Original: TOAST: T OPOLOGICAL A LGORITHM FOR S INGULARITY T RACKING
Revised: TOAST: T OPOLOGICAL A LGORITHM FOR S INGULARITY T RACKING

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: The manifold hypothesis, which assumes that data lie on or close to an unknownmanifold of low intrinsic dimensionality, is a staple of modern machine learningresearch.
Revised: The manifold hypothesis, which assumes that data lie on or close to an unknownmanifold of low intrinsic dimensionality, is a staple of modern machine learningresearch.

Original: However, recent work has shown that real-world data exhibit distinctnon-manifold structures, which result in singularities that can lead to erroneousconclusions about the data.
Revised: However, recent work has shown that real-world data exhibit distinctnon-manifold structures, which result in singularities that can lead to erroneousconclusions about the data.

Original: Detecting such singularities is therefore crucial as aprecursor to interpolation and inference tasks.
Revised: Detecting such singularities is therefore crucial as aprecursor to interpolation and inference tasks.

Original: We address detecting singularitiesby developing (i) persistent local homology , a new topology-driven frameworkfor quantifying the intrinsic dimension of a data set locally, and (ii) Euclidicity , atopology-based multi-scale measure for assessing the ‘manifoldness’ of individualpoints.
Revised: We address detecting singularitiesby developing (i) persistent local homology , a new topology-driven frameworkfor quantifying the intrinsic dimension of a data set locally, and (ii) Euclidicity , atopology-based multi-scale measure for assessing the ‘manifoldness’ of individualpoints.

Original: We show that our approach can reliably identify singularities of complexspaces, while also capturing singular structures in real-world data sets.
Revised: We show that our approach can reliably identify singularities of complexspaces, while also capturing singular structures in real-world data sets.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: The ever-increasing amount and complexity of real-world data necessitate the development of newmethods to extract less complex—but still meaningful —representations of the underlying data.
Revised: The ever-increasing amount and complexity of real-world data necessitate the development of newmethods to extract less complex—but still meaningful —representations of the underlying data.

Original: Oneapproach to this problem is via dimensionality reduction techniques, where the data is assumed tobe of strictly lower dimension than its number of features.
Revised: Oneapproach to this problem is via dimensionality reduction techniques, where the data is assumed tobe of strictly lower dimension than its number of features.

Original: Traditional algorithms in this field suchas PCA are restricted to linear descriptions of data, and are therefore of limited use for complex,non-linear data sets that often appear in practice.
Revised: Traditional algorithms in this field suchas PCA are restricted to linear descriptions of data, and are therefore of limited use for complex,non-linear data sets that often appear in practice.

Original: By contrast, non-linear dimensionality reduction algorithms, such as UMAP (McInnes et al., 2018), t -SNE (van der Maaten & Hinton, 2008),or autoencoders (Kingma & Welling, 2019) share one common assumption: the underlying data issupposed to be close to a manifold with small intrinsic dimension, i.e. while the input data may havea large ambient dimension N , there is a n -dimensional manifold with n ≪ N that best describes thedata.
Revised: By contrast, non-linear dimensionality reduction algorithms, such as UMAP (McInnes et al., 2018), t -SNE (van der Maaten & Hinton, 2008),or autoencoders (Kingma & Welling, 2019) share one common assumption: the underlying data issupposed to be close to a manifold with small intrinsic dimension, i.e. while the input data may havea large ambient dimension N , there is a n -dimensional manifold with n ≪ N that best describes thedata.

Original: For some data sets, this manifold hypothesis is appropriate: certain natural images are knownto be well-described by a manifold, for instance (Carlsson, 2009), enabling the use of specialisedautoencoders for visualisation (Moor et al., 2020).
Revised: For some data sets, this manifold hypothesis is appropriate: certain natural images are knownto be well-described by a manifold, for instance (Carlsson, 2009), enabling the use of specialisedautoencoders for visualisation (Moor et al., 2020).

Original: However, recent research shows evidence thatthe manifold hypothesis does not necessarily hold for complex data sets (Brown et al., 2022), andthat manifold learning techniques tend to fail for non-manifold data (Rieck & Leitte, 2015; Scoccola& Perea, 2022).
Revised: However, recent research shows evidence thatthe manifold hypothesis does not necessarily hold for complex data sets (Brown et al., 2022), andthat manifold learning techniques tend to fail for non-manifold data (Rieck & Leitte, 2015; Scoccola& Perea, 2022).

Original: These failures are typically the result of singularities , i.e. regions of a space thatviolate the properties of a manifold.
Revised: These failures are typically the result of singularities , i.e. regions of a space thatviolate the properties of a manifold.

Original: For example, the ‘pinched torus,’ an object obtained by compressing a neighbourhood of a random point in a torus to a single point, fails to satisfy the manifoldhypothesis at the ‘pinch point:’ this point, unlike all other points of the ‘pinched torus,’ does nothave a neighbourhood homeomorphic to R 2 (see Fig.
Revised: For example, the ‘pinched torus,’ an object obtained by compressing a neighbourhood of a random point in a torus to a single point, fails to satisfy the manifoldhypothesis at the ‘pinch point:’ this point, unlike all other points of the ‘pinched torus,’ does nothave a neighbourhood homeomorphic to R 2 (see Fig.

Original: 1 for an illustration).
Revised: 1 for an illustration).

Original: Since singularities—unlike outliers that arise from incorrect labels, for example—may carry relevantinformation (Jakubowski et al., 2020), we address the shortcomings of existing dimensionality reduction methods by assuming an agnostic view on any given data set.
Revised: Since singularities—unlike outliers that arise from incorrect labels, for example—may carry relevantinformation (Jakubowski et al., 2020), we address the shortcomings of existing dimensionality reduction methods by assuming an agnostic view on any given data set.

Original: Instead of trying to prescribethe rigid requirements of a manifold, we consider intrinsic dimensionality to be a fundamentallylocal phenomenon : we permit dimensionality to vary across points in the data set, and, more importantly, across the scale of locality to be considered.
Revised: Instead of trying to prescribethe rigid requirements of a manifold, we consider intrinsic dimensionality to be a fundamentallylocal phenomenon : we permit dimensionality to vary across points in the data set, and, more importantly, across the scale of locality to be considered.

Original: The only assumption we make is that thedata is of significantly lower dimension than the dimension of the ambient space.
Revised: The only assumption we make is that thedata is of significantly lower dimension than the dimension of the ambient space.

Original: This perspectiveenables us to assess the deviation of individual points from idealised non-singular spaces, resultingin a measure of the Euclidicity of a point.
Revised: This perspectiveenables us to assess the deviation of individual points from idealised non-singular spaces, resultingin a measure of the Euclidicity of a point.

Original: Our method is based on a local version of topological dataanalysis (TDA), a method from computational topology that is capable of quantifying the shape ofa data set on multiple scales (Edelsbrunner & Harer, 2010).
Revised: Our method is based on a local version of topological dataanalysis (TDA), a method from computational topology that is capable of quantifying the shape ofa data set on multiple scales (Edelsbrunner & Harer, 2010).

Original: [Figure]
Revised: [Figure]

Original: Our contributions.
Revised: Our contributions.

Revised: We present a universal framework for detecting singular regions in data.

Revised: Thisframework is agnostic with respect to geometric or stochastic properties of the underlying data andonly requires a notion of intrinsic dimension of neighbourhoods.

Original: We present persistent local homology (PLH) as a multi-parameter tool thatdetects the shape of local neighbourhoods of a given point in the data set, making use of multiplescales of locality.
Revised: Our approach is based on a novelformulation of persistent local homology (PLH), a multi-parameter tool that detects the shape oflocal neighbourhoods of a given point in the data set, making use of multiple scales of locality.

Original: The purpose of this framework is twofold: (i)
Revised: We employ PLH in two different capacities: (i)

Original: We use PLH to estimate the intrinsicdimension of a point locally.
Revised: We use PLH to estimate the intrinsic dimensionof a point locally.

Original: Thisperspective enables us to assess how complex a given data set is,both in terms of the magnitude of the intrinsic dimension and in terms of the variance of its intrinsicdimension across individual points.
Revised: This enables us to assess how complex a given data set is, both in terms of themagnitude of the intrinsic dimension and in terms of the variance of its intrinsic dimension acrossindividual points.

Original: (ii)
Revised: (ii)

Original: Given the intrinsic dimension of the neighbourhood of apoint, we use PLH to measure Euclidicity , i.e. the deviation of a point from beingEuclidean.
Revised: Given the intrinsic dimension of the neighbourhood of a point, we use PLH tomeasure Euclidicity , a novel quantity that we define to measure the deviation of a point from being

Original: Weachieve this by comparing PLH of the given data with appropriate samples from a Euclidean space. Our framework is the first to describe approximations to (persistent) local homology in a multiparameter setting, enabling us to assess the Euclidicity of points at multiple scales. We thus obtaina universal framework for detecting singular regions in data.
Revised: Euclidean.

Original: While our approach is agnostic withrespect to geometric or stochastic properties of the underlying data and only requires a notion ofintrinsic dimension of neighbourhoods, we provide theoretical guarantees on  approximation qualityfor certain classes of spaces. Finally, we show the utility of our proposed method experimentally onseveral data sets.
Revised: We also provide theoretical guarantees on the approximation quality for certain classes ofspaces and show the utility of our proposed method experimentally on several data sets.

Original:  B ACKGROUND : P ERSISTENT HOMOLOGY AND STRATIFIED SPACES
Revised:  B ACKGROUND : P ERSISTENT HOMOLOGY AND STRATIFIED SPACES

Original: We first provide an overview of persistent homology and stratified spaces, as well as their relation tolocal homology .
Revised: We first provide an overview of persistent homology and stratified spaces, as well as their relationto local homology .

Original: The former concept constitutes a generic framework for assessing complex data atmultiple scales, while the latter will subsequently serve as a theoretical justification for our proposedmethod. Homology will be understood to have Z / 2 Z coefficients, unless mentioned otherwise.
Revised: The former concept constitutes a generic framework for assessing complex dataat multiple scales by measuring its topological characteristics such as ‘holes’ and ‘voids,’ while thelatter will subsequently serve as a general setting to describe singularities, in which our frameworkadmits advantageous properties.

Original: 2. P ERSISTENT HOMOLOGY
Revised: Persistent homology.

Original: Persistent homology is a method to compute topological features at different scales of resolution,where more persistent features arise over a broader range of spatial scale parameters.
Revised: Persistent homology is a method for computing topological features at different scales, capturing an intrinsic notion of relevance in terms of spatial scale parameters.

Original: Given a finitemetric space ( X , d) , the Vietoris–Rips complex at step t is defined to be the abstract simplicialcomplex V ( X , t ) , in which an abstract k -simplex ( x 0 , . . .
Revised: Givena finite metric space ( X , d) , the Vietoris–Rips complex at step t is defined as the abstract simplicialcomplex V ( X , t ) , in which an abstract k -simplex ( x 0 , . . .

Original: , x k ) of points in X is spanned if and onlyif d( x i , x j ) ≤ t for all 0 ≤ i ≤ j ≤ k . 

Revised: , x k ) of points in X is spanned if and onlyif d( x i , x j ) ≤ t for all 0 ≤ i ≤ j ≤ k .

Original: For t 1 ≤ t 2 , the inclusions V ( X , t 1 ) (cid:44) → V ( X , t 2 ) yield 69 a filtration, i.e. a sequence of nested simplicial complexes, which we denote by
Revised: For t 1 ≤ t 2 , the inclusions V ( X , t 1 ) (cid:44) → V ( X , t 2 ) yield 66 a filtration, i.e. a sequence of nested simplicial complexes, which we denote by

Original: V ( X , • ) .
Revised: V ( X , • ) .

