#  Deep Mapper Graph and its Application to Visualize Plausible Pathways on
High-Dimensional Distribution with Small Time-Complexity

Ziyad Oulhaj111 This was a joint-work with Fujitsu, and the mentor was H.
Kurhihara.  Yoshiyuki Ishii  Kento Ohga  Kimihiro Yamazaki  Mutsuyo Wada
Yuhei Umeda  Takashi Kato  Yuichiro Wada  Hiroaki Kurihara
<EMAIL_ADDRESS>
###### Abstract

Mapper is a topology based data analysis method that extracts topological
features from high-dimensional data. The Mapper algorithm requires a filter
function that maps the dataset to a Euclidian space and a clustering method,
that is performed on the original dataset. This produces a graph which
represents the shape of the original data. In this work, we use Mapper to
uncover the conformational change of protein structures and we choose the
filter function from a parameterized family, based on a deep neural network
architecture. By optimizing its parameters with respect to an Energy-loss
function derived from a theoretical background, we produce a Mapper graph that
unveils the conformational pathways undertaken by the studied protein. Our
method tackles conformational pathway detection in a unsupervised manner and
therefore greatly reduces the manual and time costs necessary in this task.

###### keywords:

Mapper Graph , Topological Data Analysis , Deep Learning , Conformational
Pathway , Protein Pathway , Cryo-EM

[fr-institute]organization=Nantes Université, École Centrale Nantes, CNRS,
Laboratoire de Mathématiques Jean Leray, UMR 6629,addressline=2 Chemin de la
Houssinière, city=Nantes, postcode=F-44000, country=France

[fujitsu]organization=Fujitsu Limited,addressline=4-1-1 Kamikodanaka,
Nakahara-ku, Kawasaki-shi, city=Kanagawa, postcode=211-8588, country=Japan

[riken]organization=RIKEN AIP,addressline=Nihonbashi 1-chome Mitsui Building,
15th floor, 1-4-1 Nihonbashi, Chuo-ku,, city=Tokyo, postcode=103-0027,
country=Japan

##  1 Introduction

Acquiring a summary of plausible paths (a.k.a. ridgelines) on a high-
dimensional distribution is beneficial in several domains such as food and
drug discovery. In those domains, acquiring plausible conformational paths
with chemical structural change can enhance the efficiency for creating a new
product.

Despite the demands, there only exists methods to compute one plausible path
with long running time, i.e., there exists no method to summarize such paths
with short running time, to the best of our knowledge. The example methods to
compute one plausible path are Barroso da Silva et al. (2020); Wu et al.
(2022); Kinman et al. (2023).

As a brief review of aforementioned references, in food domain, a combined
method of Molecular Dynamics (MD) and Quantitative Structure-Activity
Relationships (QSARs) can assist in constructing a plausible conformational
path (Barroso da Silva et al., 2020), where QSARs is an approach that predicts
principles by correlating physico-chemical properties with biological
properties. The conformational path leads to the elucidation of mechanisms
such as pH-controlled immobilization and taste receptors. However, the method
is based on MD, and therefore it takes a long time to compute a plausible
conformational path.

In drug discovery, a protocol by Kinman et al. (2023) guides the construction
of a plausible path with a protein structural change, utilizing an auto-
encoder named cryoDRGN (Zhong et al., 2021). Here, the structure is not
expressed by an all-atom model of MD but a $3$D density map, and the path is
represented by a sequence of $3$D density maps. In addition, cryoDRGN is
trained by a set of $2$D projection protein images collected by cryo-Electron
Microscopy (cryo-EM) (Earl et al., 2017), and a trained cryoDRGN predicts the
corresponding protein $3$D density map from a latent variable. According to
Kinman et al. (2023), it takes $2$ to $3$ days to compute one plausible path,
since the pathway needs to be carefully constructed from latent variables with
manual labor and extra biological tools. This implies that it takes much
longer time to acquire a summary of multiple plausible paths, if we naively
use the protocol.

In recent years, Kato et al. (2020) have proposed an auto-encoder named
RaDOGAGA which can construct an isometric latent space to the original space,
taking advantage of manifold assumptions (Chapelle et al., 2010). Thereafter,
based on RaDOGAGA, Yamazaki et al. (2023) have proposed an auto-encoder called
cryoTWIN that can construct an isometric latent space to a space of $3$D
density maps. Similar to cryoDRGN, cryoTWIN is trained on a set of cryo-EM
images. In the numerical experiments in Yamazaki et al. (2023), using the
isometric space, the authors computed a plausible conformational path of
$50$S-ribosomal dataset222The authors in Zhong et al. (2020) publicly share
this dataset with the estimated pose orientation by cryoSPARC. with much
shorter running time than Kinman et al. (2023).

However, in our preliminary experiments, we still take several hours to
acquire a summary of the plausible protein pathways, since many random samples
in the latent space are generated to construct one pathway in Algorithm 1 in
Yamazaki et al. (2023). We describe details of the preliminary experiments in
Section 2.1.3.

Considering the background above, our goal of this study is to propose a
method to visualize a summary of plausible paths on a high-dimensional
distribution with short running time, utilizing an isometric latent space
introduced by an auto-encoder such as cryoTWIN. To achieve this goal, we focus
on the mapper algorithm, which outputs a graph called a Mapper graph. It takes
a set of points and a continuous real valued function, called a filter
function, defined on those points as input, as well as other parameters. The
graph constitutes a visualization of the topological structure (loops,
branches, connected components) of the data. The algorithm usually handles
datasets with several tens of dimension, and the running time to return the
graph is sufficiently short. Although there exists a wide range of
applications of the mapper algorithm, the choice of the filter function has
hindered many of them, since summarizing complex data structure can suffer
from using naive filter functions. Then, in order to answer the problem,
Oulhaj et al. (2024) have proposed a mapper algorithm, where the filter
function is parameterized, and the function is optimized to output a well-
summarized mapper graph for the complex data structure, based on a topological
criteria.

We summarize our contributions in this study, as follows:

  1. 1.

Inspired by Oulhaj et al. (2024), we propose a generic mapper algorithm, whose
filter function is parameterized by a deep neural network. This generic
algorithm returns a visualized summary of plausible paths on a high-
dimensional distribution, based on a method introducing a low-dimensional
isometric space such as cryoTWIN (Yamazaki et al., 2023). The difference
between ours and Oulhaj et al. (2024) is that, for extracting complex data
structure, we employ a neural network, which holds the universal approximation
theorem (Shalev-Shwartz and Ben-David, 2014).

  2. 2.

We design a specialized mapper algorithm to visualize a summary of plausible
paths with chemical structural changes using the generic one, based on
theoretical guarantees. Then, we apply this specialized algorithm to the
public $50$S-ribosomal dataset used in Zhong et al. (2020). As a result, we
confirm that a visualized graph summarizing the ribosomal conformational
pathways by our method is consistent with an existing graph, which was
manually constructed by Davis et al. (2016, Figure 7). Furthermore, our
running time to compute the graph is much shorter than the counterpart of
Yamazaki et al. (2023).

The present paper is organized as follows. In Section 2, we describe related
work to our method. In Section 3, we give a detailed account of our method
with theoretical guarantees. In Section 4, we show the results obtained in our
numerical experiments using a ribosomal dataset. Finally, we conclude this
study and discuss our future work in Section 5.

##  2 Related Work

In this section, we first review cryTWIN (Yamazaki et al., 2023), and then
describe details of existing mapper algorithms.

###  2.1 Revisit of cryoTWIN

Yamazaki et al. (2023) proposed a method to automatically compute a plausible
protein conformational pathway from the single particle cryo-EM images. In
their method, first, an auto-encoder named cryoTWIN is trained on the images,
and the trained auto-encoder’s latent space is isometric to a space of $3$D
density maps. Then, a plausible protein conformational pathway as a sequence
of $3$D protein density maps is computed via the isometric latent space.

####  2.1.1 CryoTWIN

CryoTWIN is an auto-encoder consisting of an encoder $\text{Enc}_{\zeta}$, a
decoder $g_{\xi}$, and a latent distributional model $P_{\psi}$, where
$\zeta,\xi,\psi$ are trainable parameters. Then the encoder outputs the latent
variable $z$ as input of a Fourier transformed cryo-EM image $x$. The latent
model is defined as a Gaussian Mixture Model (GMM)
$P_{\psi},\psi=\\{(\pi_{k},\mu_{k},\Sigma_{k})\\}_{k=1}^{\tilde{K}}$, where
$\pi_{k},\mu_{k},$ and $\Sigma_{k}$ represent $k$-th Gaussian’s weight, mean,
and variance, respectively. Let $x_{i}$ be a Fourier transformation of $i$-th
cryo-EM image. The training objective is as follows:

$\theta^{\ast},\phi^{\ast},\psi^{\ast}=\arg\min_{\theta,\phi,\psi}\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}_{\varepsilon}\biggl{[}\underbrace{\left\|W\odot\left(x_{i}-\hat{x}_{{z}_{i}+\varepsilon}\right)\right\|_{2}^{2}}_{\text{\tiny{distortion}}}+\beta\underbrace{(-\log
P_{\psi}({z}_{i}))}_{\text{\tiny{rate}}}\biggr{]},$

where the symbols $N,\varepsilon,\odot$, and $\beta$ are the number of cryo-EM
images, random noise, Hadamard product, and the positive hyper-parameter,
respectively. In addition,
$\hat{x}_{{z}_{i}+\varepsilon}=g_{\xi}(z_{i}+\varepsilon,\hat{R}_{i})$,
$z_{i}=\text{Enc}_{\zeta}(x_{i})$, $\hat{R}_{i}$ is the corresponding
estimated pose orientation to $x_{i}$, and $\hat{x}_{{z}_{i}+\varepsilon}$ is
the predicted $2$D Fourier image in the $3$D Fourier volume. The symbol $W$ is
a weight matrix for introducing the isometricity; see Yamazaki et al. (2023,
Appendix A). This objective is inspired by rate-distortion theory (Cover,
1999), like RaDOGAGA (Kato et al., 2020). After the training, for a latent
variable $z$, the corresponding $3$D density map $\hat{V}_{z}$ is
reconstructed by the trained decoder $g_{\xi^{\ast}}$.

