Original: Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set
Revised: P ERSISTENT H OMOLOGY C APTURES THE G ENERALIZATION OF N EURAL N ETWORKS W ITHOUT A V ALIDATION S ET

Original: Anonymous Author(s)
Revised: Anonymous authors

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

Original: Input "This is a text"
Revised: Input "This is a text"

Original: Training
Revised: Training

Original: Network Status by Batch
Revised: Network Status by Batch

Original: Batch N Batch N- (Weighted and directed) Graph Representation
Revised: Batch N Batch N- (Weighted and directed) Graph Representation

Original: Distance Evolution
Revised: Distance Evolution

Original: Persistent Homology
Revised: Persistent Homology

Original: Batch NBatch N (n * (birth, death, dimension))
Revised: Batch NBatch N (n * (birth, death, dimension))

Original: Batch NBatch N (Filtering ϵ =0.01)
Revised: Batch NBatch N (Filtering ϵ =0.01)

Original: Discretization
Revised: Discretization

Original: D
Revised: D

Original: Diference
Revised: Diference

Original: Abstract
Revised: A BSTRACT

Original: The training of neural networks is usually monitored with a validation (holdout)set to estimate the generalization of the model.
Revised: The training of neural networks is usually monitored with a validation (holdout) set to estimate the generalization of the model.

Original: This is done instead of measuringintrinsic properties of the model to determine whether it is learning appropriately.
Revised: This is done instead of measuring intrinsic properties of the model to determine whether it is learning appropriately.

Original: In this work, we suggest studying the training of neural networks with Algebraic
Revised: In this work, we suggest studying the training of neural networks with Algebraic

Original: Topology, speciﬁcally Persistent Homology (PH).
Revised: Topology, speciﬁcally Persistent Homology (PH).

Original: Using simplicial complex representations of neural networks, we study the PH diagram distance evolution on theneural network learning process with different architectures and several datasets.
Revised: Using simplicial complex representations of neural networks, we study the PH diagram distance evolution on the neural network learning process with different architectures and several datasets.

Original: Results show that the PH diagram distance between consecutive neural networkstates correlates with the validation accuracy, implying that the generalization errorof a neural network could be intrinsically estimated without any holdout set.
Revised: Results show that the PH diagram distance between consecutive neural network states correlates with the validation accuracy, implying that the generalization error of a neural network could be intrinsically estimated without any holdout set.

Original:  Introduction
Revised:  I NTRODUCTION

Original: Generalization is what makes a machine learning model useful; the uncertainty of its behaviour withunseen data is what makes it potentially dangerous.
Revised: Generalization is what makes a machine learning model useful; the uncertainty of its behaviour with unseen data is what makes it potentially dangerous.

Original: Thus, understanding the generalization error of amodel can be considered one of the holy grails of the entire machine learning ﬁeld.
Revised: Thus, understanding the generalization error of a model can be considered one of the holy grails of the entire machine learning ﬁeld.

Original: Machine learning practitioners typically monitor some metrics of the model to estimate its generalization error and stop the training even before the numerical convergence to prevent the overﬁtting ofthe model.
Revised: Machine learning practitioners typically monitor some metrics of the model to estimate its generalization error and stop the training even before the numerical convergence to prevent the overﬁtting of the model.

Original: Usually, the error measure or the metric relevant to the task is computed for a holdoutset, the validation set.
Revised: Usually, the error measure or the metric relevant to the task is computed for a holdout set, the validation set.

Original: Since these data have not been directly used for updating the parameters, itis assumed that the performance of the model on the validation set can be used as a proxy of the generalization error, provided it is representative of the data that will be used in inference.
Revised: Since these data have not been directly used for updating the parameters, it is assumed that the performance of the model on the validation set can be used as a proxy of the generalization error, provided it is representative of the data that will be used in inference.

Original: Onecan, though, potentially overﬁt to this holdout set if is repeatedly used for guiding a hyperparametersearch.
Revised: One can, though, potentially overﬁt to this holdout set if is repeatedly used for guiding a hyperparameter search.

Original: Instead of relying on an external set, though, the question of whether it could be possible to estimatethe generalization error with some intrinsic property of the model is highly relevant, and it has beenbarely explored in the literature.
Revised: Instead of relying on an external set, though, the question of whether it could be possible to estimate the generalization error with some intrinsic property of the model is highly relevant, and it has been barely explored in the literature.

