AuToMATo: An Out-Of-The-Box Persistence-Based Clustering Algorithm
Authors: Marius Huber
Sara Kalisnik Hintz
Patrick Schnider
Venue: Accepted by TMLR ()

Abstract: We present AuToMATo, a novel clustering algorithm based on persistent homology. While AuToMATo is not parameter-free per se, we provide default choices for its parameters that make it into an out-of-the-box clustering algorithm that performs well across the board. AuToMATo combines the existing ToMATo clustering algorithm with a bootstrapping procedure in order to separate significant peaks of an estimated density function from non-significant ones. We perform a thorough comparison of AuToMATo (with its parameters fixed to their defaults) against many other state-of-the-art clustering algorithms. We find not only that AuToMATo compares favorably against parameter-free clustering algorithms, but in many instances also significantly outperforms even the best selection of parameters for other algorithms. AuToMATo is motivated by applications in topological data analysis, in particular the Mapper algorithm, where it is desirable to work with a clustering algorithm that does not need tuning of its parameters. Indeed, we provide evidence that AuToMATo performs well when used with Mapper. Finally, we provide an open-source implementation of AuToMATo in Python that is fully compatible with the standard scikit-learn architecture.

Published in Transactions on Machine Learning Research (10/2025)
Au ToMATo: An The--Bo Persistence-Based Clus- tering Algorithm
Marius Huber
marius.huberShuah.c
Deportment of Computational Linguistics University of Zürich
Sara Kališnik
skalisnik@@psu. edu
Deportment Mathematics
Pennsylvania State University
Patrick Schnider
putrickk.chniderr@in...c ethz..
Department of Mathematics ond Computer Science
University of Basel Department of Computer Science
ETH Zürich
Reviewed on OpenReview: https: openrviiew. net/ forum? id= bzz
Abstract
We present AuToMATo, novel clustering algorithm based on persistent homology. While AuToMATo is not parameter-free per se, provide default choices for its parameters that make it into out-of-the-box clustering algorithm that performs well across the board. AuToMATo combines the existing ToMATo clustering algorithm with a boolstrapping pro- cedure in order to separate significant peaks of an estimated density function from non- significant ones. We perform a thorough comparison of AuToMATo (with its parameters fixed to their defaults) against many other state-of-the-arl clustering algorithms. We find not only that AoToMATo compares favorably against parameter-free clustering algorithms, but in many instances also significantly outperforms even the best selection of parameters for other algorithms. AuToMATo is motivated by applications in topological data analysis, in particular the Mapper algorithm, where it is desirable to work with clustering algorithm that does not need tuning of its parameters. Indeed, we provide evidence that AuToMMTTo performs well when used with Mapper. Finally, we provide an open-source implementation of AuToMATo Python that fully compatible with the standard ceikit-leary architecture.
1 Introduction
Clustering techniques play a central role in understanding and interpreting data in a variety of fields. The idea is to divide a heterogeneous group of objects into groups based on a notion of similarity. This simi- larity often measured with distance or metric on a data set. There exist. many different. clustering techniques (Anderberg] 1973 Duda et al., (200),, including hicrarchical, centroid-based and density-basod. techniques, as well techniques arising from probabilistic generative models. Each of these methods is proficient at finding clusters of particular nature. Many of the most commonly used clustering algorithms require selection of parameters, a process which poses considerable challenge when applying clustering to real-world problems.
In this work, we present and implement AuToMATo (Automated Topological Mode Analysis Tool), a novel clustering algorithm based on the topological clustering algorithm ToMATo (Chazal et al. 2013). The latter summarizes the prominences peaks of a density function in so-called persistence diagram. The user then selects prominence threshold and retains all peaks whose prominence is above this threshold, which