####  2.1.2 Pathway Computing Algorithm

Since the latent space of the trained cryoTWIN is locally isometric to a space
of $3$D density maps, plausible conformational pathways can be computed via
the trained GMM $P_{\psi^{\ast}}$ using the decoder $g_{\xi^{\ast}}$; see
Algorithm 1 in Yamazaki et al. (2023). This algorithm requires two means
$\mu_{i}^{\ast}$ and $\mu_{j}^{\ast}$ as start and end points of the pathway.
Then, a Max-Flux path (Huo and Straub, 1997) on $P_{\psi^{\ast}}$ between
$\mu_{i}^{\ast}$ and $\mu_{j}^{\ast}$ is approximately computed using greedy
optimization technique. The following minimization problem is used in the
greedy optimization at $t+1$-th iteration to compute the $t+1$-th latent
variable ${z}^{i\to j}(t+1)$:

${z}^{i\to
j}(t+1)=\underset{\\{\textrm{z}\\}}{\operatorname{argmin}}\frac{1}{P_{\psi^{\ast}}(\textrm{z})}\|\textrm{z}-{z}^{i\to
j}(t)\|_{2},$

where $\\{\textrm{z}\\}$ is a set of the random latent variables. The path on
$P_{\psi^{\ast}}$ is expressed as a sequence of the latent variables:
$(\mu_{i}^{\ast},\dots,z^{i\to j}(t),\dots,\mu_{j}^{\ast})$. Thereafter, by
decoding each element in the sequence $(\mu_{i}^{\ast},\dots,z^{i\to
j}(t),\dots,\mu_{j}^{\ast})$ using $g_{\xi^{\ast}}$, the algorithm outputs a
sequence of $3$D density maps
$(\hat{V}_{\mu_{i}^{\ast}},\dots,\hat{V}_{z^{i\to
j}(t)},\dots,\hat{V}_{\mu_{j}^{\ast}})$.

####  2.1.3 Preliminary Experiments

In the numerical experiments in Yamazaki et al. (2023), the authors showed
several plausible conformational pathways of the $50$S-ribosomal dataset,
while focusing on significant $\mu_{i}^{\ast}$. According to Yamazaki et al.
(2023), the number of significant means $\mu^{\ast}$ are $30$. Thus, the
number of pairs $(\mu_{i}^{\ast},\mu_{j}^{\ast})$ is around 450. In our
preliminary experiments, we first employ the same hyper-parameters and
computational resources with Yamazaki et al. (2023, Section 4). Then, we
compute all 450 conformational pathways to summarize the pathways. As a
result, the running time based on the parallel computing is around 240
minutes.

###  2.2 Review of the Mapper Algorithm

In this section, we introduce the definition of the Mapper algorithm first.
Thereafter, we introduce existing mapper techniques, in particular, which are
applied to biological research domains.

####  2.2.1 Definition of Mapper Algorithm

Let $X$ be a topological space and let $f\colon X\to\mathbb{R}$ be a
continuous function called a filter function. We define an equivalence
relation between two elements $x$ and $y$ in $X$ by $x\sim_{f}y$ if and only
if $x$ and $y$ are in the same connected component of $f^{-1}(z)$ for some $z$
in $f(X)$. We then define the Reeb graph as follows.

######  Definition 1.

(Reeb Graph) The Reeb graph $\mathcal{R}_{f}(X)$ of a topological space $X$
computed with a filter function $f$ is the quotient space $X/\sim_{f}$.

The Mapper, introduced in Singh et al. (2007), is a statistical version of the
Reeb graph consisting of a computable approximation. It is discrete in the
sense that it is computed on a finite metric space
$(\mathbb{X}_{n}=\\{x_{0},\dots,x_{n}\\}\subseteq X,d_{X})$ with a continuous
filter function $f$, where $d_{X}$ is a metric on $X$. The procedure to
compute a Mapper graph is as follows (Chazal and Michel, 2021):

  1. 1.

Cover the range of values $f(\mathbb{X}_{n})$ with a set of consecutive
intervals $(I_{s})_{1\leq s\leq S}$ that overlap.

  2. 2.

Apply a clustering algorithm to each pre-image $f^{-1}(I_{s})$,
$s\in\\{1,\dots,S\\}$. This produces a cover
$\mathscr{C}=\\{\mathcal{C}_{1,1},\dots,\mathcal{C}_{1,K_{1}},\dots,\mathcal{C}_{S,1},\dots,\mathcal{C}_{S,K_{S}}\\}$
of $\mathbb{X}_{n}$ called the pullback cover, where $\mathcal{C}_{s,k}$ is
the $k$-th cluster of $f^{-1}(I_{s})$.

  3. 3.

The Mapper graph is the $1$-skeleton of the nerve complex of $\mathcal{C}$. It
is a graph with a vertex $v_{s,k}$ for each $\mathcal{C}_{s,k}$, and an edge
between two vertices $v_{s,k}$ and $v_{s^{\prime},k^{\prime}}$ if and only if
$\mathcal{C}_{s,k}\cap\mathcal{C}_{s^{\prime},k^{\prime}}\neq\emptyset$.

See the complementary information in A.

####  2.2.2 Existing Mapper Algorithms

In this subsection, we introduce existing Mapper techniques focusing on
biological research areas since the range of applications of Mapper techniques
are fairly vast. In Wang et al. (2018), the Mapper algorithm is compared to
other visualization methods, like t-SNE (van der Maaten and Hinton, 2008) and
PCA, in the context of single-cell RNA sequencing data analysis. Notably, a
filter function based on gene co-expression network analysis is used. Mapper
graphs are shown to conserve the continuous nature in gene expression
profiles, while also differentiating different cell types successfully. In
Dehghanpoor et al. (2021), the Mapper algorithm is used to evaluate a
preexisting protein conformational space exploration tool, that combines
Rapidly-exploring Random Trees algorithm and Monte Carlo criteria. Regular PCA
directions are used as a multidimensional filter function. The resulting graph
shows clusters that are identified as known conformations, while its topology
is found to respect an energy criteria. Mapper graphs represent topological
structure of high-dimensional data as graphs which can be regarded as a
1-dimensional simplicial complex. In Imoto and Hiraoka (2023), the authors
adopted the Mapper algorithm to data with velocity, and associated flow on
edges of the resulting Mapper graph. They applied their method to single-cell
gene expression and combined their method with the Hodge decomposition on a
graph to enhance the interpretation of the flow on the mapper graph.

The Mapper graph can also capture various features in data which has a
complicated structure. It is also known that Mapper algorithm is less
sensitive the distance metric than non-linear dimensional reduction or
geometric embedding method. The authors of Yao et al. (2009) applied Mapper
algorithm to characterize transient intermediate or transition state which are
quite crucial for the description of biomolecular folding pathways. In Nicolau
et al. (2011), a Disease-Specific Genomic Analysis is first performed on a
breast cancer dataset to produce measures of deviation between tumor and
normal tissue. This information is then introduced to the Mapper algorithm as
a filter function. Due to its ability to conserve topological information in
the dataset, the resulting graph reveals a region that corresponds to a unique
mutational profile, that is otherwise scattered across different clusters in a
regular clustering analysis.

A relaxed and more general version of the Mapper graph, that enjoys improved
stratification properties, is introduced in Oulhaj et al. (2024). It is then
used to optimize parameterized filter functions for regular Mapper graphs,
with respect to a topological risk based on persistent homology. This is shown
to produce quality Mapper graphs for 3-dimensional shapes and single cell RNA-
sequencing data. The filter functions that are investigated in these
experiments belong to a linear family and, as such, to the best of our
knowledge, our work constitutes the first use of a deep neural network as a
Mapper filter function.

Input:

$\\{z_{i}\\}_{i=1}^{N}$: a set of latent variables,

$P_{\psi}$: a trained GMM with
$\psi=\\{(\pi_{k},\mu_{k},\Sigma_{k})\\}_{k=1}^{\tilde{K}}$,

$\tilde{K}$: the number of influential mean vectors in $P_{\psi}$,

$\bar{K}(<\tilde{K})$: a natural number,

$\kappa$: the number of neighbors,

$r\in(0,1)$: an overlap rate,

$K$: the maximum number of clusters,

$f_{\theta}$: a filter function parameterized by trainable parameters
$\theta$,

$\lambda$: a positive value.

Output: An optimized mapper graph.

1 Using $\kappa$, build a kNN graph $G=(\mathcal{V},\mathcal{E})$ on
$\\{z_{i}\\}_{i=1}^{N}$, where $\mathcal{V}=\\{z_{i}\\}_{i=1}^{N}$ and
$\mathcal{E}$ is the set of the edges.