Original: On the other hand, Algebraic Topology has recently been gainingmomentum as a mathematical tool for studying graphs, machine learning algorithms, and data.
Revised: On the other hand, Algebraic Topology has recently been gaining momentum as a mathematical tool for studying graphs, machine learning algorithms, and data.

Original: In this work, we have the goal of, once having characterized neural networks as weighted, acyclicgraphs, represented as Algebraic Topology objects (following previous works), computing distancesbetween consecutive neural network states.
Revised: In this work, we have the goal of, once having characterized neural networks as weighted, acyclic graphs, represented as Algebraic Topology objects (following previous works), computing distances between consecutive neural network states.

Original: More speciﬁcally, we can calculate the Persistent
Revised: More speciﬁcally, we can calculate the Persistent

Original: Homology (PH) diagram distances between a give state (i.e., when having a speciﬁc weights duringthe training process) and the next one (i.e., after having updated the weights in a training step) (see Figure 1.
Revised: Homology (PH) diagram distances between a give state (i.e., when having a speciﬁc weights during the training process) and the next one (i.e., after having updated the weights in a training step), as depicted in Figure 1.

Original: We observe that during the training procedure of neural networks we can measure thisdistance in each learning step, and show that there exists a high correlation with the correspondingvalidation accuracy of the model.
Revised: We observe that during the training procedure of neural networks we can measure this distance in each learning step, and show that there exists a high correlation with the corresponding validation accuracy of the model.

Original: We do so in a diverse set of deep learning benchmarks and modelhyperparameters.
Revised: We do so in a diverse set of deep learning benchmarks and model hyperparameters.

Original: This shines light on the question of whether the generalization error could beestimated from intrinsic properties of the model, and opens the path towards a better theoreticalunderstanding of the dynamics of the training of neural networks.
Revised: This shines light on the question of whether the generalization error could be estimated from intrinsic properties of the model, and opens the path towards a better theoretical understanding of the dynamics of the training of neural networks.

Original: In summary, our contributions are as follows:
Revised: In summary, our contributions are as follows:

Original: • Based on principles of Algebraic Topology, we propose measuring the distances (Silhouetteand Heat) between the PH persistence diagrams obtained from a given state of a neuralnetwork during the training procedure and the one in the immediately previous weightsupdate.•
Revised: • Based on principles of Algebraic Topology, we propose measuring the distances (Silhouette and Heat) between the PH persistence diagrams obtained from a given state of a neural network during the training procedure and the one in the immediately previous weights update.

Original:  We empirically show that the evolution of these measures during training correlate withthe accuracy in the validation set.
Revised: • We empirically show that the evolution of these measures during training correlate with the accuracy in the validation set.

Original: We do so in diverse benchmarks (MNIST, CIFAR10,CIFAR100, Reuters text classiﬁcation), and models (MLPs in MNIST and Reuters, MLPsand CNNs in CIFAR100 and CIFAR100).• We thus provide empirical proof of the fact that valuable information related to the learningprocess of neural networks can be obtained from PH distances between persistence diagrams( homological convergence ).
Revised: We do so in diverse benchmarks (MNIST, CIFAR10, CIFAR100, Reuters text classiﬁcation), and models (MLPs in MNIST and Reuters, MLPs and CNNs in CIFAR100 and CIFAR100). • We thus provide empirical proof of the fact that valuable information related to the learning process of neural networks can be obtained from PH distances between persistence diagrams (we will call this process homological convergence ).

Original: In particular, we show that homological convergence is relatedto learning process and the generalization properties of neural networks.•
Revised: In particular, we show that homological convergence is related to learning process and the generalization properties of neural networks.

Original:  In practice, we provide a new tool for monitoring the training of neural networks, and openthe path to estimating their generalization error without a validation set.
Revised: • In practice, we provide a new tool for monitoring the training of neural networks, and open the path to estimating their generalization error without a validation set.

Original: The remainder of this article is as follows.
Revised: The remainder of this article is as follows.

Original: In Section 2 we describe the theoretical background of ourproposal in terms of Algebraic Topology, while in Section 3 we go through the related work.
Revised: In Section 2 we describe the theoretical background of our proposal in terms of Algebraic Topology, while in Section 3 we go through the related work.

Original: Then, in
Revised: Then, in

Original: Section 4 we formalize our method.
Revised: Section 4 we formalize our method.

Original: Finally, in sections 6 and 7 we present and discuss our empiricalresults, respectively.
Revised: Finally, in sections 6 and 7 we present and discuss our empirical results, respectively.