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results in the final clustering. A simple heuristic to select is to sort the peaks by decreasing prominence, and to look for the largest gap between two consecutive prominence values (Chazal et al. 2013). While yielding reasonable results in general, this procedure is not very robust to small changes in the prominence values.
A more robust and sophisticated method is to perform a bottleneck bootstrap on the persistence diagram produced by ToMATo, which is precisely what AuToMATo does. That is, given persistence diagram obtained by running ToMATo on point cloud, AuToMATo produces confidence region for that diagram with respect to the bottleneck distance, which translates into choice of 7 that determines the final clustering. While AuToMATo is not paramcter-froe per sc, we provide default choices that make it perform well across the board. Unless stated otherwise, AuToMATo will henceforth refer to our algorithm with its parameters set to these defaults. We experimentally analyze the clustering performance of AuToMATo and we find that it not only outperforms paramcter-fire clustering algorithms, but often also even the best choice of hyperparameters for many parametric clustering algorithms. Paramoter-fre algorithms building on ToMATo exist in the literature, for example, in Cotsakis et al. (202T) the final clustering is determined by fitting a curve to the values of prominence, and in Bois et al. (2024) significant values are separated from non-significant ones by adapting the process that produces the persistence diagrams. Indeed, the former algorithm is one of those that AuToMATo is shown to outperform.
We envision one important application of AuToMATo to be to the Mapper algorithm, introduced in Singh et al. (2007). Mapper constructs a graph that captures the topological structure of a data set. It relies on many parameters, one of them being a clustering algorithm applied to various chunks of the data, Algorithms that depend heavily on a good choice of tunable hyperparameter are generally not good candidates for usage with Mapper, as the best choice for the hyperparameter can vary significantly over the different chunks, and manually choosing a different hyperparameter for each may not be possible in practice. Thus, most choices of hyperparameter will generally perform badly on some of the subsets, leading to undesired results of Mapper. Thus, AuToMATo can be seen as progress towards finding optimal paramcters for Mapper, which is an active area of research (Carrière et al.y 2018: Chalapathi et al.) 2021; Rosen el al.] 2023). Running examples for Mapper with AuToMATo,, we see that is indeed a good choice for a clustering algorithm in this application
when compared to parametric clustering algorithm such as DBSCAN.
2 Background
2.1
Persistence and the ToMATo clustering algorithm
Both ToMATo and AuToMATo rely on the theory of persistence (EdeIsbrunner et al. 2002 Zomorodian &
Carlsson 2005
Carlsson 2014) to quantify the prominence of peaks of (an estimate of) density function, and to build a. hierarchy of peaks. Given a topological space X equipped with a density function f: X - Rgo. the first step of persistence is to build a filtration from X.
Definition 2.1. Let X be a topological space, and let f: X R be continuous. The superlevel set filtration of (X, f) is the family of superlevel sets (X2: |t E R), where f
In the following, we assume for case of exposition that all local extrema of f have distinct values. The idea underlying ToMATo is to track the evolution of (the number of) connected components of X2r as ranges from +o0 to -C. In that process, the number of connected components of X2 remains constant, unless passes through the value
local extremum of f. As passes through the value of a local maximum, a new connected component is "born" and added to the superlevel set X21. Similarly, as passes through the value of local minimum, two connected components of X2, are merged into one. ToMATo builds hierarchy of local maxima of f by declaring that, as two components get merged, the component corresponding to the local maximum with higher value absorbs the other one and persists, whereas the component corresponding to the local maximun with lower value "dies". Therefore, to each local maximum we associate pair (b, d) where b denotes the birth and d the death time, respectively. The cvolution of the connected components can be concisely recorded in a persistence diagram.
Definition 2.2. Let {brr, di)) denote the birth and death times of connected components of the superlevel set filiration X2thes associated to the densily I: X - R. The associated persistence diagram, denoted
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by Dgm(X,f) f), the multiset in the extended plane Rº, = R U f consisting of the points {(b,, dih C K (counted with multiplicity) and the diagonal A {(x, x)) | E [ (where each point A has infinite multiplicity). For given local marimum of with birth time by and death time di, we refer to the difference di - bt as ils prominence or lifetime.
The reason for working in the extended plane that, provided that f has a global maximum, the superlevel set filtration X21 will have connected component that never dies, that is, has death time equal to -00. See the red graph in Figure for an illustration.
The persistence diagram Dgm(X, f) provides a summary of f. The points of Dgm(X, f) are in one-to-one correspondence with the local maxima of f. and twice the disstinance of a point to the diagonal A (that is, its Euclidean vertical distance) equals its prominence.
We now outline how the ToMATo clustering algorithm works. Given a point cloud X ToMATo relies on the assumption that the points of X were sampled according to some unknown density function f. In nutshell, ToMATo infers information about the local maxima of by applying the above procedure to an estimate of f. ToMATo takes as input:
A neighborhood graph 9 on the points of X. Chazal et al. mostly use the d-Rips graph and the k-nearest neighbor graph
A density estimator f. Each vertex e of G is assigned a non-negative value f(v) that corresponds to the estimated density at v. Chazal et al. propose two possible density estimators: the truncated Gaussian kernel density estimator and the distance-to-measuree density, originally introduced in Biau et al. (2011)
A merging parameter 2 0. This is a threshold that the prominence of local maximum of the estimated density ff must clear for that local maximum to be deemed a feature.
Given the inputs above, ToMATo procoeds as follows.
Estimate the underlying density function f at the points of X.
2. Apply a hill-climbing algorithm on G. Construct the neighborhood graph G on the points of X, and construct directed subgraph g' of G as follows: at cach vertex e of g, place directed edge from to its neighbor with highest value of f, provided that that value is higher than f(v). If all neighbors of v have lower values, is a peak of f. This yields a collection of directed edges that form a spanning forest of the graph G, consisting of one tree for each local maximun of f. In particular, these trees yield a partition of the elements of X into pairwise disjoint sets that serves as a candidate clustering on X.
3. Construct the persistence diagram. Construct the persistence diagram Dgm(G, f) associated to the superlevel set filtration of f: GG
4. Merge non-significant clusters. Iteratively merge every cluster of prominence less than T of the candidate clustering found in Step 2 into its parent cluster, that is, into the cluster corresponding to the local maximum that it gets merged into in the superlevel set filtration of f: 9 - TR. ToMATo outputs the resulting clustering of points of X, in which every cluster has prominence at least by construction.
The reason why we expect the persistence diagram of the approximated density to be "close" to the original one stems from the stability of persistence diagrams under the bottleneck distance (explained in Sention 2.2 This is illustrated in Figure]] [
In practice, the user must run ToMATo twice. First, ToMATo is run with т = too which is equivalent to computing the birth and death time of each local maximum of of and hence the persistence diagram
GGiven point cloud, both of these undirected graphs have the set of data points as their vertex set. the case of the 6-Rips graph, two vertices are connected iff they are at a distance of at most apart, whereas in the k-ncarest neighbor graph, data point counected to another the latter is among the k-ncarest neighbors the first 2For a smoothing parameter m E (0, 1), and a given data point æ, its empirical (unnormelized) distance-to-moasur density
is given by f(x) = ( (a()) || - v12) where & [mn]] Nz(2)) denotes the set of the k nearest neighbors of and the cardinality of the data set.
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f(z)
d
P
8y
Pb
Figure 1: A function f: K R, K C R, in red, and estimate f of f in blue (left), with corresponding persistence diagrams Dgm(K,) f) and Dgm(G, f) consisting of the red and blue dots, respectively, together with dashed line separating noise from features (right).
Dg, From the diagram Dg f) the user then determines a merging parameter 7 by visually identifying large gap in Dgm(G, f) separating, say, C points corresponding highly prominent peaks from the rest of the points. Then, ToMATo is run second time with 7 set to that value, which results in the final clustering of X into C' clusters.
2.2
The bottleneck bootstrap
The bottleneck bootstrap, introduced in Chazal et al. 20117 Section 6), is used to separate significant features in persistence diagrams from non-significant ones. While it may be used more general settings, we will restrict ourselves to the scenario of Section 2.1
We first review the bottleneck distance, which is the standard distance measure between persistence dia- grams Eddesbrunner & Harer 2010 Chazal et al. 2016).
Definition 2.3. Let Dgm, and Dging be two persistence diagrams that have finitely many points off the diagonal. Let π denote the set of bijections v: Dgm, + Dgm2. Given points x = (x1, and = (y1, in R², let || y|1oo max{[æ1 vıl. |x2 - y223 denote their -isddisance, where we set (+oo) - (+00) (-oo) (-oo)-(-o0) (-00) 0. Then, the bottleneck distance between Dgm, and Dgm, is defined as
W (Dgm,, Dgm,) = inf sup || =(x)]loo-
VET #EDmm
Note that a bijection v: Dgm, + Dgm, is allowed to match an off-diagonal point of Dgm, to the diagonal of Dgm,, and vice versa.
We now outline the bottleneck bootstrap. relies on the following theorem, which summarizes the relevant results of Chazal et al. (2017) Section 6).
Theorem 2.4 (Chazal et al. (207))) Let X NRN be sample consisting of n. data points drawn according to probability density function K + 10, K c RN. Denote by D (,g((Kf)) and D = Dgm(X, the correspording unknown. and estimated, respectively, persislence. diagroms of superlevel sels. Given confidence level a E 10, define qa by
P(VnW (D, S 4m) 1-
Then a consistent estimator for qa is given by qa, which in turn is defined by P(VnWo (Di D) =1
Here, D' = Dgm(X", denotes the random persistence diagram constructed from a sample X* of size n draum according the empirical measure Pn, on X, where P,, defined as the probability measure on that assigns the probability mass 1 1/n to each data point in X.
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In setting, the theorem above is applied follows. Given the sample X R we estimate and the connectivity of K with density estimator and a neighborhood graph, respectively (as explained in Scction 2.1 This allows us to compute D Dg, f), which, in turn, serves estimate of D. The approximation of the unknown probability measure f, and using this, Theorem 2.4 allows us to approximate the distribution
empirical measure P. X serves
F(z) = Ptv=W 2)
with the distribution
P(vT= (D- D) S 2))
where Dr = Dgm(X*, f) is random quantity. Note that X may be thought of sample drawn from X with replacement.
Like the distribution F, the distribution F is still not explicitly computable, but, unlike F. it can be approximated by Monte Carlo as follows. We draw B samples ,i...., X; of size n from Pe. and for each of these B samples, we compute the persistence diagram D = Dgm(X; and the quantity T = Vn (D;, D), I,..., B. Finally, we the function
F(z)
Σ (T)) B
as an approximation of F, and hence of F. Using this, we
aa = inf F(z) (-))
to be our estimate of la- This estimate is asymptotically consistent by Theorem 2.4 that is, To qa. In conclusion, the true, unknown persistence diagram is at bottleneck distance of at most / from D with probability at least a-a. Hence, points of D that are at -isstiitncce at most qa/ / from the diagonal could be matched to the diagonal under the bottleneck distance, and thus a point of D is declared to be significant feature iff it is at 1,9ddistance of at least To/Vá to the diagonal, that is, iff its prominence is at least 2 0a/ /T.
3 Methodology and implementation of AuToMATo
3.1 Methodology of AuToMATo
AuToMATo builds upon the ToMATo clustering scheme introdueed in Chazal et al. 2013 and implemented in Glisse 20025). AuToMATo automates the step of visual inspection the persistence diagram by means
of the bottleneck bootstrap, thus promoting ToMATo to a clustering scheme that does not rely on human input.
More precisely, given point cloud X to perform the clustering on, AuToMATo takes as input
an instance of ToMATo with fixed neighborhood graph and density function estimators;
a confidence level a E (0, 1); and
a number of bootstrap iterations B Z21-
Remark 3.1. We point out that our implementation of AuToMATo comes with default values for each of
the objects. Each of these values can, of course, be adjusted by the user. For details on these default values,
see Subsection
To apply the bottleneck bootstrap as described in Section 2.2 AuToMATo generates B bootstrap subsamples xf....,X X3 of X, cach of the same cardinality as X, where X the data set whose points are be clustered. Then the underlying ToMATo instance with = too and its neighborhood graph and density function estimators is used to compute the persistence diagram for X and each of Xi,..., X yielding
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persistence diagrams D and Di...., Dis. respectively. Using the bootstrapped diagrams Di...., Dis bottleneck bootstrap performed on D. This yields value Ta that (asymptotically as - oo) satisfies
Po D) S 7o,)
a;
where D denotes the persistence diagram of the true, unknown density function from which X was sampled. Thus, points of D of prominence least 2- v are declared to be significant features of D, and AuToMATo outputs its underlying ToMATo instance with prominence threshold set to = 2. / Vn. This procedure is schematically depicted in Figure 2
ToMATo PD
ToMATo PD with threshold
Data set X
ToMATo PDs
Bootstrap samples x X3
Figure 2: Schematic of the methodology of AuToMATo: from data set X, the usual ToMATo persistence diagram (with 5 = +oo) is computed. Additionally, the analogous persistence diagrams are computed for the bootstrap samples xi....,y; which are created from X by drawing with replacement. Finally, the bootstrap procedure (indicated by 8) is used to compute prominence threshold for the original persistence diagram.
When computing the values points in Dj and D with finite lifetimes. The reason for this choice is that we consider peaks with infinite lifetime to be significant a priori. Moreover, some of the bootstrapped diagrams among the D Dj have different number of points with infinite lifetime than the reference diagram D. In these cases, the bottleneck distance of the bootstrapped diagram to the reference diagram is infinite, which heavily distorts the distribution Fzz). This choice is justified by experiments.
L,..., B, in the bottleneck bootstrap, we only consider
3.2 Implementation of AuToMATo
We implemented AuToMATo in Python, and all code with documentation is publicly available 3 For description of AuToMATo in pscudocode,
Algorithm [ The algorithm has worst-case complexity of ((B(nd + n log(n) + 1 log N)), where d is the dimensionality of the data and N is the maximal number of off-diagonal points across all relevant persistence diagrams (which is generally much smaller than n); see below for details. Note that the factor of B can be significantly decreased through parallclization.
While the input parameters may be adjusted by the user, the implementation provides default values whose choices discuss presently.
Choice of ToMATo parameters: Our implementation of AuToMATo is such that the user can directly pass parameters to the underlying ToMATo instance. If no such arguments are provided AuToMATo uses the default choices for those parameters, as determined by the inplementation of ToMATo given in Glisse (2025). In particular, AuToMATo uses the k-nearest neighbor graph and the (logarithm of the) distance- to-measure density estimators by default, each with k 10. Of course, the persistence diagrams produced
Thee code archived Zenodo wtt.oor//1...../zemme. 17279741 and developed openly GitHub ggithub.com/
a-a-huber/automato patppppppaooottppprp......