Original: Applying 70 the i th homology functor to this collection of spaces and inclusions between them induces mapson the homology level f t 1 ,t 2 i : H i ( V ( X , t 1 ))
Revised: Applying 67 the i th homology functor to this collection of spaces and inclusions between them induces mapson the homology level f t 1 ,t 2 i : H i ( V ( X , t 1 ))

Original: → H i ( V ( X , t 2 )) for any t 1 ≤ t 2 .
Revised: → H i ( V ( X , t 2 )) for any t 1 ≤ t 2 .

Original: The i th persistent 72 homology (PH) of X with respect to the Vietoris-Rips construction is defined to be the collection ofall these i th homology groups, together with the respective induced maps between them, and denotedby PH i ( X ; V ) .
Revised: The i th persistent 69 homology (PH) of X with respect to the Vietoris-Rips construction is defined to be the collectionof all these i th homology groups, together with the respective induced maps between them, anddenoted by PH i ( X ; V ) .

Original: PH can therefore be viewed as a tool that keeps track of topological features such as holes and voids on multiple scales.
Revised: PH can therefore be viewed as a tool that keeps track of topological features such as holes and voids on multiple scales.

Original: For a more comprehensive introduction to PH, in particular inthe context of machine learning, see Hensel et al.
Revised: For a more comprehensive introduction to PH inthe context of machine learning, see Hensel et al.

Original: The so-called ‘creation’ and ‘destruction’times of these features are summarised in a persistence diagram D ⊂ R × R ∪ {∞} , where  any78 point ( b, d ) ∈ D corresponds to a homology class that arises at filtration step b , and lasts untilfiltration step d .
Revised: The so-called ‘creation’ and ‘destruction’ times of these features are summarised in a persistence diagram D ⊂ R × R ∪ {∞} , where 75 any point ( b, d ) ∈ D corresponds to a homology class that arises at filtration step b , and lasts until filtration step d .

Original: The difference | d − b | is referred to as the lifetime or eponymous persistence ofthis homology class.
Revised: The difference | d − b | is referred to as the lifetime or eponymous persistenceof this homology class.

Original: There are several distance measures for comparing persistence diagrams, oneof them being the bottleneck distance d B .
Revised: There are several distance measures for comparing persistence diagrams,one of them being the bottleneck distance d B .

Original: For two persistence diagrams D , D ′ , it is defined asd 
Revised: For two persistence diagrams D , D ′ , it is defined asd B

Original: [Equation] ( D , 
Revised: ( D , D ′ ) : =

Original: [Equation]
Revised: [Equation]

Original: ) := inf γ sup x ∈D ∥ x − γ ( x ) ∥ ∞ , where γ ranges over all bijections between D and D ′ .
Revised: , where γ ranges over all bijections between D and D ′ .

Original: [Figure]
Revised: [Figure]

Original:  Manifolds are widely studied and particularly well-behavedtopological spaces: they locally resemble Euclidean spacenear any point.
Revised: Stratified spaces. Manifolds are widely studied and particularly well-behaved topological spaces: they locally resemble Euclidean space near any point.

Original: However, spaces that arise naturally often violate this local homogeneity condition, for example due tothe occurrence of singularities (see Fig.
Revised: However, spaces that arisenaturally often violate this local homogeneity condition, forexample due to the occurrence of singularities (see Fig.

Original: 2 for an example), orsince the space is of mixed dimensions.
Revised: 2 foran example), or since the space is of mixed dimensions.

Original: Stratified spaces generalise the concept of a manifold such that singular spaces arealso addressed.
Revised: Stratified spaces generalise the concept of a manifold such thatsingular spaces are also addressed.

Original: Large classes of singular spaces can be formulated as stratified spaces, including (i) complex algebraicvarieties, (ii) spaces that are disjoint unions of a finite number of manifolds of arbitrary dimensions, and (iii) spaces thatadmit isolated singularities.
Revised: Large classes of singular spaces can be formulated as stratified spaces, including(i) complex algebraic varieties, (ii) spaces that are disjointunions of a finite number of manifolds of arbitrary dimensions, and (iii) spaces that admit isolated singularities.

Original: Being thus intrinsically capableof describing a wider class of spaces, we argue that stratified spaces are the right tool to analysereal-world data.
Revised: Beingthus intrinsically capable of describing a wider class of spaces, we argue that stratified spaces arethe right tool to analyse real-world data.

Original: Subsequently, we define stratified spaces in the setting of simplicial complexes.
Revised: Subsequently, we define stratified spaces in the setting ofsimplicial complexes.

Original: Astratified simplicial complex 2 of dimension 0 is a finite set of points with the discrete topology.
Revised: A stratified simplicial complex 2 of dimension 0 is a finite set of points withthe discrete topology.

Original: Astratified simplicial complex of dimension n is an n -dimensional simplicial complex X , togetherwith a filtration of closed subcomplexes X =
Revised: A stratified simplicial complex of dimension n is an n -dimensional simplicialcomplex X , together with a filtration of closed subcomplexes X =

Original: X
Revised: X

Original: n ⊃ X n − 1 ⊃ X n − 2 ⊃ · · · ⊃  X − 1 = ∅ such [-
-]
Revised: n ⊃ X n − 1 ⊃ X n − 2 ⊃ · · · ⊃ 97 X − 1 = ∅ such [-
-]that X

Original: that X i \ X i − 1 is an i -dimensional manifold for all i , and such that every point x ∈ X possesses adistinguished local neighbourhood U ∼ =
Revised: i \ X i − 1 is an i -dimensional manifold for all i , and such that every point x ∈ Xpossesses a distinguished local neighbourhood U ∼ =

Original: R k × c ◦ L in X , where L is a compact stratified simplicialcomplex of dimension n − k − 1 and c ◦ refers to the open cone construction (see Appendix A.1).
Revised: R k × c ◦ L in X , where L is a compact stratifiedsimplicial complex of dimension n − k − 1 and c ◦ refers to the open cone construction (see Appendix A.1).

Original: If Xis a manifold, then independently of the point under consideration, L is given by a sphere since fora manifold, any point admits a local neighbourhood that is homeomorphic to R n .
Revised: If X is a manifold, then independently of the point under consideration, L is given bya sphere since for a manifold, any point admits a local neighbourhood that is homeomorphic to R n .

Original: This observationwill serve as the primary motivation for our Euclidicity measure in Section 4.3.
Revised: This observation will serve as the primary motivation for our Euclidicity measure in Section 4.2.

Original: 2. L OCAL HOMOLOGY
Revised: Local homology.

Original: Local homology serves as a tool to quantify topological properties of infinitesimal small neighbourhoods of a fixed point.
Revised: Local homology serves as a tool to quantify topological properties of infinitesimal small neighbourhoods of a fixed point.

Original: For a topological space X and x ∈ X , its i th local homology group is definedas H i ( X, X \ x ) :=
Revised: For a topological space X and x ∈ X , its i th localhomology group is defined as H i ( X, X \ x ) :=

Original: lim −→ U H i ( X, X \ U ) , where the direct system is given by the induced maps onhomology that arise via the inclusion of (small) neighbourhoods of x .
Revised: lim −→ U H i ( X, X \ U ) , where the direct system isgiven by the induced maps on homology that arise via the inclusion of (small) neighbourhoods ofx .

Original: When X is a simplicial complex, we may view x as a vertex in X , using subdivision if necessary.
Revised: When X is a simplicial complex, we may view x as a vertex in X , using subdivision if necessary.

Original: Its star St( x ) is defined to bethe union of simplices in X that have x as a face, whereas its link Lk( x ) consists of all simplicesin St( x ) that do not have x as a face.
Revised: Its star St( x ) is defined to be the union of simplices in X that have x as a face, whereas its link Lk( x ) consists of all simplices in St( x ) that do not have x as a face.

Original: Using excision and the long exact homology sequence (see Appendix A.2), we have
Revised: Using excision and the longexact homology sequence (see Appendix A.3), we have

Original: [Equation]
Revised: [Equation]

Original: The key takeaway here is that the homology of Lk( x ) will usually differ from the homology ofa sphere, once Lk( x ) is not homotopy-equivalent to a sphere.
Revised: The key takeaway here is that the homology of Lk( x ) will usually differ from the homology ofa sphere, once Lk( x ) is not homotopy-equivalent to a sphere.

Original: For example, when x is an isolatedsingularity in a stratified simplicial complex X of dimension n , then its distinguished neighbourhoodis given by U ∼ =
Revised: For example, when x is an isolatedsingularity in a stratified simplicial complex X of dimension n , then its distinguished neighbourhoodis given by U ∼ =

Original: c ◦ L .
Revised: c ◦ L .

Original: Thus, Lk( x ) =
Revised: Thus, Lk( x ) =

Original: L and H i ( X, X \ x ) =
Revised: L and H i ( X, X \ x ) =

Original: ˜H i − 1 ( L ) by Eq.
Revised: ˜H i − 1 ( L ) by Eq.

Original: (1), which is usuallydifferent from ˜H i − 1 ( S n − 1 ) , when x does not admit a Euclidean neighbourhood.
Revised: (1), which is usuallydifferent from ˜H i − 1 ( S n − 1 ) , when x does not admit a Euclidean neighbourhood.

Original: This observationmotivates and justifies using local homology for detecting non-Euclidean neighbourhoods.
Revised: This observationmotivates and justifies using local homology for detecting non-Euclidean neighbourhoods.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Methods from topological data analysis have recently attracted much attention in machine learning,particularly due to persistent homology, which captures global topological properties of the underlying data set on different scales.
Revised: Methods from topological data analysis have recently attracted much attention in machine learning,particularly due to persistent homology, which captures global topological properties of the underlying data set on different scales.

Original: We give a brief overview of existing methods in the emerging fieldof topology-driven singularity detection, outlining the differences to our approach below.
Revised: We give a brief overview of existing methods in the emerging fieldof topology-driven singularity detection, outlining the differences to our approach below.

Original: Severalworks already assume a local perspective on homology to detect information about the intrinsic dimensionality of the data or the presence of certain singularities.
Revised: Severalworks already assume a local perspective on homology to detect information about the intrinsic dimensionality of the data or the presence of certain singularities.

Original: Rieck et al.
Revised: Rieck et al.

Original: (2020) define persistentintersection homology via known stratifications, whereas Fasy & Wang (2016) and Bendich (2008),for instance, both present persistent versions of local homology.
Revised: (2020) define persistentintersection homology via known stratifications, whereas Fasy & Wang (2016) and Bendich (2008),for instance, both present persistent versions of local homology.

Original: By contrast, Stolz et al.
Revised: By contrast, Stolz et al.

Original: (2020)follow a different approach, where local homology is approximatedas the absolute homology ofa small annulus of a given neighbourhood, resulting in an algorithm for geometric anomaly detection (which requires knowing the intrinsic dimension of the data set).
Revised: (2020)follow a different approach, where local homology is approximated the absolute homology of a small annulus of a given neighbourhood, resulting in an algorithm for geometric anomaly detection (which requires knowing the intrinsic dimension of the data set).

Original: Bendich et al.
Revised: Bendich et al.

Original: (2007) employpersistence vineyards, i.e. continuous families of persistence diagrams, to assess the local homologyof a point in a stratified space, whereas Dey et al.
Revised: (2007) employpersistence vineyards, i.e. continuous families of persistence diagrams, to assess the local homologyof a point in a stratified space, whereas Dey et al.

Original: (2014) use local homology to estimate the (global)intrinsic dimension of hidden, possibly noisy manifolds.
Revised: (2014) use local homology to estimate the (global)intrinsic dimension of hidden, possibly noisy manifolds.

Original: While manifold learning is concerned withthe development of algorithms that extract geometric information under the assumption that thegiven data lie on a manifold, Brown et al.
Revised: While manifold learning is concerned withthe development of algorithms that extract geometric information under the assumption that thegiven data lie on a manifold, Brown et al.