2Find top $\bar{K}$ largest values in $P_{\psi}(\mu_{k}),k=1,\dots,\tilde{K}$,
and define the corresponding Gaussian indexes as $k_{j}\;(j=1,\dots,\bar{K})$,
where $P_{\psi}(\mu_{k_{j}})\leq P_{\psi}(\mu_{k_{j+1}})$.

3for  _each epoch_ do

4 Construct a Mapper graph on $\\{z_{i}\\}_{i=1}^{N}$ based on $f_{\theta}$
and a specific cluster algorithm.

5 Compute a loss $L_{\theta}$ based on the resulting Mapper graph $M_{\theta}$

6 Compute a regularization term
$\mathrm{Reg}(G)\coloneqq\lambda\sum_{(i,j)\in\mathcal{E}}(f_{\theta}(z_{i})-f_{\theta}(z_{j}))^{2}$
based on $G$.

7 $L_{\theta}\leftarrow L_{\theta}+\mathrm{Reg}(G)$.

8 Minimize $L_{\theta}$ with respect to the parameters $\theta$.

return the final Mapper graph.

Algorithm 1 Generic computation of a Mapper graph

##  3 Proposed Method

In this section, we introduce a generic entire procedure to compute an
optimized Mapper graph which contains plausible comformational pathways. After
formulating the method, we then introduce an Energy-loss based on existing
reaction path optimization methods (see Huo and Straub (1997)), that serves as
an optimization objective in our Mapper setting.

###  3.1 General Mapper Graph Optimization

Our approach focuses on the Mapper algorithm to produce pathways between
conformations. We firstly show a generic algorithm for a computation of a
Mapper graph in Algorithm 1. Algorithm 1 takes a set of latent variables as an
input. Then we compute a loss from a Mapper graph which is constructed with a
parameterized filter function. For the construction of the Mapper graph, note
that one can apply arbitrary clustering algorithm. As a technique to
incorporate topological information into the loss, we build a kNN graph on top
of the input dataset, and add a regularization term to the loss. Finally, we
update the parameters so as the value of the loss decrease. In this way, we
obtain an optimized Mapper graph as the output.

###  3.2 Specialized Mapper Graph Optimization

The Mapper algorithm is applied to a large sample of latent space variables
with a fixed number of intervals and a fixed overlap. Two main features are
introduced in our Mapper based method:

  1. 1.

The clustering method is based on the GMM distribution that is assumed to be
that of the latent variables and estimated beforehand. The exact method is
described in Algorithm 3. This approach guarantees that the nodes in the
Mapper correspond to clusters that regroup conformations which are
structurally similar.

  2. 2.

The filter function is given by a neural network that acts on a feature
transform of the latent space. Specifically, the filter function has the form
of

$f_{\theta}(z)=DNN_{\theta}(T(z)),$

where $DNN_{\theta}$ is a deep neural network. Let
$\\{\mu_{i},\Sigma_{i}\\}_{i\in I}$ denote Gaussian distribution parameters
corresponding to notable conformations in the latent space. We also define a
transformation $T$ of a latent variable $z$ with respect to $\mu_{i}$ and
$\Sigma_{i}$ as follows:

$T\colon
z\mapsto\left((z-\mu_{i})^{\top}\cdot\Sigma_{i}\cdot(z-\mu_{i})\right)_{i\in
I}.$

The neural network is introduced for its ability to approximate a large class
of functions, allowing for a filter function that stratifies the latent space
in a way that detects conformational pathways. The parameters $\theta$ are
then optimized to minimize the total Energy-loss inside the pathways described
by the Mapper graph. More details about the specific loss used are given in
Algorithm 5. The parameters $\\{\mu_{i},\Sigma_{i}\\}_{i\in I}$ can be taken
from the Gaussian mixture model that is fit on the latent variables during the
training of the auto-encoder. Components that correspond to conformations of
interest can be selected with respect to their scale in the GMM distribution.
By means of the order with which the filter function stratifies the latent
space, the resulting Mapper graph topology and the cluster representatives are
all controlled by the deep neural network. Depending on the complexity of the
chosen neural network architecture, conformational paths of interest are
produced by our method. More details on this are given in Section 4. In order
to approach the continuous Energy-loss associated to the paths described by a
Mapper graph built on the latent space dataset, we compute a modified discrete
Energy-loss where distances are given locally by variations in the filter
values, by taking into account the gradient of the filter and using a first
order Taylor approximation; see Algorithm 4. The total loss associated to a
set of parameters $\theta$ is

$L_{\theta}=\text{Energy-loss}(M_{\theta},f_{\theta},n_{\text{steps}}).$

This total loss serves as an optimization objective in our setting; see
Algorithm 5.

We now give a specialized method to optimize a Mapper graph which contains
meaningful conformational pathways in Algorithm 2. All the algorithms used in
Algorithm 2 can be found in Algorithms 3, 4, and 5. The step count
$n_{\text{steps}}$ in Algorithm 5 is chosen to be fairly large without
impacting running time, in accordance with the theoretical guarantees relating
to convergence given in Section 3.3. It should also be noted that choosing
$n_{\text{steps}}$ to be too large can result in poor results, as the pathways
that are used to compute the loss become too long. The loss $L_{\theta}$ is
minimized using the Adam variant of stochastic gradient descent (Kingma and
Ba, 2015).

1

Input:

$\\{z_{i}\\}_{i=1}^{N}$: set of latent variables,

$P_{\psi}$: trained GMM with
$\psi=\\{(\pi_{k},\mu_{k},\Sigma_{k})\\}_{k=1}^{\tilde{K}}$,

$\kappa$: number of neighbors,

$\bar{K}(<\tilde{K}),S,B$: natural numbers,

$r\in(0,1)$: overlap rate,

$K$: maximum number of clusters,

$f_{\theta}$: filter function defined by a deep neural network parameterized
by trainable parameters $\theta$,

$\lambda$: positive value.

Output: An optimized conformational mapper graph.

2

3Using $\kappa$, build a kNN graph $G=(\mathcal{V},\mathcal{E})$ on
$\\{z_{i}\\}_{i=1}^{N}$, where $\mathcal{V}=\\{z_{i}\\}_{i=1}^{N}$ and
$\mathcal{E}$ is the set of the edges.

4Find top $\bar{K}$ largest values with $P_{\psi}(\mu_{k}),k=1,\dots,\bar{K}$,
and define the corresponding Gaussian indexes as $k_{j}\;(j=1,\dots,\bar{K})$,
where $P_{\psi}(\mu_{k_{j}})\leq P_{\psi}(\mu_{k_{j+1}})$.

5for  _each epoch_ do

6 Using $\\{z_{i}\\}_{i=1}^{N}$, a filter function $f_{\theta}$,
$S\in\mathbb{N}$, and $r\in(0,1)$, compute a set of the intervals
$\\{I_{s}\\}_{s=1}^{S}$; see Section 2.2.1.

7 Then, conduct the GMM clustering (see Algorithm 3) for each interval.

8 Based on the above clustering result, construct the mapper graph
$M_{\theta}=(\tilde{\mathcal{V}}_{\theta},\tilde{\mathcal{E}}_{\theta})$; see
Section 2.2.1.

9 Set the energy-loss $L_{\theta}$ to zero, and update $M_{\theta}$ via
minimization of $L_{\theta}$ w.r.t. $\theta$, as follows:

10 for  _each
$(\mathcal{C},\mathcal{C}^{\prime})\in\tilde{\mathcal{E}_{\theta}}$_ do

11 First, sort the latent variables in $\mathcal{C}\cup\mathcal{C}^{\prime}$
based on the filter value in ascending order.

12 Secondly, let $\iota(b)$ denote the
$\lfloor\frac{b(\mid\mathcal{C}\cup\mathcal{C}^{\prime}\mid-1)}{B}\rfloor$-th
index of the corresponding latent variable in the order, where
$b\in\\{1,\dots,B\\}$.

13 Thirdly,
$L_{\theta}\;+=\sum_{b=1}^{B-1}\frac{1}{P_{\psi}\left(\frac{z_{\iota(b)}+z_{\iota(b+1)}}{2}\right)}\text{LDA}(z_{\iota(b)},z_{\iota(b+1)},f_{\theta},\nabla
f_{\theta}(z_{\iota(b)}))$; see the function LDA in Algorithm 4.

14

15 Compute a regularization term
$\mathrm{Reg}(G)\coloneqq\lambda\sum_{(i,j)\in\mathcal{E}}(f_{\theta}(z_{i})-f_{\theta}(z_{j}))^{2}$
based on $G$.

16 $L_{\theta}\leftarrow L_{\theta}+\mathrm{Reg}(G)$.

17 $\theta\leftarrow\arg\min_{\theta}L_{\theta}$.

18

19return the final mapper graph $M_{\theta}$.

Algorithm 2 Specialized computation of a conformational Mapper graph

Input:

$\\{z_{i}\\}_{i=1}^{N}$: dataset of latent variables,

$\\{\left(\pi_{k},\mu_{k},\Sigma_{k}\right)\\}_{k=1}^{\bar{K}}$: GMM
parameters of interest,

$K$: maximum number of clusters.

Output: An estimated label of each sample $z_{i}$.

1 for  _$1\leq k\leq\bar{K}$_ do

2 $L_{k}\leftarrow\max_{i\in\\{1,\dots,N\\}}\pi_{k}\cdot P_{k}(z_{i})$, where
$P_{k}$ is the $k$-th highest Gaussian probability density with parameters.

