# automato_an_outofthebox_persistencebased_clustering_algorithm__19a918a0.pdf

Published in Transactions on Machine Learning Research (10/2025)

Au To MATo: An Out-Of-The-Box Persistence-Based Clustering Algorithm

Marius Huber marius.huber@uzh.ch Department of Computational Linguistics University of Zürich

Sara Kališnik skalisnik@psu.edu Department of Mathematics Pennsylvania State University

Patrick Schnider patrick.schnider@inf.ethz.ch Department of Mathematics and Computer Science University of Basel Department of Computer Science ETH Zürich

Reviewed on Open Review: https: // openreview. net/ forum? id= Qd7H5m Abz V

We present Au To MATo, a novel clustering algorithm based on persistent homology. While Au To MATo 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. Au To MATo combines the existing To MATo clustering algorithm with a bootstrapping procedure in order to separate significant peaks of an estimated density function from nonsignificant ones. We perform a thorough comparison of Au To MATo (with its parameters fixed to their defaults) against many other state-of-the-art clustering algorithms. We find not only that Au To MATo compares favorably against parameter-free clustering algorithms, but in many instances also significantly outperforms even the best selection of parameters for other algorithms. Au To MATo 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 Au To MATo performs well when used with Mapper. Finally, we provide an open-source implementation of Au To MATo in Python that is fully compatible with the standard scikit-learn 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 similarity is often measured with a distance or a metric on a data set. There exist many different clustering techniques (Anderberg, 1973; Duda et al., 2000), including hierarchical, centroid-based and density-based techniques, as well as techniques arising from probabilistic generative models. Each of these methods is proficient at finding clusters of a particular nature. Many of the most commonly used clustering algorithms require a selection of parameters, a process which poses a considerable challenge when applying clustering to real-world problems.

In this work, we present and implement Au To MATo (Automated Topological Mode Analysis Tool), a novel clustering algorithm based on the topological clustering algorithm To MATo (Chazal et al., 2013). The latter summarizes the prominences of peaks of a density function in a so-called persistence diagram. The user then selects a prominence threshold τ 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 τ 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 To MATo, which is precisely what Au To MATo does. That is, given a persistence diagram obtained by running To MATo on a point cloud, Au To MATo produces a confidence region for that diagram with respect to the bottleneck distance, which translates into a choice of τ that determines the final clustering. While Au To MATo is not parameter-free per se, we provide default choices that make it perform well across the board. Unless stated otherwise, Au To MATo will henceforth refer to our algorithm with its parameters set to these defaults. We experimentally analyze the clustering performance of Au To MATo and we find that it not only outperforms parameter-free clustering algorithms, but often also even the best choice of hyperparameters for many parametric clustering algorithms. Parameter-free algorithms building on To MATo exist in the literature, for example, in Cotsakis et al. (2021) 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 Au To MATo is shown to outperform.

We envision one important application of Au To MATo 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 a 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, Au To MATo can be seen as progress towards finding optimal parameters for Mapper, which is an active area of research (Carrière et al., 2018; Chalapathi et al., 2021; Rosen et al., 2023). Running examples for Mapper with Au To MATo, we see that it 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 To MATo clustering algorithm

Both To MATo and Au To MATo rely on the theory of persistence (Edelsbrunner et al., 2002; Zomorodian & Carlsson, 2005; Carlsson, 2014) to quantify the prominence of peaks of (an estimate of) a density function, and to build a hierarchy of peaks. Given a topological space X equipped with a density function f : X R 0, 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 {X t | t R}, where X t := f 1 ([t, )).

In the following, we assume for ease of exposition that all local extrema of f have distinct values. The idea underlying To MATo is to track the evolution of (the number of) connected components of X t as t ranges from + to . In that process, the number of connected components of X t remains constant, unless t passes through the value of a local extremum of f. As t passes through the value of a local maximum, a new connected component is born and added to the superlevel set X t. Similarly, as t passes through the value of a local minimum, two connected components of X t are merged into one. To MATo builds a 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 maximum with lower value dies . Therefore, to each local maximum we associate a pair (b, d) where b denotes the birth and d the death time, respectively. The evolution of the connected components can be concisely recorded in a persistence diagram. Definition 2.2. Let {(bl, dl)}l denote the birth and death times of connected components of the superlevel set filtration {X t}t R associated to the density f : X R. The associated persistence diagram, denoted

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by Dgm(X, f), is the multiset in the extended plane R 2 := R { } consisting of the points {(bl, dl)}l R 2

(counted with multiplicity) and the diagonal := (x, x) | x R (where each point on has infinite multiplicity). For a given local maximum of f with birth time bl and death time dl , we refer to the difference dl bl as its prominence or lifetime.

The reason for working in the extended plane is that, provided that f has a global maximum, the superlevel set filtration X t will have a connected component that never dies, that is, has death time equal to . See the red graph in Figure 1 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 L -distance of a point to the diagonal (that is, its Euclidean vertical distance) equals its prominence.

We now outline how the To MATo clustering algorithm works. Given a point cloud X To MATo relies on the assumption that the points of X were sampled according to some unknown density function f. In a nutshell, To MATo infers information about the local maxima of f by applying the above procedure to an estimate of f. To MATo takes as input:

A neighborhood graph G on the points of X. Chazal et al. mostly use the δ-Rips graph and the k-nearest neighbor graph.1

A density estimator ˆf. Each vertex v of G is assigned a non-negative value ˆf(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-measure density, originally introduced in Biau et al. (2011).2

A merging parameter τ 0. This is a threshold that the prominence of a local maximum of the estimated density ˆf must clear for that local maximum to be deemed a feature.

Given the inputs above, To MATo proceeds as follows.

1. Estimate the underlying density function ˆf 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 a directed subgraph G of G as follows: at each vertex v of G, place a directed edge from v to its neighbor with highest value of ˆf, provided that that value is higher than ˆf(v). If all neighbors of v have lower values, v is a peak of ˆf. This yields a collection of directed edges that form a spanning forest of the graph G, consisting of one tree for each local maximum of ˆf. 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, ˆf) associated to the superlevel set filtration of ˆf : G R. 4. Merge non-significant clusters. Iteratively merge every cluster of prominence less than τ 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 ˆf : G R. To MATo outputs the resulting clustering of points of X, in which every cluster has prominence at least τ by construction.

The reason why we can 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 Section 2.2). This is illustrated in Figure 1.

In practice, the user must run To MATo twice. First, To MATo is run with τ = + which is equivalent to computing the birth and death time of each local maximum of ˆf and hence the persistence diagram

1Given a point cloud, both of these undirected graphs have the set of data points as their vertex set. In the case of the δ-Rips graph, two vertices are connected iff they are at a distance of at most δ apart, whereas in the k-nearest neighbor graph, a data point is connected to another iff the latter is among the k-nearest neighbors of the first. 2For a smoothing parameter m (0, 1), and a given data point x, its empirical (unnormalized) distance-to-measure density

is given by ˆf(x) = 1 k P

y Nk(x) x y 2 1

2 , where k = mn , Nk(x) denotes the set of the k nearest neighbors of x, and

n is the cardinality of the data set.

Published in Transactions on Machine Learning Research (10/2025)

s = (sx, sy)

Figure 1: A function f : K R, K R, in red, and an estimate ˆf of f in blue (left), with corresponding persistence diagrams Dgm(K, f) and Dgm(G, ˆf) consisting of the red and blue dots, respectively, together with a dashed line separating noise from features (right).

Dgm(G, ˆf). From the diagram Dgm(G, ˆf) the user then determines a merging parameter τ by visually identifying a large gap in Dgm(G, ˆf) separating, say, C points corresponding to highly prominent peaks from the rest of the points. Then, To MATo is run a second time with τ 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. (2017, Section 6), is used to separate significant features in persistence diagrams from non-significant ones. While it may be used in 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 diagrams (Edelsbrunner & Harer, 2010; Chazal et al., 2016). Definition 2.3. Let Dgm1 and Dgm2 be two persistence diagrams that have finitely many points off the diagonal. Let π denote the set of bijections ν : Dgm1 Dgm2. Given points x = (x1, x2) and y = (y1, y2) in R 2, let x y = max{|x1 y1|, |x2 y2|} denote their L -distance, where we set (+ ) (+ ) = ( ) ( ) = 0. Then, the bottleneck distance between Dgm1 and Dgm2 is defined as

W (Dgm1, Dgm2) = inf ν π sup x Dgm1 x ν(x) .