Original: (2022) recently introduced the idea to assume data spacesto consist of a union of manifolds .
Revised: (2022) recently introduced the idea to assume data spacesto consist of a union of manifolds .

Original: Intrinsic dimension is thus allowed to vary across connectedcomponents of the data space, but singularities are excluded under this assumption, whereas ourframework detects the correct intrinsic dimension for large classes of singular spaces.
Revised: Intrinsic dimension is thus allowed to vary across connectedcomponents of the data space, but singularities are excluded under this assumption, whereas ourframework detects the correct intrinsic dimension for large classes of singular spaces.

Original: Birdal et al.(2021) define a global persistent homology dimension for describing neural networks; our persistentintrinsic dimension, by contrast, is local and may thus change across different points in the data set.
Revised: Birdal et al.(2021) define a global persistent homology dimension for describing neural networks; our persistentintrinsic dimension, by contrast, is local and may thus change across different points in the data set.

Original: Key differences to existing approaches.
Revised: Key differences to existing approaches.

Original: Our approach crucially differs from existing approachesin essential components.
Revised: Our approach crucially differs from existing approachesin essential components.

Original: In comparison to all aforementioned contributions, we capture additionallocal geometric information: we consider multiple scales of locality in a persistent framework forlocal homology.
Revised: In comparison to all aforementioned contributions, we capture additionallocal geometric information: we consider multiple scales of locality in a persistent framework forlocal homology.

Original: Concerning the overall construction, Stolz et al.
Revised: Concerning the overall construction, Stolz et al.

Original: (2020) is the closest to our method.
Revised: (2020) is the closest to our method.

Original: However, the authors assume that the intrinsic dimension is known and the proposed algorithm usesa fixed scale, whereas our approach (i) operates in a multi-scale setting, (ii) provides local estimatesof intrinsic dimensionality of the data space, and (iii) incorporates model spaces that serve as a comparison.
Revised: However, the authors assume that the intrinsic dimension is known and the proposed algorithm usesa fixed scale, whereas our approach (i) operates in a multi-scale setting, (ii) provides local estimatesof intrinsic dimensionality of the data space, and (iii) incorporates model spaces that serve as a comparison.

Original: We can thus measure the deviation from an idealised manifold, requiring fewer assumptionson the structure of the input data (Section 5.3 demonstrates the benefits of this perspective).
Revised: We can thus measure the deviation from an idealised manifold, requiring fewer assumptionson the structure of the input data (Section 5.4 demonstrates the benefits of this perspective).

Original:  M ETHODS
Revised:  M ETHODS

Original: Our framework TOAST (Topological Algorithm for Singularity Tracking) consists of two parts:(i) a method to calculate a local intrinsic dimension of the data, and (ii) Euclidicity , a measure forassessing the multi-scale deviation from a Euclidean space.
Revised: Our framework TOAST (Topological Algorithm for Singularity Tracking) consists of two parts:(i) a method to calculate a local intrinsic dimension of the data, and (ii) Euclidicity , a measure forassessing the multi-scale deviation from a Euclidean space.

Original: TOAST is based on the assumption thatthe intrinsic dimension of some given data is not necessarily constant across the data set, and isbest described by local measurements , i.e. measurements in a small neighbourhood of a given point.
Revised: TOAST is based on the assumption thatthe intrinsic dimension of some given data is not necessarily constant across the data set, and isbest described by local measurements , i.e. measurements in a small neighbourhood of a given point.

Original: However, since there is no canonical choice for the magnitude of such a neighbourhood, TOASTis fundamentally built on a multi-scale analysis of data. 
Revised: Since there is no canonical choice for the magnitude of such a neighbourhood, TOAST is built on amulti-scale analysis of data. Our main idea involves constructing a collection of local (punctured)neighbourhoods for varying locality scales, and subsequently recording their topological features. This procedure allows us to approximate local topological features (specifically, local homology)of a given point, which we use to measure the intrinsic dimensionality of a space.

Revised: Moreover, bycalculating the distance to Euclidean model spaces, we are capable of detecting singularities in alarge range of input data sets.

Original: Subsequently, we will briefly describe the‘moving parts’ of TOAST; please refer to Appendix A.1 for a terminology list.
Revised: Subsequently, we will briefly describe the ‘moving parts’ of TOAST;please refer to Appendix A.1 for a terminology list.

Original: 4. P ERSISTENT LOCAL HOMOLOGY
Revised: 4. P ERSISTENT INTRINSIC DIMENSION

Original: For a finite metric space ( X , d) and x ∈ X , let B sr ( x ) :
Revised: For a finite metric space ( X , d) and x ∈ X , let B sr ( x ) :

Original: = { y ∈ X | r ≤ d( x, y ) ≤ s } denotethe intrinsic annulus of x in X with respect to the parameters r and s .
Revised: = { y ∈ X | r ≤ d( x, y ) ≤ s } denotethe intrinsic annulus of x in X with respect to the parameters r and s .

Original: Moreover, let F denote aprocedure that takes as input a finite metric space and outputs an ascending filtration of topologicalspaces—such as a Vietoris–Rips filtration.
Revised: Moreover, let F denote aprocedure that takes as input a finite metric space and outputs an ascending filtration of topologicalspaces—such as a Vietoris–Rips filtration.

Original: By applying F to the intrinsic annulus of x , we obtaina tri-filtration ( F ( B sr ( x ) , t )) r,s,t , where t corresponds to the respective filtration step that is determined by F .
Revised: By applying F to the intrinsic annulus of x , we obtaina tri-filtration ( F ( B sr ( x ) , t )) r,s,t , where t corresponds to the respective filtration step that is determined by F .

Original: Note that this tri-filtration is covariant in s and t , but contravariant in r ; we denote it by
Revised: Note that this tri-filtration is covariant in s and t , but contravariant in r ; we denote it by

Original: F ( B •• ( x ) , • ) .
Revised: F ( B •• ( x ) , • ) .

Original: Applying i th homology to this filtration yields a tri-parameter persistent module thatwe call
Revised: Applying i th homology to this filtration yields a tri-parameter persistent module thatwe call

Original: i th persistent local homology (PLH) of x , denoted by PLH i ( x ; F ) :=
Revised: i th persistent local homology (PLH) of x , denoted by PLH i ( x ; F ) :=

Original: PH i ( F ( B •• ( x ) , • )) .
Revised: PH i ( F ( B •• ( x ) , • )) .

Original: To the best of our knowledge, this is the first time that PLH is considered as a multi-parameter persistence module.
Revised: To the best of our knowledge, this is the first time that PLH is considered as a multi-parameter persistence module.

Original: Since the Vietoris–Rips filtration is the pre-eminent filtration in TDA, we will alsouse the abbreviated notation PLH i ( x ) :=
Revised: Since the Vietoris–Rips filtration is the pre-eminent filtration in TDA, we will alsouse the abbreviated notation PLH i ( x ) :=

Original: PLH i ( x ; V ) .
Revised: PLH i ( x ; V ) .

Original: [Equation] Our PLH formulation enjoys stability properties similar to the seminal stability theorem in persistenthomology (Cohen-Steiner et al., 2007), making it robust to small parameter changes (we assessempirical stability in Section 5.1).
Revised: Our PLH formulation enjoys stability properties similar to the seminal stability theorem in persistenthomology (Cohen-Steiner et al., 2007), making it robust to small parameter changes (we assessempirical stability in Section 5.1).

Original: Theorem 1.
Revised: Theorem 1.

Original: Given a finite metric space X and x ∈ X , let B s r ( x ) and B s ′ r ′
Revised: Given a finite metric space X and x ∈ X , let B sr ( x ) and B s ′ r ′

Original: ( x ) denote twointrinsic annuli with | r − r ′
Revised: ( x ) denote twointrinsic annuli with | r − r ′

Original: | ≤ ϵ 1 and | s − s ′ | ≤ ϵ 2 .
Revised: | ≤ ϵ 1 and | s − s ′ | ≤ ϵ 2 .

Original: Furthermore, let D , D ′ denote thepersistence diagrams corresponding to PH i ( B sr ( x ); V ) and PH i ( B s ′ r ′ ( x ); V ) , respectively. Then12 d B ( D , D ′ ) ≤ max { ϵ 1 , ϵ 2 } .
Revised: Furthermore, let D , D ′ denote the

Original: Proof.
Revised: [Figure]

Original: The Hausdorff distance of two non-empty subsets A, B ⊂ X is d H (A, B  ) : = inf { ϵ ≥0 | A ⊂ B ϵ , B ⊂ A ϵ } , where A ϵ = ∪ a ∈ A { x ∈ X ; d( x, a ) ≤ ϵ } denotes the ϵ -thickening of A in X .
Revised: persistence diagrams corresponding to PH i ( B sr ( x ); V ) and PH i ( B s ′ r ′ ( x ); V ) , respectively.

Original: Set

Original: ϵ := max { ϵ 1 , ϵ 2 } . By assumption, B sr ( x ) ⊂ B s ′ r ′ ( x ) ϵ and B s ′ r ′ ( x ) ⊂ B sr ( x ) ϵ , i.e.d H ( B sr ( x ) , B s ′ r ′
Revised: Then

Original: ( x )) ≤ ϵ . Using the geometric stability theorem of persistence diagrams (Chazalet al., 2014), we have 12 d B ( D , D ′ ) ≤ d H ( B sr ( x ) , B s ′ r ′ ( x )) , which proves the claim.
Revised: [Equation]

Original: 4. P ERSISTENT INTRINSIC DIMENSION

Original: Let X ⊂ R N be a finite set ofdata points  and x ∈ X . We define the persistent intrinsic dimension(PID) of x at scale ϵ as i x ( ϵ ) :
Revised: For a finite set of points X ⊂ R N and x ∈ X , we define the persistent intrinsic dimension (PID)of x at scale ϵ as i x ( ϵ ) :

Original: = max { i ∈ N | PH i − 1 ( B sr ( x ))
Revised: = max { i ∈ N | PH i − 1 ( B sr ( x ))

Original: ̸
Revised: ̸

Original: = 0 for some r and s with s < ϵ } .
Revised: = 0 for some r and s with s < ϵ } .

Original: This measure serves as a multi-scale characterisation of the intrinsic dimension of a data set.
Revised: Thismeasure serves as a multi-scale characterisation of the intrinsic dimension of a data set.

Original: In caseour data set constitutes a manifold sample, it turns out that we can recover the correct dimension.
Revised: In case ourdata set constitutes a manifold sample, it turns out that we can recover the correct dimension.

Original: Theorem 2.
Revised: Theorem 2.

Original: Let M ⊂ R N be an n -dimensional compact smooth manifold and let X :={ x 1 , . . .
Revised: Let M ⊂ R N be an n -dimensional compact smooth manifold and let X :={ x 1 , . . .

Original: , x S } be a collection of uniform samples from M .
Revised: , x S } be a collection of uniform samples from M .

Original: For a sufficiently large S , there existconstants ϵ 1 , ϵ 2 > 0 such that i x ( ϵ ) = n for all
Revised: For a sufficiently large S , there existconstants ϵ 1 , ϵ 2 > 0 such that i x ( ϵ ) = n for all

Original: ϵ 1 <
Revised: ϵ 1 <

Original: ϵ < ϵ 2 and any point x ∈ X .
Revised: ϵ < ϵ 2 and any point x ∈ X .

Original: Moreover, ϵ 1 canbe chosen arbitrarily small by increasing S .
Revised: Moreover, ϵ 1 canbe chosen arbitrarily small by increasing S .

Original: Proof.

Original: Let x ∈ X be an arbitrary point. Since M is a manifold, x admits a Euclidean neighbourhood U . Since M is smooth, we can assume U to be arbitrarily close to being flat by shrinking it. Thus, we can find ϵ2 > 0 with B sr ( x ) ⊂ U for all r, s <
Revised: The implication of Theorem 2 is that i x ( ϵ ) computes the correct intrinsic dimension of M in a certainrange of values ϵ > 0 , provided the sample is sufficiently large.