3

4Candidates $\leftarrow$ indices of the $K$-biggest values in
$\\{L_{k}\\}_{k=1}^{\bar{K}}$

5for  _$1\leq i\leq N$_ do

6 $\text{Label}_{i}\leftarrow\mathop{\rm arg\leavevmode\nobreak\
max}\limits_{k\in\text{candidates}}\pi_{k}\cdot P_{k}(z_{i})$

return $\\{\text{Label}_{i}\\}_{i=1}^{N}$.

Algorithm 3 GMM based clustering for each interval

Input:

$z,\,z^{\prime}\in\mathbb{R}^{d}$: points,

$f$: a filter function,

$\nabla f(z)$: the gradient of $f$ at $z$.

1 for  _$i\in\\{1,\dots,d\\}$_ do

2 $\tilde{z}_{i}\leftarrow(z_{1},\dots,z^{\prime}_{i},\dots,z_{d})$, $\partial
f_{i}\leftarrow f(z)-f(\tilde{z}_{i})$.

return $\left\lVert\left(\frac{\partial f_{i}}{\nabla
f(z)_{i}}\right)_{i\in\\{1,\dots,d\\}}\right\rVert_{2}$.

Algorithm 4 Local Distance Approximation (LDA)

Input:

$(V,E)$: a Mapper graph,

$f$: a filter function,

$n_{\text{steps}}$: the number of steps.

1 for  _$(c_{i},c_{j})\in E$_ do

2 Order $c_{i}\cup c_{j}$ by filter value as $\\{z_{k}\\}_{k=0}^{\mid
c_{i}\cup c_{j}\mid-1}$ where for any $k\in\\{0,\dots,\mid c_{i}\cup
c_{j}\mid-2\\},f(z_{k})\leq f(z_{k+1})$.

3 $\text{index}(k)\leftarrow\lfloor\frac{k\cdot(\mid c_{i}\cup
c_{j}\mid-1)}{K}\rfloor$, for any $k\in\\{1,\dots,n_{\text{steps}}\\}$.

4
$l_{(c_{i},c_{j})}\coloneqq\sum_{k=1}^{n_{\text{steps}}-1}\frac{1}{P\left(\frac{z_{\text{index}(k)}+z_{\text{index}(k+1)}}{2}\right)}\text{LDA}(z_{\text{index}(k)},z_{\text{index}(k+1)},f,\nabla
f(z_{\text{index}(k)}))$.

return $\sum_{(c_{i},c_{j})\in E}l_{(c_{i},c_{j})}$.

Algorithm 5 Energy-loss

###  3.3 Theoretical Guarantees for Energy-Loss

We mathematically prove that computing plausible pathways in the latent space
is equivalent to that in the output space of cryoTWIN. Hence, the theoretical
guarantees presented in this section justify the use of random samples to
detect pathways on one hand, and gives insurance to using the latent space to
achieve this task on the other. We note that the similar statement can be
found in Yamazaki et al. (2023) without a proof.

Consider the case where we are interested in an optimal conformational path
between two protein conformation representations $z_{0}$ and $z_{\infty}$,
where $z_{0},z_{\infty}\in\mathbb{R}^{d}$ ($d\in\mathbb{N}$). We also assume
that we know a closed form $P\colon\mathbb{R}^{d}\to[0,+\infty)$ for the
probability density of the protein conformation representations, and
$P(\mathbb{R}^{d})$ is connected.

We denote by $\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$ the set of
piecewise continuous functions $f\colon[0,1]\rightarrow\mathbb{R}^{d}$, that
satisfy $f(0)=z_{0}$ and $f(1)=z_{\infty}$. A function
$f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$ will be called a
conformational path between $z_{0}$ and $z_{\infty}$. For
$f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$, we define its Energy-
loss $I_{f}$ as follows.

######  Definition 2.

(Energy-loss) Let $f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$. The
Energy-loss associated to $f$ is

$I_{f}\coloneqq\int_{0}^{1}\frac{1}{P(f(t))}\,\mathrm{d}f(t).$

The integral above, called a line integral, is defined as the following
Riemann integral

$\int_{0}^{1}\frac{1}{P(f(t))}\,\mathrm{d}f(t)\coloneqq\lim_{n\to\infty}\sum_{i=0}^{n-1}\frac{1}{P\left(\frac{f(\frac{i}{n})+f(\frac{i+1}{n})}{2}\right)}\left\|f\left(\frac{i+1}{n}\right)-f\left(\frac{i}{n}\right)\right\|_{2}.$

The optimal Energy-loss for all possible conformational paths is defined as

$S^{\ast}\coloneqq\inf_{f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)}I_{f}.$

####  3.3.1 Convergence of Energy-Loss

In our work, we wish to approach the optimal Energy-loss by using a finite $N$
sample $\\{z_{j}\\}_{j=1}^{N}$ of conformations, that independently follow a
distribution of density $P$. We then discretize the Energy-loss using
$n_{\mathrm{steps}}$ steps. Precisely, we define the following stochastic loss
as follows.

######  Definition 3.

(Stochastic Energy-loss) Let $\\{z_{j}\\}_{j=1}^{N}\subseteq\mathbb{R}^{d}$ be
a random sample drawn independently from a distribution of density $P$ and
$n_{\mathrm{steps}}\in\mathbb{N}$ such that $n_{\mathrm{steps}}>2$. The
stochastic Energy-loss with $n_{\mathrm{steps}}$ steps associated to
$\\{z_{j}\\}_{j=1}^{N}$ is

$S_{n_{\mathrm{steps}}}^{(N)}\coloneqq\min_{\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}\subseteq\\{z_{j}\\}_{j=1}^{N}}\sum_{i=0}^{n_{\mathrm{steps}}-1}\frac{1}{P\left(\frac{z_{i}+z_{i+1}}{2}\right)}\left\|z_{i+1}-z_{i}\right\|_{2},$

with the convention $z_{K}=z_{\infty}$.

The theoretical part of our work aims at proving that
$S_{n_{\mathrm{steps}}}^{(N)}$ gets arbitrarily close to $S^{\ast}$ when $N$
and $n_{\mathrm{steps}}$ go to infinity. In order to properly define this
property in terms of real sequence convergence and random variable sequence
convergence, we introduce the following quantity:

######  Definition 4.

(Discrete Optimal Energy-loss) Let $n_{\mathrm{steps}}\in\mathbb{N}$ such that
$n_{\mathrm{steps}}>2$. The discrete optimal Energy-loss with
$n_{\mathrm{steps}}$ steps is

$S_{n_{\mathrm{steps}}}\coloneqq\inf_{\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}\subseteq\mathbb{R}^{d}}\sum_{i=0}^{n_{\mathrm{steps}}-1}\frac{1}{P\left(\frac{z_{i}+z_{i+1}}{2}\right)}\left\|z_{i+1}-z_{i}\right\|_{2},$

with the same convention $z_{n_{\mathrm{steps}}}=z_{\infty}$.

We have the following result regarding the different losses that we defined
above.

######  Theorem 1.

$S_{n_{\mathrm{steps}}}\xrightarrow[n_{\mathrm{steps}}\to\infty]{}S^{\ast}.$
(1)

Moreover, for every $n_{\mathrm{steps}}\in\mathbb{N}$ such that
$n_{\mathrm{steps}}>2$, assuming that

  1. 1.

The density $P$ is continuously differentiable.

  2. 2.

The density $P$ is strictly positive almost everywhere.

  3. 3.

We now only consider discrete paths
$\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}\subseteq\mathbb{R}^{d}$ such that
$\forall i\in\\{0,\dots,n_{\mathrm{steps}}-1\\}$ $z_{i+1}\neq z_{i}$

Then we have

$S_{n_{\mathrm{steps}}}^{(N)}\xrightarrow[N\to\infty]{a.s}S_{n_{\mathrm{steps}}}.$
(2)

####  3.3.2 Relating Electronic Density Space to Latent Space with Energy-Loss

We consider the case where protein conformations can be determined by three
dimensional electronic density maps as well as by lower dimensional latent
representations. Typically, in the first case, the representations consist of
electronic density values on a three dimensional cubic grid of resolution
$\gamma$, while the latent representations are real vectors of dimension $p$.

We suppose that we have access to a decoder function
$g_{\xi}\colon\mathbb{R}^{d}\to\mathbb{R}^{\gamma^{3}}$ that decodes the
latent representations into the corresponding electronic densities. We also
suppose that all the electronic densities of interest are in the image of the
decoder $g_{\xi}(\mathbb{R}^{d})$. In practice, this setting can arise from a
Cryo-EM imagery decoding machine learning model. Given a probability density
$P_{\text{ed}}\colon g_{\xi}\left(\mathbb{R}^{d}\right)\to[0,+\infty)$ on the
space of electronic densities, we turn our interest to the optimal Energy-loss
on this space, and to how it translates to the latent space given the decoder
$g_{\xi}$. We denote

$S^{\ast}_{\mathbb{R}^{d}}\coloneqq\inf_{f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)}\int_{0}^{1}\frac{1}{P(f(t))}\,\mathrm{d}f(t),$

and

$S^{\ast}_{g_{\xi}\left(\mathbb{R}^{d}\right)}\coloneqq\inf_{g\in\mathrm{PC}^{0}\left([0,1],g_{\xi}\left(\mathbb{R}^{d}\right)\right)}\int_{0}^{1}\frac{1}{P_{\text{ed}}(g(t))}\,\mathrm{d}g(t).$

We call $S^{\ast}_{\mathbb{R}^{d}}$ and $S^{\ast}_{g_{\xi}(\mathbb{R}^{d})}$
respectively the optimal Energy objectives in the latent space and the
electronic density space respectively.

