Topological Singularity Detection at Multiple Scales

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Julius von Rohrscheidt 1 2 Bastian Rieck 1 2

Abstract

The manifold hypothesis, which assumes that data

lies on or close to an unknown manifold of low

intrinsic dimension, is a staple of modern ma-

chine learning research. However, recent work

has shown that real-world data exhibits distinct

non-manifold structures, i.e. singularities, that

can lead to erroneous ﬁndings. Detecting such

singularities is therefore crucial as a precursor to

interpolation and inference tasks. We address this

issue by developing a topological framework that

(i) quantiﬁes the local intrinsic dimension, and

(ii) yields a Euclidicity score for assessing the

‘manifoldness’ of a point along multiple scales.

Our approach identiﬁes singularities of complex

spaces, while also capturing singular structures

and local geometric complexity in image data.

1. Introduction

The ever-increasing amount and complexity of real-world

data necessitate the development of new methods to extract

less complex but still meaningful representations of the un-

derlying data. While numerous methods for approaching

this representation learning problem exist, they all share a

common assumption: the underlying data is supposed to be

close to a manifold with small intrinsic dimension, i.e. while

the input data may have a large ambient dimension

, there

is an

-dimensional manifold with

n≪N

that best de-

scribes the data. For some data sets, such as natural images,

this manifold hypothesis is appropriate (Carlsson,2009).

However, recent research shows evidence that the mani-

fold hypothesis does not necessarily hold for complex data

sets (Brown et al.,2023), and that manifold learning tech-

niques tend to fail for non-manifold data (Rieck & Leitte,

2015;Scoccola & Perea,2023). These failures are often

the result of singularities, i.e. regions of a space that viol-

ate the properties of a manifold (see Section 2for details).

Helmholtz Munich

Technical University of Munich. Cor-

respondence to: Bastian Rieck

bastian.rieck@helmholtz-

munich.de>.

Proceedings of the

40 th

International Conference on Machine

Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright

2023 by the author(s).

Input space with singularities

0.04 0.09 0.14

Euclidicity

Figure 1: Overview of our method. Given a space with

singularities, our Euclidicity score measures the deviation

from a space to a Euclidean model space. Here, Euclidicity

highlights the singularity at the ‘pinch point.’ Please refer

to Section 4for more details.

For example, the ‘pinched torus,’ an object obtained by

compressing a random region of the torus (speciﬁcally, a

meridian) to a single point, fails to satisfy the manifold hypo-

thesis at the ‘pinch point:’ this point, unlike all other points,

does not have a neighbourhood homeomorphic to

(see

Fig. 1for an illustration). Since singularities—in contrast

to outliers arising from incorrect labels, for example—often

carry relevant information (Jakubowski et al.,2020), new

tools for detecting and handling non-manifold regions in

spaces are needed.

Our contributions. We develop an unsupervised repres-

entation learning framework for detecting singular regions

in point cloud data. Our framework is agnostic with re-

spect to geometric or stochastic properties of the underlying

data and only requires a notion of intrinsic dimension of

neighbourhoods, which, crucially, is allowed to vary across

different points. Our approach is based on a novel formu-

lation of persistent local homology (PLH), a method for

assessing the shape of neighbourhoods at multiple scales

of locality. We use PLH to (i) estimate the intrinsic dimen-

sion of a point locally, and (ii) deﬁne Euclidicity, a novel

quantity that measures the deviation of a point from being

Euclidean. We also provide theoretical guarantees on the ap-

proximation quality for certain classes of spaces, including

manifolds. Euclidicity yields a complementary perspective

on data, highlighting regions where the manifold hypothesis

breaks down. We show the utility of this perspective ex-

perimentally on several data sets, ranging from spaces with

known singularities to high-dimensional image data sets.

arXiv:2210.00069v4 [cs.LG] 14 Jun 2023

Topological Singularity Detection at Multiple Scales

2. Mathematical Background

We ﬁrst provide an overview of persistent homology, and

stratiﬁed spaces, as well as their relation to local homology.

The former concept constitutes a generic framework for

assessing complex data at multiple scales by measuring to-

pological characteristics such as ‘holes’ and ‘voids’ (Edels-

brunner & Harer,2010), while the latter serves as a general

setting to describe singularities, in which our framework

admits advantageous properties.