Original: ϵ 2 such that H i( V ( B sr ( x) , t )) = 0 for alli ≥ n , and all t . Hence, PH i ( B sr ( [-x )) = 
 for all i ≥ n , and therefore-] i x ( ϵ2 ) ≤ n .
Revised: In particular, i x ( [-x )) = 
 for all i ≥ n , and therefore-]ϵ ) persists in thisrange, which suggests to consider a collection of i x ( ϵ ) for varying ϵ to analyse the intrinsic dimension of x .

Original: By contrast, for

Original: S sufficiently large, and r, s as before, there exists a parameter t such that V ( B sr ( x ) , t ) is homotopyequivalent to an ( n − 1) -sphere, and so H n − 1 ( V ( B sr ( x ) , t )) admits a generator, i.e. it is non-trivial.

Original: Consequently, PH n − 1 ( B sr ( x )) ̸ = 0 , and i x ( ϵ 2 ) = n .

Original: By further increasing S , we can ensure thatthe statement still holds when we decrease ϵ 2 , which proves the two remaining claims.

Original: The implication of Theorem 2 is that, given a sufficiently large sample from a smooth manifold

Original: M , we know that i x ( ϵ ) computes the correct intrinsic dimension of M (in a certain range of valuesϵ > 0 ).

Original: In particular, i x ( ϵ ) persists in this range, and so Theorem 2 suggests to consider the wholecollection of i x ( ϵ ) for varying ϵ in order to analyse the intrinsic dimension of x .

Original: We also havethe following corollary, which specifically addresses stratified spaces (such as the ‘pinched torus’),implying that our method can correctly detect the intrinsic dimension of individual strata.
Revised: We also have the following corollary, which specifically addresses stratified spaces (suchas the ‘pinched torus’), implying that our method can correctly detect the intrinsic dimension ofindividual strata.

Original: PID isthus capable of handling large classes of ‘non-manifold’ data sets.
Revised: PID is thus capable of handling large classes of ‘non-manifold’ data sets.

Original: Corollary 1.
Revised: Corollary 1.

Original: Let X =
Revised: Let X =

Original: X
Revised: X

Original: [Equation] ⊃ X [Equation] 2 ⊃ · · · ⊃ X − 1 = ∅ be an n -dimensional 211 compact stratified simplicial complex, s.t.
Revised: n ⊃ X n − 1 ⊃ X n − 2 ⊃ · · · ⊃ X − 1 = ∅ be an n -dimensional 196 compact stratified simplicial complex, s.t.

Original: X
Revised: X

Original: i \ X i − 1 is smooth for every i .
Revised: i \ X i − 1 is smooth for every i .

Original: For a fixed i , let  X i := { x 1 , . . .
Revised: For a fixed i , let 197 X i := { x 1 , . . .

Original: , x S } be a collection of uniform samples from X
Revised: , x S } be a collection of uniform samples from X

Original: i \ X
Revised: i \ X

Original: i − 1 .
Revised: i − 1 .

Original: For a sufficiently large
Revised: For a sufficiently large

Original: S , there are constants ϵ 1 , ϵ 2 > 0 such that i x ( ϵ )
Revised: S , there are constants ϵ 1 , ϵ 2 > 0 such that i x ( ϵ )

Original: =
Revised: =

Original: i for all ϵ 1 < ϵ <
Revised: i for all ϵ 1 < ϵ <

Original: ϵ 2 and any point x ∈ X i .
Revised: ϵ 2 and any point x ∈ X i .

Original: Moreover, ϵ 1 can be chosen arbitrarily small by increasing S .
Revised: Moreover, ϵ 1 can be chosen arbitrarily small by increasing S .

Original: 4. E UCLIDICITY
Revised: 4. E UCLIDICITY

Original: Knowledge about the intrinsic dimension of a neighbourhood is crucial for measuring to what extentsuch a neighbourhood deviates from being Euclidean.
Revised: Knowledge about the intrinsic dimension of a neighbourhood is crucial for measuring to what extentsuch a neighbourhood deviates from being Euclidean.

Original: We refer to this property as Euclidicity , withthe understanding that low values indicate Euclidean neighbourhoods while high values indicatesingular regions of a data set.
Revised: We refer to this deviation as Euclidicity , withthe understanding that low values indicate Euclidean neighbourhoods while high values indicatesingular regions of a data set.

Original: Euclidicity can be calculated without stringent assumptions on manifoldness: let X ⊂ R N be a finite data set, x ∈ X a point, and assume that we are given an estimate nof the intrinsic dimension of x .
Revised: Euclidicity can be calculated without stringent assumptions on manifoldness: let X ⊂ R N be a finite data set, x ∈ X a point, and assume that we are given an estimate nof the intrinsic dimension of x .

Original: In particular, the previously-described PID estimation procedure isapplicable in this setting and may be used to obtain n , for example by calculating statistics on theset of i x ( ϵ ) for varying locality parameters ϵ .
Revised: In particular, the previously-described PID estimation procedure isapplicable in this setting and may be used to obtain n , for example by calculating statistics on theset of i x ( ϵ ) for varying locality parameters ϵ .

Original: Euclidicity, however, can also make use of other dimensionality estimation procedures (see Camastra & Staiano (2016) for a survey).
Revised: Euclidicity, however, can also make use of other dimensionality estimation procedures (see Camastra & Staiano (2016) for a survey).

Original: To assess howfar a given neighbourhood of x is from being Euclidean, we compare it to a Euclidean model spaceby measuring the deviation of their corresponding persistent local homology features.
Revised: To assess howfar a given neighbourhood of x is from being Euclidean, we compare it to a Euclidean model spaceby measuring the deviation of their corresponding persistent local homology features.

Original: We start bydefining the Euclidean annulus E B s r ( x ) of x for parameters r and s to be a set of random uniformsamples of { y ∈ R n | r ≤ d( x, y ) ≤ s } such that | E B sr ( x ) | = | B sr ( x ) | .
Revised: We start bydefining the Euclidean annulus E B sr ( x ) of x for parameters r and s to be a set of random uniformsamples of { y ∈ R n | r ≤ d( x, y ) ≤ s } such that | E B sr ( x ) | = | B s r ( x ) | .

Original: Here, r and s correspond 229 to the inner and outer radius of the Euclidean annulus, respectively.
Revised: Here, r and s correspond 214 to the inner and outer radius of the Euclidean annulus, respectively.

Original: For r ′ ≤ r and s ≤ s ′ we extend 230 E
Revised: For r ′ ≤ r and s ≤ s ′ we extend 215 E

Original: B sr ( x ) by sampling additional points to obtain E B s ′ r ′
Revised: B sr ( x ) by sampling additional points to obtain E B s ′ r ′

Original: ( x ) with | E B s ′ r ′
Revised: ( x ) with | E B s ′ r ′

Original: ( x ) | =
Revised: ( x ) | =

Original: | B s ′ r ′
Revised: | B s ′ r ′

Original: ( x ) | .
Revised: ( x ) | .

Original: Iterating thisprocedure leads to a tri-filtration ( F ( E B sr ( x ) , t )) r,s,t for any filtration F (following our descriptionin Section 4.1).
Revised: Iterating thisprocedure leads to a tri-filtration ( F ( E B sr ( x ) , t )) r,s,t for any filtration F , following our descriptionin Section 4.1.

Original: We now define the persistent local homology of a Euclidean model space as
Revised: We now define the persistent local homology of a Euclidean model space as

Original: [Equation]
Revised: [Equation]

Original:  Again, for a Vietoris–Rips filtration V , we use a short-form notation, i.e. PLH E i ( x ) :=
Revised: [Equation] Again, for a Vietoris–Rips filtration V , we use a short-form notation, i.e. PLH E i ( x ) :=

Original: PLH E i ( x ; V ) .
Revised: PLH E i ( x ; V ) .

Original: Notice that PLH E i ( x ) implicitly depends on the choice of intrinsic dimension n , and on the samplesthat are generated randomly.
Revised: Notice that PLH E i ( x ) implicitly depends on the choice of intrinsic dimension n , and on the samplesthat are generated randomly.

Original: To remove the dependency on the samples, we consider PLH E i ( x )to be a sample of a random variable PLH E i ( x ) .
Revised: To remove the dependency on the samples, we consider PLH E i ( x )to be a sample of a random variable PLH E i ( x ) .

Original: Let D( · , · ) be a distance measure for 3-parameterpersistence modules, such as the interleaving distance .
Revised: Let D( · , · ) be a distance measure for 3-parameterpersistence modules, such as the interleaving distance .

Original: We then define the Euclidicity of x , denotedby E ( x ) , as the expected value of these distances, i.e.
Revised: We then define the Euclidicity of x , denotedby E ( x ) , as the expected value of these distances, i.e.

Original: [Equation]
Revised: [Equation]

Original: This quantity essentially assesses how far x is from admitting a regular Euclidean neighbourhood.
Revised: This quantity essentially assesses how far x is from admitting a regular Euclidean neighbourhood.

Original: 4. I MPLEMENTATION
Revised: Implementation.

Original: The implementation of Section 4.3 in practice requires different choices, namely (i) a range of locality scales, (ii) a filtration, and (iii) a distance metric between filtrations D .We will subsequently (see
Revised: Calculating E ( x ) requires different choices, namely (i) a range of locality scales,(ii) a filtration, and (iii) a distance metric between filtrations D .

Original: Section 5) discuss how to pick these parameters in practice and make one specific instantiation of ourframework publicly available.

Original: 4 Using a grid Γ of possible radii ( r, s ) with r < s , we approximate Eq.
Revised: Using a grid Γ of possible radii ( r, s )with r < s , we approximate Eq.

Original: (3) using the mean of the bottleneck distances of fibred Vietoris–Rips barcodes , i.e.
Revised: (3) using the mean of the bottleneck distances of fibred Vietoris–Ripsbarcodes , i.e.

Original: [Equation]
Revised: [Equation]

Original: where C is equal to the number of summands and PLH E i ( x ) refers to a sample from a Euclideanannulus of the same size as the intrinsic annulus around x .
Revised: where C is equal to the number of summands and PLH E i ( x ) refers to a sample from a Euclideanannulus of the same size as the intrinsic annulus around x .

Original: Eq.
Revised: Eq.

Original: (4) can be implemented usingeffective persistent homology calculation methods (Bauer, 2021), thus permitting an integration intoexisting TDA and machine learning frameworks (The GUDHI Project, 2015; Tauzin et al., 2020).
Revised: (4) can be implemented usingeffective persistent homology calculation methods (Bauer, 2021), thus permitting an integration intoexisting TDA and machine learning frameworks (The GUDHI Project, 2015; Tauzin et al., 2020).

Original: Amore detailed description of pseudocode implementations is provided in Appendix A.3.
Revised: Appendix A.4 provides pseudocode implementations, while Section 5 discusses how to pick theseparameters in practice. We make one specific instantiation of our framework publicly available.

Original: 4. P ROPERTIES
Revised: Properties.

Original: The main appeal of our formulation is that calculating both PID and Euclidicity does not requirestrong assumptions about the input data.
Revised: The main appeal of our formulation is that calculating both PID and Euclidicity doesnot require strong assumptions about the input data.

Original: Treating dimension as a local quantity that is allowed tovary across multiple scales leads to beneficial expressivity properties.
Revised: Treating dimension as a local quantity that isallowed to vary across multiple scales leads to beneficial expressivity properties.

Original: As we showed in Section 4.2,our method is guaranteed to yield the right values for manifolds and stratified simplicial complexes.
Revised: As we showedin Section 4.1, our method is guaranteed to yield the right values for manifolds and stratified simplicial complexes.