######  Lemma 1.

(Probability measure of electronic densities) Let $V$ be a random variable
valued in $g_{\xi}\left(\mathbb{R}^{d}\right)$. Assume that it admits a
probability density $P_{\text{ed}}\coloneqq
g_{\xi}\left(\mathbb{R}^{d}\right)\rightarrow[0,+\infty)$ with respect to the
measure $\mu\circ g_{\xi}^{-1}$, with $\mu$ being the Lebesgue measure in
$\mathbb{R}^{d}$. We denote by
$P\coloneqq\mathbb{R}^{d}\rightarrow[0,+\infty)$ the probability density of
$z=g_{\xi}^{-1}(V)$ with respect to $\mu$. Then we have $P_{\text{ed}}=P\circ
g_{\xi}^{-1}$.

######  Lemma 2.

(Stability of the optimal Energy-loss by composition with the decoder) Assume
that the decoder $g_{\xi}$ is a topological embedding, i.e., that it is a
homeomorphism from $\mathbb{R}^{d}$ to $g_{\xi}(\mathbb{R}^{d})$. Then we have

$\inf_{g\in\mathrm{PC}^{0}\left([0,1],g_{\xi}(\mathbb{R}^{d})\right)}\int_{0}^{1}\frac{1}{P_{\text{ed}}(g(t))}\,\mathrm{d}g(t)=\inf_{f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)}\int_{0}^{1}\frac{1}{P_{\text{ed}}(g_{\xi}\circ
f(t))}\,\mathrm{d}g_{\xi}\circ f(t).$

######  Theorem 2.

(Correspondence between Energy objectives in the latent space and the
electronic density space) Let $z$ be a random variable valued in
$\mathbb{R}^{d}$, having probability density $P$. Assume that

  1. 1.

$P_{\text{ed}}$ is the probability density of $V=g_{\xi}(z)$ with respect to
the measure $\mu\circ g_{\xi}^{-1}$,

  2. 2.

the decoder $g_{\xi}$ is a topological embedding,

  3. 3.

the decoder $g_{\xi}$ verifies the following local approximate isometry
property

$\exists C>0,\,\exists\epsilon>0,\forall
z,z^{\prime}\in\mathbb{R}^{d},\|z-z^{\prime}\|_{2}\leq\epsilon\Rightarrow\|g_{\xi}(z)-g_{\xi}(z^{\prime})\|_{2}=C\cdot\|z-z^{\prime}\|_{2}.$

Then we have that for any
$f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$,

$\int_{0}^{1}\frac{1}{P_{\text{ed}}(g_{\xi}\circ
f(t))}\,\mathrm{d}g_{\xi}\circ
f(t)=C\cdot\int_{0}^{1}\frac{1}{P(f(t))}\,\mathrm{d}f(t).$ (3)

We also have

$S^{\ast}_{g_{\xi}\left(\mathbb{R}^{d}\right)}=C\cdot
S^{\ast}_{\mathbb{R}^{d}}.$ (4)

We remark that the assumptions in Theorems 1 and 2 are verified by cryoTWIN
and were written specifically with its setting in mind. However, these
theorems are given here for arbitrary auto-encoders that verify the stated
assumptions.

##  4 Numerical Experiments

###  4.1 Setting

We use our method on the L17-depleted 50S Ribosomal intermediates dataset
(EMPIAR-10076) Davis et al. (2016). A CryoTWIN model is fit on this dataset
beforehand, which gives us access to the GMM parameters of the latent space
distribution. As described, certain Gaussian components are used to extract
learning features from the dataset and to perform clustering. Here, they
correspond to $11$ components, chosen to be all different by visual inspection
of their decoded densities and belonging to the $20$ most influential
components with respect to their scale in the GMM distribution.

###  4.2 Parameter Selection

We first fix the following parameters used to create the Mapper graph:

  * 1.

Number of intervals $S$: $25$,

  * 2.

Overlap rate $r$: $0.25$,

  * 3.

Maximum number of clusters in each interval $K$: $5$,

  * 4.

Number of steps $N_{\text{steps}}$: $10^{2}$,

  * 5.

Number of mean vectors $\bar{K}$: $24$,

  * 6.

Number of neighbors of the kNN graph $\kappa$: $15$.

We then run $300$ optimization iterations for different filter neural network
architectures with learning rate value $10^{-3}$. The neural networks used
consist of successive dense layers with rectifier activation functions (ReLU).
We refer to each architecture by the total number of layers $n_{layers}$ and
the number of units $n_{units}$ in each intermediate layer.

Figure 1: Labeled optimized Mapper graph. The label of each cluster is that of
its representative GMM mean.

We use a filter function that consists of a single layer. This corresponds to
the following filter function form:

$f_{\theta}\colon z\mapsto\left\langle T(z)\;,\theta\;\right\rangle,$

with $\theta\in\mathbb{R}^{11}$. We optimize the loss $L_{\theta}$ with
respect to the parameters $\theta$ for $300$ iterations.

###  4.3 Result and Discussion

Regarding the number of layers of the filter function, we run experiments
under several conditions. Through them, we empirically confirmed that the most
simple neural network architecture achieves the best optimization result. We
justify this by pointing at the feature transformation $T$ that is introduced
before the neural network. It gives information about the main conformations
that are present in the dataset while the neural network part of the filter
needs to map them to $\mathbb{R}$ in an order that respects the chemical
transformations possible between them. This can be achieved by one layer, as
the coefficient given to each feature can be tuned to place its corresponding
conformation arbitrarily in $\mathbb{R}$.

Experiments show that the optimization result is highly dependent on the
initial state. This is due to the roughness of the objective, which is only
smooth in subsets of the parameter space.

We evaluate the pathways given in the Mapper graph using the manually
constructed conformational graph given in Davis et al. (2016, Figure 7).

The nodes in the resulting mapper graph can be labeled in the same way their
corresponding clusters were constructed in Algorithm 3. A node is given the
conformation label of the GMM component with the highest probability of a
point in the cluster belonging to it. The GMM components are labeled
beforehand using visual inspection. The labeled Mapper graph is in Figure 1.
We constructed the labeled Mapper graph presented in Figure 1 in range of
$40$–$45$ minutes using Apple M$1$ ($16$ GB, $8$ cores).

Several points below can be drawn from the comparison of Davis’ graph and
Figure 1. We note that E3 is absent from the graph because it did not
correspond to one of the $30$ GMM components used.

  * 1.

The Mapper graph captured three accurate pathways
($\mathrm{B}\to\mathrm{D1}\to\mathrm{E1}\to\mathrm{E2}\to\mathrm{E4}\to\mathrm{E5}$,
$\mathrm{B}\to\mathrm{C2}\to\mathrm{E1}\to\mathrm{E2}\to\mathrm{E4}\to\mathrm{E5}$,
and
$\mathrm{B}\to\mathrm{C2}\to\mathrm{C3}\to\mathrm{C1}\to\mathrm{E4}\to\mathrm{E5}$),
which are given in the Davis’ graph. For the rest pathway
($\mathrm{B}\to\mathrm{D1}\to\mathrm{D2}\to\mathrm{D4}\to\mathrm{E5}$), it is
almost exact except the lack of label D3.

  * 2.

The Mapper graph captured the differentiation of label B into labels C and D.

  * 3.

The Mapper graph captured the entire distribution of labels B, C, D, and E.

  * 4.

The Mapper graph separated clusters of labels C and D, which should not be
connected each other.

##  5 Conclusion and Future Work

We introduced a method based on the Mapper algorithm for visualizing protein
conformational pathways in unsupervised manner. We based this method on
preexisting auto-encoder models that reconstruct protein 3-dimensional
electronic densities from cryo-EM images. Our method uses a parameterized
filter function by deep neural network in order to automatically detect
pathways of interest inside the latent space of trained auto-encoders, on top
of using Gaussian mixture model based clustering. We experimented on a
ribosomal intermediates dataset and found that our constructed Mapper graph
captures several features when compared to a manually created pathway graph
based on biological knowledge. While our method fails to accurately
reconstruct the connection of conformational pathways in some regions of the
latent space, its unsupervised nature and its computational efficiency make it
a fairly useful tool.