Persistent homology. Persistent homology is a method for

computing topological features at different scales, capturing

an intrinsic notion of relevance in terms of spatial scale para-

meters. Given a ﬁnite metric space

(X,d)

, the Vietoris–Rips

complex at step

is deﬁned as the abstract simplicial com-

plex

V(X, t)

, in which an abstract

-simplex

(x0, . . . , xk)

of points in

is spanned if and only if

d(xi, xj)≤t

for all

0≤i≤j≤k

For

t1≤t2

, the inclusions

V(X, t1)→ V(X, t2)

yield a ﬁltration, i.e. a sequence of

nested simplicial complexes, which we denote by

V(X,•)

Applying the

th homology functor to this collection of

spaces and inclusions between them induces maps on the

homology level

ft1,t2

i: Hi(V(X, t1)) →Hi(V(X, t2))

for

any

t1≤t2

. The

th persistent homology (PH) of

with

respect to the Vietoris-Rips construction is deﬁned to be the

collection of all these

th homology groups, together with

the respective induced maps between them, and denoted by

PHi(V(X,•))

can therefore be viewed as a tool that

keeps track of topological features such as holes and voids

on multiple scales. For a more comprehensive introduction

in the context of machine learning, see Hensel et al.

(2021). The so-called ‘creation’ and ‘destruction’ times

of these features are summarised in a persistence diagram

D ⊂ R×R∪{∞}

, where any point

(b, d)∈ D

corresponds

to a homology class that arises at ﬁltration step

, and lasts

until ﬁltration step

. The difference

|d−b|

is referred to

as the lifetime or eponymous persistence of this homology

class. There are several distance measures for comparing

persistence diagrams, one of them being the bottleneck dis-

tance, deﬁned as

dB(D,D′) := infγsupx∈D ∥x−γ(x)∥∞

where γranges over all bijections between Dand D′.

Stratiﬁed spaces. Manifolds are widely studied and par-

ticularly well-behaved topological spaces as they locally

resemble Euclidean space near any point. However, spaces

that arise naturally often violate this local homogeneity

condition (see Fig. 2for an example). Stratiﬁed spaces gen-

eralise the concept of a manifold to address singular spaces.

Being intrinsically capable of describing a wider class of

spaces, we argue that stratiﬁed spaces are the right tool to

For readers familiar with persistent homology, we depart from

the usual convention of using

as the threshold parameter since we

use it for the scale of our persistent local homology calculations.

(a) (b) (c)

Figure 2: (a): Non-manifold space. (b): Annulus around a

regular point

.(c): Annulus around a singular point. The

neighbourhood around yis different from all others.

analyse real-world data. In particular, the intrinsic dimen-

sion of points is allowed to vary in this setting, thus leading

to high ﬂexibility in comparison to manifolds. Subsequently,

we deﬁne stratiﬁed spaces in the setting of simplicial com-

plexes. A stratiﬁed simplicial complex of dimension

is a

ﬁnite set of points with the discrete topology. A stratiﬁed

simplicial complex of dimension

is an

-dimensional sim-

plicial complex

, together with a ﬁltration of closed sub-

complexes

X=Xn⊃Xn−1⊃Xn−2⊃ · · · ⊃ X−1=∅

such that

Xi\Xi−1

is an

-dimensional manifold for all

and such that every point

x∈X

possesses a distinguished

local neighbourhood

U∼

=Rk×c◦L

, where

is a com-

pact stratiﬁed simplicial complex of dimension

n−k−1

and

c◦

refers to the open cone construction (see Appendix A.1).

If there exists a neighbourhood

that is homeo-

morphic to

, we say that

is a regular point, otherwise

we call xasingularity.

is a manifold, then independently of the point under

consideration,

is given by a sphere since for a manifold,

any point is regular by deﬁnition. By contrast, a small

neighbourhood of the pinch point in a pinched torus can

be described as the open cone on the disjoint union of two

circles, and therefore the link

is given by

S1⊔S1

this case, whose homology is different from homology of

the link of any other point in the pinched torus (which is

just given by a circle for every other point). The insets in

Fig. 1(left) depict the different neighbourhoods.