Note that a bijection ν : Dgm1 Dgm2 is allowed to match an off-diagonal point of Dgm1 to the diagonal of Dgm2, and vice versa.

We now outline the bottleneck bootstrap. It relies on the following theorem, which summarizes the relevant results of Chazal et al. (2017, Section 6). Theorem 2.4 (Chazal et al. (2017)). Let X RN be a sample consisting of n data points drawn according to a probability density function f : K [0, 1], K RN. Denote by D := Dgm(K, f) and b D := Dgm(X, f) the corresponding unknown and estimated, respectively, persistence diagrams of superlevel sets. Given a confidence level α (0, 1), define qα by

P( n W (D, b D) qα) = 1 α.

Then a consistent estimator for qα is given by bqα, which in turn is defined by

P( n W ( b D , b D) bqα) = 1 α.

Here, b D := Dgm(X , f) denotes the random persistence diagram constructed from a sample X of size n drawn according to the empirical measure Pn on X, where Pn is defined as the probability measure on X that assigns the probability mass 1/n to each data point in X.

Published in Transactions on Machine Learning Research (10/2025)

In our setting, the theorem above is applied as follows. Given the sample X RN, we estimate f and the connectivity of K with a density estimator and a neighborhood graph, respectively (as explained in Section 2.1). This allows us to compute b D := Dgm(X, f), which, in turn, serves as an estimate of D. The empirical measure Pn on X serves as an approximation of the unknown probability measure f, and using this, Theorem 2.4 allows us to approximate the distribution

F(z) := P( n W (D, b D) z)

with the distribution b F(z) := P( n W ( b D , b D) z),

where b D := Dgm(X , f) is a random quantity. Note that X may be thought of as a sample drawn from X with replacement.

Like the distribution F, the distribution b F is still not explicitly computable, but, unlike F, it can be approximated by Monte Carlo as follows. We draw B samples X 1, . . . , X B of size n from Pn, and for each of these B samples, we compute the persistence diagram b D i := Dgm(X i , f) and the quantity T i := n W ( b D i , b D), i = 1, . . . , B. Finally, we use the function

e F(z) := 1

i=1 1[0,z](T i )

as an approximation of b F, and hence of F. Using this, we set

bqα := inf{z | e F(z) 1 α}

to be our estimate of qα. This estimate is asymptotically consistent by Theorem 2.4, that is, bqα n qα.

In conclusion, the true, unknown persistence diagram D is at bottleneck distance of at most bqα/ n from b D with probability at least 1 α. Hence, points of b D that are at L -distance at most bqα/ n from the diagonal could be matched to the diagonal under the bottleneck distance, and thus a point of b D is declared to be a significant feature iff it is at L -distance of at least bqα/ n to the diagonal, that is, iff its prominence is at least 2 bqα/ n.

3 Methodology and implementation of Au To MATo

3.1 Methodology of Au To MATo

Au To MATo builds upon the To MATo clustering scheme introduced in Chazal et al. (2013) and implemented in Glisse (2025). Au To MATo automates the step of visual inspection of the persistence diagram by means of the bottleneck bootstrap, thus promoting To MATo to a clustering scheme that does not rely on human input.

More precisely, given a point cloud X to perform the clustering on, Au To MATo takes as input

an instance of To MATo with fixed neighborhood graph and density function estimators; a confidence level α (0, 1); and a number of bootstrap iterations B Z 1.

Remark 3.1. We point out that our implementation of Au To MATo 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 3.2.

To apply the bottleneck bootstrap as described in Section 2.2, Au To MATo generates B bootstrap subsamples X 1, . . . , X B of X, each of the same cardinality as X, where X is the data set whose points are to be clustered. Then the underlying To MATo instance with τ = + and its neighborhood graph and density function estimators is used to compute the persistence diagram for X and each of X 1, . . . , X B, yielding

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persistence diagrams b D and b D 1, . . . , b D B, respectively. Using the bootstrapped diagrams b D 1, . . . , b D B, a bottleneck bootstrap is performed on b D. This yields a value bqα that (asymptotically as n ) satisfies

P( n W (D, b D) bqα) = 1 α,

where D denotes the persistence diagram of the true, unknown density function from which X was sampled. Thus, points of b D of prominence at least 2 bqα/ n are declared to be significant features of b D, and Au To MATo outputs its underlying To MATo instance with prominence threshold set to τ = 2 bqα/ n. This procedure is schematically depicted in Figure 2.

Bootstrap samples X 1, . . . , X B

To MATo PDs

To MATo PD with threshold

Figure 2: Schematic of the methodology of Au To MATo: from a data set X, the usual To MATo persistence diagram (with τ = + ) is computed. Additionally, the analogous persistence diagrams are computed for the bootstrap samples X 1, . . . , X B, which are created from X by drawing with replacement. Finally, the bootstrap procedure (indicated by ) is used to compute a prominence threshold for the original persistence diagram.

When computing the values n W ( b D i , b D), i = 1, . . . , B, in the bottleneck bootstrap, we only consider points in b D i and b 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 b D 1, . . . , b D B have a different number of points with infinite lifetime than the reference diagram b D. In these cases, the bottleneck distance of the bootstrapped diagram to the reference diagram is infinite, which heavily distorts the distribution e F(z). This choice is justified by experiments.

3.2 Implementation of Au To MATo

We implemented Au To MATo in Python, and all code with documentation is publicly available.3 For a description of Au To MATo in pseudocode, see Algorithm 1. The algorithm has a worst-case complexity of O(B(nd + n log(n) + N 1.5 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 parallelization.

While the input parameters may be adjusted by the user, the implementation provides default values whose choices we discuss presently.

Choice of To MATo parameters: Our implementation of Au To MATo is such that the user can directly pass parameters to the underlying To MATo instance. If no such arguments are provided Au To MATo uses the default choices for those parameters, as determined by the implementation of To MATo given in Glisse (2025). In particular, Au To MATo uses the k-nearest neighbor graph and the (logarithm of the) distanceto-measure density estimators by default, each with k = 10. Of course, the persistence diagrams produced

3The code is archived on Zenodo (doi.org/10.5281/zenodo.17279741) and developed openly on Git Hub (github.com/ m-a-huber/automato_paper).

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Algorithm 1: Au To MATo Input: point cloud X of n data points; instance tomτ of To MATo with neighborhood graph and density function estimators, and prominence threshold τ; confidence level α (0, 1); number of bootstrap iterations B Z 1.

D Dgm(tom (X)) ; // compute persistence diagram of point cloud for i = 1 to B do

Let X i be a subsample of X of size n, sampled with replacement; D i Dgm(tom (X i )) ; // compute persistence diagram of subsample di n W fin (D i , D) ; // compute bottleneck distance between finite points end Sort and reindex {D 1, . . . , D B} such that d1 d B; k (1 α) B ; bqα dk; τ 2 bqα/ n;

Output: tomτ(X) ; // copy of initial To MATo instance with prominence threshold set to τ

by To MATo, and hence the output of Au To MATo, depend on this choice. This can lead to suboptimal clustering performance of Au To MATo; see Section 6.

Choice of α and B: By default, Au To MATo performs the bootstrap on B = 1000 subsamples of the input point cloud, and sets the confidence level to α = 0.35. The choice of this latter parameter means that Au To MATo determines merely a 65% confidence region for the persistence diagram produced by the underlying To MATo instance. While in bootstrapping the confidence level is often set to, for instance,α = 0.05, the seemingly strange choice of α = 0.35 in the setting of Au To MATo is justified by experiments. The value of 65% seems to be low enough to offset some of the negative influence of using possibly non-optimized neighborhood graph and density estimators discussed in Section 6, while at the same time being high enough to yield good results when these estimators are chosen suitably. We point out that the value α = 0.35 (as well as the value B = 1000) was decided on after running an early implementation of Au To MATo on just a few synthetic data sets. In particular, the choice was made before conducting the experiments in Section 4. Au To MATo is implemented in such a way that the parameter α can be adjusted after fitting and the clustering is automatically updated.