Original:  Background
Revised:  B ACKGROUND

Original: In this section we introduce the mathematical foundations of this paper.
Revised: In this section we introduce the mathematical foundations of this paper.

Original: A detailed mathematicaldescription is included in the Supplementary Material.
Revised: A detailed mathematical description is included in the Supplementary Material.

Original: A simplicial complex is a set composed of points, line segments, triangles, and their n-dimensionalcounterparts, named simplex ( K ).
Revised: A simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts, named simplex ( K ).

Original: In particular, a simplicial complex must comply with two properties:1.
Revised: In particular, a simplicial complex must comply with two properties: 1.

Original: Every face of a simplex is also in the simplicial complex (of lower dimension).
Revised: Every face of a simplex is also in the simplicial complex (of lower dimension).

Original: The non-emptyintersection of any two simplices contained on a simplicial complex is a face of both.
Revised: The non-empty intersection of any two simplices contained on a simplicial complex is a face of both.

Original: 0,1,2,3-simplexand non simplex examples are shown in Figure 2. 
Revised: 0,1,2,3-simplex and non simplex examples are shown in Figure 2. [Figure]

Original: We can associate to an undirected graph, G =
Revised: We can associate to an undirected graph, G =

Original: ( V , E ) , a simplicial complex where all the verticesof G are the 0-simplex of the simplicial complex and the complete subgraphs with i vertices, in Gcorresponds to a ( i − 1 ) -simplex.
Revised: ( V , E ) , a simplicial complex where all the vertices of G are the 0-simplex of the simplicial complex and the complete subgraphs with i vertices, in G [Figure] corresponds to a ( i − 1 ) -simplex.

Original: This type of construction is usually called a complex clique on thegraph G, and is denoted by Cl ( G ) .
Revised: This type of construction is usually called a complex clique on the graph G, and is denoted by Cl ( G ) .

Original: Figure 3 shows a graph clique complex Cl(G) example.[Figure] [Equation] [Equation]
Revised: Figure 3 shows a graph clique complex Cl(G) example.

Original: simplex The boundary function is deﬁned as a map, froman i -simplex to an ( i − 1 ) -simplex, as the sumof its ( i − 1 ) -dimensional faces.
Revised: The boundary function is deﬁned as a map, from an i -simplex to an ( i − 1 ) -simplex, as the sum of its ( i − 1 ) -dimensional faces.

Original: A boundaryfunction sample is shown in Figure 4.
Revised: A boundary function sample is shown in Figure 4.

Original: In algebraic topology, a k -chain is a combinationof k -simplices (sometimes symbolized as a linear combination of simplices that compose thechain).
Revised: In algebraic topology, a k -chain is a combination of k -simplices (sometimes symbolized as a linear combination of simplices that compose the chain).

Original: The boundary of a k -chain is a ( k − 1 ) -chain.
Revised: The boundary of a k -chain is a ( k − 1 ) - chain.

Original: It is the linear and signed combination ofchain element boundary simplices.
Revised: It is the linear and signed combination of chain element boundary simplices.

Original: The space ofi -chains is denoted by C i ( K ) .
Revised: The space of i -chains is denoted by C i ( K ) .

Original: There are two special cases of chains that will be useful to deﬁne homology:
Revised: There are two special cases of chains that will be useful to deﬁne homology group:

Original: • Closed chain or i -cycle: i -chain with empty boundary.
Revised: • Closed chain or i -cycle: i -chain with empty boundary.

Original: An i -chain c is an i -cycle if and only if ∂ i c = 0,
Revised: An i -chain c is an i -cycle if and only if ∂ i c = 0,

Original: i
Revised: i

Original: e. c ∈ ker ( ∂ i ) .
Revised: e. c ∈ ker ( ∂ i ) .

Original: This subspace of C i ( K ) is denoted as Z i ( K ) .
Revised: This subspace of C i ( K ) is denoted as Z i ( K ) .

Original: • Exact chain or i -boundary:
Revised: • Exact chain or i -boundary:

Original: An i -chain c is an i -boundary if there exists an ( i + 1 ) -chain
Revised: An i -chain c is an i -boundary if there exists an ( i + 1 ) -chain

Original: d such that c = ∂ i + 1 ( d ) , i
Revised: d such that c = ∂ i + 1 ( d ) , i

Original: e. c ∈
Revised: e. c ∈

Original: im ( ∂ i + 1 ) .
Revised: im ( ∂ i + 1 ) .