Local homology. We now formalise the idea of tracking

the homology of a link of a point, following the previous

description. Intuitively, the local homology of a point is ob-

tained by the homology of an inﬁnitesimal small punctured

neighbourhood around the point (see Appendix A.3 for a

more rigorous description). One can show that the local

homology of a singularity usually differs from the local ho-

mology of a regular point. In particular, points that are of

different intrinsic dimensions with respect to the stratiﬁed

simplicial complex can be distinguished by local homology;

this includes the disjoint union of manifolds of possibly

varying dimensions. This observation motivates and justi-

ﬁes using local homology for detecting neighbourhoods of

singular points, and serves as the primary motivation for our

novel Euclidicity measure in Section 4.2.

Topological Singularity Detection at Multiple Scales

3. Related Work

While manifold learning is concerned with the development

of algorithms that extract geometric information under the

assumption that the given data lie on a manifold, recent

work starts to question this assumption. Brown et al. (2023),

for instance, introduce the union of manifolds hypothesis,

which augments the manifold hypothesis to spaces that can

be modelled as (disjoint) unions of manifolds. Intrinsic

dimension is thus allowed to vary across connected compon-

ents of such a space. However, singularities are excluded

under this assumption, whereas our method detects the cor-

rect intrinsic dimension for large classes of singular spaces.

We assume a fundamental ‘singularity-centric’ perspective

in this paper and argue that a multi-scale analysis of the local

geometry and topology of data is necessary. In this context,

methods from topological data analysis have started attract-

ing attention in machine learning (Hensel et al.,2021). This

is particularly due to persistent homology, which captures

geometrical and topological properties of the underlying

data set on different scales (Bubenik et al.,2020;Turke

et al.,2022). The idea of tracking objects on multiple scales

can at least be traced back to (Koenderink & van Doorn,

1986;Lindeberg,1994), with scale space theory playing an

eminent role in computer vision. However, the utility of

persistent homology in the context of geometric singularit-

ies in data only came up more recently, since early work in

persistent homology focuses predominantly on the simpliﬁc-

ation of functions on manifold domains (Edelsbrunner et al.,

2002). While some research already discusses the utility

of persistent homology for general unsupervised data ana-

lysis workﬂows (Chazal et al.,2013;Rieck & Leitte,2017;

2016;2015), it focuses more on global structures, whereas

singularities are inherently local. A notable exception is a

work by Wang et al. (2011), which analyses local branching

and global circular features. We give a brief overview of

methods in the emerging ﬁeld of topology-driven singularity

detection, outlining the differences to our approach below.

Topology-driven singularity detection. Several works as-

sume a local perspective on homology to detect information

about the intrinsic dimensionality of the data or the presence

of certain singularities. Rieck et al. (2020) use pre-deﬁned

stratiﬁcations and persistent intersection homology, a tech-

nique developed by Bendich & Harer (2011), whereas Fasy

& Wang (2016) and Bendich (2008) both develop persistent

variants of local homology. By contrast, Stolz et al. (2020)

approximate local homology as 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). Moreover,

Bendich et al. (2007) employ persistence vineyards, i.e. con-

tinuous families of persistence diagrams, to assess the local

homology of a point in a stratiﬁed space, whereas Dey et al.

(2014) use local homology to estimate the (global) intrinsic

dimension of hidden, possibly noisy manifolds.

Key differences to existing approaches. While existing

methods overall underscore the relevance of a local perspect-

ive, as well as the use of notions such as stratiﬁed spaces,

our approach differs from them in essential components.

In comparison to all aforementioned contributions, we cap-

ture additional local geometric information: we consider

multiple scales of locality in a persistent framework for

local homology. Concerning the overall construction, Stolz

et al. (2020) is the closest to our method. However, the

authors assume that the intrinsic dimension is known and

the proposed algorithm uses one global scale, whereas our

approach (i) operates in a multi-scale setting, (ii) provides

local estimates of intrinsic dimensionality of the data space,

and (iii) incorporates model spaces that serve as a compar-

ison. We can thus measure the deviation from an idealised

manifold, requiring fewer assumptions on the structure of

the input data (see Appendix A.7 for a comparison).