Complexity analysis of Algorithm 1: Recall from Chazal et al. (2013, Section 2) that, if an estimated density and a neighborhood graph are provided, To MATo has a worst-case time complexity in O(n log(n) + mα(n)), where n and m are the number of vertices and edges of the neighborhood graph, respectively, and α denotes the inverse Ackermann function (note that n equals the number of data points). By default, To MATo (and hence Au To MATo) works with the k-nearest neighbor graph and distance-to-measure density estimators, where the latter itself relies on the k-nearest neighbor graph (each with k = 10). Taking into account the known complexity bound O(nd) for the creation of the k-nearest neighbor graph (where d is the dimensionality of the data), and using the fact that m O(n) for this graph, this leads to a worst-case time complexity in O(nd+n log(n)) for a single run of To MATo. Creating the bootstrap samples X i , i = 1, . . . , B, has complexity in O(Bn); computing the values n W fin (D i , D), i = 1, . . . , B, has worst-case complexity O(BN 1.5 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 Au To MATo in O(B(nd + n log(n) + N 1.5 log(N)) + B log(B)). Using that B is a constant, we obtain the runtime of O(B(nd + n log(n) + N 1.5 log N)) claimed above.

We point out that the complexity of O(N 1.5 log(N)) for the computation of the bottleneck distance between a pair of persistence diagrams is a theoretical worst-case scenario, whose validity was established in Efrat et al. (2001). In 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 Au To MATo 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 Au To MATo can be drastically decreased in practice through parallelization.

The Python package: Our Python package for Au To MATo consists of two separate modules; one for Au To MATo itself, and one for the bottleneck bootstrap. Both are compatible with the scikit-learn architecture, and the latter may also be used as a stand-alone module for other scenarios. In addition to the functionality inherited from the scikit-learn API, the implementation of Au To MATo comes with options of

adjusting the parameter α of a fitted instance of Au To MATo which automatically updates the resulting clustering without repeating the (computationally expensive) bootstrapping; plotting the persistence diagram and the prominence threshold found in the bootstrapping; setting a seed in order to make the creation of the bootstrap subsamples in Au To MATo deterministic, thus allowing for reproducible results; and parallelizing the bottleneck bootstrap for speed improvements.

Finally, our implementation of Au To MATo contains a parameter that allows the algorithm to label points as outliers. In a nutshell, a point is classified as an outlier if it is not among the nearest neighbors of more than a specified percentage of its own nearest 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 Au To MATo with its default parameters against

DBSCAN and its extension HDBSCAN; hierarchical clustering with Ward, single, complete and average linkage; the FINCH clustering algorithm (Sarfraz et al., 2019); and a clustering algorithm building on To MATo stemming from the Topology Tool Kit (TTK) suite (Tierny et al., 2018); in the following, we will refer to this as the TTK-algorithm.4

For DBSCAN, HDBSCAN and the hierarchical clustering algorithms mentioned above, we worked with their implementations in scikit-learn.5 For the FINCH clustering algorithm, we worked with the version available on Git Hub.6 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 5.2). While we included DBSCAN and HDBSCAN among the clustering algorithms to compare Au To MATo against because they are standard choices, we chose to include the hierarchical clustering algorithms because they are readily available through scikit-learn. Finally, we chose to include FINCH and the TTK-algorithm because, like Au To MATo, they are out-of-the-box (indeed, parameter-free) methods and are thus especially interesting to compare Au To MATo against.

4.2 Choice of data sets

The data sets on which we ran Au To MATo and the above clustering algorithms stem from the Clustering Benchmarks suite (Gagolewski, 2022).7 We chose this collection as it comes with a large variety of different data sets, all of which are labeled by one 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.

4For the Topology Tool Kit, see topology-tool-kit.github.io/ (BSD license). 5scikit-learn.org/stable/modules/clustering.html 6github.com/ssarfraz/FINCH-Clustering (CC BY-NC-SA 4.0 license) 7Specifically, we worked with version 1.1.0 of the benchmarking suite (Gagolewski et 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 Au To MATo determines clusters depending on connectivity, and topologically speaking, there is only one 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 in Au To MATo.

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 as follows. Given a clustering C found by an algorithm and a ground truth clustering G, one defines the Fowlkes-Mallows score as

TP TP + FN,

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 Fowlkes-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 attain any value between 0 and 1, and these extremal values correspond to the worst and best possible clustering, respectively. We chose to use the Fowlkes-Mallows score as opposed to, for instance, mutual information or any of the Rand indices, because the latter have been shown to exhibit biased behavior depending on whether the clusters in the ground truth are mostly of similar sizes or not, see, for instance, Romano et 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 Au To MATo against, whereas we are interested in comparing Au To MATo 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, we 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 Au To MATo against the best and worst performing version of these clustering algorithms. To account for the randomized component of Au To MATo, we ran it ten times, each time with a different seed.

While we restricted ourselves 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 as outliers. In order to prevent these algorithms from getting systematically low Fowlkes-Mallows scores because of these outliers, we removed all the points labeled as outliers by these algorithms, and only computed the Fowlkes-Mallows score on the remaining points, both for these clustering algorithms and for Au To MATo. This of course gives an advantage to DBSCAN and HDBSCAN over Au To MATo.

In order to allow reproducibility, we chose a fixed seed for all our experiments, which can be found in our code.8 We ran our experiments on a laptop with a 12th Gen Intel Core i7-1260P processor running at 2.10GHz.

8The code is archived on Zenodo (doi.org/10.5281/zenodo.17279741) and developed openly on Git Hub (github.com/ m-a-huber/automato_paper).

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4.4 Results and interpretation

Table 1 shows the average Fowlkes-Mallows score of each algorithm across all benchmarking data sets; for Au To MATo, 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 algorithm. 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 1 shows, Au To MATo outperforms each clustering algorithm on average across all data sets, thus showing that it is indeed a versatile and powerful out-of-the-box clustering algorithm. In particular, Au To MATo outperforms the TTK-algorithm, which also build on To MATo.

Table 1: Average clustering performance of Au To MATo vs. reference clustering algorithms

Algorithm Fowlkes-Mallows score

Au To MATo 0.8554 0.0228 DBSCAN 0.8457 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 6 in Appendix A.2. As an illustration, Figure 3 shows that the best choice of parameter for DBSCAN sometimes outperforms Au To MATo, which is to be expected. However, on most data sets where this is the case, the results from Au To MATo are still competitive, and there is a significant number of instances where Au To MATo outperforms DBSCAN for all parameter selections, in some cases by a lot.

wut_trajectories

sipu_spiral

wut_isolation

sipu_pathbased

graves_dense

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sipu_unbalance

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graves_ring

graves_ring_outliers

graves_zigzag

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wut_trapped_lovers

wut_windows

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1 clusterer

automato_mean

Figure 3: Fowlkes-Mallows score of Au To MATo and DBSCAN across benchmarking data sets. The shading of automato_mean indicates the standard deviation of the score across the ten runs.

5 Applications of Au To MATo in combination with Mapper

The goal of Mapper (Singh et al., 2007) is to approximate the Reeb graph of a manifold M based on a sample from M. The input is a point cloud P with a filter function P R; a collection of overlapping intervals U = {U1, . . . , Un} covering R; and a clustering algorithm. For each Ui U, Mapper runs the clustering

Published in Transactions on Machine Learning Research (10/2025)

algorithm on the data points in the preimage f 1(Ui), creating a 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 (Tauzin et al., 2020) on a synthetic two-dimensional data set consisting of noisy samples from two concentric circles (see Figure 4a) with projection onto the x-axis as the filter function. We ran Mapper on the same interval cover with three different choices of clustering algorithms: Au To MATo, DBSCAN, and HDBSCAN. As can be seen in Figure 4b, 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 Au To MATo is exactly the Reeb graph of two circles.

(a) (b) (c) (d)

Figure 4: (a) input data set; result of Mapper with (b) Au To MATo; (c) DBSCAN; (d) HDBSCAN

We further tested the combination of Mapper with Au To MATo on one of the standard applications of Mapper: the Miller-Reaven diabetes data set, where Mapper can be used detect two strains of diabetes that correspond to flares in the data set (see Singh et al. (2007, Section 5.1) for details).9 As can be seen in Figure 5, Au To MATo performs well in this task; the graphs show a central core of vertices corresponding to healthy patients, and two flares corresponding to the two strains of diabetes. We were not able to reproduce this using DBSCAN or HDBSCAN; Figure 5 shows the output of Mapper with these algorithms with their respective default parameters.