## References

  * Chapelle et al. (2010) Chapelle, O., Schlkopf, B., Zien, A., 2010.  Semi-Supervised Learning.  1st ed., The MIT Press.
  * Chazal and Michel (2021) Chazal, F., Michel, B., 2021.  An introduction to topological data analysis: Fundamental and practical aspects for data scientists.  Frontiers in artificial intelligence 4.  doi:10.3389/frai.2021.667963.
  * Cover (1999) Cover, T.M., 1999.  Elements of information theory.  John Wiley & Sons.
  * Davis et al. (2016) Davis, J.H., Tan, Y.Z., Carragher, B., Potter, C.S., Lyumkis, D., Williamson, J.R., 2016.  Modular assembly of the bacterial large ribosomal subunit.  Cell 167, 1610–1622.
  * Dehghanpoor et al. (2021) Dehghanpoor, R., Afrasiabi, F., Haspel, N., 2021.  Using topological data analysis and rrt to investigate protein conformational spaces.  bioRxiv , 2021–08.
  * Earl et al. (2017) Earl, L.A., Falconieri, V., Milne, J.L., Subramaniam, S., 2017.  Cryo-EM: beyond the microscope.  Curr. Opin. Struct. Biol. 46, 71–78.
  * Huo and Straub (1997) Huo, S., Straub, J.E., 1997.  The MaxFlux algorithm for calculating variationally optimized reaction paths for conformational transitions in many body systems at finite temperature.  Journal of Chemical Physics 107, 5000–5006.
  * Imoto and Hiraoka (2023) Imoto, Y., Hiraoka, Y., 2023.  V-mapper: topological data analysis for high-dimensional data with velocity.  Nonlinear Theory and Its Applications, IEICE 14, 92–105.
  * Kato et al. (2020) Kato, K., Zhou, J., Sasaki, T., Nakagawa, A., 2020.  Rate-distortion optimization guided autoencoder for isometric embedding in euclidean latent space, in: Proceedings of the 37th International Conference on Machine Learning, ICML 2020, pp. 5166–5176.
  * Kingma and Ba (2015) Kingma, D.P., Ba, J., 2015.  Adam: A method for stochastic optimization, in: International Conference on Learning Representations, ICLR 2015.
  * Kinman et al. (2023) Kinman, L., Powell, B., Zhong, E., Berger, B., Davis, J., 2023.  Uncovering structural ensembles from single-particle cryo-em data using cryodrgn.  Nature Protocols 18, 319–339.
  * van der Maaten and Hinton (2008) van der Maaten, L., Hinton, G., 2008.  Visualizing data using t-SNE.  Journal of Machine Learning Research 9, 2579–2605.
  * Nicolau et al. (2011) Nicolau, M., Levine, A.J., Carlsson, G., 2011.  Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival.  Proceedings of the National Academy of Sciences 108, 7265–7270.
  * Oulhaj et al. (2024) Oulhaj, Z., Carrière, M., Michel, B., 2024.  Differentiable mapper for topological optimization of data representation.  arXiv:2402.12854.
  * Shalev-Shwartz and Ben-David (2014) Shalev-Shwartz, S., Ben-David, S., 2014.  Understanding Machine Learning: From Theory to Algorithms.  Cambridge University Press, USA.
  * Barroso da Silva et al. (2020) Barroso da Silva, F.L., Carloni, P., Cheung, D., Cottone, G., Donnini, S., Foegeding, E.A., Gulzar, M., Jacquier, J.C., Lobaskin, V., MacKernan, D., Mohammad Hosseini Naveh, Z., Radhakrishnan, R., Santiso, E.E., 2020.  Understanding and controlling food protein structure and function in foods: Perspectives from experiments and computer simulations.  Annual Review of Food Science and Technology 11, 365–387.
  * Singh et al. (2007) Singh, G., Mémoli, F., Carlsson, G.E., et al., 2007.  Topological methods for the analysis of high dimensional data sets and 3d object recognition.  PBG@ Eurographics 2, 091–100.
  * Wang et al. (2018) Wang, T., Johnson, T., Zhang, J., Huang, K., 2018.  Topological methods for visualization and analysis of high dimensional single-cell rna sequencing data, in: BIOCOMPUTING 2019: Proceedings of the Pacific Symposium, World Scientific. pp. 350–361.
  * Wu et al. (2022) Wu, Z., Chen, E., Zhang, S., Ma, Y., Mao, Y., 2022.  Visualizing conformational space of functional biomolecular complexes by deep manifold learning.  International Journal of Molecular Sciences 23, 8872.
  * Yamazaki et al. (2023) Yamazaki, K., Wada, Y., Tokuhisa, A., Wada, M., Katoh, T., Umeda, Y., Okuno, Y., Nakagawa, A., 2023.  An auto-encoder to reconstruct structure with cryo-em images via theoretically guaranteed isometric latent space, and its application for automatically computing the conformational pathway, in: Medical Image Computing and Computer Assisted Interventions, MICCAI 2023.
  * Yao et al. (2009) Yao, Y., Sun, J., Huang, X., Bowman, G.R., Singh, G., Lesnick, M., Guibas, L.J., Pande, V.S., Carlsson, G., 2009.  Topological methods for exploring low-density states in biomolecular folding pathways.  The Journal of chemical physics 130.
  * Zhong et al. (2021) Zhong, E.D., Bepler, T., Berger, B., Davis, J.H., 2021.  CryoDRGN: reconstruction of heterogeneous cryo-EM structures using neural networks.  Nature Methods 18, 176–185.
  * Zhong et al. (2020) Zhong, E.D., Bepler, T., Davis, J.H., Berger, B., 2020.  Reconstructing continuous distributions of 3D protein structure from cryo-EM images, in: International Conference on Learning Representations, ICLR 2020.

##  Appendix A Complements of the construction of a Mapper graph

There are a number of choices to be made when computing a Mapper from a point
cloud : (i) Choice of $(I_{s})_{1\leq s\leq S}$ : a popular practice when
choosing a cover for the range of values of the filter is to choose intervals
of the same length $\ell$, where no more than two intervals can intersect at
once. The overlap rate $r$ between intervals is also constant (and is
therefore less than $1/2$). By doing this, we can tune the cover through $r$
and $g$. (ii) Choice of the clustering algorithm : It is possible to cluster
the pre-images using any existing algorithm. Another popular choice however is
to construct a $\delta$-neighborhood graph and to look at its connected
components as clusters. A $\delta$-neighborhood graph is a graph with vertices
corresponding to each point in $\mathbb{X}_{n}$ and with an edge between two
vertices if and only if the distance between the two points is less than
$\delta$.

A mesh of a human model in 3 dimensions is in Figure 2. We wish to compute the
mapper on the set of vertices of the mesh with the height coordinate as our
filter function. We choose to cover the range of height values in the manner
we described above, with $25$ intervals of the same length with an overlap of
$30\%$. For clustering, we use the KMeans algorithm with $3$ clusters. The
resulting graph is also represented in Figure 2.

Figure 2:  On the left: 3-dimensional mesh of a human shape colored in terms
of height. On the right: Mapper graph computed using the vertices of the mesh,
height as a filter, 25 equal intervals with 30% overlap and KMeans clustering
with 3 clusters.

Here, we represent the mean value of the filter function on each cluster using
the color of its vertex in the Mapper graph. The Mapper graph we computed can
be seen as a skeleton of the shape of a human : we can distinguish two legs, a
torso and two arms.

##  Appendix B Proofs

###  B.1 Proof of Equation (1)

###### Proof.

We firstly show that $\\{S_{n_{\mathrm{steps}}}\\}_{n_{\mathrm{steps}}\geq 2}$
is lower bounded by $0$ and non-increasing. Let
$\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}\subseteq\mathbb{R}^{d}$, and
consider
$\\{z^{\prime}_{i}\\}_{i=1}^{n_{\mathrm{steps}}}\subseteq\mathbb{R}^{d}$ such
that $z^{\prime}_{1}=z_{1}$ and for any
$i\in\\{2,\dots,n_{\mathrm{steps}}\\}$, $z^{\prime}_{i}=z_{i-1}$ holds. Note
that we always have
$z_{n_{\mathrm{steps}}}=z^{\prime}_{n_{\mathrm{steps}}+1}=z_{\infty}$. We then
have

$\sum_{i=0}^{n_{\mathrm{steps}}-1}\frac{1}{P\left(\frac{z_{i}+z_{i+1}}{2}\right)}\left\|z_{i+1}-z_{i}\right\|_{2}=\sum_{i=0}^{n_{\mathrm{steps}}}\frac{1}{P\left(\frac{z^{\prime}_{i}+z^{\prime}_{i+1}}{2}\right)}\left\|z^{\prime}_{i+1}-z^{\prime}_{i}\right\|_{2}.$

Accordingly, we have

$\begin{split}\left\\{\sum_{i=0}^{n_{\mathrm{steps}}-1}\frac{1}{P\left(\frac{z_{i}+z_{i+1}}{2}\right)}\left\|z_{i+1}-z_{i}\right\|_{2},\,\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}\subseteq\mathbb{R}^{d}\right\\}\\\
\subseteq\left\\{\sum_{i=0}^{n_{\mathrm{steps}}}\frac{1}{P\left(\frac{z_{i}+z_{i+1}}{2}\right)}\left\|z_{i+1}-z_{i}\right\|_{2},\,\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}}\subseteq\mathbb{R}^{d}\right\\}.\end{split}$

Therefore $S_{n_{\mathrm{steps}}+1}\leq S_{n_{\mathrm{steps}}}$ holds by the
definition of the infimum. We showed that
$\\{S_{n_{\mathrm{steps}}}\\}_{n_{\mathrm{steps}}\geq 2}$ is convergent, and
we furthermore have

$\lim_{K\to\infty}S_{K}=\inf_{K\geq 2}\\{S_{K}\\}.$

We now show that this limit is exactly $S^{\ast}$. We firstly have
$S^{\ast}\leq\lim_{K\to\infty}S_{n_{\mathrm{steps}}}$ because for any
$n_{\mathrm{steps}}\geq 2$, $S^{\ast}\leq S_{n_{\mathrm{steps}}}$ holds. Let
$n_{\mathrm{steps}}\in\mathbb{N}$ such that $n_{\mathrm{steps}}>2$, and fix a
discrete path
$\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}\subseteq\mathbb{R}^{d}$. Then let us
consider the piecewise constant function $g\colon[0,1]\to\mathbb{R}^{d}$ such
that for each $i\in\\{0,\dots,n_{\mathrm{steps}}-1\\}$ and for any
$t\in[\frac{i}{n_{\mathrm{steps}}},\frac{i+1}{n_{\mathrm{steps}}})$,
$g(t)=z_{i}$ and $g(1)=z_{\infty}$ hold. We then have

$I_{g}=\sum_{i=0}^{n_{\mathrm{steps}}-1}\frac{1}{P\left(\frac{z_{i}+z_{i+1}}{2}\right)}\left\|z_{i+1}-z_{i}\right\|_{2}.$

This is because the sequence of Riemann sums, which limit is by definition
equal to $I_{g}$, is stationary after rank $n_{\mathrm{steps}}$ and its terms
are equal to the sum above. Since
$g\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$,

$S^{\ast}\leq\sum_{i=0}^{n_{\mathrm{steps}}-1}\frac{1}{P\left(\frac{z_{i}+z_{i+1}}{2}\right)}\left\|z_{i+1}-z_{i}\right\|_{2}$

holds for all
$\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}\subseteq\mathbb{R}^{d}$. Hence we
conclude that $S^{\ast}\leq S_{n_{\mathrm{steps}}}$.