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Algorithm 1: AuToMATo
Input: point cloud X of n data points; instance tom, of ToMATo with neighborhood graph and density function estimators, and prominence threshold т; confidence level (0, 1); mumber of bootstrap iterations B E Zz1.
D < [x))((tomx(()))
// compute persistence diagran of point cloud
for i to B do
Let X be a subsample of X of size n, sampled with replacement;
D; + Dgm(tomx(XX]
// compute persistence diagram of subsample // compute bottleneck distance between finite points
d, <- / 8 (D;, ;);
end
Sort and reindex (Di ....... ) such that d1 .... S dB:
(Di...., k +- a1- B];
La + dzi
7 + 2//22///// Vn;
Output: tom,(X): // copy of initial ToMATo instance with prominence threshold set to т
by ToMATo, and hence the output of AuToMATo, depend on this choice. This can lead suboptimal clustering performance of AuToMATo; see Section
Choice of a and B: By default, AuToMATo performs the bootstrap B 1000 subsamples of the input point cloud, and sets the confidence level to 0.35. The choice of this latter parameter means that MATo determines merely a 65% confidence region for the persistence diagram produced by the underlying ToMATo instance. While in bootstrapping the confidence level is often set to, for instance,a = 0.05, the seemingly strange choice of a = 0.35 in the setting AuToMATo is justified by experiments. The value of 65% seerss to be low enough offset some of the negative influence of using possibly non-optinized neighborhood graph and density estimators discussed in Section 6 while at the same time being high enough to yield good results when these estimators (as well as the value B 1000) was decided on after running early implementation of AuToMMAo on just a few synthetic data sets. In particular, the choice was made before conducting the experiments in Section 4 AuToMATo is implemented in such a way that the parameter a can be adjusted after fitting and the clustering is automatically updated.
chosen suitably. We point out that the value
0.35
Complexity analysis of Algorithm Recall from Chazal et al. 2201 Section 2) that, if an estimated density and a neighborhood graph are provided, ToMATo has worst-case time complexity in On log(n) + ma(n)), where and are the number of vertices and edges of the ncighborhood graph, respectivcly, and denotes the inverse Ackermann function (note that equals the number of data points). By default, density
ToMATo (and hence AuToMATo) works with the k-nearest neighbor graph and distance-to-measuue estimators, where the latter itself relies on the k-nearest neighbor graph (each with A = 10). Taking into account the known complexity bound O(nd) for the creation of the k-nearest neighbor graph (where the dimensionality of the data), and using the fact that m E (n)) for this graph, this leads to worst-case time complexity in O(nd+ n log(n)) for single run of ToMATo. Creating the bootstrap samples X;, i= 1:....,, B, has complexity in O(Bn); computing the values VnWffn (D; (D;, D), i I,.... B, has worst-case complexity L5BN log(N)) (where N denotes the maximal number of off-diagonal points across all relevant persistence diagrams), and sorting them has worst-case complexity in O(B log B)). Combined, this leads to a worst-case complexity for AuToMATo in O(B(nd + n log(n) + 1 log(N) + B log(B)), Using that B is a constant, we obtain the runtime of O(B(nd + n log(n) + 11.5 log N)) claimed above.
We point out that the complexity of 1(N1.51gg(N)) for the computation of the bottlenock distance between a pair of persistence diagrans is a theoretical worst-case scenario, whose validity was established in Efrat et al. (2001). Tn practice, one typically allows for a small error in the computation of bottleneck distances
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which decreases the effective complexity (Kerber et al. 2017 Our implementation of AuToMATo makes use of this by allowing for the smallest possible error as determined by the smallest positive double (see the implementation in Godi (2025) for details). Moreover, note that the factor B appearing in the complexity of AuToMATo can be drastically decreased in practice through parallelization.
The Python package: Our Python package for AuToMATo consists of two separate modules; one for AuToMATo itself, and one for the bottleneck bootstrap. Both are compatible with the scikit-learn archi- tecture, and the latter may also be used as stand-alone module for other scenarios. In addition to the functionality inherited from the scikit-leary API, the implementation of AuToMMTo comes with options of plotting the persistence diagram and the prominence threshold found in the bootstrapping;
adjusting the parameter of fitted instance of AuToMATo which automatically updates the resulting clustering without repeating the (computationally expensive) bootstrapping:
setting seed in order to make the creation of the bootstrap subsamples in AuToMATo deterministic, thus allowing for reproducible results; and parallclizing the bottleneck bootstrap for speed improvements.
Finally, our implementation of AuToMATo contains parameter that allows the algorithm to label points outliers. In mutshell, point is classified as outlier if it is not among the nearest neighbors of more than specified percentage of its own ncarest neighbors. This feature, however, is currently experimental (and is thus turned off by default).
4 Experiments
4.1 Choice of clustering algorithms for comparison
We chose to compare AuToMATo with its default parameters against
DBSCAN and its extension HDBSCAN;
hierarchical clustering with Ward, single, complete and average linkage;
the FINCH clustoring algorithm (Sarfraz et al. 2019); and
a clustering algorithm building on ToMATo stemming from the Topology Tool (TTK)
suite (Tierny et al. 2018); in the following, we will refer to this as the TlK-algorithm
For DBSCAN, HDBSCAN and the hicrarchical clustering algorithms mentioned above, we worked with their implementations in scikit--laar For the FINCH clustering algorithm, worked with the version available on GitHub Indeed, we subclassed that version in order to make it compatible with the scikit-learn API. Similarly, we created a scikit-learn compatible version of the TTK-algorithm by combining code from TTK with the description of the algorithm given in Cotsakis et al. 2021 Section 3.2). While we included DBSCAN and HDBSCAN among the clustering algorithms to compare AuToMATo against because they are standard choices, we chose to include the hicrarchical clustering algorithms bocause they are rcadily available through scikil-learn. Finally, we chose include FINCH and the TTK-algorithm because, like AuToMATo, they are out-of out-of-the-bos box (indeed, parameter- faree methods and are thus especially interesting to compare AuToMATo against.
4.2 Choice of data sets
The data sets on which we ran AuToMATo and the above clustering algorithms stem from the Clustering Benchmarks suite iGagolewski] 2022 We chose this collection it comes with large variely of different data sets, all of which are labeled by or more ground truths, allowing for a fair and extensive comparison. The collection contains five recommended batteries of data sets from which we selected those (data set, ground truth)-pairs that we deemed reasonable for a general purpose parameter-free clustering algorithm.
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html
"Specilically we worked version 1.1.0 of the benchmarking suite GGagolewski al. 2022
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For instance, we chose to include the data set named windows that is part of the wut-battery, but not the data set named windows from the same battery (see Figure 6 in the appendix for an illustration). We chose to include the windows data set because AuToMATo determines clusters depending on connectivity, and topologically speaking, there is only onc connected component in the olympic data set. Finally, we excluded all instances where the ground truth contains data points that are labeled as outliers, as outliers creation is currently an experimental feature AuToMATo.
4.3 Methodology of the experiments
We min-max scaled each data set, fitted the clustering algorithms to them, and recorded the clustering performance of each result by computing the Fowlkes-Mallows score (Fowlkes & Mallows 1983) of the clustering obtained and the respective ground truth. While the Fowlkes-Mallows score was originally defined for hierarchical clusterings only, it may be defined for general clusterings follows. Given clustering C found by an algorithm and ground truth clustering G, one defines the Fowlkes-Mallows score as
TP V TP+FP V
TP
FMS
TP FN
where
TP is the number of pairs of data points which are in the same cluster in C and in G;
FP is the number of pairs of data points which are in the same cluster in G but not in C; and
FN is the number of pairs of data points which are not in the same cluster in G but are in the same cluster in C.
In other words, the Kowlkes-Mallows score is defined as the geometric mean of precision and recall of a classifier whose relevant elements are pairs of points that belong to the same cluster in both C and G. It may atlain any value between and 1, and these extremal values correspond to the worst and best possible clustering, respectively. We chose to use the wwwlkes-Mallowa score as opposed to, for instance, mutual infor- mation or any of the Rand indices, because the latter have been shown to exhibit biased behavior depending whether the clusters in the ground truth mostly of similar sizes not, see, for instance, Romano ct al. (2016); to the best of our knowledge, the Fowlkes-Mallows score does not suffer from such drawbacks. Moreover, we chose not to use any intrinsic measures of clustering performance since any such measure implicitly defines a further clustering algorithm to compare AuToMATo against, whereas we are interested in comparing AuToMATo against a predefined ground truth clustering.
We set the hyperparameters of the HDBSCAN, FINCH and the TTK-algorithm to their default values (as per their respective implementations). In contrast to this, let the distance threshold parameter for the DBSCAN and the hierarchical clustering algorithms vary from 0.05 to 1.00 in increments of 0.05, with the goal of comparing AuToMATo against the best and worst performing version of these clustering algorithms. To account for the randomized component of AuToMATo, we ran it len times, each time with a different seed.
While we restricted oursolves to instances where the ground truth does not contain any points labeled as outliers, some of the clustering algorithms in our list (DBSCAN and HDBSCAN) label some data points outliers. In order to prevent these algorithms from getting systematically low Fowlkos scores because of these outliers, we removed all the points labeled as outliers by these algorithms, and only computed the Fowlkes- Mallowss score on the remaining points, both for these clustering algorithms and for AuToMATo. This of course gives an advantage to DBSCAN and HDBSCAN over AuToMATo.
In order to allow reproducibility, we chose a fixed seed for all our experiments, which can found in our code B We ran our experiments on a laptop with 12th Gen Intel Core 17-1260P processor running 2. 2.10GHz.
The code archived Zenodo (doi.org/10. http:/10.528/zeeode. 17279741 and developed openly on GitHub (github.com wwa--ubbrr/aautcnttoouuuu
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4.4 Results and interpretation
Table D shows the average Fowlkes-Mallowe score of each algorithm across all benchmarking data sets; for AuToMATo, it shows the average and the standard deviation across the ten runs. For those benchmarking data sets that come with more than one ground truth, we included only the best score of the respective algo- rithm. Similarly, we included only the best performing parameter selection for those algorithms that we ran with varying distance thresholds (which, of course, skews the comparison in favor of those algorithms). As Table shows, AuTooMATo outperforms each clustering algorithm on average across all data sets, thus show- ing that it is indeed versatile and powerful out-of-the-boo clustering algorithm. In particular, AuToMATo out perforrsmss the TTK-algorithm, which also build on ToMATo.
Table 1: Average clustering performance of AuToMATo reference clustering algorithms
Algorithm
Fowlkes- Mallows score
AuToMATo
0.855440.0228
DBSCAN
0.8437
Average linkage
0.8321
HDBSCAN
0.8209
Single linkage
0.8156
TTK clustering algorithm
0.8019
Complete linkage
0.7592
Ward linkage
0.5896
FINCH
0.5074
The scores of our experiments are reported in Tables 2 through c in Appendix A.2 As an illustration, Figure g shows that the best choice of parameter for DBSCAN sometimes outperforms AuToMATo, which is to be expected. However, on most data sets where this the case, the results from AuToMATo still competitive, and there is significant number of instances where AuToMATo outperforms DBSCAN for all parameter selections, in some cases by lot.
clusterer
fis score
0.8-
dbscan, kin
dbscan
9.4-
wit
top zetra
sipu
vat x3
var Lon_isntttinn
sipu patthaed
sipu sipu.stin Jain
tops stos
wot cir.irei
other irts
sipu ri5
vat ek2
other
vot smite
sipu corpound
fops
sipu flam
tops boedt
sipu squregati
tops kota
tops Loun
graves, ring
grews ring monnonllee
graves, zi0z20
other square
wit stripts
vat Trapped tovers
was jeinttes
wit tuespl twesplestees estres
wit Labirynth
grows parsbalii ii
si00 at
App 22
22 ndt
chain
unbalaace
vinguut
Line
benchaark
Figure 3: Mallows score of AuToMATo and DBSCAN across benchmarking data sets. The shading of "automalo mean" indicates the standard devialion of the score across the ten runs.
5 Applications of AuToMATo in combination with Mapper
The goal of Mapper (Singh et al. 2007) to approximate the Reeb groph of manifold M based on a sample from M. The input point cloud with a filter function P + R; collection of overlapping intervals u (UI...., Un} covering IR; and clustering algorithm. For each U; U4 Mapper runs the clustering
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of vertices corresponding to Published in Transactions on Machine Learning Research (10/2025)
algorithm the data points in the preimage (U,), creating vertex for each cluster. Two vertices are then connected by an edge if the corresponding clusters (in different preimages) have some data points in common, yielding a graph that represents the shape of the data set.
We ran the Mapper implementation of giotto-tda (Tazin et al, 2020 on a synthetic two-dimensional data set consisting of noisy samples from two concentric circles (see Figure 4aj with projection onto the 2-axis the filter function. We ran Mapper on the same interval cover with three different choices of clustering algorithms: AuToMATo, DBSCAN, and HDBSCAN. As be seen in Figure [b] using DBSCAN, we see many unwanted edges in the graph. HDBSCAN performs better, giving two cycles with some extra loops. The output of Mapper with AuToMATo is exactly the Reeb graph of two circles.
(a)
(b)
(c)
(d)
Figure 4: (a) input data set; result of Mapper with (b) AuToMATO; (c) DBSCAN; (d) HDBSCAN
We further tested the combination Mapper with AuToMATo on one of the standard applications of Mapper: the Miller- Miller-Reaven diabetes data set, where Mapper can be used detect two strains of diabetes that correspond to "flarcs" in the data set (sce Singh et al. (2007 Section 5.1) for details) As can be seen Figure 5 AuToMATo performs well in this (ask; the graphs show central
healthy patients, and two flares corresponding to the two strains of diabetes. We were not able to reproduce this using DBSCAN or HDBSCAN; Figure shows the output of Mapper with these algorithms with their respective default parameters.
Figure 5: Mapper applied the diabetes data with AuToMATo (left); DBSCAN (center); HDBSCAN (right). Labels 0, and stand for "no ", "chemical" and "overt diabetes".
6
Discussion
We briefly outline some limitations of AuToMATo. AuToMATo comes with choice of default values for its parameters.
9The data is available as part the "lacfit" R-package Loader 2024
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Optimizing the choice of the neighborhood graph and density estimators is an aspect of AuToMATo that we plan to pursue in future work. Moreover, we plan to improve the currently experimental feature for outlier creation in AuToMATo discussed at the end of Section 3.2 Finally, it is natural to ask whether the results from Carrière et al. (2018) on optimal paramcter selection in the Mapper algorithm can be adapted to the scenario where Mapper uses AuToMATo as clustering algorithm.
Acknowledgments
The first author was supported by the Swiss National Science Foundation (project no. 209413).
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A Appendix
A.1 About the choice of data sets
we chose to include the data set named windows from the battery named wut, but not the data set named olympic from the same battery. Those are illustrated in Figure In that figure, the data points are colored according to the ground truth clustering.
As explained in Section 4.2
A.
34
0.4
03-
0.5
0.2 0.6
0th
0th
Figure 6: The data sets named windows (left) and olympic (right) from the wut-battery.
A.2 Benchmarking results
In this subsection report the Fowlkes Mallows scores coming from comparing AuToMATo to the other clustering algorithms, explained in Section 4 For those benchmarking data sets that come with more than one ground truth, we report the scores for each of those, and different ground truths are indicated by the last digit in the data set name. Moreover, each table is sorted according to increasing difference in clustering performance compared against. As is customary, we indicate the score stemming from the best performing clustering algorithm in bold. Finally, each of the table is accompanied by graph similar to the one depicted in Figure 3 Note that, in particular, that those figures indicate only the score corresponding to the ground truth on which the respective clustering algorithun performs best on.
AuToMMTo and the respective clustering algorithm that AuToMATo is being
Table Fowlkes-Mallows scores of AuToMATo DBSCAN
Dataset
automato mean dbscan max
dbscan min
sipu r15 2 wut trajectories
0.4867+0.0000
1.0000
0.5607
0.503810.0107
1.0000
0.4999
wut x3 0
0.5153:0.0000
0.9398
0.5149
wut x2_0 r15 rpp__r15___ feps tetra sipu_pathbased 0
0.5846=0.0000
0.9483
0.5779
0.5436
0.8954
0.5021
0.62611100000
0.9403
0.0000
0.651710.000
0.9569
0.5769
sipu_spiral_ 0
0.702810.0000
1.0000
0.5756
wut vut__ssoatiin_1 0
0.7256+0.0113
1.0000
0.5773
sipu pathbased 1
0..020
0.9620
0.5170
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Table 2: Fowlkes- Mallows scores of AuToMATo vs. DBSCAN
Dataset
automato mean
dbscan max
dbscan min
sipu jain 0
0.783710.0000
0.9880
0.7837
graves_ _eiissdd_______ 0
0.8377=0.1396
0.9970
0.7053
sipu 1ipu__oompund_0 feps_atom wut circles 0 feps seps_chainlink_. other iris 0 wut_mkk wut mk2
0.8616=0.0000
1.0000
0.4972
0.8694=0.0000
1.0000
0.7067
0.8857=000000
1.0000
0.4998
0.8896=000000
1.0000
0.7068
0.7715=000000
0.8721
0.0000
0.9072=0.0034
1.0000
0.5770
0.6356+000000
0.7068
0.5778
sipu_sippmeeee tal___copouddd 4
0.944210.0000
1.0000
0.5523
wut_x3_
0.6546=0.0000
0.7042
0.6546
graves zigzag other_iris5__ wut smile wut x1_ feps_tarred 0
0.6720+000000
0.7149
0.4446
0.6712+0.0000
0.7046
0.0000
0.9701
1.0000
0.5825
0.9741=0.0818
1.0000
0.5846
0.9850=000000
1.0000
0.6963
wut smile 0
00.000
0.9753
0.5471
wut mk3 0
0.0000
0.7774
0.5764
_oa__coppuudd_
0.9786=000000
0.9825
0.5715
sipu_flame_0 sipu unbalance
0.7320=0.0000
0.7341
0.5918
0.9986+10.008
1.0000
0.5339
feps dsps_twodiamonds 0
0.706710.0000
0.7067
0.7067
sipu aggregation 0
0.865210.0000
0.8652
0.4653
wut wut_stripes_0
1.0000=0.0000
1.0000
0.7070
wut trapped lovers O
1.0000=100000
1.0000
0.6632
wut wi__wiidww____
1.00001100000
1.0000
0.6753
Teps hepta
1.0000=1.00000
1.0000
0.3727
1.0000=100000
1.0000
0.6111
graves line
1.000010.0000
1.0000
0.8238
graves ring graves siin_ooutiiers 0
1.0000=0.0000
1.0000
0.7068
1.0000=0.0000
1.0000
0.6863
graves zigzag 0
1.0000+0.0000
1.0000
0.5328
other coro__squre__0 wul mk1 wut wut_twosplashes feps_wingnut, graves Liares_prrabllc____ wut wut_labirynth 0
1.000010.0000
1.0000
0.7068
0.98661000000
0.9651
0.5754
1.000010.0000
0.9649
0.7062
0.9805=00.000
0.8784
0.7068
0.691610.0000
0.5000
0.4999
0.788410.0000
0.5221
0.5221
graves _pctlappapaapa___ 0
0.98021000000
0.7068
0.7068
sipu dipu_d31_0
0.6001
0.1846
0.1787
sipu al 0
0.7499+0.0000
0.3269
0.2229
sipu_rr r15_0
0.920810.0000
0.4551
0.2552
sipu
0.9888
0.4890
0.2581
sipu a2 0
0.7555+0.0000
0.1685
0.1685
sipu a3 0
0.743410.0000
0.1410
0.1410
sipu s2 0
0.9405=0.0000
0.2581
0.2581
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clusterer
SUcOmatO BCOn
dbscan RIX
Ins-score
8.6-
0.4.
0.2
vit
tao ters
sinu simu,ppiza
vut x3
vit Losoto ation
sinu pathassd
zipu zain
Yas atos
other iris
sigu r15
other iris
wut smile
zigu conpond
teps, terget
wut nk
zipu Lsee
sinu unballane
tas twadlaronds
sipu
fap hotts
teps, Isun
grawes, ring
graees ring wuttier
graves Hig00g
other. sagirre
wot, tropped lovers
wit windoas
wut
wit teosp astes
wut tab5rynth
graves, parabalic
stpu a1
mar 92 sie
trajecttories
niz
chakalikk
Lime
benchwark
Figure Comparison of AuToMATo and DRSCAN.
Table 3: Fowlkes-Mallowe scores of AuToMMTo vs. hierarchical clustering with average linkage
Dataset
automato mean
linkage average
max
linkage
average min
sipu r15 2
0.4867+1000000
1.0000
0.3971
wut trajectories 0
0.5038
1.0000
0.3115
fops tetra 0
0.6261++00000
1.0000
0.0651
sipu r15 1
0.5436:10.0000
0.8954
0.4435
sipu d31 0
0.6001++0.0085
0.9322
0.1787
wut x3_ 0 wut x3 feps Iwodiamonds 0
0.5153=:0.0000
0.8389
0.3343
0.6546=+0.0000
0.9747
0.2693
0.7067+00.0000
0.9925
0.1287
sipu_a2_ 0 sipu al 0
0.7555=100000
0.9432
0.1685
0.7499:100000
0.9268
0.2229
a3 sipu_a3_0 0
0.7434+100000
0.8825
0.1410
sipu aggregation 0
0.8652+0.0000
0.9932
0.1785
tppu_pathbased 1
0.7322
0.8564
0.2058
graves dense 0
0.8377+10.11
0.9604
0.6633
_ou__patbbase___1 0 wut_xx2 x2_ 0 wut circles 0
0.6517+000000
0.7704
0.1848
0.5846++0.0000
0.7001
0.2782
0.8857+0.0000
1.0000
0.2369
wut mk3 0
0.7720=000000
0.8771
0.0876
wut mk2 0 sipu_ri5_0
0.6356+10.0000
0.7068
0.1421
0.9258+000000
0.9900
0.2552
graves zigzag 1
0.6720++0.0000
0.7202
0,4380
other_iris_0
0.7715+1.0000
0.8080
0.0791
other iris5 wut_xi__
0.671 4,6712+0.0000
0.7042
0.0451
0.9741+0.0818
1.0000
0.3070
sipu jain 0
0.7837+0.0000
0.7904
0.1
wut mk1 0
0.98661+0.0000
0.9933
0.2037
sipu__unpel _on__ubbllnne__0 0
0.9986++0.0008
0.9995
0.5339
teps hepta 0
1.0000+10.0000
1.0000
0.3727
sipu 0
0.7300+0.0000
0.7320
0.0913
sipu sl
0.0000
0.9821
0.2581
inutd on next page
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Table Fowlkes- Malkows scores
AuToMATo vs, hierarchical clustering with average linkage
Dataset
automato mean
linkage average max
linkage average min
sipu compound 0
0.8616=10.0000
0.8431
0.2207
sipu_compound 4
0.9442=0.0000
0.9224
0.1985
sipu_ _op__compund__ sipu_s2_0 wut smile feps_atom_ 0
0.9786=000000
0.9546
0.1922
0.9405=0000000
0.9097
0.2581
0.9701=0.0000
0.8726
0,4041
0.8694#000000
0.7491
0.2555
graves_ppaabbls
0.6916=000000
0.5708
0.2135
sipu_spiral_0
0.7028#0.0000
0.5756
0.1919
wkt_mk4 0 wut smile wut put_isolaatin_1 0
0.9072:0.0234
0.7714
0.2071
0.9681=0.0000
0.8221
0.4303
0.7256=0.0113
0.5773
0.1651
graves line geps_chainlink__ feps larget
1.0000=00.0000
0.8238
0.3047
0.8806=000000
0.7068
0.1456
0.0000
0.7986
0.3285
feps_ Losg_wingut____ fops_lsun 0 graves ring 0
0.9005
0.7739
0.1101
1.0000=0.0000
0.7896
0.1735
1.0000=00.000
0.7780
0.2638
graves parabolic 0
0.9802=100000
0.7580
0.1598
graves_ ring _otllirrserr__s 0
1.0000=+0.000
0.7767
0.2801
other_ Sther_square_0 wut labirynth 0
1.0000=000000
0.7413
0.1746
0.7884+0.0000
0.5221
0.2306
wut wut__stripe___ wut twosplashes wut wiidwwsdwws___ 0
1.0000=0..000
0.7070
0,1082
1.0000
0.7062
0.4837
1.0000=0.0000
0.6753
0.1194
wut trapped lovers 0
1.0000=0.0000
0.6632
0.1077
graves zigzag 0
1.0000+10.0000
0.6616
0.3008
clusterer
8.8
linkage average FIX
fno-score
linkage average
0.4-
tops tetra
Nit x3
sipu 22
sipu a3
sipu pathbssad
Nat x2
Not m9
not mi
other_
other iriss
sipu Jain
Net
zipu unba unbalane Lance
faps
zipu Flane
sipu
zipu corpound
sipu
Not snite
faps ates
sipu spiraa
Not
not isalation
graves line
teps, target
teps,
graves
graves ring outiters graves parabotic
Not rynth other square
Not coeop cussplashes Nit graves zig13g
NOT crappes tovers Nat siinoos
al
chuinlint
singnut
benchnark
Figure 8: Comparison of AuTOMATo and agglomerative clustering with average linkage.
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Table 4: Nowlkes-Mallowe scores of AuToMATo vs. HDBSCAN
Dataset
automato mean
automato std
hdbscan
wut trajectories 0
0.5038=10.007
0.0107
1.0000
wut x3 0
0.5153=0.0000
0.0000
0.8959
wut x2 0
0.5846+000000
0.0000
0.9344
sipu__siine _is__spira____ 0
0.7028=000000
0.0000
0.9815
sipu flame 0
0.7320-10.0000
0.0000
0.9900
sipu_d31 d31_
0.6001=000085
0.0085
0.8231
aipu_jain, 0
0.7837=000000
0.0000
0.9779
fops_tetra_
0.6261=000000
0.0000
0.8157
graves dense
0.8377++01336
0.1396
0.9894
_ou__patbbased 1
0.7322:10.000
0.0000
0.8634
feps feps_atom__ 0
0.8694+0.0000
0.0000
1.0000
sipu path-apthesedd 0
0.6517+0.0000
0.0000
0.7815
feps coss_chaaliin___ 0 sipu r15 0
0.8896+#00000
0.0000
1.0000
0.9258:100000
0.0000
0.9932
sipu al_0 0
0.7499#0.0000
0.0000
0.8081
wut x3 other iris5 0
0.6546==000000
0.0000
0.6972
0.6712
0.0000
0.7042
wul x1 0
0.9741 0.0881
0.0818
1.0000
sipu_compount 1
0.9786=000000
0.0000
1.0000
sipu sipu__cmpound__ 4
0.9442+=00000
0.0000
0.9656
leps target 0
0.9850-10.0000
0.0000
1.0000
sipu_comppund, 0
0.861610.0000
0.0000
0.8751
Sipu_unbalance 0
0.998611.0008
0.0008
1.0000
graves_zigzage
0.6720=0.0000
0.0000
0.6720
sipu__gguue sau__agregation 0
0.8652:100000
0.0000
0.8652
wut stripes 0
1.0000=10.0000
0.0000
1.0000
wut trapped lovers 0
1.0000=10.0000
0.0000
1.0000
wut__windo hot__inddws__ fops hepta_ 0
1.0000=0000000
0.0000
1.0000
1.0000=0000000
0.0000
1.0000
feps Isun 0
1.0000=0.0000
0.0000
1.0000
_iass__lne_____________ 0
1.0000=0.0000
0.0000
1.0000
graves ring 0
1.0000=0.0000
0.0000
1.0000
Laaee_riigg rring_oullirrs 0
1.0000=00.0000
0.0000
1.0000
graves zigzag 0
1.0000=10.0000
0.0000
1.0000
atber_square__
1.0000=0000000
0.0000
1.0000
other_irts
0.7715+000000
0.0000
0.7715
mut__k3___ wut mk1
0.7720=0.0000
0.0000
0.7719
0.9866=1000000
0.0000
0.9863
sipu a3 0
0.7434+0.0000
0.0000
0.7415
sipu_a2 a2_0
0.7555=0.0000
0.0000
0.7502
sipu r15 2 sipu_ r15_1 wut isolation 0
0.4007
0.0000
0.4671
0.5436=10.0000
0.0000
0.5212
0.7256:00.0113
0.0113
0.6377
feps__wipgee wingnut 0 sipu sl 0
0.0000
0.0000
0.8725
0.9088=000000
0.0000
0.8717
sipu s2 0
0.9405
0.0000
0.7410
wul mk4 0
0.9072+10.234
0.0234
0.6459
wut labirynth 0
0.788410.0000
0.0000
0.5134
Continued on next page
19