Figure 5: Mapper applied to the diabetes data set with Au To MATo (left); DBSCAN (center); HDBSCAN (right). Labels 0, 1 and 2 stand for no , chemical and overt diabetes .

6 Discussion

We briefly outline some limitations of Au To MATo. Au To MATo comes with a choice of default values for its parameters.

9The data set is available as part of the locfit R-package (Loader, 2024).

Published in Transactions on Machine Learning Research (10/2025)

Optimizing the choice of the neighborhood graph and density estimators is an aspect of Au To MATo that we plan to pursue in future work. Moreover, we plan to improve the currently experimental feature for outlier creation in Au To MATo discussed at the end of Section 3.2. Finally, it is natural to ask whether the results from Carrière et al. (2018) on optimal parameter selection in the Mapper algorithm can be adapted to the scenario where Mapper uses Au To MATo as its clustering algorithm.

Acknowledgments

The first author was supported by the Swiss National Science Foundation (project no. 209413).

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Published in Transactions on Machine Learning Research (10/2025)

A.1 About the choice of data sets

As explained in Section 4.2, 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 6. In that figure, the data points are colored according to the ground truth clustering.

Figure 6: The data sets named windows (left) and olympic (right) from the wut-battery.

A.2 Benchmarking results

In this subsection we report the Fowlkes-Mallows scores coming from comparing Au To MATo to the other clustering algorithms, as 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 of Au To MATo and the respective clustering algorithm that Au To MATo is being 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 a 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 algorithm performs best on.

Table 2: Fowlkes-Mallows scores of Au To MATo vs. DBSCAN

Dataset automato_mean dbscan_max dbscan_min

sipu_r15_2 0.4867 0.0000 1.0000 0.5607 wut_trajectories_0 0.5038 0.0107 1.0000 0.4999 wut_x3_0 0.5153 0.0000 0.9398 0.5149 wut_x2_0 0.5846 0.0000 0.9483 0.5779 sipu_r15_1 0.5436 0.0000 0.8954 0.5021 fcps_tetra_0 0.6261 0.0000 0.9403 0.0000 sipu_pathbased_0 0.6517 0.0000 0.9569 0.5769 sipu_spiral_0 0.7028 0.0000 1.0000 0.5756 wut_isolation_0 0.7256 0.0113 1.0000 0.5773 sipu_pathbased_1 0.7322 0.0000 0.9620 0.5170

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Published in Transactions on Machine Learning Research (10/2025)

Table 2: Fowlkes-Mallows scores of Au To MATo vs. DBSCAN

Dataset automato_mean dbscan_max dbscan_min

sipu_jain_0 0.7837 0.0000 0.9880 0.7837 graves_dense_0 0.8377 0.1396 0.9970 0.7053 sipu_compound_0 0.8616 0.0000 1.0000 0.4972 fcps_atom_0 0.8694 0.0000 1.0000 0.7067 wut_circles_0 0.8857 0.0000 1.0000 0.4998 fcps_chainlink_0 0.8896 0.0000 1.0000 0.7068 other_iris_0 0.7715 0.0000 0.8721 0.0000 wut_mk4_0 0.9072 0.0234 1.0000 0.5770 wut_mk2_0 0.6356 0.0000 0.7068 0.5778 sipu_compound_4 0.9442 0.0000 1.0000 0.5523 wut_x3_1 0.6546 0.0000 0.7042 0.6546 graves_zigzag_1 0.6720 0.0000 0.7149 0.4446 other_iris5_0 0.6712 0.0000 0.7046 0.0000 wut_smile_1 0.9701 0.0000 1.0000 0.5825 wut_x1_0 0.9741 0.0818 1.0000 0.5846 fcps_target_0 0.9850 0.0000 1.0000 0.6963 wut_smile_0 0.9681 0.0000 0.9753 0.5471 wut_mk3_0 0.7720 0.0000 0.7774 0.5764 sipu_compound_1 0.9786 0.0000 0.9825 0.5715 sipu_flame_0 0.7320 0.0000 0.7341 0.5918 sipu_unbalance_0 0.9986 0.0008 1.0000 0.5339 fcps_twodiamonds_0 0.7067 0.0000 0.7067 0.7067 sipu_aggregation_0 0.8652 0.0000 0.8652 0.4653 wut_stripes_0 1.0000 0.0000 1.0000 0.7070 wut_trapped_lovers_0 1.0000 0.0000 1.0000 0.6632 wut_windows_0 1.0000 0.0000 1.0000 0.6753 fcps_hepta_0 1.0000 0.0000 1.0000 0.3727 fcps_lsun_0 1.0000 0.0000 1.0000 0.6111 graves_line_0 1.0000 0.0000 1.0000 0.8238 graves_ring_0 1.0000 0.0000 1.0000 0.7068 graves_ring_outliers_0 1.0000 0.0000 1.0000 0.6863 graves_zigzag_0 1.0000 0.0000 1.0000 0.5328 other_square_0 1.0000 0.0000 1.0000 0.7068 wut_mk1_0 0.9866 0.0000 0.9651 0.5754 wut_twosplashes_0 1.0000 0.0000 0.9649 0.7062 fcps_wingnut_0 0.9805 0.0000 0.8784 0.7068 graves_parabolic_1 0.6916 0.0000 0.5000 0.4999 wut_labirynth_0 0.7884 0.0000 0.5221 0.5221 graves_parabolic_0 0.9802 0.0000 0.7068 0.7068 sipu_d31_0 0.6001 0.0085 0.1846 0.1787 sipu_a1_0 0.7499 0.0000 0.3269 0.2229 sipu_r15_0 0.9258 0.0000 0.4551 0.2552 sipu_s1_0 0.9888 0.0000 0.4890 0.2581 sipu_a2_0 0.7555 0.0000 0.1685 0.1685 sipu_a3_0 0.7434 0.0000 0.1410 0.1410 sipu_s2_0 0.9405 0.0000 0.2581 0.2581

Published in Transactions on Machine Learning Research (10/2025)

wut_trajectories

sipu_spiral

wut_isolation

sipu_pathbased

graves_dense

wut_circles

fcps_chainlink

other_iris5

sipu_compound

fcps_target

sipu_unbalance

fcps_twodiamonds

sipu_aggregation

graves_line

graves_ring

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graves_zigzag

other_square

wut_stripes

wut_trapped_lovers

wut_windows

wut_twosplashes

fcps_wingnut

wut_labirynth

graves_parabolic

1 clusterer

automato_mean

Figure 7: Comparison of Au To MATo and DBSCAN.

Table 3: Fowlkes-Mallows scores of Au To MATo vs. hierarchical clustering with average linkage

Dataset automato_mean linkage_average_max linkage_average_min

sipu_r15_2 0.4867 0.0000 1.0000 0.3971 wut_trajectories_0 0.5038 0.0107 1.0000 0.3115 fcps_tetra_0 0.6261 0.0000 1.0000 0.0651 sipu_r15_1 0.5436 0.0000 0.8954 0.4435 sipu_d31_0 0.6001 0.0085 0.9322 0.1787 wut_x3_0 0.5153 0.0000 0.8389 0.3343 wut_x3_1 0.6546 0.0000 0.9747 0.2693 fcps_twodiamonds_0 0.7067 0.0000 0.9925 0.1287 sipu_a2_0 0.7555 0.0000 0.9432 0.1685 sipu_a1_0 0.7499 0.0000 0.9268 0.2229 sipu_a3_0 0.7434 0.0000 0.8825 0.1410 sipu_aggregation_0 0.8652 0.0000 0.9932 0.1785 sipu_pathbased_1 0.7322 0.0000 0.8564 0.2058 graves_dense_0 0.8377 0.1396 0.9604 0.6633 sipu_pathbased_0 0.6517 0.0000 0.7704 0.1848 wut_x2_0 0.5846 0.0000 0.7001 0.2782 wut_circles_0 0.8857 0.0000 1.0000 0.2369 wut_mk3_0 0.7720 0.0000 0.8771 0.0876 wut_mk2_0 0.6356 0.0000 0.7068 0.1421 sipu_r15_0 0.9258 0.0000 0.9900 0.2552 graves_zigzag_1 0.6720 0.0000 0.7202 0.4380 other_iris_0 0.7715 0.0000 0.8080 0.0791 other_iris5_0 0.6712 0.0000 0.7042 0.0451 wut_x1_0 0.9741 0.0818 1.0000 0.3070 sipu_jain_0 0.7837 0.0000 0.7904 0.1736 wut_mk1_0 0.9866 0.0000 0.9933 0.2037 sipu_unbalance_0 0.9986 0.0008 0.9995 0.5339 fcps_hepta_0 1.0000 0.0000 1.0000 0.3727 sipu_flame_0 0.7320 0.0000 0.7320 0.0913 sipu_s1_0 0.9888 0.0000 0.9821 0.2581