Original: This property substantially increases the practical applicability and expressivity, enabling our framework to handle unions of manifolds of varying dimensions, for instance.
Revised: This property substantially increases the practical applicability and expressivity,enabling our framework to handle unions of manifolds of varying dimensions, for instance.

Original: We require only a basicassumption, namely that the intrinsic dimension n of the given data space is significantly lower thanthe ambient dimension N , making Euclidicity broadly applicable.
Revised: Werequire only a basic assumption, namely that the intrinsic dimension n of the given data space issignificantly lower than the ambient dimension N , making Euclidicity broadly applicable.

Original: Similar to curvature, Euclidicitymakes use of the fact that one can compare data to ‘model spaces,’ allowing for different futureadjustments.
Revised: Similarto curvature, Euclidicity makes use of the fact that one can compare data to ‘model spaces,’ allowingfor different future adjustments.

Original: One limitation of Euclidicity is its apparent computational complexity: the Vietoris–Rips complexis known to grow exponentially with increasing dimensionality.
Revised: Limitations. Our implementation of Euclidicity makes use of the Vietoris–Rips complex, which isknown to grow exponentially with increasing dimensionality.

Original: While all calculations of Eq.
Revised: While all calculations of Eq.

Original: (3) canbe performed in parallel —thus substantially improving scalability vis-`a-vis persistent homology onthe complete input data set, both in terms of dimensions and in terms of data points—the memoryrequirements for a full Vietoris–Rips complex construction might still prevent our method to beapplicable for certain high-dimensional data sets.
Revised: (3) can beperformed in parallel —thus substantially improving scalability vis-`a-vis persistent homology on thecomplete input data set, both in terms of dimensions and in terms of samples—the memory requirements for a full Vietoris–Rips complex construction may still prevent our method to be applicablefor certain high-dimensional data sets.

Original: This can be mitigated by selecting a differentfiltration (Anai et al., 2020; Sheehy, 2013);for now, our proofs do not assume a specific filtration,and we leave the treatment of filtration-specific theoretical properties for future work.
Revised: This can be mitigated by selecting a different filtration (Anaiet al., 2020; Sheehy, 2013); our proofs do not assume a specific filtration, and we leave the treatmentof filtration-specific theoretical properties for future work.

Original: Finally, weremark that the reliability of the Euclidicity score depends on the validity of the intrinsic dimension;otherwise, the comparison does not take place with respect to the appropriate model space.
Revised: Finally, we remark that the reliability ofthe Euclidicity score depends on the validity of the intrinsic dimension; otherwise, the comparisondoes not take place with respect to the appropriate model space.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We demonstrate the expressivity of our proposed TOAST procedure in different settings, empirically showing that it (i) calculates the correct intrinsic dimension, and (ii) detects singularities whenanalysing data sets with known singular points.
Revised: We demonstrate the expressivity of our proposed TOAST procedure in different settings, empirically showing that it (i) calculates the correct intrinsic dimension, and (ii) detects singularities whenanalysing data sets with known singular points.

Original: We also conduct a comparison with one-parameterapproaches, showcasing how our multi-scale approach results in more stable outcomes.
Revised: We also conduct a comparison with one-parameterapproaches, showcasing how our multi-scale approach results in more stable outcomes.

Original: Finally, weanalyse Euclidicity scores of benchmark datasets, giving evidence that our technique can be used asa measure for the geometric complexity of data.
Revised: Finally, weanalyse Euclidicity scores of benchmark datasets, giving evidence that our technique can be used asa measure for the geometric complexity of data.

Original: 5. P ARAMETER SELECTIONAND STABILITY
Revised: 5. P ARAMETER SELECTION

Original: Since Eq.
Revised: Since Eq.

Original: (3) intrinsically incorporates multiple scales of locality, we need to specify an upper boundfor the radii ( r min , r max , s min , s max ) that define the respective annuli in practice.
Revised: (3) intrinsically incorporates multiple scales of locality, we need to specify an upper boundfor the radii ( r min , r max , s min , s max ) that define the respective annuli in practice.

Original: Given a point x ,we found the following procedure to be useful in practice: we set s max , i.e. the maximum of theouter radius, to the distance to the k th nearest neighbour of a point, and r min ,i.e the minimum innerradius, to the smallest non-zero distance to a neighbour of x .
Revised: Given a point x ,we found the following procedure to be useful in practice: we set s max , i [Table] e. the maximum of theouter radius, to the distance to the k th nearest neighbour of a point, and r min ,

Original:  Finally, we set the minimum outerradius s min and the maximum inner radius r max to the distance to the ⌊ k 3 ⌋ th nearest neighbour.
Revised: i [Table] e the minimum innerradius, to the smallest non-zero distance to a neighbour of x . Finally, we set the minimum outerradius s min and the maximum inner radius r max to the distance to the ⌊ k 3 ⌋ th nearest neighbour.

Original: While we find k = 50 to yield sufficient results, spaces with a high intrinsic dimension may requirelarger values.
Revised: While we find k = 50 to yield sufficient results, spaces with a high intrinsic dimension may requirelarger values.

Original: The advantage of using such a parameter selection procedure is that it works in adata-driven manner, accounting for differences in density.
Revised: The advantage of using such a parameter selection procedure is that it works in adata-driven manner, accounting for differences in density.

Original: Since our approach is inherently local ,we need to find a balance between sample sizes that are sufficiently large to contain topologicalinformation, while at the same time being sufficiently small to retain a local perspective.
Revised: Since our approach is inherently local ,we need to find a balance between sample sizes that are sufficiently large to contain topologicalinformation, while at the same time being sufficiently small to retain a local perspective.

Original: We foundthe given range to be an appropriate choice in practice.
Revised: We foundthe given range to be an appropriate choice in practice.

Original: As for the number of steps, we discretisethe parameter range using 20 steps by default.
Revised: As for the number of steps, we discretisethe parameter range using 20 steps by default.

Original: Higher numbers are advisable when there are largediscrepancies between the radii, for instance when s max ≫ r max . 
Revised: Higher numbers are advisable when there are largediscrepancies between the radii, for instance when s max ≫ r max . [Table]

Original: Persistent intrinsic dimension (PID) is stable.
Revised: We first analyse the behaviour of persistent intrinsic dimension (PID) on samples from a space obtained by concatenating S 1 (a circle) and S 2 (a sphere) at a gluing point. Table 1 shows a comparison of PID with state-of-the-artdimensionality estimators. 6 We find that PID outperformsall estimators in terms of mean and standard deviation forthe 2D points, thus correctly indicating that the majorityof all points admit non-singular 2D neighbourhoods. Forthe 1D points, the mean of the dimensionality estimate of

Revised: PID is still close to the ground truth, while its standard deviation and maximum correctly capture the fact that some1D points are situated closer to the gluing point.

Revised: This behaviour is in line with our philosophy of considering dimensionality as an inherently local phenomenon.

Revised: In casesuch behaviour is not desirable for a specific data set, Euclidicity calculations support any dimensionality estimator; since such estimators do not come withstrong guarantees such as Theorem 2, their choice must be ultimately driven by the data set at hand.

Revised: See Appendix A.6 for a more detailed analysis of these estimates.

Original:  In practice, the sample density is not necessarilysufficiently high for Theorem 2 to be applicable.
Revised: Stability. In practice, the sample density may not be sufficiently high for Theorem 2 to apply.

Original: This means that there may appear artefact homological features in dimensions higher than the intrinsic dimension of a given space.
Revised: Thismeans that there may appear artefact homological features in dimensions higher than the intrinsicdimension of a given space.

Original: Followingthe ideas of persistent homology, we thus only consider features that exceed a certain persistencethreshold in comparison to the persistence of features of lower dimension: for any data point x andthe respective intrinsic annulus B sr ( x ) , we eliminate all topological features whose lifetimes aresmaller than the maximum lifetime of features in one dimension below.This results in markedlystable estimates of intrinsic dimension, which are less prone to overestimations (see Appendix A.5for a comparison with other state-of-the-art local dimension estimators). Euclidicity is stable. As predicted by Theorem 1, Euclidicity estimates are stable in practice.
Revised: We thus only consider features that exceed a certain persistence threshold in comparison to the persistence of features of lower dimension: for any data point x and therespective intrinsic annulus B sr ( x ) , we eliminate all topological features whose lifetimes are smallerthan the maximum lifetime of features in one dimension below.

Original: Wefirst note that Euclidicity is robust towards sampling : repeating the calculations for the ‘pinchedtorus’ over different batches results in highly similar distributions that are not distinguishable according to Tukey’s range test (Tukey, 1949) at the α = 0 . 05 confidence level. Moreover, choosinglarger locality scales still enables us to detect singularities at higher computational costs and incorporating larger parts of the point cloud. Please refer to Appendix A.4 for a more detailed discussionof this aspect.
Revised: This results in markedly stableestimates of intrinsic dimension, which are less prone to overestimations.

Original: 5. E UCLIDICITY CAPTURES SINGULARITIES
Revised: 5. E UCLIDICITY CAPTURES SINGULARITIES

Original: [Figure]
Revised: [Figure]

Original: As shown in Fig.
Revised: Fig.

Original: 1, Euclidicity is capable of detecting the singularity of the ‘pinched torus.’
Revised: 1 shows that Euclidicity is capable of detecting the singularity of the ‘pinched torus.’

Original: Of particular relevance is the fact that Euclidicity also highlights that points in the vicinity of the singularpoint are not fully regular.
Revised: Of particular relevance is the fact that Euclidicity also highlights that points in the vicinity of the singularpoint are not fully regular.

Original: This is an important property for practical applications since it impliesthat Euclidicity can detect such isolated singularities even in the presence of sampling errors.
Revised: This is an important property for practical applications since it impliesthat Euclidicity can detect such isolated singularities even in the presence of sampling errors.

Original: Besides the pinched torus, another prototypical example of singular spaces is given bythe wedgeof two n -dimensional spheres, denoted by S n ∨ S n .
Revised: Besides the pinched torus, another prototypical example of singular spaces is given by S n ∨ S n , thewedge of two n -dimensional spheres.

Original: Intuitively, S n ∨ S n is obtained by two ndimensional spheres that are glued together at a certain point.
Revised: Intuitively, S n ∨ S n is obtained by two n -dimensional spheresthat are glued together at a certain point.

Original: Denoting the gluing point by x 0 , fora suitable triangulation of X = S n ∨ S n , this space is naturally stratified by X ⊃ { x 0 } .
Revised: Denoting the gluing point by x 0 , for a suitable triangulationof X = S n ∨ S n , this space is naturally stratified by X ⊃ { x 0 } .

Original: [Figure]
Revised: Next, we apply TOAST to samples

Original: The Euclidicity of the singular point always constitutes a clear outlier.
Revised: D i m e n s i on (a) 0 .0 .(b)

Original:    In the following, we apply TOAST to samples of suchwedged spheres of dimensions 2 , 3 and 4 , calculatingtheir respective Euclidicity scores.
Revised:    (c) of such wedged spheres of dimensions 2 , 3 and 4 , calculating their respective Euclidicity scores.

Original: Since the samplesize that we need to ensure a certain density of the samples grows exponentially with the intrinsic dimensionof the space, we adapted the sample size accordingly,starting with a sample size of 20000 in dimension 2 ,and increasing the sample size consecutively by a factor of 10 .
Revised: Since larger intrinsic dimensions require higher sample sizes to maintain the same density, we startwith a sample size of 20000 in dimension 2 and increase it consecutively by a factor of 10 .

Original: We then calculated Euclidicity of 50 randomsamples in the respective data set, and additionally forthe singular point x 0 .
Revised: Wethen calculate Euclidicity of 50 random samples in the respective data set, and additionally for thesingular point x 0 .

Original: Fig.
Revised: Fig.

Original: 4 shows the results of ourexperiments.
Revised: 4a shows the results of our experiments.

Original: We observe that the singular point possesses a significantly higher Euclidicity score than therandom samples.
Revised: We observe that the singular pointpossesses a significantly higher Euclidicity score than the random samples.