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Published in Transactions on Machine Learning Research (10/2025)
Table 4: Mallows scores of AuToMATo vs, HDBSCAN
Dataset
automato mean
automato std
hdbscan
graves parabolic
0.6916=10.0000
0.0000
0.3616
feps_ tww.iamonds.. 0
0.7067=0.0000
0.0000
0,2886
wut wut_mk2__
0.6356=000000
0.0000
0.1574
wut rut_smile__0
0.9681=0.0000
0.0000
0.4000
wul smile
0.0000
0.0000
0.3714
graves paresssprabblic___
0.0002
0.0000
0.3526
wut__wwospaashes
1.0000=0.00000
0.0000
0.3074
wut_ wut_circles__
0.8857#0.0000
0.0000
0.1204
clusterer
automsto meon adbscan
Ins-score
8.6-
8.2
7400 spirst
Hot
Sipu
fas tetra
=igu pathossed
tops
fige
-a00 115
ocher triss
wit
frgs target
=400 apg
rop herpa
for
graves
graves ring puctiers
graves rigon
other
wut stripes
wut NIn20s
other tris
wat PK1
zipu
sipu s1
wot eid
fag boodlawonds
wit mk2
graves jarabolic
Hut circles Lxcrpliasten
trajectordes
chaintink
compound
trapped
Esol aolaatio
wirgnut
Labieynth
Line
benchwark
Figure 9: Comparison of AuToMATo and HDBSCAN.
Table Fowlkes-Mallows scores of AuToMATo vs. hierarchical clustering with single linkage
Datasel
automato mean
linkage single max
linkage single min
sipu r15 2
0.4867+0.0000
1.0000
0.5607
wut trajectories 0
0.5038=1001077
1.0000
0.4999
@0pu_r15__
0.5433:=000000
0.8954
0.5021
feps_tetra tetra 0
0.6261 +000000000000
0.9296
0.0829
sipu_spiral_0 wut isolation
0.7028=+000000
1.0000
0.5756
0.7256+:0.0113
1.0000
0.5773
wut_x33
0.5153+0.0000
0.7347
0.4951
sipu__aii__
0.7837+000000
0.9510
0.7837
feps atom 0 wut circles chainink 0
0.8694+000000
1.0000
0.7067
0.8857+000000
1.0000
0.4998
0. 0.8896=L0.000
1.0000
0. 7068
wut_mk44_ sipu compound
0.9072+0.0233
1.0000
0.5770
0.8616:L0.0000
0.9454
0.4972
sipu _opu_pathbased sipu pathbased graves_derme dense__e 0 mk2 0
0.6517=2000000
0.7337
0.5769
0.7322-L0.0000
0.8091
0.5170
0.8377=0.132
0.9096
0,6882
0.6356+
0.7068
0.6007
graves zigzag
0.6720+0.0000
0.7344
0.4446
Continued next page
20