We secondly show $\lim_{n_{\mathrm{steps}}\to\infty}S_{n_{\mathrm{steps}}}\leq
S^{\ast}$. Let $f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$. By
definition of $I_{f}$,

$\forall\epsilon>0,\exists n_{f,\epsilon}\in\mathbb{N}\text{ such that
}\forall n\geq n_{f,\epsilon},$
$\left|\sum_{i=0}^{n-1}\frac{1}{P\left(\frac{f(\frac{i}{n})+f(\frac{i+1}{n})}{2}\right)}\left\|f\left(\frac{i+1}{n}\right)-f\left(\frac{i}{n}\right)\right\|_{2}-I_{f}\right|\leq\epsilon.$

Hence,

$\forall\epsilon>0,\exists n_{f,\epsilon}\in\mathbb{N}\text{ such that
},S_{n_{f,\epsilon}}\leq I_{f}+\epsilon.$

Moreover, since
$\lim_{n_{\mathrm{steps}}\to\infty}S_{n_{\mathrm{steps}}}=\inf_{n_{\mathrm{steps}}\geq
2}\\{S_{n_{\mathrm{steps}}}\\}$,
$\lim_{n_{\mathrm{steps}}\to\infty}S_{n_{\mathrm{steps}}}\leq I_{f}+\epsilon$
follows for any $\epsilon>0$. Therefore, for any
$f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$,
$\lim_{n_{\mathrm{steps}}\to\infty}S_{n_{\mathrm{steps}}}\leq I_{f}.$ holds.
Therefore, we conclude that
$\lim_{n_{\mathrm{steps}}\to\infty}S_{n_{\mathrm{steps}}}\leq S^{\ast}$.
Finally we have

$S_{n_{\mathrm{steps}}}\xrightarrow[n_{\mathrm{steps}}\to\infty]{}S^{\ast}.$

∎

###  B.2 Proof of Equation (2)

###### Proof.

For the rest of the proof, we fix $n_{\mathrm{steps}}\in\mathbb{N}$ such that
$n_{\mathrm{steps}}>2$.
We firstly work at a fixed elementary event in the probability space where our
random sequences are defined. Note that
$\\{S_{n_{\mathrm{steps}}}^{(N)}\\}_{N\geq 1}$ is lower bounded by $0$ and
non-increasing. Let $N\geq n_{\mathrm{steps}}$. Since
$\\{z_{j}\\}_{j=1}^{N}\subseteq\\{z_{j}\\}_{j=1}^{N+1}$, we have that
$S_{n_{\mathrm{steps}}}^{(N+1)}\leq S_{n_{\mathrm{steps}}}^{(N)}$. Therefore,
$\\{S_{n_{\mathrm{steps}}}^{(N)}\\}_{N\geq 1}$ converges and its limit is

$\lim_{N\to\infty}S_{n_{\mathrm{steps}}}^{(N)}=\inf_{N\geq
2}\\{S_{n_{\mathrm{steps}}}^{(N)}\\}.$

We now have that $\\{z_{j}\\}_{j=1}^{N}\subseteq\mathbb{R}^{d}$ for every
$N\geq 1$. This shows that for every realization of the random sequence

$\forall N\geq 1\,:\,S_{n_{\mathrm{steps}}}\leq S_{n_{\mathrm{steps}}}^{(N)}$

and

$S_{n_{\mathrm{steps}}}\leq\lim_{N\to\infty}S_{n_{\mathrm{steps}}}^{(N)}.$

Let us consider the following function $S$ defined as

$\displaystyle S\colon{(\mathbb{R}^{d})}^{n_{\mathrm{steps}}-1}$
$\displaystyle\longrightarrow\mathbb{R},$
$\displaystyle\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}$
$\displaystyle\longmapsto\sum_{i=0}^{n_{\mathrm{steps}}-1}\frac{1}{P\left(\frac{z_{i}+z_{i+1}}{2}\right)}\left\|z_{i+1}-z_{i}\right\|_{2}.$

We fix a discrete path
$z=\\{z_{i}\\}_{i=1}^{n_{\mathrm{steps}}-1}\subseteq\mathbb{R}^{d}$. As stated
in Theorem 1, for any $i\in\\{0,\dots,n_{\mathrm{steps}}-1\\}$, $z_{i+1}\neq
z_{i}$ and $P$ is smooth. This means that $S$ is also smooth in some open ball
$B(z,\epsilon_{z})$ centered at $z$ with radius $\epsilon_{z}$. Since $S$ is
smooth in $B(z,\epsilon_{z})$, it is locally Lipschitz continuous in the
closed ball $\bar{B}(z,\frac{\epsilon_{z}}{2})$ for some constant $L_{z}$.
This comes from the mean value theorem and the fact that the gradient of $S$
is bounded on the compact set $\bar{B}(z,\frac{\epsilon_{z}}{2})$.

Let $\epsilon>0$. Then for any
$z^{\prime}\in\bar{B}(z,\frac{\epsilon_{z}}{2})\cap\bar{B}(z,\frac{\epsilon}{L_{z}})$
we have

$\displaystyle\mid S(z)-S(z^{\prime})\mid$ $\displaystyle\leq
L_{z}\|z-z^{\prime}\|_{2}$ $\displaystyle\leq\epsilon.$

Furthermore, $P$ is strictly positive almost everywhere. This means that with
the notation
$A_{z}=\bar{B}(z,\frac{\epsilon_{z}}{2})\cap\bar{B}(z,\frac{\epsilon}{L_{z}})$,

$\int_{A_{z}}P(v_{1})\cdots
P(v_{n_{\mathrm{steps}}-1})\mathrm{d}v_{1}\cdots\mathrm{d}v_{n_{\mathrm{steps}}-1}>0$

holds. Therefore, almost surely, there exists a rank $N_{z}$ after which there
exists $z^{\prime}\in A_{z}$ such that
$z^{\prime}\subseteq\\{z_{j}\\}_{j=1}^{N_{z}}$. Then the following proposition
holds:

$\forall\epsilon>0,\exists N_{z}\in\mathbb{N}\text{ almost surely such that
}S_{n_{\mathrm{steps}}}^{(N_{z})}\leq S(z)+\epsilon,$

which implies that $\lim_{N\to\infty}S_{n_{\mathrm{steps}}}^{(N)}\leq
S(z)+\epsilon$ for any $\epsilon>0$ and in a subset of measure $1$ of the
probability space. Hence, we have
$\lim_{N\to\infty}S_{n_{\mathrm{steps}}}^{(N)}\leq S(z).$ We proved the
inequality above for an arbitrary
$z\in{(\mathbb{R}^{d})}^{n_{\mathrm{steps}}-1}$. We therefore have

$\lim_{N\to\infty}S_{n_{\mathrm{steps}}}^{(N)}\leq S_{n_{\mathrm{steps}}}.$

We finally have

$S_{n_{\mathrm{steps}}}^{(N)}\xrightarrow[N\to\infty]{a.s}S_{n_{\mathrm{steps}}}.$

∎

###  B.3 Proof of Lemma 1

###### Proof.

Let $A$ be a measurable set in $\mathbb{R}^{d}$. Then we have

$\displaystyle\mathrm{Pr}\left(V\in g_{\xi}\left(A\right)\right)$
$\displaystyle=\mathrm{Pr}\left(z\in A\right)$
$\displaystyle=\int_{A}P(z)\,\mathrm{d}\mu$
$\displaystyle=\int_{g_{\xi}\left(A\right)}P\circ
g_{\xi}^{-1}(V)\mathrm{d}\mu\circ g_{\xi}^{-1}.$

This property holds for all measurable sets $A$. Hence we conclude that
$P_{\mathrm{ed}}=P\circ g_{\xi}^{-1}$ by the definition of a probability
density. ∎

###  B.4 Proof of Lemma 2

###### Proof.

Set $A=\\{g_{\xi}\circ f\mid f\in\mathrm{PC}^{0}([0,1],\mathbb{R}^{d})\\}.$
Lemma 2 can be proved by showing
$A=\mathrm{PC}^{0}\left([0,1],g_{\xi}(\mathbb{R}^{d})\right).$ We firstly
prove $A\subseteq\mathrm{PC}^{0}([0,1],g_{\xi}(\mathbb{R}^{d}))$. Let
$f\in\mathrm{PC}^{0}([0,1],\mathbb{R}^{d})$. Since $g_{\xi}$ is a topological
embedding, then it is continuous on $\mathbb{R}^{d}$. Therefore, $g_{\xi}\circ
f\in\mathrm{PC}^{0}([0,1],g_{\xi}(\mathbb{R}^{d}))$ as a composition of
continuous and piecewise continuous functions.