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Published in Transactions on Machine Learning Research (10/2025)

Table 3: Fowlkes-Mallows scores of Au To MATo vs. hierarchical clustering with average linkage

Dataset automato_mean linkage_average_max linkage_average_min

sipu_compound_0 0.8616 0.0000 0.8431 0.2207 sipu_compound_4 0.9442 0.0000 0.9224 0.1985 sipu_compound_1 0.9786 0.0000 0.9546 0.1922 sipu_s2_0 0.9405 0.0000 0.9097 0.2581 wut_smile_1 0.9701 0.0000 0.8726 0.4041 fcps_atom_0 0.8694 0.0000 0.7491 0.2555 graves_parabolic_1 0.6916 0.0000 0.5708 0.2135 sipu_spiral_0 0.7028 0.0000 0.5756 0.1919 wut_mk4_0 0.9072 0.0234 0.7714 0.2071 wut_smile_0 0.9681 0.0000 0.8221 0.4303 wut_isolation_0 0.7256 0.0113 0.5773 0.1651 graves_line_0 1.0000 0.0000 0.8238 0.3047 fcps_chainlink_0 0.8896 0.0000 0.7068 0.1456 fcps_target_0 0.9850 0.0000 0.7986 0.3285 fcps_wingnut_0 0.9805 0.0000 0.7739 0.1101 fcps_lsun_0 1.0000 0.0000 0.7896 0.1735 graves_ring_0 1.0000 0.0000 0.7780 0.2638 graves_parabolic_0 0.9802 0.0000 0.7580 0.1598 graves_ring_outliers_0 1.0000 0.0000 0.7767 0.2801 other_square_0 1.0000 0.0000 0.7413 0.1746 wut_labirynth_0 0.7884 0.0000 0.5221 0.2306 wut_stripes_0 1.0000 0.0000 0.7070 0.1082 wut_twosplashes_0 1.0000 0.0000 0.7062 0.4837 wut_windows_0 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 0.0000 0.6616 0.3008

wut_trajectories

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graves_dense

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wut_stripes

wut_twosplashes

wut_windows

wut_trapped_lovers

1 clusterer

automato_mean

linkage_average_max

linkage_average_min

Figure 8: Comparison of Au To MATo and agglomerative clustering with average linkage.

Published in Transactions on Machine Learning Research (10/2025)

Table 4: Fowlkes-Mallows scores of Au To MATo vs. HDBSCAN

Dataset automato_mean automato_std hdbscan

wut_trajectories_0 0.5038 0.0107 0.0107 1.0000 wut_x3_0 0.5153 0.0000 0.0000 0.8959 wut_x2_0 0.5846 0.0000 0.0000 0.9344 sipu_spiral_0 0.7028 0.0000 0.0000 0.9815 sipu_flame_0 0.7320 0.0000 0.0000 0.9900 sipu_d31_0 0.6001 0.0085 0.0085 0.8231 sipu_jain_0 0.7837 0.0000 0.0000 0.9779 fcps_tetra_0 0.6261 0.0000 0.0000 0.8157 graves_dense_0 0.8377 0.1396 0.1396 0.9894 sipu_pathbased_1 0.7322 0.0000 0.0000 0.8634 fcps_atom_0 0.8694 0.0000 0.0000 1.0000 sipu_pathbased_0 0.6517 0.0000 0.0000 0.7815 fcps_chainlink_0 0.8896 0.0000 0.0000 1.0000 sipu_r15_0 0.9258 0.0000 0.0000 0.9932 sipu_a1_0 0.7499 0.0000 0.0000 0.8081 wut_x3_1 0.6546 0.0000 0.0000 0.6972 other_iris5_0 0.6712 0.0000 0.0000 0.7042 wut_x1_0 0.9741 0.0818 0.0818 1.0000 sipu_compound_1 0.9786 0.0000 0.0000 1.0000 sipu_compound_4 0.9442 0.0000 0.0000 0.9656 fcps_target_0 0.9850 0.0000 0.0000 1.0000 sipu_compound_0 0.8616 0.0000 0.0000 0.8751 sipu_unbalance_0 0.9986 0.0008 0.0008 1.0000 graves_zigzag_1 0.6720 0.0000 0.0000 0.6720 sipu_aggregation_0 0.8652 0.0000 0.0000 0.8652 wut_stripes_0 1.0000 0.0000 0.0000 1.0000 wut_trapped_lovers_0 1.0000 0.0000 0.0000 1.0000 wut_windows_0 1.0000 0.0000 0.0000 1.0000 fcps_hepta_0 1.0000 0.0000 0.0000 1.0000 fcps_lsun_0 1.0000 0.0000 0.0000 1.0000 graves_line_0 1.0000 0.0000 0.0000 1.0000 graves_ring_0 1.0000 0.0000 0.0000 1.0000 graves_ring_outliers_0 1.0000 0.0000 0.0000 1.0000 graves_zigzag_0 1.0000 0.0000 0.0000 1.0000 other_square_0 1.0000 0.0000 0.0000 1.0000 other_iris_0 0.7715 0.0000 0.0000 0.7715 wut_mk3_0 0.7720 0.0000 0.0000 0.7719 wut_mk1_0 0.9866 0.0000 0.0000 0.9863 sipu_a3_0 0.7434 0.0000 0.0000 0.7415 sipu_a2_0 0.7555 0.0000 0.0000 0.7502 sipu_r15_2 0.4867 0.0000 0.0000 0.4671 sipu_r15_1 0.5436 0.0000 0.0000 0.5212 wut_isolation_0 0.7256 0.0113 0.0113 0.6377 fcps_wingnut_0 0.9805 0.0000 0.0000 0.8725 sipu_s1_0 0.9888 0.0000 0.0000 0.8717 sipu_s2_0 0.9405 0.0000 0.0000 0.7410 wut_mk4_0 0.9072 0.0234 0.0234 0.6459 wut_labirynth_0 0.7884 0.0000 0.0000 0.5134

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Published in Transactions on Machine Learning Research (10/2025)

Table 4: Fowlkes-Mallows scores of Au To MATo vs. HDBSCAN

Dataset automato_mean automato_std hdbscan

graves_parabolic_1 0.6916 0.0000 0.0000 0.3616 fcps_twodiamonds_0 0.7067 0.0000 0.0000 0.2886 wut_mk2_0 0.6356 0.0000 0.0000 0.1574 wut_smile_0 0.9681 0.0000 0.0000 0.4000 wut_smile_1 0.9701 0.0000 0.0000 0.3714 graves_parabolic_0 0.9802 0.0000 0.0000 0.3526 wut_twosplashes_0 1.0000 0.0000 0.0000 0.3074 wut_circles_0 0.8857 0.0000 0.0000 0.1204

wut_trajectories

sipu_spiral

graves_dense

sipu_pathbased

fcps_chainlink

other_iris5

sipu_compound

fcps_target

sipu_unbalance

sipu_aggregation

graves_line

graves_ring

graves_ring_outliers

graves_zigzag

other_square

wut_stripes

wut_trapped_lovers

wut_windows

wut_isolation

fcps_wingnut

wut_labirynth

fcps_twodiamonds

graves_parabolic

wut_twosplashes

wut_circles

1 clusterer

automato_mean

Figure 9: Comparison of Au To MATo and HDBSCAN.