4. Methods

Our framework TARDIS (Topological Algorithm for Robust

DIscovery of Singularities) consists of two parts: (i) a

method to calculate a local intrinsic dimension of the data,

and (ii) Euclidicity, a measure for assessing the multi-scale

deviation from a Euclidean space. TARDIS is based on the

assumption that the intrinsic dimension of data may not be

constant across the data set, and is thus best described by

local measurements, i.e. measurements in a small neighbour-

hood of a given point. Since there is no canonical choice for

the magnitude of such a neighbourhood, TARDIS analyses

data on multiple scales. Our main idea involves construct-

ing a collection of local (punctured) neighbourhoods for

varying locality scales, and calculating their topological

features. This procedure allows us to approximate local to-

pological features (speciﬁcally, local homology) of a given

point, which we use to measure the intrinsic dimensional-

ity of a space. Moreover, by calculating the distance to

Euclidean model spaces, we are capable of detecting singu-

larities in a large range of input data sets. For the subsequent

description of TARDIS, we only assume that data can be

represented as a ﬁnite metric space (i.e. as a point cloud).

4.1. Persistent Intrinsic Dimension

For a ﬁnite metric space

(X,d)

and

x∈X

, let

r(x) :=

{y∈X|r≤d(x, y)≤s}

denote the intrinsic annulus

with respect to radii

and

. Moreover, let

denote a procedure that takes as input a ﬁnite metric

space and outputs an ascending ﬁltration of topological

spaces—such as a Vietoris–Rips ﬁltration. By applying

to the intrinsic annulus of

, we obtain a tri-ﬁltration

Topological Singularity Detection at Multiple Scales

r(x)

V(Bs

r(x),t)

Figure 3: The intrinsic annulus

r(x)

around a point

in a

metric space

(X,d)

, as well as one ﬁltration step for some

choice of t. By adjusting rand s, we obtain a tri-ﬁltration.

(F(Bs

r(x), t))r,s,t

, where

corresponds to the respective

ﬁltration step that is determined by

. Note that this tri-

ﬁltration is covariant in

and

, but contravariant in

; we

denote it by

F(B•

•(x),•)

. Applying

th homology to this

ﬁltration yields a tri-parameter persistent module that we

call

th persistent local homology (PLH) of

, denoted by

PLHi(x;F) := PHi(F(B•

•(x),•))

. Fig. 3illustrates how

to obtain an annulus from a data set and depicts one step of

the ﬁltration process. To the best of our knowledge, this is

the ﬁrst time that PLH is considered as a multi-parameter

persistence module. Since the Vietoris–Rips ﬁltration is

the pre-eminent ﬁltration in TDA, we will use

PLHi(x) :=

PLHi(x;V)

as an abbreviation. Before developing ways to

detect singularities, we ﬁrst show that our PLH formulation

enjoys stability properties similar to the seminal stability

theorem in persistent homology (Cohen-Steiner et al.,2007),

making it robust to small parameter changes.

Theorem 1. Given a ﬁnite metric space

and

x∈

, let

r(x)

and

Bs′

r′(x)

denote two intrinsic annuli

with

|r−r′| ≤ 1

and

|s−s′| ≤ 2

. Furthermore,

let

D,D′

denote the persistence diagrams correspond-

ing to

PHi(V(Bs

r(x),•))

and

PHi(V(Bs′

r′(x),•))

. Then

2dB(D,D′)≤max{1, 2}.

For a ﬁnite set of points

X⊂RN

, we deﬁne the persistent

intrinsic dimension (PID) of

x∈X

at scale



ix() :=

max{i∈N| ∃ r < s <  s.t. PHi−1(F(Bs

r(x),•)) ̸= 0}

This measure characterises the intrinsic dimension of data

in a multi-scale fashion. We can also prove that we recover

the correct dimension in case our data set constitutes a

manifold sample.

Theorem 2. Let

M⊂RN

be an

-dimensional compact

smooth manifold and let

X:= {x1, . . . , xS}

be a collection

of uniform samples from

. For a sufﬁciently large

and

F=V

, there exist constants

1, 2>0

such that

ix() = n

for all

1<  < 2

and any point

x∈X

. Moreover,

1

can

be chosen arbitrarily small by increasing S.

Theorem 2implies that

ix()

computes the correct intrinsic

dimension of

in a certain range of values

 > 0

, provided

the sample size is sufﬁciently large. Moreover,

ix()

per-

sists in this range, which suggests considering a collection

ix()

for varying



to analyse the intrinsic dimension of

. We also have the following corollary, which speciﬁc-

ally addresses stratiﬁed spaces such as the ‘pinched torus,’

implying that our method can correctly detect the intrinsic

dimension of individual strata. PID is thus capable of hand-

ling large classes of ‘non-manifold’ data sets.