We secondly prove $\mathrm{PC}^{0}([0,1],g_{\xi}(\mathbb{R}^{d}))\subseteq A$.
Let $g\in\mathrm{PC}^{0}([0,1],g_{\xi}(\mathbb{R}^{d}))$. Since $g$ is valued
in $g_{\xi}(\mathbb{R}^{d})$, then $g_{\xi}^{-1}\circ g$ is well defined. As
$g_{\xi}$ is a topological embedding, $g_{\xi}^{-1}$ is continuous on
$g_{\xi}(\mathbb{R}^{d})$ which leads $g_{\xi}^{-1}\circ
g\in\mathrm{PC}^{0}([0,1],\mathbb{R}^{d})$ as a composition of continuous and
piecewise continuous functions. Furthermore, since
$g=g_{\xi}\circ\left(g_{\xi}^{-1}\circ g\right)$, then consequently $g\in A$.
Accordingly, we conclude that
$A=\mathrm{PC}^{0}\left([0,1],g_{\xi}\left(\mathbb{R}^{d}\right)\right).$ We
also have and

$\inf_{g\in\mathrm{PC}^{0}\left([0,1],g_{\xi}\left(\mathbb{R}^{d}\right)\right)}\int_{0}^{1}\frac{1}{P_{\mathrm{ed}}(g(t))}\,\mathrm{d}g(t)=\inf_{f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)}\int_{0}^{1}\frac{1}{P_{\mathrm{ed}}(g_{\xi}\circ
f(t))}\,\mathrm{d}g_{\xi}\circ f(t)$

by uniqueness of the infimum. ∎

###  B.5 Proof of Theorem 2

###### Proof.

First, we prove the equation (3). Let
$f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)$. By the definition of
piecewise continuous functions, there exists a family of compact subsets
$\\{[a_{i},b_{i}]\\}_{i\in I}$ of $[0,1]$ such that $I$ is finite and the
restriction of $f$ on each $[a_{i},b_{i}]$ can be continuously extended into a
continuous function $f_{i}\colon[a_{i},b_{i}]\to\mathbb{R}^{d}$. We also have

$\int_{0}^{1}\frac{1}{P(f(t))}\,\mathrm{d}f(t)=\sum_{i\in
I}\int_{a_{i}}^{b_{i}}\frac{1}{P(f_{i}(t))}\,\mathrm{d}f_{i}(t).$

Since $g_{\xi}$ is continuous, then $g_{\xi}\circ f_{i}$ is continuous for any
$i\in I$, and we also have

$\int_{0}^{1}\frac{1}{P_{\mathrm{ed}}(g_{\xi}\circ
f(t))}\,\mathrm{d}g_{\xi}\circ f(t)=\sum_{i\in
I}\int_{a_{i}}^{b_{i}}\frac{1}{P_{\mathrm{ed}}(g_{\xi}\circ
f_{i}(t))}\,\mathrm{d}g_{\xi}\circ f_{i}(t).$

Let $i\in I$, then it holds that

$\begin{split}&\int_{a_{i}}^{b_{i}}\frac{1}{P_{\mathrm{ed}}(g_{\xi}\circ
f_{i}(t))}\,\mathrm{d}g_{\xi}\circ f_{i}(t)\\\
&=\lim_{n\to\infty}\sum_{i=0}^{n-1}\frac{1}{P_{\mathrm{ed}}\left(g_{\xi}\circ
f_{i}\circ t_{i}(\frac{i}{n})\right)}\left\|g_{\xi}\circ f_{i}\circ
t_{i}\left(\frac{i+1}{n}\right)-g_{\xi}\circ f_{i}\circ
t_{i}\left(\frac{i}{n}\right)\right\|_{2},\end{split}$ (5)

where $t_{i}\,:\,x\mapsto a_{i}+(b_{i}-a_{i})x.$ We will refer to the final
term in the equation 5 as the Riemann sum.

Let $n\in\mathbb{N}$, $f_{i}$ is continuous on a compact set $[a_{i},b_{i}]$.
By the Heine-Cantor theorem, it is therefore uniformly continuous on
$[a_{i},b_{i}]$. Therefore, $\exists\delta>0$ such that

$\forall i\in\\{0,\dots,n-1\\},\,\left\lvert
t_{i}\left(\frac{i+1}{n}\right)-t_{i}\left(\frac{i}{n}\right)\right\rvert\leq\delta\implies\left\|f_{i}\circ
t_{i}\left(\frac{i+1}{n}\right)-f_{i}\circ
t_{i}\left(\frac{i}{n}\right)\right\|_{2}\leq\epsilon.$

However, it holds that

$\left\lvert
t_{i}\left(\frac{i+1}{n}\right)-t_{i}\left(\frac{i}{n}\right)\right\rvert=\frac{b_{i}-a_{i}}{n},$

and

$\frac{b_{i}-a_{i}}{n}\xrightarrow[n\to\infty]{}0.$

The latter point means that

$\exists N\in\mathbb{N},\,\forall n\geq N,\,\frac{b_{i}-a_{i}}{n}\leq\delta.$

Therefore, it holds that

$\exists N\in\mathbb{N},\,\forall n\geq N,\,\forall
i\in\\{0,\dots,n-1\\},\,\left\|f_{i}\circ
t_{i}\left(\frac{i+1}{n}\right)-f_{i}\circ
t_{i}\left(\frac{i}{n}\right)\right\|_{2}\leq\epsilon,$

which by the third assumption in Theorem 2 implies that

$\exists N\in\mathbb{N},\,\forall n\geq N,\,\forall i\in\\{0,\dots,n-1\\},$
$\left\|g_{\xi}\circ f_{i}\circ t_{i}\left(\frac{i+1}{n}\right)-g_{\xi}\circ
f_{i}\circ t_{i}\left(\frac{i}{n}\right)\right\|_{2}=C\cdot\left\|f_{i}\circ
t_{i}\left(\frac{i+1}{n}\right)-f_{i}\circ
t_{i}\left(\frac{i}{n}\right)\right\|_{2}.$

Moreover, by the first assumption in Theorem 2 and Lemma 1, it holds that

$P_{\mathrm{ed}}\left(g_{\xi}\circ f_{i}\circ
t_{i}\left(\frac{i}{n}\right)\right)=P\left(f_{i}\circ
t_{i}\left(\frac{i}{n}\right)\right)$

for any $n\in\mathbb{N}$ and any $i\in\\{0,\dots,n-1\\}$. By substituting
these two equalities in the Riemann sum when $n$ goes to infinity, we get

$\int_{a_{i}}^{b_{i}}\frac{1}{P_{\mathrm{ed}}(g_{\xi}\circ
f_{i}(t))}\,\mathrm{d}g_{\xi}\circ
f_{i}(t)=C\cdot\int_{a_{i}}^{b_{i}}\frac{1}{P(f_{i}(t))}\,\mathrm{d}f_{i}(t).$

Consequently, we obtain

$\int_{0}^{1}\frac{1}{P_{\mathrm{ed}}(g_{\xi}\circ
f(t))}\,\mathrm{d}g_{\xi}\circ
f(t)=C\cdot\int_{0}^{1}\frac{1}{P(f(t))}\,\mathrm{d}f(t).$

We then secondly prove the equation (4). By the second assumption in Theorem 2
and Lemma 2, we have

$\inf_{g\in\mathrm{PC}^{0}\left([0,1],g_{\xi}\left(\mathbb{R}^{d}\right)\right)}\int_{0}^{1}\frac{1}{P_{\mathrm{ed}}(g(t))}\,\mathrm{d}g(t)=\inf_{f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)}\int_{0}^{1}\frac{1}{P_{\mathrm{ed}}(g_{\xi}\circ
f(t))}\,\mathrm{d}g_{\xi}\circ f(t).$

However, as we showed, we have

$\int_{0}^{1}\frac{1}{P_{\mathrm{ed}}(g_{\xi}\circ
f(t))}\,\mathrm{d}g_{\xi}\circ
f(t)=C\cdot\int_{0}^{1}\frac{1}{P(f(t))}\,\mathrm{d}f(t).$

for any $f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right).$ Then, by
property of the infimum we conclude that

$\inf_{g\in\mathrm{PC}^{0}\left([0,1],g_{\xi}\left(\mathbb{R}^{d}\right)\right)}\int_{0}^{1}\frac{1}{P_{\mathrm{ed}}(g(t))}\,\mathrm{d}g(t)=C\cdot\inf_{f\in\mathrm{PC}^{0}\left([0,1],\mathbb{R}^{d}\right)}\int_{0}^{1}\frac{1}{P(f(t))}\,\mathrm{d}f(t).$

In other words,

$S^{\ast}_{g_{\xi}\left(\mathbb{R}^{d}\right)}=C\cdot
S^{\ast}_{\mathbb{R}^{d}}.$

∎