Table 5: Fowlkes-Mallows scores of Au To MATo vs. hierarchical clustering with single linkage

Dataset automato_mean linkage_single_max linkage_single_min

sipu_r15_2 0.4867 0.0000 1.0000 0.5607 wut_trajectories_0 0.5038 0.0107 1.0000 0.4999 sipu_r15_1 0.5436 0.0000 0.8954 0.5021 fcps_tetra_0 0.6261 0.0000 0.9296 0.0829 sipu_spiral_0 0.7028 0.0000 1.0000 0.5756 wut_isolation_0 0.7256 0.0113 1.0000 0.5773 wut_x3_0 0.5153 0.0000 0.7347 0.4951 sipu_jain_0 0.7837 0.0000 0.9510 0.7837 fcps_atom_0 0.8694 0.0000 1.0000 0.7067 wut_circles_0 0.8857 0.0000 1.0000 0.4998 fcps_chainlink_0 0.8896 0.0000 1.0000 0.7068 wut_mk4_0 0.9072 0.0234 1.0000 0.5770 sipu_compound_0 0.8616 0.0000 0.9454 0.4972 sipu_pathbased_0 0.6517 0.0000 0.7337 0.5769 sipu_pathbased_1 0.7322 0.0000 0.8091 0.5170 graves_dense_0 0.8377 0.1396 0.9096 0.6882 wut_mk2_0 0.6356 0.0000 0.7068 0.6007 graves_zigzag_1 0.6720 0.0000 0.7344 0.4446

Continued on next page

Published in Transactions on Machine Learning Research (10/2025)

Table 5: Fowlkes-Mallows scores of Au To MATo vs. hierarchical clustering with single linkage

Dataset automato_mean linkage_single_max linkage_single_min

wut_x2_0 0.5846 0.0000 0.6437 0.5105 other_iris5_0 0.6712 0.0000 0.7042 0.0451 wut_smile_1 0.9701 0.0000 1.0000 0.5825 wut_x1_0 0.9741 0.0818 0.9920 0.5846 fcps_target_0 0.9850 0.0000 1.0000 0.6963 wut_smile_0 0.9681 0.0000 0.9748 0.5471 sipu_unbalance_0 0.9986 0.0008 1.0000 0.5339 fcps_twodiamonds_0 0.7067 0.0000 0.7067 0.7067 wut_x3_1 0.6546 0.0000 0.6546 0.6140 sipu_aggregation_0 0.8652 0.0000 0.8652 0.4653 wut_stripes_0 1.0000 0.0000 1.0000 0.7070 wut_trapped_lovers_0 1.0000 0.0000 1.0000 0.6632 wut_windows_0 1.0000 0.0000 1.0000 0.6753 fcps_hepta_0 1.0000 0.0000 1.0000 0.3727 graves_line_0 1.0000 0.0000 1.0000 0.8238 graves_ring_0 1.0000 0.0000 1.0000 0.7068 graves_ring_outliers_0 1.0000 0.0000 1.0000 0.6863 graves_zigzag_0 1.0000 0.0000 1.0000 0.5381 sipu_flame_0 0.7320 0.0000 0.7320 0.4598 other_iris_0 0.7715 0.0000 0.7715 0.1223 other_square_0 1.0000 0.0000 0.9990 0.7068 fcps_lsun_0 1.0000 0.0000 0.9983 0.6111 wut_twosplashes_0 1.0000 0.0000 0.9850 0.7062 sipu_compound_1 0.9786 0.0000 0.9180 0.5715 sipu_compound_4 0.9442 0.0000 0.8824 0.5523 wut_mk1_0 0.9866 0.0000 0.8866 0.5754 fcps_wingnut_0 0.9805 0.0000 0.8087 0.7068 graves_parabolic_1 0.6916 0.0000 0.5000 0.4979 wut_mk3_0 0.7720 0.0000 0.5764 0.5314 wut_labirynth_0 0.7884 0.0000 0.5221 0.5221 graves_parabolic_0 0.9802 0.0000 0.7068 0.7040 sipu_d31_0 0.6001 0.0085 0.1846 0.1787 sipu_a1_0 0.7499 0.0000 0.3269 0.2229 sipu_r15_0 0.9258 0.0000 0.4551 0.2552 sipu_a2_0 0.7555 0.0000 0.1685 0.1685 sipu_a3_0 0.7434 0.0000 0.1410 0.1410 sipu_s1_0 0.9888 0.0000 0.3695 0.2581 sipu_s2_0 0.9405 0.0000 0.2581 0.2579

Table 6: Fowlkes-Mallows scores of Au To MATo vs. TTK clustering algorithm

Dataset automato_mean automato_std ttk

wut_trajectories_0 0.5038 0.0107 0.0107 0.8682 fcps_tetra_0 0.6261 0.0000 0.0000 0.9043 wut_x3_0 0.5153 0.0000 0.0000 0.7818 wut_isolation_0 0.7256 0.0113 0.0113 0.9416 sipu_a1_0 0.7499 0.0000 0.0000 0.9143 wut_x2_0 0.5846 0.0000 0.0000 0.7283

Continued on next page

Published in Transactions on Machine Learning Research (10/2025)

Table 6: Fowlkes-Mallows scores of Au To MATo vs. TTK clustering algorithm

Dataset automato_mean automato_std ttk

sipu_flame_0 0.7320 0.0000 0.0000 0.8562 sipu_aggregation_0 0.8652 0.0000 0.0000 0.9692 graves_zigzag_1 0.6720 0.0000 0.0000 0.7698 other_iris_0 0.7715 0.0000 0.0000 0.8639 sipu_jain_0 0.7837 0.0000 0.0000 0.8182 graves_dense_0 0.8377 0.1396 0.1396 0.8615 sipu_r15_0 0.9258 0.0000 0.0000 0.9374 wut_mk3_0 0.7720 0.0000 0.0000 0.7755 wut_labirynth_0 0.7884 0.0000 0.0000 0.7884 fcps_chainlink_0 0.8896 0.0000 0.0000 0.8896 wut_smile_0 0.9681 0.0000 0.0000 0.9681 wut_stripes_0 1.0000 0.0000 0.0000 1.0000 graves_ring_outliers_0 1.0000 0.0000 0.0000 1.0000 wut_smile_1 0.9701 0.0000 0.0000 0.9701 sipu_unbalance_0 0.9986 0.0008 0.0008 0.9951 sipu_s1_0 0.9888 0.0000 0.0000 0.9843 sipu_s2_0 0.9405 0.0000 0.0000 0.9311 other_iris5_0 0.6712 0.0000 0.0000 0.6612 fcps_atom_0 0.8694 0.0000 0.0000 0.8472 wut_x3_1 0.6546 0.0000 0.0000 0.6312 sipu_d31_0 0.6001 0.0085 0.0085 0.5667 fcps_hepta_0 1.0000 0.0000 0.0000 0.9594 graves_parabolic_1 0.6916 0.0000 0.0000 0.6473 sipu_r15_2 0.4867 0.0000 0.0000 0.4322 sipu_pathbased_0 0.6517 0.0000 0.0000 0.5947 sipu_spiral_0 0.7028 0.0000 0.0000 0.6422 sipu_r15_1 0.5436 0.0000 0.0000 0.4827 sipu_pathbased_1 0.7322 0.0000 0.0000 0.6668 fcps_target_0 0.9850 0.0000 0.0000 0.9185 wut_x1_0 0.9741 0.0818 0.0818 0.8960 wut_twosplashes_0 1.0000 0.0000 0.0000 0.9140 wut_mk4_0 0.9072 0.0234 0.0234 0.8050 wut_mk2_0 0.6356 0.0000 0.0000 0.5302 graves_parabolic_0 0.9802 0.0000 0.0000 0.8653 sipu_compound_4 0.9442 0.0000 0.0000 0.8145 fcps_wingnut_0 0.9805 0.0000 0.0000 0.8497 wut_circles_0 0.8857 0.0000 0.0000 0.7543 wut_mk1_0 0.9866 0.0000 0.0000 0.8148 fcps_twodiamonds_0 0.7067 0.0000 0.0000 0.5251 sipu_compound_0 0.8616 0.0000 0.0000 0.6728 sipu_compound_1 0.9786 0.0000 0.0000 0.7892 graves_line_0 1.0000 0.0000 0.0000 0.7917 fcps_lsun_0 1.0000 0.0000 0.0000 0.7897 wut_trapped_lovers_0 1.0000 0.0000 0.0000 0.7859 other_square_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_0 0.7555 0.0000 0.0000 0.4641 wut_windows_0 1.0000 0.0000 0.0000 0.5853

Continued on next page

Published in Transactions on Machine Learning Research (10/2025)

