Kwangho Kim
email: kwanghk@cmu.edu

Jisu Kim
email: jisu.kim@inria.fr

Manzil Zaheer
email: manzilzaheer@google.com

Joon Sik Kim
email: joonsikk@cs.cmu.edu

Frederic Chazal
email: frederic.chazal@inria.fr

Larry Wasserman

PLLay: Efficient Topological Layer based on Persistence Landscapes

We propose PLLay, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit the underlying topological features of the input data structure. In this work, we show differentiability with respect to layer inputs, for a general persistent homology with arbitrary filtration. Thus, our proposed layer can be placed anywhere in the network and feed critical information on the topological features of input data into subsequent layers to improve the learnability of the networks toward a given task. A taskoptimal structure of PLLay is learned during training via backpropagation, without requiring any input featurization or data preprocessing. We provide a novel adaptation for the DTM function-based filtration, and show that the proposed layer is robust against noise and outliers through a stability analysis. We demonstrate the effectiveness of our approach by classification experiments on various datasets.

Introduction

With its strong generalizability, deep learning has been pervasively applied in machine learning. To improve the learnability of deep learning models, various techniques have been proposed. Some of them have achieved an efficient data processing method through specialized layer structures; for instance, inserting a convolutional layer greatly improves visual object recognition and other tasks in computer vision [e.g., [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF][START_REF] Lecun | Lenet-5, convolutional neural networks[END_REF]. On the other hand, a large body of recent work focuses on optimal architecture of deep network [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF][START_REF] He | Deep residual learning for image recognition[END_REF][START_REF] Szegedy | Going deeper with convolutions[END_REF][START_REF] Albelwi | Automated optimal architecture of deep convolutional neural networks for image recognition[END_REF].

In this paper, we explore an alternative way to enhance the learnability of deep learning models by developing a novel topological layer which feeds the significant topological features of the underlying data structure in an arbitrary network. The power of topology lies in its capacity which differentiates sets in topological spaces in a robust and meaningful geometric way [START_REF] Carlsson | Topology and data[END_REF][START_REF] Ghrist | Barcodes: the persistent topology of data[END_REF]. It provides important insights into the global "shape" of the data structure via persistent homology [ [START_REF] Zomorodian | Computing persistent homology[END_REF]. The use of topological methods in data analysis has been limited by the difficulty of combining the main tool of the subject, persistent homology, with statistics and machine learning. Nonetheless, a series of recent studies have reported notable successes in utilizing topological methods in data analysis [e.g., [START_REF] Zhu | Persistent homology: An introduction and a new text representation for natural language processing[END_REF][START_REF] Meryll Dindin | Topological data analysis for arrhythmia detection through modular neural networks[END_REF][START_REF] Nanda | Simplicial models and topological inference in biological systems[END_REF][START_REF] Christopher | quasi) periodicity quantification in video data, using topology[END_REF][START_REF] Lee M Seversky | On time-series topological data analysis: New data and opportunities[END_REF][START_REF] Gamble | Exploring uses of persistent homology for statistical analysis of landmark-based shape data[END_REF][START_REF] Cássio | Persistent homology for time series and spatial data clustering[END_REF][START_REF] Umeda | Time series classification via topological data analysis[END_REF][START_REF] Liu | Applying topological persistence in convolutional neural network for music audio signals[END_REF][START_REF] Venkataraman | Persistent homology of attractors for action recognition[END_REF][START_REF] Emrani | Persistent homology of delay embeddings and its application to wheeze detection[END_REF] There are at least three benefits of utilizing the topological layer in deep learning; 1) we can efficiently extract robust global features of input data that otherwise would not be readily accessible via traditional feature maps, 2) an optimal structure of the layer for a given task can be easily embodied via backpropagation during training, and 3) with proper filtrations it can be applied to arbitrarily complicated data structure even without any data preprocessing.

Related Work. The idea of incorporating topological concepts into deep learning has been explored only recently, mostly via feature engineering perspective where we use some fixed, predefined features that contain topological information [e.g., Dindin et al., 2020[START_REF] Umeda | Time series classification via topological data analysis[END_REF][START_REF] Liu | Applying topological persistence in convolutional neural network for music audio signals[END_REF]. [START_REF] William | On characterizing the capacity of neural networks using algebraic topology[END_REF], [START_REF] Rieck | Neural persistence: A complexity measure for deep neural networks using algebraic topology[END_REF] proposed a complexity measure for neural network architectures based on topological data analysis. [START_REF] Carlsson | Topological approaches to deep learning[END_REF] applied topological approaches to deep convolutional networks to understand and improve the computations of the network. [START_REF] Hofer | Deep learning with topological signatures[END_REF] first developed a technique to input persistence diagrams into neural networks by introducing their own topological layer. [START_REF] Mathieu Carrière | Perslay: A neural network layer for persistence diagrams and new graph topological signatures[END_REF] proposed a network layer for persistence diagrams built on top of graphs. [START_REF] Poulenard | Topological function optimization for continuous shape matching[END_REF], [START_REF] Brüel Gabrielsson | A topology layer for machine learning[END_REF], [START_REF] Hofer | Connectivity-optimized representation learning via persistent homology[END_REF], [START_REF] Moor | Topological autoencoders[END_REF] also proposed various topology loss functions and layers applied to deep learning. Nevertheless, all the previous approaches suffer from at least one or more of the following limitations: 1) they rely on a particular parametrized map or filtration, 2) they lack stability results or the stability is limited to a particular type of input data representation, and 3) most importantly, the differentiability of persistent homology is not guaranteed with respect to the layer's input therefore we can not place the layer in the middle of deep networks in general.

Contribution. This paper presents a new topological layer, PLLay (Persistence Landscape-based topological Layer: see Figure 1 for an illustration), that does not suffer from the above limitations. Our topological layer does not rely on a particular filtration or a parametrized mapping but still shows favorable theoretical properties. The proposed layer is designed based on the weighted persistence landscapes to be less prone to extreme topological distortions. We provide a tight stability bound that does not depend on the input complexity, and show the stability with respect to input perturbations. We also provide a novel adaptation for the DTM function-based filtration, and analyze the stability property. Importantly, we guarantee the differentiability of our layer with respect to the layer's input.

Reproducibility. The code for PLLay is available at https://github.com/jisuk1/pllay/.

Background and definitions

Topological data analysis (TDA) is a recent and emerging field of data science that relies on topological tools to infer relevant features for possibly complex data [START_REF] Carlsson | Topology and data[END_REF]. In this section, we briefly review basic concepts and main tools in TDA which we will harness to develop our topological layer in this paper. We refer interested readers to [START_REF] Chazal | An introduction to topological data analysis: fundamental and practical aspects for data scientists[END_REF], [START_REF] Hatcher | Algebraic Topology[END_REF], [START_REF] Edelsbrunner | Computational topology: an introduction[END_REF], [START_REF] Chazal | Proximity of persistence modules and their diagrams[END_REF]Chazal et al. [ , 2016b] ] for details and formal definitions.

Simplicial complex, persistent homology, and diagrams

When inferring topological properties of X, a subset of R d , from a finite collection of samples X, we rely on a simplicial complex K, a discrete structure built over the observed points to provide a topological approximation of the underlying space. Two common examples are the Čech complex and the Vietoris-Rips complex. The Čech complex is the simplicial complex where k-simplices correspond to the nonempty intersection of k + 1 balls centered at vertices. The Vietoris-Rips (or simply Rips) complex is the simplicial complex where simplexes are built based on pairwise distances among its vertices. We refer to Appendix A for formal definitions.

A collection of simplicial complexes

F = {K a ⊂ K : a ∈ R} satisfying K a ⊂ K b whenever a ≤ b is called a filtration of K.
A typical way of setting the filtration is through a monotonic function on the simplex. A function f :

K → R is monotonic if f (ς) ≤ f (τ ) whenever ς is a face of τ . If we let K a := f -1 (-∞, a],
then the monotonicity implies that K a is a subcomplex of K and K a ⊂ K b whenever a ≤ b. In this paper, we assume that the filtration is built upon a monotonic function.

Persistent homology is a multiscale approach to represent the topological features of the complex K, and can be represented in the persistence diagram. For a filtration F and for each nonnegative k, we keep track of when k-dimensional homological features (e.g., 0-dimension: connected component, 1-dimension: loop, 2-dimension: cavity,. . .) appear and disappear in the filtration. If a homological feature α i appears at b i and disappears at d i , then we say α i is born at b i and dies at d i . By considering these pairs (b i , d i ) as points in the plane, one obtains the persistence diagram defined as follows.

Definition 2.1 Let R 2 * := {(b, d) ∈ (R ∪ ∞) 2 : d > b}. A persistence diagram D is a finite multiset of {p : p ∈ R 2 * }.
We let D denote the set of all such D's.

We will use D X , D X as shorthand notations for the persistence diagram drawn from the simplicial complex constructed on original data source X, X, respectively.

Lastly, we define the following metrics to measure the distance between two persistence diagrams.

Definition 2.2 (Bottleneck and Wasserstein distance) Given two persistence diagrams D and D , their bottleneck distance (d B ) and q-th Wasserstein distance (W q ) for q ≥ 1 are defined by

d B (D, D ) = inf γ∈Γ sup p∈ D p -γ(p) ∞ , W q (D, D ) = inf γ∈Γ p∈ D p -γ(p) q ∞ 1 q , ( 1

Note that for all

q ∈ [1, ∞), d B (D X , D Y ) ≤ W q (D X , D Y ) for any given D X , D Y .
As q tends to infinity, the Wasserstein distance approaches the bottleneck distance. Also, see Appendix B for a further relationship between the bottleneck distance and Wasserstein distance.

Persistence landscapes

A persistence diagram is a multiset, which is difficult to be used as inputs for machine learning methods (due to the complicated space structure, cardinality issues, computationally inefficient metrics, etc.). Hence, it is useful to transform the persistent homology into a functional Hilbert space, where the analysis is easier and learning methods can be directly applied. One good example is the persistence landscape [START_REF] Bubenik | Statistical topological data analysis using persistence landscapes[END_REF][START_REF] Bubenik | The persistence landscape and some of its properties[END_REF][START_REF] Bubenik | A persistence landscapes toolbox for topological statistics[END_REF]. Let D denote a persistence diagram that contains N off-diagonal birth-death pairs. We first consider a set of piecewise-linear functions {Λ p (t)} p∈D for all birth-death pairs p = (b, d) ∈ D as

Λ p (t) = max{0, min{t -b, d -t}}.
Then the persistence landscape λ of the persistence diagram D is defined as a sequence of functions {λ k } k∈N , where

λ k (t) = kmax p Λ p (t), t ∈ R, k ∈ N,
(2) Hence, the persistence landscape is a set of real-valued functions and is easily computable. Advantages for this kind of functional summaries are discussed in Chazal et al. [2014b], [START_REF] Berry | Functional summaries of persistence diagrams[END_REF].