Original: Moreover, we find that Euclidicityscores of non-singular points exhibit a high degree ofvariance across the data, which is caused by the factthat the sampled data does not perfectly fit the underlying space the points are being sampled from.
Revised: Moreover, we find that Euclidicity scores of non-singular points exhibit a high degree of variance across the data, whichis caused by the fact that the sampled data does not perfectly fit the underlying space the pointsare being sampled from.

Original: This strengthens our main argument: assessing whether a specific point is Euclidean does not requirea binary decision butshould incorporate a continuous measure such as Euclidicity. 
Revised: This strengthens our main argument: assessing whether a specific point is Euclidean does not require a binary decision but a continuous measure such as Euclidicity. Stability.

Original: 5. E UCLIDICITY IS MORE EXPRESSIVE THAN SINGLE - PARAMETER APPROACHES
Revised: As predicted by Theorem 1, Euclidicity estimates are stable in practice.

Original: [Figure]
Revised: We first note that

Original: Euclidicity (a) results in a clearly-delineated singular region when compared to a single-parameter score (b). In contrast to single-parameter approaches such asthe one suggested by Stolz et al.
Revised: Euclidicity is robust towards sampling : repeating the calculations for the ‘pinched torus’ over different batches results in highly similar distributions that are not distinguishable according to Tukey’srange test (Tukey, 1949) at the α = 0 .

Original: (2020), our novel
Revised: 05 confidence level. Moreover, choosing larger locality scalesstill enables us to detect singularities at higher computational costs and incorporating larger parts ofthe point cloud. Please refer to Appendix A.5 for a more detailed discussion of this aspect.

Original:  Euclidicity measureexhibits advantageous properties:we observed that it leads to significantly more stable results than a comparable one-parameter approachfor geometry-based anomaly detection. More precisely, Fig.
Revised: Our Euclidicity measure leads to significantly more stable results than a comparable one-parameterapproach for geometry-based anomaly detection (Stolz et al., 2020): Fig. 4b and Fig.

Original: 5 shows the comparison of one-parameterand multi-parameter Euclidicity for 20000 samplesof S 2 ∨ S 2 .
Revised: 4c comparemulti-parameter Euclidicity with one-parameter Euclidicity for 20000 samples of S 2 ∨ S 2 .

Original: The constant-scale approach results inmany points with high anomaly scores that in factadmit a Euclidean neighbourhood.
Revised: Theconstant-scale approach results in many points with high anomaly scores that in fact do admit a Euclidean neighbourhood.

Original: We quantify thisby analysing the empirical distributions of anomalyscores of the two data spaces (see Appendix A.7 formore details), with the one-parameter method exhibiting a much larger variance than our multi-parameter Euclidicity measure.
Revised: We quantify this by analysing the empirical distributions of anomaly scoresof the two data spaces (see Appendix A.8 for more details), with the one-parameter method exhibiting a much larger variance than our multi-parameter Euclidicity measure.

Original: The multi-parameterdistribution shows that the mass is concentrated around the mean, but also contains outliers withhigh Euclidicity scores.
Revised: The multi-parameterdistribution shows that the mass is concentrated around the mean, but also contains outliers withhigh Euclidicity scores.

Original: These outliers correspond to points in the data space whose distance to the singular point is small.
Revised: These outliers correspond to points in the data space whose distance to thesingular point is small.

Original: We thus conclude that Euclidicity scores increase once one approaches thesingularity—which is not the case for single-parameter methods with a fixed locality scale.
Revised: We thus conclude that Euclidicity scores increase once one approaches thesingularity—which is not the case for single-parameter methods with a fixed locality scale.

Original: In fact,the main advantage of Euclidicity is that it implicitly incorporates information about the scale onwhich a given data point admits a Euclidean neighbourhood.
Revised: In fact,the main advantage of Euclidicity is that it implicitly incorporates information about the scale onwhich a given data point admits a Euclidean neighbourhood.

Original: 5. E UCLIDICITY CAPTURES GEOMETRIC COMPLEXITY OF HIGH - DIMENSIONAL SPACES
Revised: 5. E UCLIDICITY CAPTURES GEOMETRIC COMPLEXITY OF HIGH - DIMENSIONAL SPACES

Original: [Figure] To test TOAST in an unsupervised setting, we calculate Euclidicity scores for the MNIST and F ASHION MNIST datasets, selecting mini-batches of 1000 samples from a subsample of 10000 random images of these data sets.
Revised: To test TOAST in an unsupervised setting, we calculate Euclidicity scores for the MNIST and F ASHION MNIST data sets, selecting mini-batches of 1000 samples from a subsample of 10000random images of these data sets.

Original: Following Pope et al.
Revised: Following Pope et al.

Original: (2021), we assume an intrinsic dimension of 10 ;moreover, we use k = 50 neighbours for local scale estimation.
Revised: (2021), we assume an intrinsic dimensionof 10 ; moreover, we use k = 50 neighbours for local scale estimation.

Original: To ensure that our results are representative, we repeat all calculations for five different subsamples.
Revised: To ensure that our resultsare representative, we repeat all calculations for five different subsamples.

Original: Euclidicity scores range from [1 . 1 , 5 . 3] for MNIST, and [1 . 3 , 5 . 6] for F ASHION MNIST.
Revised: Euclidicity scores rangefrom [1 . 1 , 5 . 3] for MNIST, and [1 . 3 , 5 . 6] for F ASHION MNIST.

Original: The scores of the two datasets appear to befollowing different distributions (see Appendix A.6 for a visualisation and a more detailed depiction of the distributions).
Revised: The scores of the two datasetsappear to be following different distributions (see Appendix A.7 for a visualisation and a moredetailed depiction of the distributions).

Original: Fig.
Revised: Fig.

Original: 6 shows a selection of 9 images, corresponding to thelowest, median, and highest Euclidicity scores, respectively.
Revised: 5 shows a selection of 9 images, corresponding to the [Figure] lowest, median, and highest Euclidicity scores, respectively.

Original: We observe that high Euclidicity scores correspond to imageswith a high degree of non-linearity, whereas low Euclidicityscores correspond to images that exhibit less complex structures: for MNIST, these are digits of ‘1.’
Revised: We observe that high Euclidicity scorescorrespond to images with a high degree of non-linearity, whereas low Euclidicity scores correspondto images that exhibit less complex structures: for MNIST, these are digits of ‘1.’

Original: Interestingly, weobserve the same phenomenon for F ASHION MNIST, where images with low Euclidicity (‘pants’)possess less geometric complexity in contrast to images with high Euclidicity.
Revised: Interestingly, weobserve the same phenomenon for F ASHION MNIST, where images with low Euclidicity (‘pants’)possess less geometric complexity in contrast to images with high Euclidicity.

Original: Since low Euclidicitycan also be seen as an indicator of how close a neighbourhood is to being locally linear , this findinghints at the existence of simple substructures in such data sets.
Revised: Since low Euclidicitycan also be seen as an indicator of how close a neighbourhood is to being locally linear , this findinghints at the existence of simple substructures in such data sets.

Original: Euclidicity could thus be used as anunsupervised measure of geometric complexity.
Revised: Euclidicity could thus be used as anunsupervised measure of geometric complexity.

Original:  D ISCUSSION
Revised:  D ISCUSSION

Original: We presented TOAST, a novel framework for locally estimating the intrinsic dimension (via PID, thepersistent intrinsic dimension) and the ‘manifoldness’ (via Euclidicity, a multi-scale measure of thedeviation from Euclidean space) of point clouds.
Revised: We presented TOAST, a novel framework for locally estimating the intrinsic dimension (via PID, thepersistent intrinsic dimension) and the ‘manifoldness’ (via Euclidicity, a multi-scale measure of thedeviation from Euclidean space) of point clouds.

Original: Our method is based on a novel formulation of persistent local homology as a multi-parameter approach, and we provide theoretical guarantees for itin a dense sample setting.
Revised: Our method is based on a novel formulation of persistent local homology as a multi-parameter approach, and we provide theoretical guarantees for itin a dense sample setting.

Original: Our experiments showed significant improvements of stability comparedto geometry-based anomaly detection methods with fixed locality scales, and we found that Euclidicity can detect singular regions in data sets with known singularities.
Revised: Our experiments showed significant improvements of stability comparedto geometry-based anomaly detection methods with fixed locality scales, and we found that Euclidicity can detect singular regions in data sets with known singularities.

Original: Using high-dimensionalbenchmark data sets, we also observed that Euclidicity can serve as an unsupervised measure ofgeometric complexity .
Revised: Using high-dimensionalbenchmark data sets, we also observed that Euclidicity can serve as an unsupervised measure ofgeometric complexity .

Revised: For future work, we envision two relevant research directions.

Revised: First and foremost will be the inclusion of Euclidicity into machine learning models to make them ‘singularity-aware.’

Revised: In light of ourexperiments in Section 5.5, we believe that Euclidicity could be particularly useful in unsupervisedscenarios, or provide an additional weight in classification settings (to ensure that singular examplesare being given lower confidence scores).

Revised: Moreover, Euclidicity could be used in the detection ofadversarial samples—a task for which knowledge about the underlying topology of a space is knownto be crucial (Jang et al., 2020).

Revised: As a second direction, we want to further improve the properties of

Original: For future work, we plan to investigate if the incorporation of custom distance measures for threeparameter persistence modules, i.e. different metrics for Eq.
Revised: Euclidicity itself. To this end, we plan to investigate if incorporating custom distance measures forthree-parameter persistence modules, i.e. different metrics for Eq.

Original: (4), lead to improved results in termsof stability, expressivity, or computational efficiency.
Revised: (4), lead to improved results interms of stability, expressivity, or computational efficiency.

Original: Moreover, we envision that replacing the Vietoris–Rips filtration by other constructions could prove beneficial; for instance, the use of landmarks and witness complexes (de Silva & Carlsson, 2004) can substantially reduce the number ofsamples that are needed to detect the correct intrinsic dimension and calculate Euclidicity.
Revised: Moreover, we hypothesise that replacingthe Vietoris–Rips filtration by other constructions such as landmarks or witness complexes (de Silva& Carlsson, 2004) could prove beneficial by substantially reducing the number of samples that areneeded to detect the correct intrinsic dimension and calculate Euclidicity.

Original: Alongthese lines, we also plan to derivemore theoretical results that relate specific filtrations and the expressivity of the corresponding Euclidicity measure.
Revised: Along these lines, wealso plan to derive theoretical results that relate specific filtrations and the expressivity of the corresponding Euclidicity measure.

Original: Another direction for future research concernsthe approximation of a manifold from inherently singular data, i.e. finding the best manifold approximation to a given data set with singularities.
Revised: Another direction for future research concerns the approximation ofa manifold from inherently singular data, i.e. finding the best manifold approximation to a given dataset with singularities.

Original: This way, singularities could be resolved duringtraining, provided an appropriate loss function exists.
Revised: This way, singularities could be resolved during the training phase of models,provided an appropriate loss function exists.

Original: Euclidicity may thus serve as a metric forassessing data sets, paving the way towards more trustworthy and faithful embeddings.
Revised: Euclidicity may thus serve as a metric for assessingdata sets, paving the way towards more trustworthy and faithful embeddings.

Original: R EPRODUCIBILITY STATEMENT
Revised: R EPRODUCIBILITY STATEMENT

Original: We provide our code as part of the supplementary materials.
Revised: We provide our code as part of the supplementary materials.

Original: All dependencies are listed in the 408 pyproject.toml file, and the README discusses how to install our package and run our experiments.
Revised: All dependencies are listed in the respective pyproject.toml file, and the README discusses how to install our package and run ourexperiments.

Original: Our implementation leverages multiple CPUs if available but has no specific hardwarerequirements otherwise.
Revised: Our implementation leverages multiple CPUs if available but has no specific hardwarerequirements otherwise.