Table 6: Fowlkes-Mallows scores of Au To MATo vs. TTK clustering algorithm

Dataset automato_mean automato_std ttk

sipu_a3_0 0.7434 0.0000 0.0000 0.1882

Table 7: Fowlkes-Mallows scores of Au To MATo vs. hierarchical clustering with complete linkage

Dataset automato_mean linkage_complete_max linkage_complete_min

sipu_r15_2 0.4867 0.0000 1.0000 0.2256 wut_trajectories_0 0.5038 0.0107 1.0000 0.1706 sipu_r15_1 0.5436 0.0000 0.8954 0.2516 wut_x3_1 0.6546 0.0000 0.9740 0.2004 fcps_tetra_0 0.6261 0.0000 0.9356 0.0651 sipu_d31_0 0.6001 0.0085 0.8733 0.2717 wut_x3_0 0.5153 0.0000 0.7842 0.2477 sipu_a3_0 0.7434 0.0000 0.8979 0.2294 wut_mk3_0 0.7720 0.0000 0.9207 0.0711 wut_x2_0 0.5846 0.0000 0.7298 0.1964 sipu_a2_0 0.7555 0.0000 0.8992 0.2642 sipu_jain_0 0.7837 0.0000 0.9116 0.1288 wut_circles_0 0.8857 0.0000 1.0000 0.1761 sipu_r15_0 0.9258 0.0000 0.9799 0.3372 sipu_a1_0 0.7499 0.0000 0.8040 0.3092 graves_zigzag_1 0.6720 0.0000 0.7119 0.3039 sipu_aggregation_0 0.8652 0.0000 0.9030 0.1246 other_iris_0 0.7715 0.0000 0.8064 0.0680 wut_x1_0 0.9741 0.0818 1.0000 0.2326 sipu_unbalance_0 0.9986 0.0008 0.9988 0.5774 fcps_hepta_0 1.0000 0.0000 1.0000 0.4321 fcps_twodiamonds_0 0.7067 0.0000 0.7060 0.0916 sipu_flame_0 0.7320 0.0000 0.7276 0.0834 sipu_compound_0 0.8616 0.0000 0.8472 0.1567 sipu_s1_0 0.9888 0.0000 0.9563 0.3672 sipu_pathbased_0 0.6517 0.0000 0.6022 0.1384 sipu_pathbased_1 0.7322 0.0000 0.6709 0.1539 graves_parabolic_1 0.6916 0.0000 0.6168 0.1482 sipu_compound_1 0.9786 0.0000 0.9023 0.1366 sipu_compound_4 0.9442 0.0000 0.8652 0.1408 graves_dense_0 0.8377 0.1396 0.7584 0.3538 wut_mk1_0 0.9866 0.0000 0.8950 0.1591 wut_smile_1 0.9701 0.0000 0.8697 0.3562 graves_parabolic_0 0.9802 0.0000 0.8610 0.1088 wut_mk2_0 0.6356 0.0000 0.5032 0.1096 other_iris5_0 0.6712 0.0000 0.5305 0.0451 fcps_atom_0 0.8694 0.0000 0.7278 0.1364 wut_smile_0 0.9681 0.0000 0.8206 0.3793 sipu_s2_0 0.9405 0.0000 0.7642 0.3114 wut_mk4_0 0.9072 0.0234 0.7282 0.1297 fcps_target_0 0.9850 0.0000 0.7881 0.1934 fcps_lsun_0 1.0000 0.0000 0.7668 0.1377 fcps_wingnut_0 0.9805 0.0000 0.7406 0.0816

Continued on next page

Published in Transactions on Machine Learning Research (10/2025)

Table 7: Fowlkes-Mallows scores of Au To MATo vs. hierarchical clustering with complete linkage

Dataset automato_mean linkage_complete_max linkage_complete_min

graves_ring_0 1.0000 0.0000 0.7589 0.1849 graves_ring_outliers_0 1.0000 0.0000 0.7528 0.1995 wut_labirynth_0 0.7884 0.0000 0.4893 0.1426 fcps_chainlink_0 0.8896 0.0000 0.5889 0.0919 sipu_spiral_0 0.7028 0.0000 0.3512 0.1424 wut_isolation_0 0.7256 0.0113 0.3397 0.1153 graves_line_0 1.0000 0.0000 0.5972 0.1909 other_square_0 1.0000 0.0000 0.5846 0.1142 graves_zigzag_0 1.0000 0.0000 0.5505 0.2042 wut_twosplashes_0 1.0000 0.0000 0.5310 0.2771 wut_stripes_0 1.0000 0.0000 0.5136 0.0706 wut_trapped_lovers_0 1.0000 0.0000 0.4790 0.0579 wut_windows_0 1.0000 0.0000 0.4349 0.0781

Table 8: Fowlkes-Mallows scores of Au To MATo vs. hierarchical clustering with Ward linkage

Dataset automato_mean linkage_ward_max linkage_ward_min

wut_x3_0 0.5153 0.0000 0.8537 0.2203 sipu_d31_0 0.6001 0.0085 0.9223 0.2766 sipu_a3_0 0.7434 0.0000 0.9377 0.2889 sipu_a2_0 0.7555 0.0000 0.9360 0.2653 sipu_a1_0 0.7499 0.0000 0.9166 0.2464 wut_x2_0 0.5846 0.0000 0.7219 0.2076 sipu_r15_2 0.4867 0.0000 0.5893 0.1868 graves_zigzag_1 0.6720 0.0000 0.7358 0.2693 sipu_r15_0 0.9258 0.0000 0.9832 0.4072 sipu_r15_1 0.5436 0.0000 0.5993 0.2082 wut_x3_1 0.6546 0.0000 0.7090 0.1795 wut_x1_0 0.9741 0.0818 1.0000 0.2429 sipu_unbalance_0 0.9986 0.0008 1.0000 0.2063 fcps_hepta_0 1.0000 0.0000 1.0000 0.4314 sipu_s1_0 0.9888 0.0000 0.9844 0.2453 sipu_pathbased_0 0.6517 0.0000 0.6251 0.1370 sipu_s2_0 0.9405 0.0000 0.9085 0.2177 sipu_pathbased_1 0.7322 0.0000 0.6844 0.1523 fcps_tetra_0 0.6261 0.0000 0.5622 0.0651 graves_dense_0 0.8377 0.1396 0.7454 0.2684 wut_trajectories_0 0.5038 0.0107 0.3933 0.0981 other_iris_0 0.7715 0.0000 0.6377 0.0680 fcps_atom_0 0.8694 0.0000 0.7272 0.1016 graves_parabolic_1 0.6916 0.0000 0.5260 0.1301 fcps_target_0 0.9850 0.0000 0.7759 0.1503 other_iris5_0 0.6712 0.0000 0.4508 0.0451 sipu_aggregation_0 0.8652 0.0000 0.6064 0.1215 sipu_flame_0 0.7320 0.0000 0.4555 0.0819 sipu_compound_0 0.8616 0.0000 0.5846 0.1523 wut_mk3_0 0.7720 0.0000 0.4911 0.0711 fcps_twodiamonds_0 0.7067 0.0000 0.3967 0.0861

Continued on next page

Published in Transactions on Machine Learning Research (10/2025)

Table 8: Fowlkes-Mallows scores of Au To MATo vs. hierarchical clustering with Ward linkage

Dataset automato_mean linkage_ward_max linkage_ward_min

sipu_jain_0 0.7837 0.0000 0.4668 0.1201 wut_mk1_0 0.9866 0.0000 0.6564 0.1473 fcps_lsun_0 1.0000 0.0000 0.6659 0.1270 sipu_compound_4 0.9442 0.0000 0.5793 0.1368 sipu_spiral_0 0.7028 0.0000 0.3100 0.1384 wut_labirynth_0 0.7884 0.0000 0.3749 0.0961 sipu_compound_1 0.9786 0.0000 0.5648 0.1327 wut_twosplashes_0 1.0000 0.0000 0.5817 0.2046 wut_smile_1 0.9701 0.0000 0.5246 0.2352 wut_smile_0 0.9681 0.0000 0.5179 0.2505 wut_mk2_0 0.6356 0.0000 0.1814 0.0997 graves_zigzag_0 1.0000 0.0000 0.5448 0.1809 wut_isolation_0 0.7256 0.0113 0.2309 0.0800 graves_line_0 1.0000 0.0000 0.5045 0.1667 wut_circles_0 0.8857 0.0000 0.3681 0.1323 fcps_chainlink_0 0.8896 0.0000 0.3407 0.0894 wut_mk4_0 0.9072 0.0234 0.3557 0.1079 graves_parabolic_0 0.9802 0.0000 0.4184 0.0935 graves_ring_0 1.0000 0.0000 0.4135 0.1427 graves_ring_outliers_0 1.0000 0.0000 0.4082 0.1515 fcps_wingnut_0 0.9805 0.0000 0.3229 0.0727 other_square_0 1.0000 0.0000 0.3347 0.1007 wut_windows_0 1.0000 0.0000 0.2833 0.0663 wut_trapped_lovers_0 1.0000 0.0000 0.2552 0.0471 wut_stripes_0 1.0000 0.0000 0.1922 0.0552

