020 computing OT with neural nets (Makkuva et al.) 2020 Korotin et al. 2023). Likewise, the computation of OT in high dimensions can be donc, for instance, through modifications the original OT objective, such as subspace robust OT (Paty Cuturi 2019).
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A Additional Details on Experiments
A.1 Detailed results on AdBench
Table 1: Comparison of AUC-ROC results on ADBench.
08.0
08.07 072
0206-0 13.55
343
11:00
18.08 144
12.0 2.2
2-1
ALCE
0100
18000 81.
800
333
18.20-3 0:33
245
0.1 211
15322 8:31
08.93
SUS
WDC
0119
88-2 214
Wil
1125
815 211
-218
2-14
0805
08.3 15T
04.22 1124
0.35 221
551
08.0
510 2317
Table 2: Comparison of AUC-PR results on ADBench.
1556 823
3.31.
1641
1846 K5
20472041
350 X55
200 1100 725
1105
1274 1.29
143
15.0
255
23
17:0--20 345
1036
201 815
5037
821-0m
2.10
1609
1000
110
18:00-4 -11
1237
1031 165
1475
1622
OML
0512 316
1800 417
014
1527 L1
1
HAL
472
1504
636
120 51
120
sandica
<22 311
Tat 1.
13.52 239
01.0
12:0
1121
150
1.0
2215
2506 L1D
118-112 122
LO
203 14
200 134
15.5 L/
24T
512
220
(3.5 835
2315
15.6
MC 2M
2222
132
9.6
2570
2/
2145
21.0
112-1-3
130-11 11:
ATV
215
589
171 11
729
734
50328D
154
V/8
144
1.7
414
524
M14
6.77
19

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Published in Transactions on Machine Learning Research (05/2025)
A.2 Tennessee Eastman Process
The TE process was first proposed by Downs & Vogel (1993), as large-scale chemical process for bench- marking control algorithms, as well as fault detection and diagnosis methods. We follow the description of Reinartz et al. (2021). This chemical process consists of 4 chemical reactions, for the production of 2 liquid liquid components, denoted G and E, and 4 gaseous reactants, denoted A, B, C and D. The reactions are as follows,
A(g) + ((g) D(g) G(liq)
Product 1, Product 2,
A(g) + (gg) + E()) - H (liq)
(12)
Agg + Egg) - F(liq)
Byproduct,
3D(g) - 2F(()
Byproduct.
The chemical plant is composed by 5 processing units: reactor, product condenser, vapor-liquid separator, recycle compressor and product stripper. We refer readers to Reinartz et al. (2021) and Montesuma et al. (202bb) for further details how these components work together in the system. As described by Downs & Vogel (1993). based on equation 12 there are 6 different modes of operation, corresponding to different product mass ratio and product rate. We detail these in Table 3
Table 3:
TE process operation modes in terms of G/H mass ratio and production rate.
Mode
Mass Ratio
Production rate
1
50/50
7038 kg h G and 7038 kg h- H
2
10/90
1408 kg h G and 12,669 kg h- H 10,000 kg h-1 G and 1111 kg h- H maximum production rate maximum production rate maximum production rate
3
90/10
4
50/50
5
10 10/90
6
00/10
From the chemical plant that performs the equations in 12 there are 54 variables divided into measured (XME) and manipulated (XMV). We show these in Table 4 Out of the 54 variables, we do in Reinartz
et al. 2021), that is, we use subset of 34 variables, boldened in Table A In the simulations of Reinartz
et al. 20021, the TE process is simulated for 100 hours, with sampling rate of 3 minutes. As a result, each simulation is represented as a time scries x E 000 where x(t) E R
Table 4: Process variables used in the TE process, divided into measurements (XME) and manipulated (XMV). Bold rows indicate that the variable is used in our experiments.