Corollary 1. Let

X=Xn⊃Xn−1⊃Xn−2⊃ · · · ⊃

X−1=∅

be an

-dimensional compact stratiﬁed simplicial

complex, s.t.

Xi\Xi−1

is smooth for every

. For a ﬁxed

let

Xi:= {x1, . . . , xS}

be a collection of uniform samples

from

Xi\Xi−1

. For a sufﬁciently large

and

F=V

there are constants

1, 2>0

such that

ix() = i

for all

1<  < 2

and any point

x∈Xi

. Moreover,

1

can be

chosen arbitrarily small by increasing S.

4.2. Euclidicity

Knowledge about the intrinsic dimension of a neighbour-

hood is crucial for measuring to what extent such a neigh-

bourhood deviates from being Euclidean. We refer to this

deviation as Euclidicity, with the understanding that low

values indicate Euclidean neighbourhoods while high val-

ues indicate singular regions of a data set. Euclidicity can

be calculated without stringent assumptions on manifold-

ness, requiring only an estimate of the intrinsic dimension

. The previously-described PID estimation procedure

is applicable in this setting and may be used to obtain

for example by calculating statistics on the set of

ix()

for

varying locality parameters



.Euclidicity can also use other

dimension estimation procedures, which is advantageous

when additional knowledge about the expected structures is

available (see Camastra & Staiano (2016) for a survey).

The main idea of Euclidicity involves assessing how far a

given neighbourhood of a point

is from being Euclidean.

To this end, we compare it to a Euclidean model space,

measuring the deviation of their corresponding persistent

local homology features. We ﬁrst deﬁne the Euclidean annu-

lus

EBs

r(x)

for parameters

and

to be a set of random

uniform samples of

{y∈Rn|r≤d(x, y)≤s}

such that

|EBs

r(x)|=|Bs

r(x)|

. Here,

and

correspond to the inner

and outer radius of the annulus, respectively. For

r′≤r

and

s≤s′

we extend

EBs

r(x)

by sampling additional points

to obtain

EBs′

r′(x)

with

|EBs′

r′(x)|=|Bs′

r′(x)|

. Iterating

this procedure leads to a tri-ﬁltration

(F(EBs

r(x), t))r,s,t

for any ﬁltration

, following our description in Section 4.1.

We now deﬁne the persistent local homology of a Euclidean

model space as

PLHE

i(x;F) := PHi(F(EB•

•(x),•)).(1)

Again, for a Vietoris–Rips ﬁltration

, we use a short-

form notation, i.e.

PLHE

i(x) := PLHE

i(x;V)

. Notice that

Topological Singularity Detection at Multiple Scales

PLHE

i(x)

implicitly depends on the choice of intrinsic di-

mension

, and on the samples that are generated randomly.

To remove the dependency on the samples, we consider

PLHE

i(x)

to be a sample of a random variable

PLHE

i(x)

Let

D(·,·)

be a distance measure for 3-parameter persist-

ence modules, such as the interleaving distance (Lesnick,

2015). We then deﬁne the Euclidicity of

, denoted by

E(x), as the expected value of these distances, i.e.

E(x) := EhDPLHn−1(x),PLHE

n−1(x)i.(2)

This quantity essentially assesses how far

is from admit-

ting a regular Euclidean neighbourhood. The choice of

and

D(·,·)

leaves us multiple ways of implementing Eq. (2)

in practice. In the following, we describe one particular

implementation with beneﬁcial robustness properties.

Implementation. Calculating

E(x)

requires different

choices, namely (i) a range of locality scales, (ii) a ﬁltration,

and (iii) a distance metric between ﬁltrations

. Using a

grid

of possible radii

(r, s)

with

r < s

, we approximate

Eq. (2) using the mean of the bottleneck distances of ﬁbred

Vietoris–Rips barcodes, i.e.