Table 9: Fowlkes-Mallows scores of Au To MATo vs. FINCH

Dataset automato_mean automato_std finch

wut_x3_0 0.5153 0.0000 0.0000 0.7970 sipu_d31_0 0.6001 0.0085 0.0085 0.8657 sipu_a3_0 0.7434 0.0000 0.0000 0.8306 wut_x2_0 0.5846 0.0000 0.0000 0.6671 wut_mk3_0 0.7720 0.0000 0.0000 0.8503 other_iris5_0 0.6712 0.0000 0.0000 0.7008 sipu_a2_0 0.7555 0.0000 0.0000 0.7635 wut_x3_1 0.6546 0.0000 0.0000 0.6619 sipu_unbalance_0 0.9986 0.0008 0.0008 0.9998 sipu_r15_0 0.9258 0.0000 0.0000 0.9083 wut_mk1_0 0.9866 0.0000 0.0000 0.9655 other_iris_0 0.7715 0.0000 0.0000 0.7477 sipu_a1_0 0.7499 0.0000 0.0000 0.7124 sipu_r15_2 0.4867 0.0000 0.0000 0.4156 graves_zigzag_1 0.6720 0.0000 0.0000 0.5965 graves_dense_0 0.8377 0.1396 0.1396 0.7615 sipu_r15_1 0.5436 0.0000 0.0000 0.4641 sipu_s1_0 0.9888 0.0000 0.0000 0.8728 fcps_hepta_0 1.0000 0.0000 0.0000 0.8794

Continued on next page

Published in Transactions on Machine Learning Research (10/2025)

Table 9: Fowlkes-Mallows scores of Au To MATo vs. FINCH

Dataset automato_mean automato_std finch

fcps_atom_0 0.8694 0.0000 0.0000 0.7319 fcps_tetra_0 0.6261 0.0000 0.0000 0.4680 sipu_s2_0 0.9405 0.0000 0.0000 0.7282 wut_x1_0 0.9741 0.0818 0.0818 0.7607 graves_parabolic_1 0.6916 0.0000 0.0000 0.4446 sipu_flame_0 0.7320 0.0000 0.0000 0.4767 sipu_pathbased_0 0.6517 0.0000 0.0000 0.3440 sipu_compound_0 0.8616 0.0000 0.0000 0.5390 sipu_pathbased_1 0.7322 0.0000 0.0000 0.3828 wut_trajectories_0 0.5038 0.0107 0.0107 0.1503 fcps_lsun_0 1.0000 0.0000 0.0000 0.6206 sipu_compound_4 0.9442 0.0000 0.0000 0.5493 sipu_jain_0 0.7837 0.0000 0.0000 0.3824 sipu_compound_1 0.9786 0.0000 0.0000 0.5356 sipu_spiral_0 0.7028 0.0000 0.0000 0.2553 wut_mk4_0 0.9072 0.0234 0.0234 0.4331 wut_mk2_0 0.6356 0.0000 0.0000 0.1478 sipu_aggregation_0 0.8652 0.0000 0.0000 0.3674 fcps_twodiamonds_0 0.7067 0.0000 0.0000 0.1837 wut_smile_0 0.9681 0.0000 0.0000 0.4452 wut_smile_1 0.9701 0.0000 0.0000 0.4181 fcps_target_0 0.9850 0.0000 0.0000 0.4297 wut_labirynth_0 0.7884 0.0000 0.0000 0.2209 wut_isolation_0 0.7256 0.0113 0.0113 0.1440 wut_twosplashes_0 1.0000 0.0000 0.0000 0.4162 graves_zigzag_0 1.0000 0.0000 0.0000 0.4094 graves_parabolic_0 0.9802 0.0000 0.0000 0.3343 graves_line_0 1.0000 0.0000 0.0000 0.3379 fcps_chainlink_0 0.8896 0.0000 0.0000 0.2224 fcps_wingnut_0 0.9805 0.0000 0.0000 0.3110 graves_ring_0 1.0000 0.0000 0.0000 0.2770 wut_trapped_lovers_0 1.0000 0.0000 0.0000 0.2767 other_square_0 1.0000 0.0000 0.0000 0.2393 graves_ring_outliers_0 1.0000 0.0000 0.0000 0.2248 wut_circles_0 0.8857 0.0000 0.0000 0.0976 wut_windows_0 1.0000 0.0000 0.0000 0.1166 wut_stripes_0 1.0000 0.0000 0.0000 0.0867

Published in Transactions on Machine Learning Research (10/2025)

wut_trajectories

sipu_spiral

wut_isolation

wut_circles

fcps_chainlink

sipu_pathbased

graves_dense

other_iris5

fcps_target

sipu_unbalance

fcps_twodiamonds

sipu_aggregation

graves_line

graves_ring

graves_ring_outliers

graves_zigzag

wut_stripes

wut_trapped_lovers

wut_windows

other_square

wut_twosplashes

sipu_compound

fcps_wingnut

wut_labirynth

graves_parabolic

1 clusterer

automato_mean

linkage_single_max

linkage_single_min

Figure 10: Comparison of Au To MATo and agglomerative clustering with single linkage.

wut_trajectories

wut_isolation

sipu_aggregation

graves_dense

wut_labirynth

fcps_chainlink

graves_ring_outliers

wut_stripes

sipu_unbalance

other_iris5

sipu_spiral

sipu_pathbased

fcps_target

wut_twosplashes

graves_parabolic

fcps_wingnut

wut_circles

sipu_compound

fcps_twodiamonds

graves_line

wut_trapped_lovers

other_square

graves_zigzag

graves_ring

wut_windows

1 clusterer

automato_mean

Figure 11: Comparison of Au To MATo and the TTK-algorithm.

wut_trajectories

wut_circles

sipu_aggregation

sipu_unbalance

fcps_twodiamonds

sipu_pathbased

sipu_compound

graves_dense

graves_parabolic

other_iris5

fcps_target

fcps_wingnut

graves_ring

graves_ring_outliers

graves_zigzag

wut_labirynth

fcps_chainlink

sipu_spiral

wut_isolation

graves_line

other_square

wut_twosplashes

wut_stripes

wut_trapped_lovers

wut_windows

1 clusterer

automato_mean

linkage_complete_max

linkage_complete_min

Figure 12: Comparison of Au To MATo and agglomerative clustering with complete linkage.

Published in Transactions on Machine Learning Research (10/2025)

sipu_unbalance

sipu_pathbased

graves_dense

wut_trajectories

fcps_target

other_iris5

sipu_aggregation

graves_zigzag

fcps_twodiamonds

sipu_spiral

sipu_compound

wut_labirynth

wut_twosplashes

graves_parabolic

wut_isolation

graves_line

wut_circles

fcps_chainlink

graves_ring

graves_ring_outliers

fcps_wingnut

other_square

wut_windows

wut_trapped_lovers

wut_stripes

1 clusterer

automato_mean

linkage_ward_max

linkage_ward_min

Figure 13: Comparison of Au To MATo and agglomerative clustering with Ward linkage.

other_iris5

sipu_unbalance

graves_dense

sipu_pathbased

wut_trajectories

graves_zigzag

sipu_compound

sipu_spiral

sipu_aggregation

fcps_twodiamonds

graves_parabolic

fcps_target

wut_labirynth

wut_isolation

wut_twosplashes

graves_line

fcps_chainlink

fcps_wingnut

graves_ring

wut_trapped_lovers

other_square

graves_ring_outliers

wut_circles

wut_windows

wut_stripes

1 clusterer

automato_mean

Figure 14: Comparison of Au To MATo and FINCH.