Distance to measure (DTM) function

The Distance to measure (DTM) [START_REF] Chazal | Geometric inference for probability measures[END_REF][Chazal et al., , 2016a] ] is a robustified version of the distance function. More precisely, the DTM d µ,m0 : R d → R for a probability distribution µ with parameter m 0 ∈ (0, 1) and r ≥ 1 is defined by

d µ,m0 (x) = 1 m 0 m0 0 (δ µ,m (x)) r dm 1/r , where δ µ,m (x) = inf{t > 0 : µ(B(x, t)) > m} when B(x, t
) is an open ball centered at x with radius t. If not specified, r = 2 is used as a default. In practice, we use a weighted empirical measure

P n (x) = n i=1 i 1(X i = x) n i=1 i
, with weights i 's for µ. In this case, we define the empirical DTM by

dm0 (x) = d Pn,m0 (x) = Xi∈N k (x) i X i -x r m 0 n i=1 i 1/r , (3)
where

N k (x) is the subset of {X 1 , . . . , X n } containing the k nearest neighbors of x, k is such that Xi∈N k-1 (x) i < m 0 n i=1 i ≤ Xi∈N k (x) i , and i = Xj ∈N k (x) j -m 0 n j=1 j
if at least one of X i 's is in N k (x) and i = i otherwise. Hence the empirical DTM behaves similarly to the k-nearest distance with k = m 0 n . For i.i.d cases, we typically set i = 1 but the weights can be flexibly determined in data-driven way. The parameter m 0 determines how much topological/geometrical information should be extracted from the local or global structure. A brief guideline on DTM parameter selection can be found in Appendix F (see [START_REF] Chazal | Geometric inference for probability measures[END_REF] for more details). Since the resulting persistence diagram is less prone to input perturbations and has nice stability properties, people often prefer using the DTM as their filtration function.

A novel topological layer based on weighted persistence landscapes

In this section, we present a detailed algorithm to implement PLLay for a general neural network. Let X, D X , h top denote our input, corresponding persistence diagram induced from X, the proposed topological layer, respectively. Broadly speaking, the construction of our proposed topological layer consists of two steps: 1) computing a persistence diagram from the input, and 2) constructing the topological layer from the persistence diagram.

Computation of diagram: X → D X

To compute the persistence diagram from the input data, we first need to define the filtration which requires a simplicial complex K and a function f : K → R. There are several options for K and f . We are in general agnostic about which filtration to use since it is in fact problem-dependent; in practice, we suggest using ensemble-like methods that can adapt to various underlying topological structures. One popular choice is the Vietoris-Rips filtration. When there is a one-to-one correspondence between X i and each fixed grid point Y i , one obvious choice for f could be just interpreting X as a function values, so f (Y i ) = X i . We refer to [START_REF] Chazal | An introduction to topological data analysis: fundamental and practical aspects for data scientists[END_REF] for more examples.

As described in Section 2.3, one appealing choice for f is the DTM function. Due to its favorable properties, the DTM function has been widely used in TDA [START_REF] Anai | Dtm-based filtrations[END_REF][START_REF] Xu | Finding cosmic voids and filament loops using topological data analysis[END_REF], and has a good potential for deep learning application. Nonetheless, to the best of our knowledge, the DTM function has not yet been adopted in previous studies. In what follows, we detail two common scenarios for the DTM adaptation: when we consider the input X as 1) data points or 2) weights.

• If the input data X is considered as the empirical data points, then the empirical DTM in (3) with weights i 's becomes

dm0 (x) = Xi∈N k (x) i X i -x r m 0 n i=1 i 1/r , (4)
where k and i are determined as in (3).

• If the input data X is considered as the weights corresponding to fixed points {Y 1 , . . . , Y n }, then the empirical DTM in (3) with data points Y i 's and weights X i 's becomes

dm0 (x) = Xi∈N k (x) X i Y i -x r m 0 n i=1 X i 1/r , (5)
where k and i are determined as in (3). Figure 2 provides some real data examples (which will be used in Section 5) of the persistence diagrams and the corresponding persistence landscapes based on the DTM functions. As shown in Figure 3, the topological features are expected to be robust to external noise or corruption.

Construction of topological layer: D X → h top

Our topological layer is defined based on a parametrized mapping which takes the persistence diagram D to be projected onto R, by harnessing persistence landscapes. Our construction is less afflicted by the artificial bending due to a particular transformation procedure as in [START_REF] Hofer | Deep learning with topological signatures[END_REF], yet still guarantees the crucial information in the persistence diagram to be well preserved as will be seen in Section 4. Insignificant points with low persistence are likely to be ignored systematically without introducing additional nuisance parameters [START_REF] Bubenik | A persistence landscapes toolbox for topological statistics[END_REF].

Let R +0 denote [0, ∞). Given a persistence diagram D ∈ D, we compute the persistence landscape of order k in (2), λ k (t), for k = 1, ..., K max . Then, we compute the weighted average λ ω (t) := Kmax k=1 ω k λ k (t) with a weight parameter ω = {ω k } k , ω k > 0, k ω k = 1. Next, we set a domain [T min , T max ] and a resolution ν := T /(m-1), and sample m equal-interval points from [T min , T max ] to obtain Λ ω = λ ω (T min ), λ ω (T min + ν), ..., λ ω (T max ) ∈ R +0 m . Consequently, we have defined a mapping Λ ω : D → R +0 m which is a (vectorized) finite-sample approximation of the weighted persistence landscapes at the resolution ν, at fixed, predetermined locations. Finally, we consider a parametrized differentiable map g θ : R +0 m → R which takes the input Λ ω and is differentiable with respect to θ as well. Now, the projection of D with respect to the mapping Algorithm 1 Implementation of single structure element for PLLay Input:

persistence diagram D ∈ D 1. compute λ k (t) (2) on t ∈ [0, T ] for every k = 1, ..., K max 2. compute the weighted average λ ω (t) := Kmax k=1 ω k λ k (t), ω k > 0, k ω k = 1 3. set ν := T m-1
, and compute Λ ω = (λ ω (T min ), λ ω (T min + ν), ..., λ ω (T max )) ∈ R m 4. for a parametrized differentiable map g θ : R m → R, define S θ,ω = g θ • Λ ω Output: S θ,ω : D → R S θ,ω := g θ • Λ ω defines a single structure element for our topological input layer. We summarize the procedure in Algorithm 1.

The projection S θ,ω is continuous at every t ∈ [T min , T max ]. Also, note that it is differentiable with respect to ω and θ, regardless of the resolution level ν. In what follows, we provide some guidelines that might be useful to implement Algorithm 1.

ω: The weight parameter ω can be initialized uniformly, i.e. ω k = 1/K max for all k, and will be re-determined during training through the softmax layer in a way that a certain landscape conveying significant information has more weight. In general, lower-order landscapes tend to be more significant than higher-order landscapes, but the optimal weights may vary from task to task. θ, g θ : Likewise, some birth-death pairs, encoded in the landscape function, may contain more crucial information about the topological features of the input data structure than others. Roughly speaking, this is equivalent to say certain mountains (or their ridge or valley) in the landscape are especially important. Hence, the parametrized map g θ should be able to reflect this by its design. In general, it can be done by affine transformation with scale and translation parameter, followed by an extra nonlinearity and normalization if necessary. We list two possible choices as below.

• Affine transformation: with scale and translation parameter σ i ,

µ i ∈ R m , g θi (Λ ω ) = σ i (Λ ω -µ i ) and θ i = (σ i , µ i ).
• Logarithmic transformation: with same

θ i = (σ i , µ i ), g θi (Λ ω ) = exp -σ i Λ ω -µ i 2 .
Note that other constructions of g θ , θ, ω are also possible as long as they satisfy the sufficient conditions described above. Finally, since each structure element corresponds to a single node in a layer, we concatenate many of them, each with different parameters, to form our topological layer.

Definition 3.1 (Persistence landscape-based topological layer (PLLay)) For n h ∈ N, let η i = (θ i , ω i ) denote the set of parameters for the i-th structure element and let η = (η i )

n h i=1 .
Given D and resolution ν, we define PLLay as a parametrized mapping with η of D → R n h such that

h top : D → S η i (D; ν) n h i=1 . (6)
Note that this is nothing but a concatenation of n h topological structure elements (nodes) with different parameter sets (thus n h is our layer dimension).

Remark 1 Our PLLay considers only K max top landscape functions. For a given persistence diagram, the points near the diagonal are not likely to appear at K max top landscape functions, and hence not considered in PLLay. And hence PLLay automatically filters out the noisy features.

Differentiability

This subsection is devoted to the analysis of the differential behavior of PLLay with respect to its input (or output from the previous layer), by computing the derivatives

∂htop ∂X . Since ∂htop ∂X = ∂htop ∂D X • ∂D X
∂X , this can be done by combining two derivatives ∂D X ∂X and ∂htop ∂D X . We have extended [START_REF] Poulenard | Topological function optimization for continuous shape matching[END_REF] so that we can compute the above derivatives for general persistent homology under arbitrary filtration in our setting. We present the result in Theorem 3.1.

Theorem 3.1 Let f be the filtration function. Let ξ be a map from each birth-death point (b i , d i ) ∈ D X to a pair of simplices (β i , δ i ). Suppose that ξ is locally constant at X, and f (β i ) and f (δ i ) are differentiable with respect to X j 's. Then, h top is differentiable with respect to X and

∂h top ∂X j = i ∂f (β i ) ∂X j m l=1 ∂g θ ∂x l Kmax k=1 ω k ∂λ k (lv) ∂b i + i ∂f (δ i ) ∂X j m l=1 ∂g θ ∂x l Kmax k=1 ω k ∂λ k (lv) ∂d i .
The proof is in Appendix E.1. Note that ∂λ k ∂bi , ∂λ k ∂di are piecewise constant and are easily computed in explicit forms. Also ∂g θ ∂x l can be easily realized by an automatic differentiation framework such as tensorflow or pytorch. Our PLLay in Definition 3.1 is thus trainable via backpropagation at an arbitrary location in the network. In Appendix D, we also provide a derivative for the DTM filtration.

Stability Analysis

A key property of PLLay is stability; its discriminating power should remain stable against nonsystematic noise or perturbation of input data. In this section, we shall provide our theoretical results on the stability properties of the proposed layer. We first address the stability for each structure element with respect to changes in persistence diagrams in Theorem 4.1.

Theorem 4.1 Let g θ be • ∞ -Lipschitz, i.e. there exists L g > 0 with |g θ (x) -g θ (y)| ≤ L g x -y ∞ for all x, y ∈ R m . Then for two persistence diagrams D, D , |S θ,ω (D; ν) -S θ,ω (D ; ν)| ≤ L g d B (D, D ).
Proof of Theorem 4.1 is given in Appendix E.2. Theorem 4.1 shows that S θ,ω is stable with respect to perturbations in the persistence diagram measured by the bottleneck distance (1). It should be noted that only the Lipschitz continuity of g θ is required to establish the result.

Next, Corollary 4.1 shows that under certain conditions our approach improves the previous stability result of [START_REF] Hofer | Deep learning with topological signatures[END_REF].

Corollary 4.1 For t > 0, let n t ∈ N be satisfying that, for any two diagrams D t , D t with d B (D, D t ) ≤ t and d B (D , D t ) ≤ t, either D t \D t or D t \D t has at least n t points. Then, the ratio of our stability bound in Theorem 4.1 to that in [START_REF] Hofer | Deep learning with topological signatures[END_REF] is upper bounded by

C g θ /(1 + (2t/d B (D, D )) × (n t -1))
, where C g θ is a constant to be specified in the proof.