Visible
Deacippion
Viciable
Descripiin
Vasiale
Dosciption
Vasiable
Dosrijtion
XMB(()
Feud (lacmh)
XME(15)
Stripper
Compusser
(sud 96)
XMVV(2)
(%)
XME(21
(kx/b)
XME(16)
Stripper Pressure (kPu gauge)
X3EE/ 301
Cauporen
(aud 56)
XMVV(3
XME(3)
(kx//k)
XME(17)
Stripper Underflow [m (Ba)
Couporu Parsi (aut 56)
XXLV44
A
XME(4)
(ka/h)
XME(15)
Steipper Trmp (CC)
XMB
Cupportt Paage (wnd 36)
XMVVSS
Compressor recycle walve (56)
XME(S)
Recycle Flow (Bsendl
XME(19)
Stelpper Steum Flow (ke/hs)
XNBB
Componat Purge [axt 56)
XMV(6)
Purge valve (%)
Reactor (karmh))
(kscmah)
XME(20)
Compresso
Componet Purge [unt 56)
XMV
Separator lquid (%)
XMR(21)
Coolant Temp (*U)
Compornt Pergo (mol 96)
Striper ligidd
Segarator
Componest TTmp. (*C)
Composeer
Striper
(%)
Theperrttoon (*C)
XMR/21
Omppone
(%)
XMR(10)
Purgs Temp
Compoone
Compornnt
Conlant
XMR(11)
Compooet
XMB
XMVV(1)
Speed (%)
XMB(
Product Sup
Component
XME
Crnpaneza Product
XME(13)
Product Sep (IPP gauge)
XME/27
XMBE/1
in Product
(M4
Product Undeefoo (ix"
XME
Oopposest
XMIV()
Feod
The simulations in Reinartz et al. (021) are faulty normal. For faulty simulations, different fault type is introduced between 28 possible categories at to = 30h. We refer readers to the original paper for more information about the different categories. In this paper, we construct 6 different AD datasets, one for each mode of operation. On each mode of operation, we use 100 normal simulat ions, and faulty simulation for each fault type. This process results in 128 time series per mode of operation.
For applying AD algorithm, we further pro-process the time series obtained from the simulation of the TE process. We decompose each time series in windows of 20 hours, i.e., 60 samples. Within each window, we
20

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Published in Transactions on Machine Learning Research (05/2025)
compute the mean and standard-deviation per sensor, that is,
1 6+00
1 f+00
Mtj 60
Zijs and, orj
(ii Hey)
59
where is the index of the window. We use the concatenation (14, 0t) E R as the representation for the t-th window. As result of this process, extract 30 windows for each time series. This results on 6 datasets with 128 x 30 = 3840 samples cach. We then proceed to apply AD algorithms at the level of windows. Here, we note an important detail. For OT algorithms we compare samples based on the Euclidean distance. Computing this distance over x = (,,o) and y = (4',o') boils down to,
|x; x1112 2 = || - 1a 1l2 + la - 12,,
which is the squared 2- Wasserstein distance W2(P) Q)² between axis-aligned Gaussians P P PN oI) and Q =N(wd',
A.2.1 Results per anomaly percentage
In Figure 14 we show overview of the AUC-ROC of different methods per percentage of anomalies.
Overall, our MROT outperforms OT with the regularized Coulomb cost on all modes. Furthermore, our method outperforms or stays competitive with other methods on most modes, such as 1, 3 and 6.
90
87.5
87.5
85.0
85
AUC-RDC
AUC-ROC
82.5
AUC-HOC
82.5
80
80.0
80.0
MROT OT
1-M8
MERVT
MIR7T
L-N
1:P0 PCA
DIFE-C
77.5
1OP
DTT
PCA
77.5
OT
1GP
PCA
75
75.0
ECOD
CBLOF
BCOD
75.0
OCSVM
72.5
OCSVM
10
20
Pescentage Ancmalless
Pecceutage Anomallies
Percenttgee Ansmalles
(a) Mode
(b) Mode 2.
(c) Mode 3.
90
90
85
AUC-ROC
AUC-ROC
AUCCBOC
80
80
80
PCA
LOE
CREOF bod
DIE-C ROOD
75
LOF
CRLOP
DILCC BOOD
75
LOF CRLOPP
FOOD
CORNO
bel
COPOD
Bwll
CYPOD
20
Peroentage of Ancmalies
Percentage of Anomalies
Percentage of Anomalies
(d) Mode 4.
(e) Mode 5.
(I) Mode 6.
Figure 14: AUC-ROC per percentage of anomalies the TE benchmark.