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Published in Transactions on Machine Learning Research (10/2025)
Table 5: Moolkes-Malloww scores of AuToMATo vs. hierarchical clustering with single linkage
Dataset
automato mean
linkage Linkage_sinngleeee max
linkage single min
wut x2 0
0.5846+000000
0.6437
0.5105
other iris5
0.6712=0.0000
0.7042
0.0451
wut _nt_smmle__1
0.9701=0.0000
1.0000
0.5825
wut_xl
0.9741=0.0818
0.9920
0.5846
Teps Larget
0.9850=10.0000
1.0000
0.6963
wut wut_ssmile_ 0
0.9601#0.0000
0.9748
0,5471
sipu_umbalance
0.9986=0000008
1.0000
0.5339
fcps_twodiamonds wut
0.7067+100000
0.7067
0.7067
0.65466:0000000
0.6546
0.6140
sipu aggregation
0.8652:10.0000
0.8652
0,4653
wut_stripes wut trapped lovers
1.0000=00.000
1.0000
0.7070
1.0000=1000000
1. 1.0000
0.6632
wut nut__windows__
1.0000=1000000
1.0000
0.6753
Teps hepta
1.0000:10.0000
1.0000
0.3727
graves_ims line
1.0000+000000
1.0000
0. 0.8238
graves rines_ring___ graves ring outliers
1.0000=10.0000
1.0000
0.7068
1.0000:10.0000
1.0000
0.6863
graves vigzag 0
1.0000=10.0000
1.0000
0.5381
sipu flame_0 _
0.7320+100000
0.7320
0,4598
other_iris.
0.7715+10.0000
0.7715
0.1223
other square fops__sun wut rut__tosplashes 0
1.0000110.0000
0.9990
0.7068
1.0000+1.00000
0.9983
0.6111
1.0000110.0000
0.9850
0.7062
sipn_comppund 1
0.9786=10.0000
0.9180
0,5715
sipu componnd 4
0.9442+10.000
0.8824
0.5523
wut @it__mk___ feps wingnut 0 sraves_parabolic. wut__kk3 wut labirynth graves
0.9866110.0000
0.8866
0,5754
0.9805=10.0000
0.8087
0.7068
0.6916++0.0000
0.5000
0.4979
0.7720+0.0000
0.5764
0.5314
0.7884110.0000
0.5221
0.5221
parabolic graves_paraabllic
0.9802+100000
0.7068
0.7040
d31 sipu_d31_ 0 sipu al
0.600111000085
0.1846
0,1787
0.74991000000
0.3269
0.2229
sipu r15 0
0.92581100000
0.4551
0.2552
sipu a2_
0.7555+0.00000
0.1685
0,1685
sipu
0.743410.0000
0.1410
0.1410
sipu sl
0.9888+0.0000
0.3695
0.2581
sipu s2
0.9405=1000000
0.2581
0.2579
Table 6: Fowlkes-Mallows scores of AuToMATo vs. TTK clustering algorithm
Dataset
automato mean
automato std
ttk
wut. trajectories
0.5038-10.0107
0.0107
0.8682
feps _eps_
0.6261-0.0000
0.0000
0.9043
wut x3 O
0.5153=100000
0.0000
0.7818
wut isolation 0
0.7256-10.0113
0.0113
0.9416
sipu al wut x2 x2_0
0.7499++0.000
0.0000
0.9143
0.5846-10.0000
0.0000
0.7283
Continued next page
21