E(x)≈DPLHi(x),PLHE

i(x):= 1

(r,s)∈Γ

dBr,s (3)

where

is equal to the cardinality of the grid, and

dBr,s := dB(PHi(V(Bs

r(x),•)),PHi(V(EBs

r(x),•)))

with

PLHE

i(x)

referring to a sample from a Euclidean an-

nulus of the same size as the intrinsic annulus around

Eq. (3) can be implemented using effective persistent homo-

logy calculation methods (Bauer,2021), thus permitting an

integration into existing TDA and machine learning frame-

works (The GUDHI Project,2015;Tauzin et al.,2020).

Appendix A.4 provides pseudocode implementations, while

Section 5discusses how to pick these parameters in practice.

We make our framework publicly available.2

As the following theorem shows, our approximation of

Eq. (3) is justiﬁed in the sense that for smooth manifolds,

E(x)

tends to be arbitrarily small in a large-sample regime.

Theorem 3. Let

M⊂RN

be a smooth

-dimensional

manifold and let

X⊂M

be a ﬁnite sample of size

S:= |X|

For a given

 > 0

, sufﬁciently large

and a point

x∈X

there exists

sϵ>0

that only depends on



and the curvature

of x (w.r.t.

), such that the approximation of

E(x)

via

Eq. (3)is bounded above by



, for any grid

with maximum

outer radius sϵ.

Properties. The main appeal of our formulation is that cal-

culating both PID and Euclidicity does not require strong as-

sumptions about the input data: we only assume that the in-

trinsic dimension

of the data is signiﬁcantly lower than the

See the supplementary materials for the code and experiments.

ambient dimension

. Treating dimension as a local quant-

ity that may vary across multiple scales makes Euclidicity

broadly applicable. Moreover, as we showed in Section 4.1,

our method is guaranteed to yield the right values for man-

ifolds and stratiﬁed simplicial complexes. This increases

both the practical applicability and expressivity, enabling

our framework to handle unions of manifolds of varying

dimensions, for instance. Euclidicity thus generalises to a

larger class of spaces than existing approaches (Brown et al.,

2023), permitting a more ﬁne-grained structural assessment.

Limitations. Our implementation of Euclidicity makes

use of the Vietoris–Rips complex, which is known to grow

exponentially with increasing dimensionality. While all

calculations of Eq. (2) can be performed in parallel—thus

substantially improving scalability vis-

a-vis persistent ho-

mology on the complete input data set, both in terms of

dimensions and in terms of samples—the memory require-

ments for a full Vietoris–Rips complex construction may

still prevent our method to be applicable for some high-

dimensional data sets. This can be mitigated by using a

different ﬁltration (Anai et al.,2020;Sheehy,2013). Our

proofs do not assume a speciﬁc ﬁltration, and we leave

the derivation of ﬁltration-speciﬁc theoretical properties for

future work. Finally, we remark 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.

5. Experiments

We demonstrate the expressivity of TARDIS in different

settings, showing that it (i) calculates the correct intrinsic

dimension, and (ii) detects singularities when analysing data

sets with known singularities. We also conduct a brief com-

parison with one-parameter approaches, showcasing how

our multi-scale approach results in more stable outcomes.

Finally, we analyse Euclidicity scores of benchmark and

real-world datasets, giving evidence that our technique can

be used as a measure for the geometric complexity of data.

5.1. Parameter Selection

Since Eq. (2) intrinsically incorporates multiple scales

of locality, we need to specify an upper bound for the

radii (

rmin, rmax, smin, smax

) that deﬁne the respective an-

nuli in practice. Given a point

, we found the following

procedure to be useful in practice: we set

smax

, i.e. the

maximum of the outer radius, to the distance to the

nearest neighbour of a point, and

rmin

, i.e the minimum in-

ner radius, to the smallest non-zero distance to a neighbour

. Finally, we set the minimum outer radius

smin

and

the maximum inner radius

rmax

to the distance to the

⌊k

3⌋

nearest neighbour. While we ﬁnd

k= 50

to yield sufﬁcient

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TopologicalSingularityDetectionatMultipleScalesJuliusvonRohrscheidt12BastianRieck12AbstractThemanifoldhypothesis,whichassumesthatdataliesonorclosetoanunknownmanifoldoflowintrinsicdimension,isastapleofmodernma-chinelearningresearch.However,recentworkhasshownthatreal-worlddataexhibitsdistinctnon-manif...

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Topological Singularity Detection at Multiple Scales

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