See Appendix E.3 for the proof. Corollary 4.1 implies that for complex data structures where each D contains many birth-death pairs (for fixed t, in general n t grows with the increase in the number of points in D), our stability bound is tighter than that of [START_REF] Hofer | Deep learning with topological signatures[END_REF] at polynomial rates.

In particular, when we use the DTM function-based filtration proposed in (4) and ( 5), Theorem 4.1 can be turned into the following stability result with respect to our input X.

Theorem 4.2 Suppose r = 2 is used for the DTM function. Let a differentiable function g θ and resolution ν be given, and let P be a distribution. For the case when X j 's are data points, i.e. when (4) is used as the DTM function of X, let P n be the empirical distribution defined by

P n = n i=1 iδX i n i=1 i
. For the case when X j 's are weights, i.e. when (5) is used as the DTM function of X, let P n be the empirical distribution defined by

P n = n i=1 Xiδ Y i n i=1
Xi . Let D P be the persistence diagram of the DTM filtration of P , and D X be the persistence diagram of the DTM filtration of X. Then,

|S θ,ω (D X ; ν) -S θ,ω (D P ; ν)| ≤ L g m -1/2 0 W 2 (P n , P ).
The proof is given in Appendix E.4. Theorem 4.2 implies that if the empirical distribution P n induced from the given input X well approximates the true distribution P with respect to the Wasserstein distance, i.e. having small W 2 (P n , P ), then PLLay constructed on observed data is close to the one as if we were to know the true distribution P .

Theorem 4.1 and 4.2 suggest that the topological information embedded in the proposed layer is robust against small noise, data corruption, or outliers. We have also discussed the stability result for the Vietoris-Rips and the Čech complex in Appendix C.

Experiments

To demonstrate the effectiveness of the proposed approach, we study classification problems on two different datasets: MNIST handwritten digits and ORBIT5K. To fairly showcase the benefits of using our proposed method, we keep our network architecture as simple as possible so that we can focus on the contribution from PLLay. In the experiments, we aim to explore the benefits of our layer through the following questions: 1) does it make the network more robust and reliable against noise, etc.? and 2) does it improve the overall generalization capability compared to vanilla models? In order to address both of these questions, we first consider the corruption process, a certain amount of random omission of pixel values or points from each raw example (so we will have less information), and the noise process, a certain amount of random addition of uniformly-distributed noise signals or points to each raw example. An example is given in Figure 3. Then we fit a standard multilayer perceptron (MLP) and a convolutional neural network (CNN) with and without the augmentation of PLLay across various noise and corruption rates given to the raw data, and compare the results.

The guideline for choosing the TDA parameters in this experiment is described in Appendix F. We intentionally use a small number of training data (∼1000) so that the convergence rates could be included in the evaluation criteria. Each simulation is repeated 20 times. We refer to Appendix G for details about each simulation setup and our model architectures.

MNIST handwritten digits

We classify handwritten digit images from MNIST dataset. Each digit has distinctive topological information which can be encoded into the Persistence Landscape as in Figure 2.

Topological layer. We add two parallel PLLays in Definition 6 at the beginning of MLP and CNN models, based on the empirical DTM function in ( 5), where we define fixed 28 × 28 points on grid and use a set of grayscale values X as a weight vector for the fixed points. We used m 0 = 0.05 and m 0 = 0.2 for each layer, respectively (referred to MLP+P, CNN+P(i), respectively). Particularly for the CNN model, it is likely that the output of the convolutional layers might carry significant information about (smoothed) geometry of the input data shape. So we additionally place another PLLay after each convolutional layer, directly taking the layer output as 2D-function values and using the sublevel filtration (CNN+P).

Baselines. As our baseline methods, we employ 2-layer vanilla MLP, 2-layer CNN, and the topological signature method by [START_REF] Hofer | Deep learning with topological signatures[END_REF] based on the empirical DTM function proposed in (5) (which we will refer to as SLay). The SLay is augmented at the beginning of MLP and CNN, referred to as MLP+S and CNN+S. See Appendix G.1 for more details.

Result. In Figure 4, we observe that PLLay augmentation consistently improves the accuracy of all the baselines. Interestingly, as we increase the corruption and noise rates, the improvement on CNN increases up to the moderate level of corruption and noise (∼ 15%), then starts to decrease. We conjecture that this is because although DTM filtration is able to robustly capture homological signals as illustrated in Figure 2, if the corruption and noise levels become too much, then the topological structure starts to dissolve in the DTM filtration.

Orbit Recognition

We classify point clouds generated by 5 different dynamical systems from ORBIT5K dataset [START_REF] Adams | Persistence images: a stable vector representation of persistent homology[END_REF][START_REF] Mathieu Carrière | Perslay: A neural network layer for persistence diagrams and new graph topological signatures[END_REF]. The detailed data generating process is described in Appendix G.2.

Topological layer. The setup remains the same as in the previous MNIST case, except that 1) PLLay at the beginning of each network uses the empirical DTM function in (4), and 2) we set m 0 = 0.02.

Baselines & Simulation. All the baseline methods remain the same. For noiseless case, we added PointNet [START_REF] Charles | Pointnet: Deep learning on point sets for 3d classification and segmentation[END_REF], a state-of-the-art in point cloud classification, and PersLay [START_REF] Mathieu Carrière | Perslay: A neural network layer for persistence diagrams and new graph topological signatures[END_REF], a state-of-the-art in TDA-utilized classification.

Result. In Figure 4, we observe that PLLay improves upon MLP and MLP+S by a huge margin (42% ∼ 60%). In particular, without augmenting PLLay, MLP and MLP+S remain at almost a random classifier, which implies that the topological information is indeed crucial for the ORBIT5K classification task, and it would otherwise be very challenging to extract meaningful features. PLLay improves upon CNN or CNN+S consistently as well. Moreover, it appears that CNN suffers from high variance due to the high complexity of ORBIT5K dataset. On the other hand, PLLay can effectively mitigate this problem and make the model more stable by utilizing robust topological information from DTM function. Impressively, for the noiseless case, PLLay has achieved better performance than all the others including the current state-of-the-art PointNet and PersLay by a large margin.

Discussion

In this study, we have presented PLLay, a novel topological layer based on the weighted persistence landscape where we can exploit the topological features effectively. We provide the differentiability guarantee of the proposed layer with respect to the layer's input under arbitrary filtration. Hence, our study offers the first general topological layer which can be placed anywhere in the deep learning network. We also present new stability results that verify the robustness and efficiency of our approach.

It is worth noting that our method and analytical results in this paper can be extended to silhouettes [START_REF] Chazal | Subsampling methods for persistent homology[END_REF][Chazal et al., , 2014b]. In the experiments, we have achieved the new state-of-the-art accuracy for ORBIT5K dataset based on the proposed method. We expect our work to bridge the gap between modern TDA tools and deep learning research.

The computational complexity depends on how PLLay is used. Computing the DTM is O(n+m log n) when m 0 ∝ 1/n and k-d tree is used, where n is the input size and m is the grid size. Computing the persistence diagram is O(m 2+ ) for any small > 0 when the simplicial complex K in Section 3.1 grows linearly with respect to the grid size such as cubical complex or alpha complex [START_REF] Chen | An output-sensitive algorithm for persistent homology[END_REF] and Theorem 4.4, 5.6 of [START_REF] Boissonnat | Geometric and topological inference[END_REF]). Computing the persistence landscape grows linearly with respect to the number of homological features in the persistence diagram, which is the topological complexity of the input and does not necessarily depend on n or m.

For our experiments, we consider fixed grids of size 28 × 28 and 40 × 40 as in Appendix G, so the computation is not heavy. Also, if we put PLLay only at the beginning of the deep learning model, then PLLay can be pre-computed and needs not to be calculated at every epoch in the training.

There are several remarks regarding our experiments. First, we emphasize that SLay in Section 5 is rather an intermediate tool designed for our simulation and not completely identical to the topological signature method by [START_REF] Hofer | Deep learning with topological signatures[END_REF]. For example, SLay combines the method by Hofer et al.

[2017] and the DTM function in (4) and (5) that have not appeared in the previous study. So we cannot exclude the possibility that the comparable performance of SLay for certain simulations is due to the contribution by the DTM function filtration. Moreover, for CNN, placing extra PLLay after each convolutional layer appears to bring marginal improvement in accuracy in our experiments.

Exploring the optimal architecture with our PLLay, e.g., finding the most accurate and efficient PLLay network for a given classification task, would be an interesting future work.

The source code of PLLay is publicly available at https://github.com/jisuk1/pllay/.

Broader Impact

This paper proposes a novel method of adapting tools in applied mathematics to enhance the learnability of deep learning models. Even though our methodology is generally applicable to any complex modern data, it is not tuned to a specific application that might improperly incur direct societal/ethical consequences. So the broader impact discussion is not needed for our work.

APPENDIX

A Simplicial complex, Persistent homology, and Distance between sets on metric spaces

Throughout, we will let X denotes a subset of R d , and X denotes a finite collection of points from an arbitrary space X.

A simplicial complex can be seen as a high dimensional generalization of a graph. Given a set V , an (abstract) simplicial complex is a set K of finite subsets of V such that α ∈ K and β ⊂ α implies β ∈ K. Each set α ∈ K is called its simplex. The dimension of a simplex α is dim α = cardα -1, and the dimension of the simplicial complex is the maximum dimension of any of its simplices. Note that a simplicial complex of dimension 1 is a graph.

When approximating the topology of the underlying space by observed samples, a common choice is the Čech complex, defined next. Below, for any x ∈ X and r > 0, we let B X (x, r) denote the open ball centered at x and radius r > 0 intersected with X.

Definition A.1 ( Čech complex) Let X ⊂ X be finite and r > 0. The (weighted) Čech complex is the simplicial complex

Čech X X (r) := {σ ⊂ X : ∩ x∈σ B X (x, r) = ∅}. ( 7
)
The superscript X will be dropped when understood from the context.

Another common choice is the Vietoris-Rips complex, also referred to as Rips complex, where simplexes are built based on pairwise distances among its vertices.

Definition A.2 (Vietoris-Rips complex) Let X ⊂ X be finite and r > 0. The Vietoris-Rips complex Rips X (r) is the simplicial complex defined as Rips X (r) := {σ ⊂ X : d(x i , x j ) < 2r, ∀x i , x j ∈ σ}.

Note that from ( 7) and (8), the Čech complex and Vietoris-Rips complex have the following interleaving inclusion relationship Čech X (r) ⊂ Rips X (r) ⊂ Čech X (2r).

In particular, when X ⊂ R d is a subset of a Euclidean space of dimension d, then the constant 2 can be tightened to 2d d+1 (e.g., see Theorem 2.5 in de Silva and [START_REF] De | Coverage in sensor networks via persistent homology[END_REF]):

Čech X (r) ⊂ Rips X (r) ⊂ Čech X 2d d + 1 r .
Persistent homology [START_REF] Barannikov | The framed Morse complex and its invariants[END_REF], Zomorodian and Carlsson, 2005[START_REF] Edelsbrunner | Topological persistence and simplification[END_REF], Chazal et al., 2014a] is a multiscale approach to represent topological features of the complex K. A filtration F is a collection of subcomplexes approximating the data points at different resolutions, formally defined as follows.