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Published in Transactions on Machine Learning Research (10/2025)
Table 6: moolkes-Mallows scores of AuToMATo vs. TTK clustering algorithm
Dataset
automato mean
automato_ std
ttk
sipu flame 0
0.7320=10.0000
0.0000
0.8562
sipu__ggee tuu__agrrgatiinn
0.8652=000000
0.0000
0.9692
graves zigzag
0.6720=0.0000
0.0000
0.7698
Lilor_iri____
0.7715=100000
0.0000
0.8639
sipu jain graves dense_0
0.7837++0.000
0.0000
0.8182
0.8377=10.396
0.1396
0.8615
sipu_r15 wut 1___mm3300 wut. labirynth 0
0.9258=0.0000
0.0000
0.9374
0.7720=0.0000
0.0000
0.7755
0.788410.0000
0.0000
0.7884
feps chainlink 0 wut_smilee_ wut stripes 0
0.8896=10.0000
0.0000
0.8896
0.9681=0.0000
0.0000
0.9681
1.0000=+000000
0.0000
1.0000
graves_riin _os__ring____ wut smile 1 _au__unbalane_____
las__rinn_ouutirrr_________ outliers
1.0000=0.0000
0.0000
1.0000
0.9701=100000
0.0000
0.9701
0.99861000008
0.0008
0.9951
sipu s1 sipu s2 0
0.9088=000000
0.0000
0.9843
0.9405=1000000
0.0000
0.9311
other iris5 0 feps_atom__ feps atom
0.6712=0.0000
0.0000
0.6612
0.8694+000000
0.0000
0.8472
wut_x3_
0.6546=10.0000
0.0000
0.6312
sipu d31
0.6001
0.0085
0.5667
pops_hepta___
1.0000+100000
0.0000
0.9594
graves parabolic
0.6916110.0000
0.0000
0.6473
sipu_ri5__ 2 sipu hatp_ppthhaseed 0
0.486710.0000
0.0000
0.4322
0.651740.0000
0.0000
0.5947
spiral_ sipu_spiral_ sipu sipu r15 1 sipu _ipu__athbased__ feps _org_t_ wut x1 0 wut twwsplashsssssss... 0
0
0.702810.0000
0.0000
0.6422
0.0000
0.0000
0.4827
0.7022+0.0000
0.0000
0.6668
0.985010.0000
0.0000
0.9185
0.9741
0.0818
0.8960
1.0000=000000
0.0000
0.9140
wut mk4 0 wut wut_mk2__0 graves parabolic 0
0.90721100234
0.0234
0.8050
0.6356+000000
0.0000
0.5302
0.9002+000000
0.0000
0.8653
sipu_compound 4
0.9442+0.0000
0.0000
0.8145
feps_wingnme wut wut__ircles_00 wut mk1 0
0.9005+000000
0.0000
0.8497
0.885710.0000
0.0000
0.7543
0.986610.0000
0.0000
0.8148
feps twodiamonds
0.70671000000
0.0000
0.5251
sipu sipu_compound
0.8616+0.0000
0.0000
0.6728
sipu compound
0.978611000000
0.0000
O. 0.7992
graves graves_line_0
1.000010.0000
0.0000
0.7917
leps Isun
1.0000+000000
0.0000
0.7897
wut tttp_reppedd__ lovers
1.0000+000000
0.0000
0.7859
other other_square___ square____.e 0
1.0000+0.0000
0.0000
0.7774
graves zigzag 0
1.0000+0.0000
0.0000
0.7686
graves ring 0
1.0000+0.0000
0.0000
0.7278
sipu a2 wut. hott_windww___
0.7555+0.0000
0.0000
0.4641
1.0000=10.0000
0.0000
0.5853
Continued next page
22

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Published in Transactions on Machine Learning Research (10/2025)
Table 6: moelkes-Mallows scores of AuToMATo vs. TTK clustering algorithm
Dataset
automato mean
automato std ttk
sipu a3
0.743411.0000
0.0000
0.1882
Table 7: Fowlkes-Mallows scores of AuToMATo vs. hierarchical clustering with complete linkage
Dataset
automato mean
minkage_complin max
linkage_complette min
sipu r15_
0.4807:1000000
1.0000
0.2256
"rut_trajcctoric 0
0.5038
1.0000
0.1706
r15 sipu_ri5_
0.5436:100000
0.8954
0.2516
wut x3
0.6546==0.0000
0.9740
0.2004
feps tetra 0
0.6261=1000000
0.9356
0.0651 1
sipu_d31 _d__331____ 0
0.6001#0.0085
0.8733
0.2717
wut__33__ x3_ 0
0.5153++.0000
0.7842
0.2477
sipu_a3
0.7434+0.0000
0.8979
0.2294
wut mk3 0
0.7720=t0.0000
0.9207
0.0711
wut x2_ 0
0.5846=100000
0.7298
0.1964
sipu_a22 a2_
0.7555=+0.0000
0.8992
0.2642
sipu jain 0 wut cit__circles__ 0
0.7837+100000
0.9116
0.1288
0.8857+00.000
1.0000
0.1761
sipu_r15 r1__ 0
0.9258++0.0000
0.9799
0.3372
sipu_al_0
0.7490=00.000
0.8040
0.3092
graves zigzag 1
0.7119
0.3039
sipu_aggregale aggregation 0
0.8652:10.0000
0.9030
0.1246
other iris wut xi 0 _ttu_umbalance__ 0
0.7015+100000
0.8064
0.0680
0.9741=10.011
1.0000
0.2326
0.9986+0000008
0.9988
0.5774
feps hepta 0
1.0000110.0000
1.0000
0.4321
feps pops_ttroiamssss_o 0
0.7007
0.7060
0.0916
0.732010.0000
0.7276
0.0834
sipu _ou___ompuund___ 0
0.8616+0.0000
0.8472
0.1567
sipu sl
0.9888+0.0000
0.9563
0.3672
sspu__athbased 0
0.6517=0.0000
0.6022
0.1384
sipu pathbast 1
0.732210.0000
0.6709
0.1539
graves_parabolie
0.6916=000000
0.6168
0.1482
sipu__ompound 1
0.978610.0000
0.9023
0.1366
sipu_compound 4
0.9442+000000
0.8652
0.1408
graves graves_dense__ dense_0
0.8377#0.1396
0.7584
0.3538
wut__mkl 0
0.9866+0.0000
0.8950
0.1591
wut__smile_
0.9701+0.0000
0.8697
0.3562
graves parabolic 0
0.98021000000
0.8610
0.1088
wut_mkk2_ 0 other_iris5_1
0.6356+10.0000
0.5032
0.1096
0.6712+0.0000
0.5305
0.0451
feps_atom.
0.8694+0.0000
0.7278
0.1364
wut. smile
0.0000
0.8206
0.3793
sipu s2 0
0.000000000000
0.7642
0.3114
0
0.907220.0234
0.7282
0.1297
feps target 0
0.9850=0.0000
0.7881
0.1934
fops Isun 0
1.0000+0.0000
0.7668
0.1377
saps_wingnut 0
0.9000+00.0000
0.7406
0.0816
Coutinued on next page
23