21


Reviews:
[{"review_id": "poRKjHymCL", "reviewer": "Reviewer_Ey1z", "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. Innovation: A novel OT problem (MROT) is proposed, which achieves anomaly detection through a repulsive cost function. This idea is original in the field of OT. The combination of OT theory and anomaly detection problems provides a new perspective for unsupervised anomaly detection.\n2. Experimental Design: The experiments cover a variety of benchmark datasets (such as AdBench), demonstrating the universality and practicality of the method.\n\nWeakness:\n1. Clarity of Method Details The core ideas of MROT (such as the design of the repulsive cost function and the definition of the \"repulsion region\") are described rather abstractly in the text, lacking intuitive mathematical or geometric explanations.\n2. Depth of Theoretical Analysis: There is a lack of discussion on the theoretical properties of MROT (such as convergence and uniqueness), especially in comparison with classical OT problems.\n3. The resolution of Figures 4-7 is low. It is recommended to provide clearer vector graphics.\nThe labels on the bar chart in Figure 8 are too dense. They can be simplified or displayed separately.\nReferences: The citation format of some references is not uniform (such as \"Montesuma et al. ()\" missing the year).", "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 introduces a novel approach to anomaly detection called Mass Repulsing Optimal Transport (MROT), which modifies classical Optimal Transport (OT) by forcing samples to redistribute their mass away from their original positions while minimizing transportation cost.  The paper's innovation and experimental design are commendable, yet there is room for improvement in methodological details, theoretical analysis, and computational efficiency.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See weaknesses.", "broader_impact_concerns": "See contributions.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ODbIekGVzb", "reviewer": "Reviewer_8CYE", "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\nInnovative Approach: The engineered ground cost and the formulation of MROT are quite original. The idea of using mass repulsion to highlight anomalies is refreshing and mathematically interesting.\n\nComprehensive Experiments: The paper presents extensive empirical evaluations over multiple benchmarks, comparing with state-of-the-art methods. The inclusion of both classical and modern techniques adds weight to the experimental claims.\n\nInterpretability: The use of a CDF transformation for the anomaly score helps in making the outputs more understandable and actionable, which is much appreciated in practical deployments.\n\n\n\nWeaknesses:\n\nScalability Concerns: As the authors themselves note, OT’s O(n²) storage and O(n³ log n) computational complexities may limit the practical use of the method on very large datasets without further approximation or batching strategies.\n\nHigh-Dimensional Data: The method seems to struggle with high-dimensional features (as seen in some NLP datasets), and while this is not uncommon in OT-based approaches, it remains a limitation.\n\nRegression Step Ambiguity: The process to generalize the anomaly score via regression could use further clarification. Details about model selection, training stability, and potential overfitting are somewhat lacking.\n\nHyper-Parameter Sensitivity: Although a robustness analysis is provided, more insight into choosing the exclusion zone parameter (number of neighbors) and entropic regularization parameter would benefit the reader.", "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 introduces a novel anomaly detection framework by reformulating optimal transport (OT) into what the authors call Mass Repulsing Optimal Transport (MROT). The key idea is to modify the ground cost so that samples are forced to “repel” their own mass away from a predefined exclusion zone (typically defined by k‐nearest neighbors). By doing so, the method accentuates the differences between normal and anomalous points—since anomalies, often lying in sparse regions, incur a higher transportation cost. The paper also provides a principled way to transform these costs into interpretable anomaly scores, and demonstrates the efficacy of the approach across several benchmarks including AdBench and the Tennessee Eastman process.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I do not have any suggested adjustments.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "HhKZSqgl7c", "reviewer": "Reviewer_aamh", "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 application of OT for AD is an interesting and promising direction.\n- The concept of using repulsive cost to identify anomalies sounds novel.\n\nWeaknesses\n\n- The literature on AD is huge, but the related works seem to overlook many important AD studies, particularly DNN-based approaches such as GAN-based and AutoEncoder-based AD, which are widely used in industry. Most of the discussed related works focus on traditional ML methods. I suggest providing a more comprehensive discussion of the AD literature. While I understand that the authors primarily focus on OT, the AD problem itself is the main target. Regardless of whether OT is used, it is essential to consider other AD techniques.\n\n- Due to the limited discussion on modern DNN-based AD methods, the experiments appear relatively weak, as there is insufficient evidence to demonstrate that the proposed method is truly practical and competitive with state-of-the-art AD techniques.\n\n- When using a repulsive cost, is there any difficulty in solving the OT problem? It seems that a similar approach to computing Sinkhorn could be applied. If not, the technical novelty may be relatively limited, although the conceptual contribution is interesting. Please kindly provide more clarifications.", "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 introduces Mass Repulsing Optimal Transport (MROT), a novel approach to anomaly detection that modifies Optimal Transport (OT) by enforcing mass displacement while minimizing effort. Anomalies, located in low-density regions, incur higher transportation costs, while inliers shift minimally. This formulation yields a new anomaly score, demonstrating superior performance over existing methods in benchmark and fault detection tasks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see weaknesses.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers agree that the proposed method is novel and interesting, and the authors have adequately responded to most of the reviewers' concerns. The remaining concern that an in-depth comparison with neural anomaly detection methods is missing is valid, but in the light of the editorial policy of TMLR (as I understand it) it does not pregnant publication. 

I suggest some minor editorial changes: The text had occasional grammar mistakes and typos, probably due to editing. Similarly, the caption off figure 5 refers to (a), which is probably figure 4. I suggest to carefully read the paper before submitting final files. Furthermore, in figure 3, the engineered cost is not largest for points close to $x_0$. I suggest redrawing this figure for a larger $L$, or adjusting the claim in the text (if I understood the claim correctly).
Decision: Accept (poster)