Definition A.3 (Filtration) A filtration F = {K a ⊂ K} a∈R is a collection of subcomplexes of K such that a ≤ b implies that K a ⊂ K b .
For a filtration F and for each k ∈ N 0 = N ∪ {0}, the associated persistent homology P H k F is an ordered collection of k-th dimensional homologies, one for each element of F.

Definition A.4 (Persistent homology) Let F be a filtration and let k ∈ N 0 . The associated k-th persistent homology P H k F is a collection of groups {H k (K a )} a∈R of each subcomplex K a in F equipped with homomorphisms {ı a,b k } a≤b , where H k (K a ) is the k-th dimensional homology group of K a and ı a,b k :

H k K a → H k K b is the homomorphism induced by the inclusion K a ⊂ K b .
For the k-th persistent homology P H k F, the set of filtration levels at which a specific homology appears is always an interval [b, d) ⊂ [-∞, ∞], i.e. a specific homology is formed at some filtration value b and dies when the inside hole is filled at another value d > b. To be more formally, the image of a specific homology class α in H k (K a ) is nonzero if and only if b ≤ a < d. We often say that α is born at b and dies at d. By considering these pairs as points in the plane, one obtains the persistence diagram as below. When topological information of the underlying space is approximated by the observed points, it is often needed to compare two sets with respect to their metric structures. Here we present two distances on metric spaces, Hausdorff distance and Gromov-Hausdorff distance. We refer to [START_REF] Burago | A course in metric geometry[END_REF] for more details and other distances.

The Hausdorff distance [Burago et al., 2001, Definition 7.3.1] is on sets embedded in the same metric spaces. This distance measures how two sets are close to each other in the embedded metric space. When S ⊂ X, we denote by U r (S) the r-neighborhood of a set S in a metric space, i.e. U r (S) = x∈S B X (x, r).

Definition A.6 (Hausdorff distance) Let X be a metric space, and X, Y ⊂ X be a subset. The Hausdorff distance between X and Y , denoted by d H (X, Y ), is defined as

d H (X, Y ) = inf{r > 0 : X ⊂ U r (Y ) and Y ⊂ U r (X)}.
The Gromov-Hausdorff distance measures how two sets are far from being isometric to each other. To define the distance, we first define a relation between two sets called correspondence.

Definition A.7 Let X and Y be two sets. A correspondence between X and Y is a set C ⊂ X × Y whose projections to both X and Y are both surjective, i.e. for every x ∈ X, there exists y ∈ Y such that (x, y) ∈ C, and for every y ∈ Y , there exists x ∈ X with (x, y) ∈ C.

For a correspondence, we define its distortion by how the metric structures of two sets differ by the correspondence.

Definition A.8 Let X and Y be two metric spaces, and C be a correspondence between X and Y . The distortion of C is defined by

dis(C) = sup {|d X (x, x ) -d Y (y, y )| : (x, y), (x , y ) ∈ C} .
Now the Gromov-Hausdorff distance [Burago et al., 2001, Theorem 7.3.25] is defined as the smallest possible distortion between two sets. Definition A.9 (Gromov-Hausdorff distance) Let X and Y be two metric spaces. The Gromov-Hausdorff distance between X and Y , denoted as d GH (X, Y ), is defined as

d GH (X, Y ) = 1 2 inf C dis(C),
where the infimum is over all correspondences between X and Y .

B Bottleneck distance and Wasserstein distance

Our stability bound in Theorem 4.1 is based on the bottleneck distance, while the stability bound in [START_REF] Hofer | Deep learning with topological signatures[END_REF] is based on Wasserstein distance. Hence to compare these bounds, we need to understand the relationship between the bottleneck distance and Wasserstein distance. We already know that the Wasserstein distance is lower bounded by the bottleneck distance. Here, we will find a tighter lower bound for the ratio of the Wasserstein distance to the bottleneck distance.

Before analyzing the relationship between them, we first show a claim. Then, the ratio of q-Wasserstein distance to the bottleneck distance is bounded as

Claim

W q (D, D ) d B (D, D ) ≥ 1 + 2t d B (D, D ) q (n t -1) 1 q .

C Stability for Vietoris-Rips and Cech filtration

When we use Vietoris-Rips or Čech filtration, our result can be turned into the stability result with respect to points in Euclidean space. Let X, Y ⊂ R d be two bounded sets. The next corollary re-states our stability theorem with respect to points in R d .

Corollary C.1 Let X, Y be any -coverings of X, Y, and let D X , D Y denote persistence diagrams induced from the Vietoris-Rips or Čech filtration on X, Y respectively. Then we have

|S θ,ω (D X ; ν) -S θ,ω (D Y ; ν)| ≤ 2L g (d GH (X, Y) + 2 ) . ( 9
)
The proof is given in Appendix E.7. Corollary C.1 implies that if we assume our observed data points are sufficiently decent quality in the sense that → 0, then our topological layers constructed on those observed points are stable with respect to small perturbations of the true representation under proper persistent homologies. Here, could be interpreted as uncertainty from incomplete sampling. This means the topological information embedded in the proposed layer is robust against small sampling noise or data corruption by missingness.

Moreover, since Gromov-Hausdorff distance is upper bounded by Hausdorff distance, the result in Corollary C.1 also holds when we use d H (X, Y ) in place of d GH (X, Y ) in RHS of (9).

Remark 2 In fact, when we have very dense data that have been well-sampled uniformly over the true representation so that → 0, our result in (9) converges to the following:

|S θ,ω (D X ; ν) -S θ,ω (D Y ; ν)| ≤ 2L g d GH (X, Y).

D Differentiability of DTM function

Here we provide a specific example of computing ∂f (ς) ∂Xj when f is the DTM filtration which has not been explored in previous approaches. We first consider the case of (4) where X j 's are data points, as in Proposition D.1. See Appendix E.8 for the proof. Proposition D.1 When X j 's and ς satisfy that Xi∈N k (y) i X iy l r are different for each

y l ∈ ς, then f (ς) is differentiable with respect to X j and ∂f (ς) ∂X j = j X j -y r-2 (X j -y)I(X j ∈ N k (y)) dm0 (y) r-1 m 0 n i=1 i
, where I is an indicator function and y = arg max z∈ς dm0 (z). In particular, f is differentiable a.e. with respect to Lebesgue measure on X.

Similarly, we consider the case of ( 5) where X j 's are weights, as in Proposition D.2. See Appendix E.9 for the proof.

Proposition D.2 When X j 's and ς satisfy that Yi∈N k (y) X i Y iy l r are different for each

y l ∈ ς, then f (ς) is differentiable with respect to X j and ∂f (ς) ∂X j = Y j -y r I(Y j ∈ N k (y)) -m 0 dm0 (y) r r dm0 (y) r-1 m 0 n i=1 X i
, where y = arg max y∈ςi dm0 (y). In particular, f is differentiable a.e. with respect to Lebesgue measure on X and Y .

Computation of ∂htop ∂µ i , ∂htop ∂ς i are simpler and can be done in a similar fashion. In the experiments, we set r = 2.

E Proofs

E.1 Proof of Theorem 3.1

For computing ∂htop ∂Xj , note that it can be expanded using the chain role as

∂h top ∂X j = i ∂h top ∂b i ∂b i ∂X j + i ∂h top ∂d i ∂d i ∂X j , ( 10
)
and hence we need to compute

∂D X ∂X = ∂bi ∂Xj , ∂di ∂Xj (bi,di)∈D X ,Xj ∈X and ∂htop ∂D X = ∂htop ∂bi , ∂htop ∂di (bi,di)∈D X to compute ∂htop ∂Xj .
We first compute ∂D X ∂X . Let K be the simplicial complex, and suppose all the simplices are ordered in the filtration so that the values of f are nondecreasing, i.e. if ς comes earlier than τ then f (ς) ≤ f (τ ). Note that the map ξ from each birth-death point (b i , d i ) ∈ D X to a pair of simplices (β i , δ i ) is simply the pairing returned by the standard persistence diagram [START_REF] Carlsson | Persistence barcodes for shapes[END_REF]. Let γ be the homological feature corresponding to (b i , d i ), then the birth simplex β i is the simplex that forms γ in K bi = f -1 (-∞, b i ], and the death simplex δ i is the simplex that causes γ to collapse in

K di = f -1 (-∞, d i ].
For example, if γ were to be a 1-dimensional feature, then β i is the edge in K bi that forms the loop corresponding to γ, and δ i is the triangle in K di which incurs the loop corresponding to γ can be contracted in

K di . Now, f (ξ(b i )) = f (β i ) = b i and f (ξ(d i )) = f (δ i ) = d i , and from ξ being locally constant on X, ∂b i ∂X j = ∂f (ξ(b i )) ∂X j = ∂f (β i ) ∂X j , ∂d i ∂X j = ∂f (ξ(d i )) ∂X j = ∂f (δ i ) ∂X j . (11)
Therefore, the derivatives of the birth value and the death value are the derivatives of the filtration function evaluated at the corresponding pair of simplices. And

∂D X ∂X = ∂bi ∂Xj , ∂di ∂Xj (bi,di)∈D X ,Xj ∈X
is the collection of these derivatives, hence applying (11) gives

∂D X ∂X = ∂b i ∂X j , ∂d i ∂X j (bi,di)∈D X ,Xj ∈X = ∂f (β i ) ∂X j , ∂f (δ i ) ∂X j ξ -1 (βi,δi)∈D X ,Xj ∈X . ( 12
)
Now, we compute

∂htop ∂D X = ∂htop ∂bi , ∂htop ∂di (bi,di)∈D X
. Computing ∂htop ∂bi can be done by applying the chain role on

h top = S θ,ω = g θ • Λ ω as ∂h top ∂b i = ∂S θ,ω ∂b i = ∂(g θ • Λ ω ) ∂b i = ∇g θ • ∂Λ ω ∂b i = m l=1 ∂g θ ∂x l ∂λ ω (lν) ∂b i , (13)
where we use x l as the shorthand notation for the input of the function g θ . Then, applying λ ω (lν) = Kmax k=1 ω k λ k (lν) to (13) gives

∂h top ∂b i = m l=1 ∂g θ ∂x l Kmax k=1 ω k ∂λ k (lv) ∂b i . (14)
Similarly, ∂htop ∂di can be computed as

∂h top ∂d i = m l=1 ∂g θ ∂x l Kmax k=1 ω k ∂λ k (lv) ∂d i . (15)
And therefore, ∂htop ∂D X is the collection of these derivatives from ( 14) and ( 15), i.e.,

∂h top ∂D X = m l=1 ∂g θ ∂x l Kmax k=1 ω k ∂λ k (lv) ∂b i , m l=1 ∂g θ ∂x l Kmax k=1 ω k ∂λ k (lv) ∂d i (bi,di)∈D X . (16)
Hence, ∂htop ∂X can be computed by applying ( 12) and ( 16) to (10) as