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Published in Transactions on Machine Learning Research (10/2025)
Table Fowlkes-Mallows scores
AuToMATo vs. hierarchical clustering with complete linkage
Dataset
automato mean
faakage_complete max
linkage_complete min
graves ring 0
1.000011.0000
0.7589
0.1849
graves ring outliers 0
1.0000=0.0000
0.7528
0.1995
wut Nut__abirynthh_ pyss_chainlin___ sipu spiral 0
0.7884=0.0000
0.4893
0.1426
0.8896=000000
0.5889
0.0919
0.7028=100000
0.3512
0.1424
wut wut_ iut__isllttion_1 graves_ line other other_square graves zigzag 0
0.7256=000013
0.3397
0.1153
1.0000=000000
0.5972
0.1909
1.0000=0.0000
0.5846
0.1142
1.0000=0.0000
0.5505
0.2042
wut twosplashes
1.0000=10.000
0.5310
0.2771
wut wut_strrppe
1.0000=0.0000
0.5136
0.0706
wut trapped lovers 0
1.0000=000000
0.4790
0.0579
wut windows_0
1.0000#0.0000
0.4349
0.0781
Table 8: Fowlkes-Mallows scores of AuToMATo vs. hierarchical clustering with Ward linkage
Dataset
automato mean
linkage ward max
linkage ward min
wut x3 0
0.5153+000000
0.8537
0.2203
sipu d31 0
0.6001#000085
0.9223
0,2766
sipu a3
0.743440.0000
0.9377
0,2889
sipu
0.7555++0.0000
0.9360
0.2653
sipu
0.7499=000000
0.9166
0.2464
x2 0
0.5846:10.0000
0.7219
0.2076
sipu sipu_r15 r15 graves sigzag sipu r15 0 sipu_ri5 r15_ wut x3 wut x1 0 sipu unbalance 0
0.4867+000000
0.5893
0.1868
0.6720+000000
0.7358
0.2693
0.000810000000
0.9832
0.4072
0.5436=+000000
0.5993
0,2082
0.6546=1000000
0.7090
0,1795
0.9741#0.0818
1.0000
0.2429
0.9986+L000008
1.0000
0.2063
feps _ee_hepta___ sipu sl 0 sipu @opu_pathbased_11 sipu s2 0 jipu_pathbaseee
1.0000+0.0000
1.0000
0.4314
0.9888:10.0000
0.9844
0.2453
0.6517+000000
0.6251
0,1370
0.9405=10.0000
0.9085
0.2177
0.7002+10.0000
0.6844
0,1523
fops_ttetra dense 0
0.6261+0.0000
0.5622
0.0651
0.8377:001396
0.7454
0.2684
wut vut_trajectories_
0.5038110.0107
0.3933
0.0981
other iris 0
0.7015110.000
0.6377
0.0680
feps__aton
0.869440.0000
0.7272
0,1016
graves Laaress_rrbboic____
0.6916:10.0000
(.5260
0.1301
fcps__tars cpp__target__
0.9850=10.0000
0.7759
0.1503
other iris5
0.600
0.4508
0.0451
sipu agnggggggttt
0.8652+000000
0.6064
0.1215
sipu dipu_flam__1 0
0.7320+0.0000
0.4555
0.0819
sipu compound 0
0.8616-10.0000
0.5846
0.1523
wut wk3_ 0
0.7720-0.0000
0.4911
0.0711
feps twsdiimddsss
0.70671000000
0.3967
0.0861
Continued next page
24

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Published in Transactions on Machine Learning Research (10/2025)
Table 8: MowloossMallows scores of AuToMATo vs. hierarchical clustering with Ward linkage
Dataset
automato mean
linkage Linage__ward max
linkage_ ward min
sipu jain
0.783711000000
0.4668
0.1201
wut 1___m11000
0.9866=1000000
0.6564
0.1473
feps [as__sum___ 0
1.0000=0.0000
0.6659
0.1270
sipu sipu__ompound_11
0.9442=10.0000
0.5793
0.1368
sipu spiral 0 wut wut_labiryuth. sipu__omppund_1 wut wut_ttooppashes smile wut smile 0
0.7028:1000000
0.3100
0.1384
0.7884+000000
0.3749
0,0961
0.9786=1000000
0.5648
0.1327
1.0000=0000000
0.5817
0,2046
0.9701=10.0000
0.5246
0.2352
0.9681+10.0000
0.5179
0,2505
wut wut_mke 2___m22__ graves vigzag 0 wut wut__ssolatl _it___illatio____ graves line 0
0.6356+0000000
0.1814
0.0997
1.0000=10.0000
0.5448
0.1809
0.7256::0.0113
0.2309
0.0800
1.0000=10.0000
0.5045
0.1667
wut circles 0
0.8857=000000
0.3681
0.1323
hop__chainlink__ wut mk4 0
0.8896=10.0000
0.3407
0.0894
0.9072:10.0234
0.3557
0.1079
graves parabolic 0
0.9502+10.0000
0.4184
0.0935
graves graves_ring__ ring 0
1.0000=10.0000
0.4135
0,1427
Laavos_ring_ore outlicrs_
1.0000=10.0000
0.4082
0.1515
leps wingnut
0.9805+0.0000
0.3229
0.0727
other other_square square_L 0 wut windows
1.0000+0.0000
0.3347
0.1007
1.0000+1.00000
0.2833
0.0663
wut trappea [ooppes__ovees______ wut stripes 0
lovers
1.0000+100000
0.2552
0.0471
1.0000+10.0000
0.1922
0.0552
Table 9: Powlkes-Mallows scores of AuToMATo vs. FINCH
Dataset
automato mean
automato std
finch
wut x3 0
0.5153-10.0000
0.0000
0.7970
sipu d31 0
0.6001=0.0083
0.0085
0.8657
sipu a3
0.7434=0.0000
0.0000
0.8306
wut_x22__
0.5846=10.0000
0.0000
0.6671
wut. mk3 0
0.7720=+0.0000
0.0000
0.8503
other iris5
0.6712:10.0000
0.0000
0.7008
a2_
0.7555+0.0000
0.0000
0.7635
wut x3 sipu__uneee unbalance
0.6546-0.00000
0.0000
0.6619
0.9986=000008
0.0008
0.9998
sipu r15 0 wut 101m000000 other iris 0 sipu_al al sipu r15 graves graves_zigzag graves dense 0
0.92001000000
0.0000
0.9083
0.9866+000000
0.0000
0.9655
0.7715+000000
0.0000
0.7477
0.749910.0000
0.0000
0.7124
0.4007+000000
0.0000
0.4156
0.6720+000000
0.0000
0.5965
0.837720.1396
0.1396
0.7615
sipu r15 1
0.5436+0.0000
0.0000
0.4641
sipu sl 0
0.9880+000000
0.0000
0.8728
feps hepta 0
1.0000+10.0000
0.0000
0.8794
Continued next page
25

---

Published in Transactions on Machine Learning Research (10/2025)
Table 9: Fowlkes-Mallows scores of AuToMATo vs. FINCH
Dataset
automato mean
automato std
finch
leps atom 0
0.86941100000
0.0000
0.7319
feps tetra 0
0.6261=0.0000
0.0000
0.4680
sipu s2
0.9405=000000
0.0000
0.7282
wut_xl_0
0.9741=0.0818
0.0818
0.7607
graves parabolic 1
0.6000=000000
0.0000
0.4446
sipu flame_lmm__ sipu sipu__ptthbasod
0.7320=0.0000
0.0000
0.4767
0.6517=000000
0.0000
0.3440
sipu_compount 0
0.86161000000
0.0000
0.5390
sipu pathbased
0.7002=100000
0.0000
0.3828
wut trajectories
0.5038+10.000
0.0107
0.1503
feps_isun_
1.0000=000000
0.0000
0.6206
sipu compound 4
0.9442+000000
0.0000
0.5493
sipu_jain__
0.7837=000000
0.0000
0.3824
sipu compound
0.0000+000000
0.0000
0.5356
sipu_spprall 0
0.702810.0000
0.0000
0.2553
wut wut_ 0
0.907210.0234
0.0234
0.4331
wut mk2 0
0.6000=000000
0.0000
0.1478
sipu aggregation 0
0.8652+000000
0.0000
0.3674
tosddtdddisssss wut wut__smil__ wut smile 1
0.7067=0.0000
0.0000
0.1837
0.9681+0.0000
0.0000
0.4452
0.970110.0000
0.0000
0.4181
feps tere_ttrge___ _
0.985010.0000
0.0000
0.4297
wut labirynth 0
0.788410.0000
0.0000
0.2209
wut _nt__isllation__ 0
0.725610.0113
0.0113
0.1440
wut. twosplashes
1.0000=0.0000
0.0000
0.4162
graves_ zigzag.iga.... 0
1.0000+100000
0.0000
0.4094
graves parabolic
0.9002+000000
0.0000
0.3343
graves graves_line__ line cos__chailiink.. feps wingnut
1.0000+0.00000
0.0000
0.3379
0.80001000000
0.0000
0.2224
0.9805=10.0000
0.0000
0.3110
foove__ring 0
1.0000+100000
0.0000
0.2770
wut _tt_tta lovers
1.0000+000000
0.0000
0.2767
1.0000=000000
0.0000
0.2393
graves ring outliers 0
1.0000+0.0000
0.0000
0.2248
wut wut_circles__
0.8857+0.0000
0.0000
0.0976
wut_windowe
1.000010.0000
0.0000
0.1166
wut wut_stripes_0
1.00001000000
0.0000
0.0867
26