Original: A A PPENDIX
Revised: PPENDIX

Original: A.
Revised: A.

Original: N OTATION
Revised: N OTATION

Original: Symbol
Revised: Symbol

Original: Meaning ϵ local annulus scale parameter R real numbers
Revised: Meaning ϵ local annulus scale parameter R real numbers

Original: H i
Revised: H i

Original: i th (ordinary) homology functor ˜
Revised: i th (ordinary) homology functor (with Z / 2 Z coefficients)

Original:  Hi
Revised: ˜ H

Original: i  th reduced homology functor  inf infimum sup supremum | · | ∞ uniform (infinity) norm n intrinsic dimension of the space under consideration
Revised: i i th reduced homology functor (with Z / 2 Z coefficients) inf infimum sup supremum | · | ∞ uniform (infinity) norm n intrinsic dimension of the space under consideration

Original: N ambient dimension of the space under consideration lim
Revised: N ambient dimension of the space under consideration lim

Original: −→ (categorical) colimit
Revised: −→ (categorical) colimit

Original: [Equation]

Original:  k -dimensional sphere 
Revised: S k k -dimensional sphere c

Original: [Equation]
Revised: [Equation]

Original: open cone of a topological space X
Revised: open cone of a topological space X

Original: A. A DDITIONAL PROOFS
Revised: A. P ROOFS OF THE MAIN STATEMENTS IN THE PAPER We restate the theorems from the main paper for the convenience of readers, along with their proofs,which were removed for space reasons. We first prove the stability theorem, first stated on p. 5 in themain text, which shows that our method enjoys stability properties with respect to radius changes ofthe intrinsic annuli.

Revised: Theorem 1.

Revised: Given a finite metric space X and x ∈ X , let B sr ( x ) and B s ′ r ′

Revised: ( x ) denote twointrinsic annuli with | r − r ′

Revised: | ≤ ϵ 1 and | s − s ′ | ≤ ϵ 2 .

Revised: Furthermore, let D , D ′ denote thepersistence diagrams corresponding to PH i ( B sr ( x ); V ) and PH i ( B s ′ r ′ ( x ); V ) , respectively.

Revised: Then12 d B ( D , D ′ )

Revised: ≤ max { ϵ 1 , ϵ 2 } .

Revised: Proof.

Revised: The Hausdorff distance of two non-empty subsets A, B ⊂ X is d H ( A, B ) :

Revised: = inf { ϵ ≥0 | A ⊂ B ϵ , B ⊂

Revised: A ϵ } , where A ϵ = ∪ a ∈ A { x ∈ X ; d( x, a ) ≤ ϵ } denotes the ϵ -thickening of

Revised: A in X .

Revised: Set

Revised: ϵ := max { ϵ 1 , ϵ 2 } .

Revised: By assumption, B sr ( x ) ⊂ B s ′ r ′

Revised: ( x ) ϵ and B s ′ r ′ ( x )

Revised: ⊂ B sr ( x ) ϵ , i.e.d H ( B sr ( x ) , B s ′ r ′

Revised: ( x )) ≤ ϵ .

Revised: Using the geometric stability theorem of persistence diagrams (Chazalet al., 2014), we have 12 d B ( D , D ′ )

Revised: ≤ d H ( B sr ( x ) , B s ′ r ′

Revised: ( x )) , which proves the claim.

Revised: Next, we prove that our persistent intrinsic dimension (PID) measure is capable of capturing thedimension of manifolds correctly, provided sufficiently many samples are present.

Revised: This theoremwas first stated on p. 5.

Revised: Theorem 2.

Revised: Let M ⊂ R N be an n -dimensional compact smooth manifold and let X :={ x 1 , . . .

Revised: , x S } be a collection of uniform samples from M .

Revised: For a sufficiently large S , there existconstants ϵ 1 , ϵ 2 > 0 such that i x ( ϵ ) = n for all

Revised: ϵ 1 <

Revised: ϵ < ϵ 2 and any point x ∈ X .

Revised: Moreover, ϵ 1 canbe chosen arbitrarily small by increasing S .

Revised: Proof.

Revised: Let x ∈ X be an arbitrary point.

Revised: Since M is a manifold, x admits a Euclidean neighbourhood U .

Revised: Since M is smooth, we can assume U to be arbitrarily close to being flat by shrinking it.

Revised: Thus, we can find ϵ 2 > 0 with B sr ( x ) ⊂ U for all r, s <

Revised: ϵ 2 such that H i ( V ( B sr ( x ) , t ))

Revised: = 0 for alli ≥ n , and all t .

Original: {+Hence, PH i ( B sr ( x )) = 
+}
Revised: {+Hence, PH i ( B sr ( x )) = 
+}

Revised: for all i ≥ n , and therefore i x ( ϵ 2 ) ≤ n .

Revised: By contrast, for

Revised: S sufficiently large, and r, s as before, there exists a parameter t such that V ( B sr ( x ) , t ) is homotopyequivalent to an ( n − 1) -sphere, and so H n − 1 ( V ( B sr ( x ) , t )) admits a generator, i.e. it is non-trivial.

Revised: Consequently, PH n − 1 ( B sr ( x )) ̸ = 0 , and i x ( ϵ 2 ) = n .

Revised: By further increasing S , we can ensure thatthe statement still holds when we decrease ϵ 2 , which proves the two remaining claims.

Revised: A. A DDITIONAL PROOFS

Original: To make this paper self-contained, we provide a brief proof of Eq.
Revised: To make this paper self-contained, we provide a brief proof of Eq.

Original: By the excision axiom forhomology, we have
Revised: By the excision axiom forhomology, we have

Original: [Equation]
Revised: [Equation]

Original: Since St( x ) is contractible , the long exact reduced homology sequence of the pair (St( x ) , St( x ) \ x )records exactness of
Revised: Since St( x ) is contractible , the long exact reduced homology sequence of the pair (St( x ) , St( x ) \ x )records exactness of

Original: [Equation]
Revised: [Equation]

Original:  (St( x )) [Equation] St [Equation]
Revised: for all i , and therefore H i (St( x ) , St( x ) \ x ) ∼ = ˜H

Original: x ) , [Equation] St [Equation]
Revised: i

Original: St

Original: [Equation]
Revised: [Equation]

Original: x

Original: [Equation]

Original: [Equation]

Original: for all i , and therefore H i (St( x ) , St( x ) \ x ) ∼ = ˜H

Original: [-i − 
-]
Revised: [-i − 
-]

Original: (St( x ) \ x ) .
Revised: (St( x ) \ x ) .

Original: Eq.
Revised: Eq.

Original: (1) now follows from theobservation that St( x ) \ x deformation retracts to Lk( x ) .
Revised: (1) now follows from theobservation that St( x ) \ x deformation retracts to Lk( x ) .

Original: A. P SEUDOCODE
Revised: A. P SEUDOCODE

Original: We provide brief pseudocode implementations of the algorithms discussed in Section 4.
Revised: We provide brief pseudocode implementations of the algorithms discussed in Section 4.

Original: In the following, we use # Bar i ( X ) to denote the number of i
Revised: In the following, we use # Bar i ( X ) to denote the number of i

Original: -dimensional
Revised: -dimensional

Original: persistent barcodes of X (w.r.t.the  filtration, but any other choice of filtration affords the same description).
Revised: persistent barcodes of X (w.r.t.the Vietoris–Rips filtration, but any other choice of filtration affords the same description).

Original: Algorithm 1 explains the calculation of persistent intrinsic dimension (see Section 4.2 in the main paperfor details).
Revised: Algorithm 1 explains the calculation of persistent intrinsic dimension (see Section 4.1 in the main paperfor details).

Original: For the subsequent algorithms, we assume that the estimated dimension of the intrinsicdimension of the data is n .
Revised: For the subsequent algorithms, we assume that the estimated dimension of the intrinsicdimension of the data is n .

Original: We impose no additional requirements on this number; it can, in fact,be obtained by any choice of intrinsic dimension estimation method.
Revised: We impose no additional requirements on this number; it can, in fact,be obtained by any choice of intrinsic dimension estimation method.

Original: As a short-hand notation, forp i = PH n − 1 ( V ( E B •• ( x ) , • )) w.r.t.
Revised: As a short-hand notation, forp i = PH n − 1 ( V ( E B •• ( x ) , • )) w.r.t.

Original: some sample of { y ∈ R n | r ≤ d( x, y ) ≤ s } , we denote by 505 p r,s i =
Revised: some sample of { y ∈ R n | r ≤ d( x, y ) ≤ s } , we denote by 519 p r,s i =

Original: PH n − 1 ( V ( E B sr ( x ) , • ))
Revised: PH n − 1 ( V ( E B sr ( x ) , • ))

Original: the respective fibred persistent local homology barcode (calculatedw.r.t.
Revised: the respective fibred persistent local homology barcode (calculatedw.r.t.

Original: the same sample).
Revised: the same sample).

Original: Algorithm 2 then shows how to calculate the Euclidicity values, following
Revised: Algorithm 2 then shows how to calculate the Euclidicity values, following

Original: Eq.
Revised: Eq.

Original: (3) and one of its potential implementations, given in Eq.
Revised: (3) and one of its potential implementations, given in Eq.

Revised: Require: x ∈ X , s max , ℓ .

Revised: 1: for s ∈ Γ do ▷ Iterate over the parameter grid 2: i x ( s ) ←3: for r < s ∈ Γ do 4: for i = 1 , . . .

Revised: , N − 1 do 5:

Revised: Calculate # Bar i

Revised: ( B sr ( x )) 6: if # Bar i ( B sr ( x )) > 0 then 7: i x ( s ) ← i +8: end if 9: end for 10: end for 11: return i x ( s ) 12: end for

Original: Require: x ∈ X , s max , ℓ , n , { p 1 , . . .
Revised: Require: x ∈ X , s max , ℓ , n , { p 1 , . . .

Original: , p m } .
Revised: , p m } .

Original: 1: for j = 1 , . . .
Revised: 1: for j = 1 , . . .

Original: , m do 2: for k = j + 1 , . . .
Revised: , m do 2: for k = j + 1 , . . .

Original: , m do 3: for s ∈ Γ do 4: for r ∈ Γ , r < s do 5:
Revised: , m do 3: for s ∈ Γ do 4: for r ∈ Γ , r < s do 5:

Original: Calculate d B ( p r,sj , p r,sk ) ▷ Calculate bottleneck distance 6: return d B ( p r,sj , p r,sk ) 7: end for 8: end for 9:
Revised: Calculate d B ( p r,sj , p r,sk ) ▷ Calculate bottleneck distance 6: return d B ( p r,sj , p r,sk ) 7: end for 8: end for 9:

Original: Calculate D( p j , p k ) ▷ Evaluate Eq.
Revised: Calculate D( p j , p k ) ▷ Evaluate Eq.

Original: (4) 10: return D( p j , p k ) 11: end for 12: end for
Revised: (4) 10: return D( p j , p k ) 11: end for 12: end for

Original: Require: x ∈ X , s max , ℓ . 1: for s ∈ Γ do ▷ Iterate over the parameter grid 2: i x ( s ) ←3: for r < s ∈ Γ do 4: for i = 1 , . . . , N − 1 do 5: Calculate # Bar i
Revised: [Figure]

Original: ( B sr ( x )) 6: if # Bar i ( B sr ( x )) > 0 then 7: i x ( s ) ← i +8: end if 9: end for 10: end for 11: return i x ( s ) 12: end for A. S TABILITY OF E UCLIDICITY ESTIMATES
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure] Sample [Figure]
Revised: While each sample invariably exhibits some degree of geometric variation, we are able to reliably identify the singularity and its neighbourhood

Original: Fig.
Revised: Fig.