∂h top ∂X j = i ∂h top ∂b i ∂b i ∂X j + i ∂h top ∂d i ∂d i ∂X j = i ∂f (β i ) ∂X j m l=1 ∂g θ ∂x l Kmax k=1 ω k ∂λ k (lv) ∂b i + i ∂f (δ i ) ∂X j m l=1 ∂g θ ∂x l Kmax k=1 ω k ∂λ k (lv) ∂d i .
E.2 Proof of Theorem 4.1

Let D and D be two persistence diagrams and let λ and λ be their persistence landscapes. All the quantities derived from D are denoted by a variable name with the superscript hereafter (e.g.,

λ k (t), Λ ω ).
For the stability of the structure element S θ,ω , we first expand the difference between S θ,ω (D; ν) and S θ,ω (D ; ν) using

S θ,ω = g θ • Λ ω as |S θ,ω (D; ν) -S θ,ω (D ; ν)| = g θ Λ ω -g θ Λ ω . (17)
Then, RHS of ( 17) is bounded by applying the Lipschitz condition of the function g θ as

g θ Λ ω -g θ Λ ω ≤ L g Λ ω -Λ ω ∞ . (18)
Then for

Λ ω -Λ ω ∞ , note that Λ ω , Λ ω ∈ R m , the L ∞ difference of Λ ω and Λ ω is bounded as Λ ω -Λ ω ∞ = max 0≤i≤m-1 λ ω (T min + iν) -λ ω (T min + iν) ≤ sup t∈[0,T ] λ ω (t) -λ ω (t) = m 1/2 λ ω -λ ω ∞ . (19)
Now, for bounding λ ωλ ω ∞ , we first consider the pointwise difference |λ ω (t)λ ω (t)|. For all t ∈ [0, T ], the difference between λ ω (t) and λ ω (t) is bounded as

λ ω (t) -λ ω (t) = 1 k ω k Kmax k=1 ω k λ k (t) - 1 k ω k Kmax k=1 ω k λ k (t) ≤ 1 k ω k Kmax k=1 ω k |λ k (t) -λ k (t)| ≤ sup 1≤k≤Kmax,t∈[0,T ] |λ k (t) -λ k (t)| = max 1≤k≤Kmax λ k -λ k ∞ . (20)
And hence

λ ω -λ ω ∞ is bounded by max 1≤k≤Kmax λ k -λ k ∞ as well, i.e., λ ω -λ ω ∞ = sup t∈[0,T ] λ ω (t) -λ ω (t) ≤ max 1≤k≤Kmax λ k -λ k ∞ . (21)
Then for all k = 1, . . . , K max , the ∞-landscape distance λ kλ k ∞ is bounded by the bottleneck distance d B (D, D ) from Theorem 13 in [START_REF] Bubenik | Statistical topological data analysis using persistence landscapes[END_REF], i.e.

λ k -λ k ∞ ≤ d B (D, D ). (22)
Hence, applying ( 18), ( 19), ( 21), ( 22) to ( 17) gives the stated stability result as

|S θ,ω (D; ν) -S θ,ω (D ; ν)| = g θ Λ ω -g θ Λ ω ≤ L g Λ ω -Λ ω ∞ ≤ L g λ ω -λ ω ∞ ≤ L g max 1≤k≤Kmax λ k -λ k ∞ ≤ L g d B (D, D ).

E.3 Proof of Corollary 4.1

First note that the result of [START_REF] Hofer | Deep learning with topological signatures[END_REF] used W 1 Wasserstein distance with L r norm for ∀r ∈ N, which will be denoted by W Lr 1 in this proof. That is,

W Lr 1 (D, D ) := inf γ p∈D X p -γ(p) r
where γ ranges over all bijections D → D (i.e., W L∞

1 corresponds to W 1 in our definition 2.2). Then, • r ≥ • ∞ implies that W Lr 1 is lower bounded by W 1 , i.e. W Lr 1 (D, D ) ≥ W 1 (D, D ). (23)
c g θ d B (D, D ) c K W Lr 1 (D, D ) ≥ c g θ c K d B (D, D ) W 1 (D, D ) ≥ c g θ c K 1 1 + 2t d B (D,D ) (n t -1)
.

Finally, we define C g θ ,T,ν := c g θ ,T ,ν c K , and the result follows. It should be noted that the bound is actually very loose. However, we can still conclude that our bound is tighter than that of [START_REF] Hofer | Deep learning with topological signatures[END_REF] at polynomial rates.

E.4 Proof of Theorem 4.2

We first bound the difference between S θ,ω (D X ; ν) and S θ,ω (D P ; ν) using Theorem 4.1 as

|S θ,ω (D X ; ν) -S θ,ω (D P ; ν)| ≤ L g d B (D X , D P ). (24)
It is left to further bound the bottleneck distance d B (D X , D P ). The bottleneck distance between two diagrams D X and D P is bounded by the stability theorem of persistent homology as

d B (D X , D P ) ≤ d Pn,m0 -d P,m0 ∞ . (25)
Then, from r = 2 in the DTM function, the L ∞ distance between d Pn,m0 and d P,m0 is bounded by the stability of DTM function (Theorem 3.5 from [START_REF] Chazal | Geometric inference for probability measures[END_REF]) as

d Pn,m0 -d P,m0 ∞ ≤ m -1/2 0 W 2 (P n , P ). (26)
Hence, combining ( 24), (25), and (26) altogether gives the stated stability result as Hence from the definition of n t , either |D t \D t | ≥ n t or |D t \D t | ≥ n t holds. Now, note that

|S θ,ω (D X ; ν) -S θ,ω (D P ; ν)| ≤ L g m -1/2 0 W 2 (P n , P ).
D t \D t ⊂ S and D t \D t ⊂ S .
And |S|=|S |, and hence we get the claimed result as |S| ≥ n t .

E.6 Proof of Proposition B.1

We consider a bijection γ * that realizes the q-Wasserstein distance between D and D : i.e. γ * = arginf γ p∈D pγ(p) q ∞ . Then we have that

d B (D, D ) q ≤ sup p∈D p -γ * (p) q ∞ . (27)
On the other hand, if we let p * = argsup p∈D pγ * (p) ∞ , we have

W q (D, D ) q = p∈D p -γ * (p) q ∞ = sup p∈D p -γ * (p) q ∞ + p =p * p -γ * (p) q ∞ .
Note that from Claim B.1, p ∈ D : pγ * (p) ∞ > 2t ≥ n t . And hence W q (D, D ) q can be lower bounded as

W q (D, D ) q = sup p∈D p -γ * (p) q ∞ + p =p * p -γ * (p) q ∞ (28) ≥ sup p∈D p -γ * (p) q ∞ + (2t) q (n t -1). (29)
Now, we lower bound the ratio Wq(D,D ) q d B (D,D ) q . By ( 27) and ( 29), this can be done as follows.

W q (D, D ) q d B (D, D ) q ≥ sup p∈D p -γ * (p) q ∞ + (2t) q (n t -1) d B (D, D ) q ≥ 1 + 2t d B (D, D ) q (n t -1).
And hence the ratio of the Wasserstein distance to thw bottleneck distance (D,D ) is correspondingly lower bounded as

Wq(D,D ) d B
W q (D, D ) d B (D, D ) ≥ W q (D, D ) q d B (D, D ) q 1 q ≥ 1 + 2t d B (D, D ) q (n t -1) 1 q . E.7 Proof of Corollary C.1
The difference between S θ,ω (D X ; ν) and S θ,ω (D Y ; ν) is bounded by Theorem 4.1 as

|S θ,ω (D X ; ν) -S θ,ω (D Y ; ν)| ≤ L g d B (D X , D Y ) , (30)
hence it suffices to show

d B (D X , D Y ) < 2 (d GH (X, Y) + 2 ) . (31)
E.9 Proof of Proposition D.2

From (5), note that for any y ∈ ς, dm0 (y) is expanded as

dm0 (y) = Xi∈N k (y) X i Y i -y r m 0 n i=1 X i 1/r , ( 39
)
where k is such that Yi∈N k-1 (y) X i < m 0 n i=1 X i ≤ Yi∈N k (y) X i , and X i = Xj ∈N k (y) X jm 0 n j=1 X j for one of Y i 's that is k-th nearest neighbor of y and X i = X i otherwise. Hence, by letting y = arg max z∈ς dm0 (z) and applying to (39), the filtration function f X at simplex ς becomes

f X (ς) = dX,m0 (y) = Xi∈N k (y) X i Y i -y r m 0 n i=1 X i 1/r , (40)
where the notations f X and dX,m0 are to clarify the dependency of f on X. And from the condition, dm0 (y) > dm0 (z) holds for all z ∈ ς. Hence for sufficiently small > 0 and for any Z = {Z 1 , . . . , Z n } with Z j -X j < , (40) becomes

f Z (ς) = dZ,m0 (y) = Xi∈N k (y) Z i Y i -y r m 0 n i=1 Z i 1/r . ( 41
)
Hence by differentiating (41), the derivative of f with respect to X is calculated as

∂f (ς) ∂X j = 1 r Xi∈N k (y) X i Y i -y r m 0 n i=1 X i 1 r -1 × Y j -y r I(Y j ∈ N k (y)) (m 0 n i=1 X i ) -m 0 Xi∈N k (y) X i Y i -y r (m 0 n i=1 X i ) 2 = Y j -y r I(Y j ∈ N k (y)) -m 0 dm0 (y) r r dm0 (y) r-1 m 0 n i=1 X i .

F Guideline for choosing TDA parameters

PLLay has several TDA parameters to choose: K max , T min , T max , m, and m 0 if DTM filtration is used. One can try grid search but it could be too time-consuming. More affordable approach is to compute the DTM filtration and the persistence diagram for some data and choose appropriate parameters that can reveal the topological and geometrical information of the data. Figure 5 illustrates one example of the digit 8 in MNIST data. Figure 5(a) shows the contour plot of the chosen data.

When using a DTM filtration, we need to choose m 0 first. DTMs with different m 0 values extract different topological and geometrical information. When m 0 is small, a DTM filtration aggregates the data more locally, and the geometrical and homological information formed from the local structure is extracted. When m 0 is large, a DTM filtration aggregates the data more globally, and the geometrical and homological information formed from the global structure is extracted. From the digit 8, we would first like to see the two-loop structure. And if we choose m 0 = 0.05, then as can be seen in Figure 5(b) and (c), the 1st persistent homology extracts the two-loop structure, which is more directly expected from the contour plot of the data itself in Figure 5(a). However, if we choose m 0 = 0.2, then as can be seen in Figure 5(d) and (e), the two-loop structure disappears, since the two-loop structure is coming from more local geometry of the data. Meanwhile, as the DTM filtration aggregates the data more globally, the global geometry information that three points on the digit 8(top, center, bottom) being close to neighboring points and being centers of local clusters is extracted in the 0th persistent homology. For MNIST data, DTM filtrations with m 0 = 0.05 and m 0 = 0.2 extract different topological and geometrical information of the data. Hence for MNIST data, we used two parallel PLLays with m 0 = 0.05 and m 0 = 0.2, respectively.