---

Published in Transactions on Machine Learning Research (10/2025)
clusterer
automsto rean
Linkage single Liniage
frs-score
8.6-
0.2
wit, to..0ra/o001e
togs, tetra
Not sot_ 1so lntion
sipu join
teps stos
feps Pheie
sipu pettiaeeed
sipu rts
sat 22
other tries
Eegs target
sipu unbalance
feps boddiaronds
sipu Apprsaation
tcpa hopta
sraves
graves ring
sraves riso outtiers
draves zippag
wit trapeed tovers
sipu flawe
other
ether square
for
wit tecrplaches
wit at:
fos vinguut
wrt ald
graves graves_paraolle parabalse
sipu 11
sipu st
sipu
sipu
sipu K2
spirs.
mid
smile
stripes
windows
compound
Labirynth
benchwark
Figure 10: Comparison of AuToMATo and agglomerative clustering with single linkage.
clusterer tIk
fns-score
9.8
0.4-
Nor. trajecctrrie
Tips
wit Isolation
Not x2
Not .3
sipu
sipu 300 regrttttioon
other
sipu soin
graves
sipu
wot mi
Nat Sali rysth
faps chainlie
vot snile
graves ring outtiers
wit strapes
sipu st.
sipu 52
other
tops stoe
tap hesto
sipu spi rat
sipu pathoazed
tops, farget
wort sverp bshes
wor phd
Net
graves parabbtii
wot circies
sipu compound
Net
tops seandss
taps tsin
sat traeped Lovers
graves riigeg
sipu 22
Not Vindons
SIDU 33
siingntt
benchnark
Figure 11: Comparison of AuToMATo and the TTK-algorithm.
clusterer
autorato nean
Fns -core
8.8-
Linage complete nin
linkage complete
8.2
top
sip CBL
sip 22
sip
wtt circtes
sipe
sipu al
sips iepregation
other
set
zips unbalance
top hopta
fag twodi teddlaonns
sipe flam
sim petibased
sipu
praves ders
grawes parabelic
wit ak2
othe r triss
taps
sipu 22
top taross
tepa vingue
graee ring out
graves
wit tabirymch
topo chain
Aillation sipu salrat
teosptester other r square growns Fine
set stripes
sat windiws
patte
trapped tovers
ring
benchwark
Figure 12: Comparison of AuToMATo and agglomerative clustering with complete linkage.
27

---

Published in Transactions on Machine Learning Research (10/2025)
clusterer
aean
0.8
Linkoge.ward wardg rax
Ins-score
Lekkoge_anrd_ntn
0.4-
1at 22
ston a1
2100 125
simu unbelance
sizu st
sinu parttosse
graaes
wut trojectorie
other
Tape aton
reps, torget
atter per_arrss ariss
gravis zigrog
wur mi3
sipu. mai
wit sict
zizu spirat
wit. labiryeth
wot,
wot seile
wat
rut 1oo ation
wat circless graves Line
faps, chainiin
wut a64
graves ring
grawes ring outtlors
other Square
wUt tropped Lovers
tetre
rogregattinn
Evodlamonds
compound
benchnark
Figure 13: Comparison of AuToMATo and agglomerative clustering with Ward linkage.
clusterer rinch
8.8
0.4-
Stol 403
5500
wit m8
other 1rru
stro Unbo lance
wit mk1
other iiter.irrs
graves dense
Teas hesta
frps
stru 52
steo flare
wut trajectorios
topz Isun
graves zigog
vipu compond
sipu sod
wit pic
stou ag3 approppcco
wit
graves pa parabolie
wit Labt Labt rynth
wit Isolation
wit teoop trooplaahee
graves Line
taps wd winguut
graws rinng
wut trosped lovers
other SQuaTe
wit circtes
wut siidoos
wut stripes
38
a2
etoe
pathiesed
chaintion
suttiors
benchaark
Figure 14: Comparison of AuToMATo and FINCH.
28


Reviews:
[{"review_id": "qHRDYzGMj7", "reviewer": "Reviewer_Woow", "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": "", "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 AuToMATo, a clustering algorithm that combines the Topological Mode Analysis Tool (ToMATo) with a bootstrapping-based significance test to automatically identify and merge meaningful density modes. The main goal is to provide an out-of-the-box clustering method that requires no user parameter tuning, yet performs competitively or better than tuned baseline algorithms.\nIt is shown empirically that AuToMATo achieves competitive or superior performance across diverse datasets compared to both parameter-free and optimally tuned clustering algorithms.\n\nStrengths\n1. Clever integration of persistent homology with bootstrapping to automate parameter selection.\n2. Removing parameter tuning lowers the barrier to applying topological clustering in real applications.\n\nWeaknesses\n1. Bootstrapping may introduce significant overhead; scalability on large datasets is not fully addressed.\n2. The universality of default parameters is assumed but not yet clearly stress-tested on more challenging edge cases (e.g., overlapping clusters, noisy data).", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper presents extensive empirical results comparing AuToMATo to both parameter-free baselines and optimally tuned algorithms. The performance benchmarks, as reported, show that AuToMATo is indeed competitive across multiple datasets.", "audience_yn": "Yes", "audience_explanation": "The proposed method, AuToMATo, directly contributes to unsupervised learning, which is a central theme in the machine learning community. Its emphasis on being parameter-free and out-of-the-box usable addresses a long-standing challenge in clustering.", "requested_changes": "The presentation of the paper can be further improved. Specifically, \n\n1. It would be helpful to include an additional figure that visually summarizes the overall procedure of the proposed method. This would improve accessibility and make the workflow clearer to readers.\n\n2. The theoretical guarantee on the confidence associated with the persistence diagram, currently in Section 3.1, could be more formally presented as a theorem. This would strengthen the rigor and readability of the contribution.\n\n3. The complexity analysis currently placed in Appendix A.2 should be moved to the main text. In addition, it would be beneficial to expand the discussion with more detail on how the bottleneck distance can be efficiently estimated in practice.\n\n4. An empirical comparison of computational complexity against baseline clustering methods should be provided to better justify the practical benefits of the proposed automatic algorithm.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "mMLvsSalOV", "reviewer": "Reviewer_PMEd", "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": "", "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 presents AuToMATo, a novel clustering algorithm that uses persistent homology to automate parameter selection for the ToMATo clustering algorithm. By replacing the manual threshold selection in ToMATo with a bootstrapping procedure, AuToMATo identifies significant peaks in the persistence diagram, providing an \"out-of-the-box\" solution. The algorithm demonstrates competitive performance compared to leading clustering methods across a range of benchmark datasets. Additionally, AuToMATo performs effectively as a clustering component within the Mapper algorithm.", "claims_and_evidence_yn": "Yes", "claims_explanation": "1. The paper compares AuToMATo against multiple clustering algorithms on benchmark datasets. Results show AuToMATo often outperforms parameter-free methods and even matches tuned versions of parametric algorithms.\n\n2. The bootstrap's role in automating threshold selection is well-motivated.\n\n3. Results align with the claim that AuToMATo's automation benefits Mapper.", "audience_yn": "Yes", "audience_explanation": "The paper presents a method that bridges topological data analysis and practical machine learning, making it relevant to multiple sub-communities within TMLR's audience.", "requested_changes": "1.  How sensitive is AuToMATo's performance to the choice of $\\alpha$ and $B$? Could adaptive methods for selecting these parameters be explored?\n\n2. For large-scale datasets, how does AuToMATo compare in runtime to other methods?\n\n3. Could you provide more details on the outlier detection feature? How does it compare to methods like DBSCAN's noise handling?\n\n4. Could you provide more details about the density estimation? How does the algorithm perform in high-dimensional clustering scenarios?", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bvzgRDAPjo", "reviewer": "Reviewer_KmGP", "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": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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:\nThe paper introduces AuToMATo, a novel clustering algorithm based on persistent homology, which extends the ToMATo algorithm by incorporating a bootstrapping procedure to distinguish significant density peaks from noise. The key contributions include:\n1. Robust default settings to make it an \"out-of-the-box\" implementation of ToMATo\n2. Use bottleneck bootstrapping to select prominence thresholds, eliminating the need to visually identify large gaps\n3. A Python package compatible with scikit-learn is provided, ensuring accessibility and reproducibility.\n\nIt's worth noting that AuToMATo is not a clustering algorithm in the traditional sense, but rather a method for selecting prominence thresholds that can be used with ToMATo. Clustering is performed by ToMATo.\n\nStrengths:\n1. This work enhances the applicability of ToMATo by providing a more robust and automated way to select prominence thresholds.\n2. The associated source code is well-engineered and well-documented. It's among the best implementations of ML research papers I've ever seen.\n\nWeaknesses:\n1. This work is undoubtedly valuable, but I personally don't find it to be a significant contribution to ML research. The major contribution lies in the bootstrapping procedure rather than in the clustering algorithm itself. Moreover, the bootstrapping procedure is proposed by another research paper, and this work just applies it to ToMATo (with some adaptations). Hence, this is more of an engineering contribution than a research one. Please keep in mind that this is a personal opinion, and I generally don't give high scores to this kind of work unless it is very well done.\n2. No theoretical guarantees are provided for the default parameters. Despite empirical results showing that the default parameters work well, these benchmarks are simple and small, hence not comprehensive enough to cover all possible scenarios.", "claims_and_evidence_yn": "No", "claims_explanation": "Claims:\n1. We present AuToMATo, a novel clustering algorithm based on persistent homology\n2. we provide default choices for its parameters that make it into an out-of-the-box clustering algorithm that performs well across the board\n\nJustifications:\n1. AuToMATo is not a clustering algorithm, but a heuristic to select prominence thresholds for ToMATo\n2. See W2", "audience_yn": "Yes", "audience_explanation": "The proposed AuToMATo might be of interest to people who want to use ToMATo", "requested_changes": "Providing theoretical analyses for the default parameters would strengthen the work. However, it's unlikely to secure my recommendation for acceptance given the overall style of this work.", "broader_impact_concerns": "no ethical implications", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)