Original: 7 shows that Euclidicity is robust undersampling; repeating the calculations for smallerbatches of the ‘pinched torus’ data set ( 500 pointseach) still lets us detect the singularity and itsneighbours reliably.
Revised: 6 shows that Euclidicity is robust undersampling; repeating the calculations for smallerbatches of the ‘pinched torus’ data set ( 500 pointseach) still lets us detect the singularity and itsneighbours reliably.

Original: This robustness is an important property in practice where we are dealingwith samples from an unknown data set whoseshape properties we want to capture.
Revised: This robustness is an important property in practice where we are dealingwith samples from an unknown data set whoseshape properties we want to capture.

Original: Euclidicity enables us to perform these calculations in arobust manner.
Revised: Euclidicity enables us to perform these calculations in arobust manner.

Original: Following the brief discussionin Section 5.1, we show the results of varyings max , the outer radius of the local annulus, forthe ‘pinched torus’ data set.
Revised: Following the brief discussionin Section 5.1, we show the results of varyings max , the outer radius of the local annulus, forthe ‘pinched torus’ data set.

Original: Fig.
Revised: Fig.

Original: 8 depicts pointclouds of 1000 samples; we observe that the singularity, i.e. the ‘pinch point,’ is always detected.
Revised: 7 depicts pointclouds of 1000 samples; we observe that the singularity, i.e. the ‘pinch point,’ is always detected.

Original: For larger radii, however, this detection becomesprogressively more global , incorporating largerparts of the point cloud.
Revised: For larger radii, however, this detection becomesprogressively more global , incorporating largerparts of the point cloud.

Original: Fig.
Revised: Fig.

Original: 9 depicts the corresponding histograms; we observe the same shiftin probability mass towards the tail end of the distribution.
Revised: 8 depicts the corresponding histograms; we observe the same shiftin probability mass towards the tail end of the distribution.

Original: For extremely large annuli, we estimatethat we lose a clear distinction between singular values and non-singular values.
Revised: For extremely large annuli, we estimatethat we lose a clear distinction between singular values and non-singular values.

Original: Our data-drivenparameter selection procedure is thus to be preferred in practice since it incorporates data density.
Revised: Our data-drivenparameter selection procedure is thus to be preferred in practice since it incorporates data density.

Original: [Figure]

Original: [Figure]
Revised: [Figure]

Original: A. C OMPARISON OF PID WITH OTHER DIMENSION ESTIMATES
Revised: A. C OMPARISON OF PID WITH OTHER DIMENSION ESTIMATES

Original: In order to assess the quality of PID, we decided to test its performance on a space that is both singular and has non-constant dimension.
Revised: In order to assess the quality of PID, we decided to test its performance on a space that is both singular and has non-constant dimension.

Original: The data space we chose consists of 2000 samples of S 1 ∨ S 2 ,i.e. a 1 -sphere glued together with a 2 -sphere at a certain concatenation point.
Revised: The data space we chose consists of 2000 samples of S 1 ∨ S 2 ,i.e. a 1 -sphere glued together with a 2 -sphere at a certain concatenation point.

Original: We then applied the
Revised: We then applied the

Original: PID procedure for a maximum locality scale that was given by the k nearest neighbour distances, fork ∈ { 25 , 50 , 75 , 100 , 125 , 150 , 175 , 200 } .
Revised: PID procedure for a maximum locality scale that was given by the k nearest neighbour distances, fork ∈ { 25 , 50 , 75 , 100 , 125 , 150 , 175 , 200 } .

Original: We assigned to each point the average of the PID scoresat the respective scales that are less than or equal to the k nearest neighbour bound.
Revised: We assigned to each point the average of the PID scoresat the respective scales that are less than or equal to the k nearest neighbour bound.

Original: Subsequently,we compared the results with other local dimension estimates for the respective number of neighbours.
Revised: Subsequently,we compared the results with other local dimension estimates for the respective number of neighbours.

Original: The methods that were chosen for comparison include lpca , twoNN , KNN , and DANCo ; weused the respective implementation from the scikit-dimension Python package.
Revised: The methods that were chosen for comparison include lpca , twoNN , KNN , and DANCo ; weused the respective implementation from the scikit-dimension Python package.

Original: Fig.
Revised: Fig.

Original: 10a shows the PID results for a maximum locality scale of 200 neighbours, with colours showing the estimated dimension values for each point.
Revised: 9a shows the PID results for a maximum locality scale of 200 neighbours, with colours showingthe estimated dimension values for each point.

Original: Overall, the correct intrinsic dimension is detected for most of the points.
Revised: Overall, the correct intrinsic dimension is detectedfor most of the points.

Original: However, points that lie close to the singular point show a PID valuebetween 1 and 2 .
Revised: However, points that lie close to the singular point show a PID value between 1 and 2 .

Original: Similarly to what we already discussed for Euclidicity, PID should therefore alsobe interpreted as a measure that incorporates the intrinsic dimension of a point on multiple scalesof locality.
Revised: Similarly to what we already discussed for Euclidicity, PID should therefore alsobe interpreted as a measure that incorporates the intrinsic dimension of a point on multiple scalesof locality.

Original: For real-world data, the dimension will generally change when changing the localityscale.
Revised: For real-world data, the dimension will generally change when changing the localityscale.

Original: However, since there is no canonical choice of scale, we believe that any such scale providesvaluable information about the intrinsic dimension that is worth being measured.
Revised: However, since there is no canonical choice of scale, we believe that any such scale providesvaluable information about the intrinsic dimension that is worth being measured.

Original: We therefore arguethat a multi-scale approach like ours is appropriate in practice, especially in a regime that is agnosticwith respect to the underlying intrinsic dimension.
Revised: We therefore arguethat a multi-scale approach like ours is appropriate in practice, especially in a regime that is agnosticwith respect to the underlying intrinsic dimension.

Original: By contrast, Fig. 10b shows the correspondingdimension estimates for twoNN, where we observe less stable and reliable results across the dataset.
Revised: By contrast, Fig. 9b shows the correspondingdimension estimates for twoNN, where we observe less stable and reliable results across the dataset.

Original: Fig.
Revised: Fig.

Original: 11a shows boxplots of the distributions of the dimension estimates, for all points that lie onthe 1D-sphere.
Revised: 10a shows boxplots of the distributions of the dimension estimates, for all points that lie onthe 1D-sphere.

Original: We see that for PID, the mass is concentrated at a value of 1 .
Revised: We see that for PID, the mass is concentrated at a value of 1 .

Original: Although there areoutliers present, these correspond to points that are close to the singularity, as it was expected.
Revised: Although there areoutliers present, these correspond to points that are close to the singularity, as it was expected.

Original: Wenote that other methods like KNN and lpca might highly overestimate the dimension, and that the [Figure] [Figure] [Figure]
Revised: Wenote that other methods like KNN and lpca might highly overestimate the dimension, and that theinterquartile range is significantly higher for twoNN and KNN .

Original: interquartile range is significantly higher for twoNN and KNN .

Original: Fig.
Revised: Fig.

Original: 11b shows the same distributionsfor the points that lie on the 2D-sphere.
Revised: 10b shows the same distributionsfor the points that lie on the 2D-sphere.

Original: Again, lpca highly overestimates the dimension since themedian lies at a value of 3 .
Revised: Again, lpca highly overestimates the dimension since themedian lies at a value of 3 .

Original: Again, the interquartile range of PID is the tightest, and the estimatesare closest to the ground truth.
Revised: Again, the interquartile range of PID is the tightest, and the estimatesare closest to the ground truth.

Original: Moreover, the lower-value outliers again correspond to points thatare close to the singular gluing point.
Revised: Moreover, the lower-value outliers again correspond to points thatare close to the singular gluing point.

Original: Fig. 12a and Fig.
Revised: Fig. 11a and Fig.

Original: 12b show average dimension estimate scores of all investigated methods for varying values of k , both for points on the 1 -sphere and the 2 -sphere.
Revised: 11b show average dimension estimate scores of all investigated methods for varying values of k , both for points on the 1 -sphere and the 2 -sphere.

Original: We note that on average, only 564 twoNN and DANCo lead to results which are comparable with the reliability of our method.
Revised: We note that on average, only 578 twoNN and DANCo lead to results which are comparable with the reliability of our method.

Original: However, as we already saw in Fig.11a and Fig.
Revised: However, as we already saw in Fig.

Original: 11b, the variance of the scores of our method is significantly lower, leading to more reliable outputs for each of the points. 
Revised: 10a and Fig. 10b, the variance of the scores of our method is significantly lower, leading to more reliable outputs for each of the points. [Figure] [Figure] [Figure]

Revised: [Figure]

Original: A. E UCLIDICITY OF MNIST AND F ASHION MNIST
Revised: A. E UCLIDICITY OF MNIST AND F ASHION MNIST

Original: [Figure]
Revised: Fig.

Original: Fig. 13 and Fig.
Revised: 12 and Fig.

Original: 14 show the Euclidicity results for the 4 additional runs on both the MNIST and
Revised: 13 show the Euclidicity results for the 4 additional runs on both the MNIST and

Original: F ASHION MNIST data sets.
Revised: F ASHION MNIST data sets.

Original: Again, we depicted the 9 images with lowest (left), medium (middle),and highest (right)
Revised: Again, we depicted the 9 images with lowest (left), medium (middle),and highest (right)

Original: Euclidicity scores for the two datasets.
Revised: Euclidicity scores for the two datasets.

Original: Moving from left to right, the imagesexhibit increases in the complexity of the local geometry, giving evidence for the reproducibility ofthe observation we remarked in Section 5.4.
Revised: Moving from left to right, the imagesexhibit increases in the complexity of the local geometry, giving evidence for the reproducibility ofthe observation we remarked in Section 5.5.

Original: Finally, as Fig.
Revised: Finally, as Fig.

Original: 15 shows, the empirical distributions of the calculated Euclidicity scores differ significantly for the MNIST and F ASHION MNIST data sets, with the distribution for MNIST exhibitinga bimodal behaviour, whereas the F ASHION MNIST Euclidicity value distribution is unimodal.
Revised: 14 shows, the empirical distributions of the calculated Euclidicity scores differ significantly for the MNIST and F ASHION MNIST data sets, with the distribution for MNIST exhibitinga bimodal behaviour, whereas the F ASHION MNIST Euclidicity value distribution is unimodal.

Original: Wehypothesise that this corresponds to regions of simple complexity—and locally linear structures—inthe MNIST data set, which are absent in the F ASHION MNIST data set.
Revised: Wehypothesise that this corresponds to regions of simple complexity—and locally linear structures—inthe MNIST data set, which are absent in the F ASHION MNIST data set.

Original: A. O NE - PARAMETER VERSUS MULTI - PARAMETER E UCLIDICITY FOR WEDGED SPHERES
Revised: A. O NE - PARAMETER VERSUS MULTI - PARAMETER E UCLIDICITY FOR WEDGED SPHERES

Original: Fig.
Revised: Fig.

Original: 16 shows the empirical distributions of Euclidicity scores for fixed locality parameters (left) andfor our proposed multi-scale locality approach (right).
Revised: 15 shows the empirical distributions of Euclidicity scores for fixed locality parameters (left) andfor our proposed multi-scale locality approach (right).

Original: We see that the variance is significantly lowerin the multi-scale regime, indicating more stable and robust results.
Revised: We see that the variance is significantly lowerin the multi-scale regime, indicating more stable and robust results.

Original: Moreover, the ratio of maximumand mean is higher in the multi-parameter setting, where high Euclidicity scores correspond to datapoints that lie close to the singularity, resulting in more reliable outcomes.
Revised: Moreover, the ratio of maximumand mean is higher in the multi-parameter setting, where high Euclidicity scores correspond to datapoints that lie close to the singularity, resulting in more reliable outcomes.

Original: [Figure]
Revised: [Figure]

Original: [Figure] [Figure][Figure]
Revised: [Figure] [Figure]