After choosing m 0 , choosing other TDA parameters K max , T min , T max , m is more straightforward. One can choose parameters so that the desired topological features are well extracted in the landscape.

For m 0 = 0.05, as can be seen from Figure 5(c), choosing K max = 2, T min = 0.06, T max = 0.3, m = 25 will extract two 1-dimensional features of the persistence diagram in the corresponding landscape. For m 0 = 0.2, as can be seen from Figure 5(e), choosing K max = 3, T min = 0.14, T max = 0.4, m = 27 will extract two 1-dimensional features of the persistence diagram in the corresponding landscape.

G Experiment Details.

All the experiments were implemented using GUDHI The GUDHI Project [2020] and Tensorflow library in Python and TDA package [START_REF] Fasy | Introduction to the R package TDA[END_REF] in R. We use mean and standard deviation across 20 runs of simulations with different network initializations. We remark that the basic purpose of our experiment design is to highlight the prospects and possibilities of using topological layer, not to win state-of-the-art performances. G.1 MNIST handwritten digits.

For MNIST handwritten digits, we use MNIST dataset. Raw input data is a 784 dimensional vector (reshaped from 28 by 28) of real values, each value being the pixel intensity. We use 1000 random samples for the training set and 10000 samples for the test set. Cross-entropy loss was used to train the network for 100 epochs, using Adam optimizer with mini-batches of size 16.

Topological layer. For MLP+P and CNN+P(i), we use two parallel PLLays at the beginning of MLP and CNN models with 32 nodes each and affine transformation, which are concatenated to the raw input to either MLP or CNN. We used the empirical DTM filtration in (5), where we define fixed 28 × 28 points on grid on [-1, 1] 2 and use X as a weight vector for the fixed points. For one PLLay, we used m 0 = 0.05, K max = 2, T min = 0.06, T max = 0.3, m = 25, and for the other PLLay, we used m 0 = 0.2, K max = 3, T min = 0.14, T max = 0.4, m = 27. For CNN+P, we additionally use one PLLay after the convolutional layer, with K max = 3, T min = 0.05, T max = 0.95, m = 18.

Baselines. For the baselines, models were designed to have simple structures for quick comparisons:

• Vanilla MLP: one hidden layer with 64 units with ReLU activations.

• CNN: two convolution layers followed by two fully connected layers.

• SLay: for comparison with PLLay, two SLays are used with 10 nodes each, which are concatenated to the raw input to either MLP or CNN. We used the value ν = 0.005 and ν = 0.01 for the hyperparameter of each SLay, respectively.

Result. The Accuracy results for MNIST data in Figure 4 is represented with 1 standard errors in Table 2 and Figure 6. In Figure 6, the results for MLP, MLP+S, MLP+P are in Figure 6(a), and the results for CNN, CNN+S, CNN+P, CNN+P(i) are in Figure 6(b). We can see that PLLay consistently improves the accuracies of all baselines. In particular from Table 2 and Figure 6(b), the improvement on CNN is 1.7% ∼ 2.8% when the corruption and noise is 0% ∼ 5%, and then the improvement goes up to 3.3% when the corruption and noise becomes 10% ∼ 15%, and then starts to decrease as the corruption and noise further increases. As discussed in Section 5, this is because although the DTM filtration can robustly capture homological signals up to a moderate amount of corruption and noise, as seen in Figure 2, when the corruption and noise become too much, the topological structure starts to dissolve in the DTM filtration. Also, the accuracies for CNN+P are consistently higher than the accuracies for CNN+P(i), meaning that adding PLLay in the middle of the network indeed further improves the accuracy. In each cell, the top number corresponds to the average accuracy of the model at the noise probability, and the bottom number corresponds to the 1 standard deviation of the accuracies. At each column, the model with the best accuracy is bolded.

G.2 Orbit recognition.

For orbit recognition, we use ORBIT5K dataset [START_REF] Adams | Persistence images: a stable vector representation of persistent homology[END_REF][START_REF] Mathieu Carrière | Perslay: A neural network layer for persistence diagrams and new graph topological signatures[END_REF], a synthetic dataset used as a benchmark in Topological Data Analysis. It consists of a point cloud generated by the following discrete dynamical system: given an initial point (x 1 , y 1 ) ∈ [0, 1] 2 and a parameter r > 0, we generate a point cloud {(x n , y n ) ∈ [0, 1] 2 : n = 1, . . . , N } as

x n+1 = x n + ry n (1 -y n )
mod 1, y n+1 = y n + rx n+1 (1x n+1 ) mod 1.

For comparison with [START_REF] Adams | Persistence images: a stable vector representation of persistent homology[END_REF][START_REF] Mathieu Carrière | Perslay: A neural network layer for persistence diagrams and new graph topological signatures[END_REF], we use parameters r = 2.5, 3.5, 4.0, 4.1, 4.3, with random initialization of (x 1 , y 1 ) and N = 1000 points in each simulated orbit. We generated 1000 orbits per each value of r, and randomly split the 5000 observations in 70% -30% training-test sets as in [START_REF] Mathieu Carrière | Perslay: A neural network layer for persistence diagrams and new graph topological signatures[END_REF]. Cross-entropy loss was used to train the network for 100 epochs, using Adam optimizer with mini-batches of size 16. For the noiseless case, the experiment for PointNet is repeated 5 times, and the experiment result for PersLay is from Carrière et al. [2020].

Topological layer. For MLP+P and CNN+P(i), we use one PLLay at the beginning of MLP and CNN models with 64 nodes and affine transformation, which is solely used as the input to MLP or concatenated to the raw input to CNN. We used the empirical DTM filtration in (4), where we define fixed 40 × 40 points on grid on [0.0125, 0.9875] 2 and use X as the empirical data points. We used m 0 = 0.01, K max = 2, T min = 0.03, T max = 0.1, m = 17. For CNN+P, we additionally use one PLLay after the convolutional layer, with K max = 2, T min = 0.05, T max = 0.95, m = 18.

Baselines. For the baselines, models were designed to have simple structures for quick comparisons:

• Vanilla MLP: one hidden layer with 32 units with ReLU activations.

• CNN: two convolution layers followed by two fully connected layers.

• SLay: for comparison with PLLay, one SLay is used with 16 nodes, which is concatenated to the raw input to either MLP or CNN. We used the value ν = 0.01 for the hyperparameter of SLay.

Result. The accuracy results for ORBIT5K data in Figure 4 is represented with 1 standard errors in Table 3 and Figure 7. In Figure 7, the results for MLP, MLP+S, MLP+P are in Figure 7(a), and the results for CNN, CNN+S, CNN+P, CNN+P(i) are in Figure 7(b). From Figure 7(a), we observe that PLLay improves over MLP and MLP+S by a huge margin (42% ∼ 60%). In particular, without PLLay, MLP and MLP+S remain at random classifiers, which implies that the topological information is indeed critical for ORBIT5K. In Figure 7(b), PLLay improves over CNN or CNN+S consistently as well. Moreover, due to the high complexity of ORBIT5K, CNN suffers from high variance at corruption and noise probability 0.2, 0.35, while PLLay can effectively reduce the variance at those simulations and make the models more stable by utilizing robust topological information from the DTM function. Also, the accuracies for CNN+P are almost always higher than the accuracies for CNN+P(i), meaning that adding PLLay in the middle of the network indeed further improves the accuracy.

Figure 1 :

 1 Figure 1: Illustration of PLLay, a novel topological layer based on weighted persistence landscapes. Information in the persistence diagram is first encoded into persistence landscapes as a form of vectorized function, and then a deep learning model determines which components of the landscape (e.g., particular hills or valleys) are important for a given task during training. PLLay can be placed anywhere in the network.

  ) respectively, where • ∞ is the usual L ∞ -norm, D = D ∪ Diag and D = D ∪ Diag with Diag being the diagonal {(x, x) : x ∈ R} ⊂ R 2 with infinite multiplicity, and the set Γ consists of all the bijections γ : D → D .

Figure 2 :

 2 Figure 2: The topological features encoded in the persistence diagram & persistence landscapes for MNIST and ORBIT5k sample. In the MNIST example, two loops (1-dimensional feature) in '8' are clearly identified and encoded into the 1st and 2nd order landscapes. The ORBIT5k sample shows more involved patterns.

Figure 3 :

 3 Figure 3: The significant point (inside green-dashed circle) in the persistence diagram remains almost unchanged even after corrupting pixels and adding noise to the image.

Definition A. 5 (

 5 Persistence diagram) Let R 2 * := {(b, d) ∈ (R∪∞) 2 : d > b}. Let F be a filtration and let k ∈ N 0 . The corresponding k-th persistence diagram Dgm k (F) is a finite multiset of R 2 * , consisting of all pairs (b, d), where [b, d) is the interval of filtration values for which a specific homology class appears in P H k F. b is called a birth time and d is called a death time.

  Now, let c K denote the Lipschitz constant in Hofer et al. [2017, Theorem 1] and c g θ denote the constant term in our result in Theorem 4.1, i.e. c g θ = L g T ν 1/2 . We want to upper bound the ratio cg θ d B (D,D ) c K W Lr 1 (D,D ) . This directly comes from (23) and Proposition B.1 as

E. 5

 5 Proof of Claim B.1 Let γ : D → D be any bijection and let S := p ∈ D : pγ(p) ∞ > 2t . Then for p ∈ D with pγ(p) ∞ ≤ 2t, there exists β(p) ∈ R 2 * such that pβ(p) ∞ ≤ t and β(p)γ(p) ∞ ≤ t. Now, define two diagrams D t , D t as follows: D t = S ∪ β(p) : p ∈ D\S \Diag, D t = S ∪ β(p) : p ∈ D\S \Diag, where S := γ(p) : p ∈ D . Then, d B (D, D t ) ≤ t and d B (D , D t ) ≤ t from the construction.

  Diagram of DTM filtration, m0 = 0.2.

Figure 5 :

 5 Figure 5: One example of the digit 8 in MNIST data, its contour plots and persistence diagrams of DTM filtration at m 0 = 0.05 and m 0 = 0.2. When m 0 = 0.05, DTM filtration aggregates more locally, and the 1st persistent homology extracts two loop structures of the digit 8. When m 0 = 0.2, DTM filtration aggregates the digit 8 more globally, and the 0th persistent homology extracts three connected component structures of the digit 8.

  Test accuracy in ORBIT5K data for MLP, MLP+S, MLP+P.

  accuracy in ORBIT5K data for CNN, CNN+S, CNN+P, CNN+P(i).

Figure 7 :

 7 Figure 7: Test in ORBIT5K experiments. PLLay contributes to consistent improvement in accuracy and robustness against noise and corruption. In particular in (b), when the corruption and noise probability is 0.1, 0.25, 0.35, PLLay effectively reduces the variance of classification accuracy.

  Figure 4: Test accuracy in MNIST and ORBIT5K experiments. PLLay consistently improves the accuracy and the robustness against noise and corruption. In particular, in many cases it effectively reduces the variance of the classification accuracy on ORBIT5K.

				Accuracy for MNIST data			Accuracy for ORBIT5K data		Model	Accuracy
						0.9				PointNet	0.708
	Accuracy	0.7 0.8			Accuracy	0.7 0.8			MLP MLP+S	PersLay	(±0.285) 0.877 (±0.010)
		0.6				0.6 0.2			MLP+P CNN	CNN	0.915 (±0.088)
	sd	0.05 0.10 0.15	0.0	0.1 Corrupt and noise probability 0.2 0.3 Sd for MNIST data	sd	0.05 0.10 0.15	0.0	0.1 Corrupt and noise probability 0.2 0.3 Sd for ORBIT5K data	CNN+S CNN+P CNN+P(i)	CNN+ SLay CNN+ PLLay	0.943 (±0.014) 0.950 (±0.016)
		0.00				0.00				Table 1: Comparison of dif-
										ferent methods for ORBIT5K
										including the current state-
										of-the-art PersLay. The pro-
										posed method achieves the
										new state-of-the-art accuracy.

  B.1 Let D, D be two persistence diagrams. For t > 0, let n t ∈ N be satisfying the followings: for any two diagrams D t , D t with d B (D, D t ) ≤ t and d B (D , D t ) ≤ t, either |D t \D t | ≥ n t or |D t \D t | ≥ n t holds. Then for any bijection γ : D → D , the number of paired points with being at least 2t apart in L ∞ distance is greater or equal to n t , i.e., p ∈ D : pγ(p) ∞ > 2t ≥ n t . And then, we get a lower bound for the ratio of Wasserstein distance to the bottleneck distance. Let D, D be two persistence diagrams. For t > 0, let n t ∈ N be satisfying the followings: for any two diagrams D t , D t with d B (D, D t ) ≤ t and d B (D , D t ) ≤ t, either |D t \D t | ≥ n t or |D t \D t | ≥ n t holds.

	Proposition B.1

Table 2 :

 2 Test accuracy in MNIST experiments. In each cell, the top number corresponds to the average accuracy of the model at the corruption and noise probability, and the bottom number corresponds to the 1 standard deviation of the accuracies. At each column, the model with the best accuracy is bolded.

  Test accuracy in MNIST data for CNN, CNN+S, CNN+P, CNN+P(i).Figure 6: Test accuracy in MNIST experiments. PLLay contributes to consistent improvement in accuracy and robustness against noise and corruption. In particular, the improvement on CNN increases up to the moderate level of corruption and noise (∼ 15%), and then start to decrease.

	0.6 0.7 0.8 0.9 0.6 0.7 0.8 0.9 MLP Accuracy Accuracy MLP+S MLP+P CNN CNN+S CNN+P CNN+P(i)	0.0 0.0	0.1 Corrupt and noise probability 0.2 Accuracy for MNIST data, MLP based 0.1 0.2 0.3 0.3 Corrupt and noise probability Accuracy for MNIST data, CNN based 0.05 0.10 0.15 0.20 0.2001 0.1997 0.1994 0.1998 0.2003 0.25 (0.0014) (0.0031) (0.0020) (0.0029) (0.0009) (0.0010) (0.0016) (0.0011) MLP MLP+S MLP+P CNN CNN+S CNN+P CNN+P(i) 0.00 0.30 0.35 0.2000 0.2004 0.1999 0.2054 0.2028 0.2171 0.2171 0.2121 0.2159 0.2115 0.2057 (0.0126) (0.0129) (0.0364) (0.0364) (0.0236) (0.0301) (0.0193) (0.0180) 0.8082 0.7906 0.7660 0.7456 0.7181 0.6942 0.6545 0.6218 (0.0103) (0.0082) (0.0115) (0.0104) (0.0100) (0.0130) (0.0110) (0.0102) 0.9466 0.9247 0.9053 0.8791 0.8224 0.8323 0.7963 0.7401 (0.0116) (0.0152) (0.0195) (0.0255) (0.1474) (0.0298) (0.0331) (0.1293) 0.9412 0.8881 0.8142 0.8142 0.8197 0.7777 0.6580 0.7195 (0.0182) (0.1612) (0.1900) (0.1900) (0.1473) (0.1875) (0.2622) (0.1778) 0.9511 0.9249 0.9095 0.8941 0.8619 0.8480 0.8087 0.7668 (0.0140) (0.0308) (0.0329) (0.0305) (0.0366) (0.0173) (0.0396) (0.0319) (b) Noise probability 0.9449 0.9319 0.8965 0.8873 0.8577 0.8285 0.7954 0.7543 (0.0343) (0.0290) (0.0471) (0.0143) (0.0349) (0.0515) (0.0516) (0.0553)

(a) Test accuracy in MNIST data for MLP, MLP+S, MLP+P.

Table 3 :

 3 Test accuracy in ORBIT5K experiments.

Preprint.

Acknowledgments and Disclosure of Funding

During the last 36 months prior to the submission, Jisu Kim received Samsung Scholarship, and Joon Sik Kim received Kwanjeong Fellowship. Freédéric Chazal was supported by the ANR AI chair TopAI.

To show (31), we first apply the triangle inequality as

And note that since X, Y, X, Y are all bounded in Euclidean space, they are totally bounded metric spaces. Thus by Theorem 5.2 in Chazal et al. [2014a], the bottleneck distance between any two diagrams is bounded by Gromov-Hausdorff distance, and in particular,

And then since the Gromov-Hausdorff distance is bounded by the Hausdorff distance,

And the Hausdorff distance between X and X or Y and Y is bounded by by the assumption that X, Y are -coverings of X, Y, respectively, i.e.,

Hence combining ( 32), ( 33), (34), and ( 35) gives (31) as

Now, the results follows from ( 30) and ( 31).

E.8 Proof of Proposition D.1

From (4), note that for any y ∈ ς, dm0 (y) is expanded as

where k is such that Xi∈N k-1 (y) i < m 0 n i=1 i ≤ Xi∈N k (y) i , and i = Xj ∈N k (y) jm 0 n j=1 j for one of X i 's that is k-th nearest neighbor of y and ω i = ω i otherwise. Hence, by letting y = arg max z∈ς dm0 (z) applying to (36), the filtration function f X at simplex ς becomes

where the notations f X and dX,m0 are to clarify the dependency of f on X. And from the condition, dm0 (y) > dm0 (z) holds for all z ∈ ς. Hence for sufficiently small > 0 and for any Z = {Z 1 , . . . , Z n } with Z j -X j < , (37) becomes

Hence by differentiating (38), the derivative of f with respect to X is calculated as

× j X jy r-2 (X jy)I(X j ∈ N k (y)) m 0 n i=1 i = j X jy r-2 (X jy)I(X j ∈ N k (y)) dm0 (y)

r-1 m 0 n i=1 i .


A Guertin

C Duchemin

E Garrido

F Haddad

V Metivier

N Michel

Subatech

Nuclear Data for Charged-particle Monitor Reactions and Medical Isotope Production Requirements for improved and extended excitation functions for monitor reactions

Motivations  Experimental set-up and data measurements  Results and comparisons  Conclusions and outlooks Hyatt Regency,

Stacked-foil technique:

-Target/monitor/degrader pattern -Thin foils:

E loss small and constant -One cross section value per foil Activity and cross section:

σ = Act . A χ. Φ . 𝒩𝒩 A . ρ . e . (1 -e -λ.t ) σ = σ ′ . 𝜒𝜒 ′ . Act . A . ρ′ . e′ . (1 -e -λ ′ .t ) 𝜒𝜒. Act′ . A′ . ρ . e . (1 -e -λ.t )

Use of a monitor foil:

 error on e, e': ≤ 1%  error on t: negligible

  Additional monitor reactions proposed producing44m Sc, 47 Sc Independent new data to constrain shape and amplitude of the recommended cross section curveHalf lives interesting and gamma rays easily detectable

*-From

  Production of Medical Isotopes from a Thorium Target Irradiated by Light Charged Particles up to 70 MeV, C. Duchemin, Subatech, August 27, 4:55 p.m., Crystal A IAEA recommended cross sections: -8 reactions available for protons 27 Al (2), nat Ni, nat Ti and nat Cu (5) -5 reactions available for deuterons 27 Al (2), nat Fe, nat Ni and nat Ti Use of a Faraday cup: -Beam dump placed at the end of the stack to measure the intensity during the irradiation nat Ti(d,x) 48 V, E < 35 MeV nat Ni(p,x) 57 Ni, 15 MeV < E < 50 MeV nat Cu(p,x) 62 Zn, 50 MeV < E < 60 MeV nat Cu(p,x) 56 Co, 60 MeV < E Gamma spectroscopy: HPGe coaxial detector γ spectra recorded on 8192 channels -Dead time: < 10% (sum peak) -FWHM:1.04 keV at 122 keV ( 57 Co) -Activity values: FitzPeaks 1.97 keV at 1332 keV ( 60 Co) -T 1/2 , E γ , I γ : Lund/LBNL, NNDC -Energy and efficiency calibrations: Co and Eu obtained to fulfill the need of coherent and reliable nuclear data for nuclear medicine  For both proton and deuteron beams  Strengthen the existing IAEA recommended monitor reactions  Complete the range in energy  Open the door to new monitor reactions TALYS 1.6 calculations : reasonable agreement in this mass range Outlooks: Innovative radio-isotope program for PET imaging, β -and α targeted radiotherapy at ARRONAX is ongoing: 82 Sr/ 82 Rb -44 Sc -64 Cu, 47 Sc -67 Cu, 186 Re, 211 At, 230 Pa…  Performing cross section measurements and production calculation (TTY)  For beam monitor reactions and isotopes of medical interest  Producing a global set of data to constrain theoretical models TALYS parameter tuning: best combination for whole mass range -optical models (5) -level density models (5) -pre-equilibrium models (4) -…

extraction lines: stripper or electrostatic deflector Extracted Energy (MeV) Max. current (µA)

	H +	30 -70	2 x 375
	D +	15 -35	2 x 50
	He 2+	68	70
	HH +	17	50

,HH + : supernanogan ECR -2 * Medium Energy Accelerators for Isotope Production in Europe, F. Haddad, Subatech, GIP ARRONAX, August 27, 3:40 p.m., Crystal A Hyatt Regency, Chicago, 26/08/

14 8ICI -ND for MI 6 Experimental set-up and data measurements Th* Ti Al

The ARRONAX cyclotron is a project promoted by the Regional Council of Pays de la Loire financed by local authorities, the French government and the European Union. This work has been, in part, supported by a grant from the French National Agency for Research called "Investissements d 'Avenir", and Labex n°ANR-11-LABX-0018-01


TRENDS IN PARASITOLOGY 2021 vol 37 num 1 pp 6-8

Booming Omics in Schistosoma Yohann Le Govic, 1,2 Benjamin Gourbal, 3 and Jérôme Boissier 3, * 1-Department of Medical Parasitology and Mycology, Amiens University Hospital, Amiens, France 2-Host-Pathogen Interaction Study Group (GEIHP, EA 3142), UNIV Angers, UNIV Brest, Federative Structure of Research '-Cellular Interactions and Therapeutic Applications', SFR 4208 ICAT, Angers, France 3-IHPE, University of Montpellier, CNRS, Ifremer, University of Perpignan Via Domitia, Perpignan, France *Correspondence: boissier@univ-perp.fr (J. Boissier). https://doi.org/10.1016/j.pt.2020. 10.010 Efforts to eliminate schistosomiasis are hindered by incomplete efficacy of the only FDA-approved antischistosomal drug, praziquantel. By using postgenomic technologies, Wendt et al. and Wang et al. deciphered the function of several genes required for worm survival and pathogenesis, which opens the way for the development of innovative parasite-targeted therapies.

Schistosomiasis -also known as bilharziasis -is one of the most devastating of the world's neglected tropical diseases that is responsible for considerable human morbidity. The etiological agents, schistosomes, are dioecious flatworms that reside permanently in copula within the host's blood vessels. While blood feeding, the females lay eggs of which the majority become encysted in surrounding host tissues and organs (especially the liver), causing pathology [START_REF] Colley | Human schistosomiasis[END_REF]. Despite increased efforts to eradicate schistosomiasis over the past five decades, more than 200 000 people still die each year from schistosomeassociated complications [START_REF] Chitsulo | Schistosomiasis[END_REF]. This apparent failure to control disease transmission is, in part, due to the incomplete parasiticidal efficacy of praziquantel, which is ineffective on immature worms and does not prevent reinfection or the emergence of drugresistant parasites. Accordingly, in the recently published roadmap, aiming at eliminating schistosomiasis by 2030, the World Health Organization (WHO) highlights that new antischistosomal treatments are urgently needed [START_REF]Ending the Neglect to Attain the Sustainable Development Goals -A Road Map for Neglected Tropical Diseases 2021-2030[END_REF].

Hitherto, the development of innovative therapies has been hampered by the paucity of molecular and genetic tools for exploring schistosome biology. In their two recent enlightening articles, Wang et al. [START_REF] Wang | Large-scale RNAi screening uncovers therapeutic targets in the parasite Schistosoma mansoni[END_REF] and Wendt et al. [START_REF] Wendt | A single-cell RNA-seq atlas of Schistosoma mansoni identifies a key regulator of blood feeding[END_REF] provide new insights into the functional biology of schistosomes that will undoubtedly unleash the discovery of new potential drug targets. Indeed, despite tremendous advances in both genome resources and genetics in the past few years (Figure 1), much of the molecular and cellular processes that underpin the tissue maintenance and survival of adult schistosomes remain poorly understood. To gain a better insight into these processes, Wendt and colleaguesjoining fluorescence-activated cell sorting (FACS), single-cell transcriptomics, and in situ hybridizationsuccessfully generate an unbiased classification of cellular subtypes in Schistosoma mansoni male, mature and immature female worms. This approach enabled them to produce a precious atlas of ~43 000 high-quality cells (freely available at SchistoCyte Atlas i ), representing 68 transcriptionally distinct cell categories [START_REF] Wendt | A single-cell RNA-seq atlas of Schistosoma mansoni identifies a key regulator of blood feeding[END_REF]. These clusters notably included neoblasts, which are motile stem cells involved in tissue turnover and regeneration [START_REF] Collins | Adult somatic stem cells in the human parasite Schistosoma mansoni[END_REF]. Although previous studies indicated that neoblasts of schistosomes predominantly serve for tegument maintenance, there is evidence that other flatworms (e.g., planarians) also have 'specialized' neoblasts that are progenitors for specific differentiated tissues [START_REF] Reddien | The cellular and molecular basis for planarian regeneration[END_REF]. Here, Wendt et al. pointed out the existence of a previously undescribed subpopulation of neoblasts that cluster with gut cells, the latter being required for digestion of host blood. By combining RNA interference (RNAi) and parasitological analyses, it was shown that this neoblast 'gut lineage' expresses a homolog of hepatocyte nuclear factor 4 (hnf4), a gut marker, in planarians, which is required for gut preservation and hematophagous behavior. Indeed, RNA silencing of hnf4 resulted in a sizeable drop in the number of luminal microvilli, leading to morphological abnormalities of the gut and subsequent inability of the parasites to digest red blood cells and absorb nutrients. Considering the importance of blood feeding for the production of eggs, which are the hallmarks of schistosomiasis pathology, Wendt et al. further evaluated the pathogenicity of the worms whose hnf4 gene was knocked down through RNAi. In a striking in vivo experiment, it was demonstrated that the loss of hnf4 confers reduced pathology (i.e., egg-induced granulomas) following surgical transplantation into Schistosoma-naive mice. Moreover, RNAitreated parasites were significantly shorter than nontreated controls, indicating that hnf4 is also required for normal growth. Taken together, these data suggest that targeting the blood-feeding machinery may constitute a relevant alternative strategy to praziquantel.

Another study, conducted by the same research group, has successfully used highthroughput RNAi screening for profiling the essential gene landscape and identification of novel therapeutic targets in S. mansoni [START_REF] Wang | Large-scale RNAi screening uncovers therapeutic targets in the parasite Schistosoma mansoni[END_REF]. By this approach, Wang and colleagues present a dataset of over 2000 genes (~20% of the protein-coding genes), of which 261 were assigned to the parasite's critical gene repertoire. From the list of top candidates, a couple of serine/threonine kinases (TAO and STK25) were found to be essential in maintaining neuromuscular function. Indeed, RNA silencing of either of these kinase-encoding genes resulted in impaired locomotion, tetanic contraction of the body of the worm, and detachment from the culture substrate. Most importantly, loss of tao or stk25 gene expression impeded schistosome capacity to reside within the host blood vessels, resulting in an attenuated schistosome egg-induced pathology. Another noteworthy feature of this work is the elegant approach used to characterize a large portion of the 195 genes designated as essential for parasite attachment. Indeed, the transcriptomic analysis revealed substantial numbers of genes related to the ubiquitinproteasome pathway (UPP), which maintains protein homeostasis and normal cellular functions in eukaryotes [START_REF] Collins | The logic of the 26S proteasome[END_REF]. Strikingly, RNA silencing of virtually all UPP elements led to tissue degeneration in vitro, worm immobilization, and even death [START_REF] Wang | Large-scale RNAi screening uncovers therapeutic targets in the parasite Schistosoma mansoni[END_REF]. This underscores a central role for UPP in the maintenance of tegument integrity and survival in schistosomes. Considering the importance of UPP in both larval and adult worm stages [START_REF] Bibo-Verdugo | The proteasome as a drug target in the metazoan pathogen, Schistosoma mansoni[END_REF][START_REF] Nabhan | The 26S proteasome in Schistosoma mansoni: bioinformatics analysis, developmental expression, and RNA interference (RNAi) studies[END_REF], Wang et al. then sought to evaluate its druggability. By filtering databases and chemical libraries, a series of compounds were therefore selected by their potential to inhibit UPP components. This last and most exciting approach allowed the identification of p97, a central component in the UPP, as a novel target for drug development. In sum, this article sheds light on the usefulness of large-scale RNAi for unraveling complex cellular pathways and exploring their potential benefit in the field of schistosomiasis.

Altogether, these two illuminating articles provide an unprecedented foundation for a better understanding of multiple aspects of schistosome biology and, as a consequence, unveil hidden drug targets. Notably, these findings have highlighted some pre-existing pharmacological agents, thus accelerating the discovery of unexpected highly effective anthelmintic compounds. It is also important to underline the considerable value of the state-of-the-art methods used here in order to identify other clinical drug candidates with selective activity on these worms (and thus with limited side effects). One future avenue for investigation may concern reproductive biology (i.e., egg production) of schistosomes because such therapeutic intervention would combine the advantage of reducing the pathology with a limitation in transmission. The latest exciting developments in functional genomic technologies (especially CRISPR-Cas9) will undoubtedly stimulate further fruitful research into this important area of schistosome biology and greatly help disease control and the development of innovative therapeutic strategies

Figure 1 .

 1 Figure 1. Timeline of Omics Milestones and Tendencies for Schistosoma Research. Abbeviations: EST, expressed sequence tag; FISH, fluorescent in situ hybridization; ORESTS, open reading frame expressed sequence tags.


Motivations  Experimental set-up and data measurements  Results and comparisons  Conclusions and outlooks Sheraton, New York, 05/03/13 ND2013 -HD 7 3 Motivations Innovative radionuclides: PET imaging, β -and α targeted radiotherapy α RIT: binding an α emitter to an antibody to target and destroy tumor cells ᴥ high LET ᴥ important cytotoxic effect ᴥ minimizing damage 226 Th: novel therapeutic nuclide, a more potent α emitter for leukemia than 213 Bi (46 min) ᴥ decay: 4 α cascade, 27.7 MeV ᴥ generator system: 230 U(21 d)/ 226 Th(31 min) Production routes: 231 Pa(p,2n) 230 U 232 Th(p,3n) 230 Pa → 230 U 232 Th(d,4n) 230 Pa → 230 U

-optical models (5) -level density models (5) -pre-equilibrium models (4)

Innovative radio-isotope program for PET imaging, β -and α targeted radiotherapy at ARRONAX is ongoing: 82 Sr/ 82 Rb -44 Sc -64 Cu, 47 Sc -67 Cu, 211 At…

-Sheraton

  Φ . 𝒩𝒩 A . ρ . e . (1 -e -λ.t ) σ = σ ′ . Act . A . ρ′ . e′ . (1 -e -λ ′ .t ) Act′ . A′ . ρ . e . (1 -e -λ.t ) Use of a monitor foil:  error on e, e': ≤ 1%  error on t: negligible IAEA recommended cross sections: -5 reactions available for deuteron beam  27 Al (2), nat Fe (1), nat Ni (1) and nat Ti(d,x) 48 V  error on σ: ≤ 12% points: 18.95 -20.86 -22.50 -24.35 -27.53 -29.11 MeV for 230,232,233 Pa HPGe coaxial detector γ spectra recorded on 8192 channels -Dead time: < 10% (sum peak) -FWHM:1.04 keV at 122 keV ( 57 Co) -Activity values: FitzPeaks 1.97 keV at 1332 keV ( 60 Co) -T 1/2 , E γ , I γ : Lund/LBNL, NNDC -Energy and efficiency calibrations : Co and Eu -Target and monitor:  counted twice  during > 24 hrs  2 week delay γ rays used to extract production cross section of 230,232,233 Pa -TALYS 1.4, default parameters -Very low cross sections, far from the experimental data -Eγ 312.17 keV Iγ 38-Difference explained by the Iγ update in database since 1986 -TALYS 1.4, default parameters -Not in agreement with data even if the shape is not too bad ray 75 keV + 136 Cs 819 keV) -Same trend -Difference explained by γ line choosen and the I γ update in database since 1986 1.4, default parameters -Neither the shape nor the maximum value are reproduced -Same trend but a small energy shift and a maximum value slightly higher -Difference not explained by the Iγ update in database since 1986 New data sets obtained: 232 Th(d,xn) 230,232,233 Pa -compared with J. Rama Rao et al.  232,233 Pa: same trends, influence of the branching ratio values  230 Pa: same trend, higher amplitude, shift in energy -compared with TALYS calculations  default parameters, no satisfactory results  Also new data sets extracted from the monitors: nat Ti(d

The ARRONAX cyclotron is a project promoted by the Regional Council of Pays de la Loire financed by local authorities, the French government and the European Union