Original: This subspace of C i ( K ) , the set of all such i-boundaries forms, is denoted by B
Revised: This subspace of C i ( K ) , the set of all such i-boundaries forms, is denoted by B

Original: i ( K ) .
Revised: i ( K ) .

Original: Now, if we think in the i -cyclesthat do not bound an ( i + 1 ) -simplicial complex, this is the deﬁnitioni -th homology  of the simplicial complex K .
Revised: Now, if we consider i -cycles not bounding an ( i + 1 ) -simplicial complex, this is the deﬁnition of an i -th homology group of the simplicial complex K .

Original: The precise deﬁnition is the quotient space of B i ( K ) asubspace of Z i ( K ) (see Supplementary Material).
Revised: The precise deﬁnition is the quotient group of B i ( K ) module Z i ( K ) (i.e. B i ( K ) / Z i ( K ) , see Supplementary Material).

Original: The number of non equivalent i -cycles (Figure 5)[Figure] is the dimension of the homology group H i
Revised: The number of non equivalent i -cycles (Figure 5) is the dimension of the homology group H i

Original: ( K ) , also named Betti numbers.
Revised: ( K ) , also named Betti numbers.

Original: [Figure] We can create a nested family of simplicial com- [Figure] plexes, K ε , where at each step t , K ε t is embedded 
Revised: We can create a nested family of simplicial complexes, K ε , where at each step t , K ε t is embedded in the simplicial complex K ε t + 1 .

Original: [Figure]

Original: in the simplicial complex K ε t + 1 .

Original: We call this set[Figure] a simplicial complex ﬁltration.
Revised: We call this set a simplicial complex ﬁltration.

Original: [Figure]

Original: For each ﬁltration simplicial complex, we can[Figure] calculate the homology groups.Then, we can
Revised: For each ﬁltration simplicial complex, we can calculate the homology groups.

Original: [Figure]

Original:  look at the birth, that is, when a homology class[Figure] appears, and death, the time when the homology [Figure]
Revised: Then, we can look at the birth, that is, when a homology class appears, and death, the time when the homology class disappears.

Original: class disappears. The PH treats the birth and the[Figure] death of these homological features in K ε for 
Revised: The PH treats the birth and the death of these homological features in K ε for different ε values.

Original: [Figure]

Original: different ε values.

Original: The lifespan of each homo-

Original: [Figure]

Original: logical feature can be represented as an interval[Figure] ( birth , death ) , of the homological class.Given
Revised: The lifespan of each homological feature can be represented as an interval ( birth , death ) , of the homological class.

Original: [Figure] a ﬁltration, this collection of intervals is named[Figure] a Persistence Diagram (PD) [5].
Revised: Given a ﬁltration, this collection of intervals is named a Persistence Diagram (PD) Carlsson (2009).

Original: [Figure]

Original: It is possible to compare two PDs using speciﬁc distances (Wasserstein and Bottleneck).
Revised: It is possible to compare two PDs using speciﬁc distances (Wasserstein and Bottleneck).

Original: To efﬁcientlyperform this operation, due to the size of these diagrams, it is sometimes necessary to simplify themby means of a discretization process (such as Weighted Silhouette and Heat vectorizations).
Revised: To efﬁciently perform this operation, due to the size of these diagrams, it is sometimes necessary to simplify them by means of a discretization process (such as Weighted Silhouette and Heat vectorizations).

Original:  Related Work
Revised:  R ELATED W ORK

Original: Algebraic Topology and Machine Learning
Revised: Algebraic Topology and Machine Learning

Original: The use of Algebraic Topology in the ﬁelds ofdata science and machine learning has been gaining momentum in recent years (see Carlsson [5]).
Revised: The use of Algebraic Topology in the ﬁelds of data science and machine learning has been gaining momentum in recent years (see Carlsson (2009)).

Original: Speciﬁcally in the case of neural networks, some works have applied topology for improving thetraining procedure of the models [15, 8], or pruning the model afterwards [30].
Revised: Speciﬁcally in the case of neural networks, some works have applied topology for improving the training procedure of the models Hofer et al. ; Clough et al.

Original:  Other works havefocused on analyzing the capacity of neural networks [14, 26, 17] or the complexity of input data[17].
Revised: (2020), or pruning the model afterwards Watanabe & Yamana (2020b). Other works have focused on analyzing the capacity of neural networks Guss & Salakhutdinov (2018a); Rieck et al. (2019b) ; Konuk & Smith (2019) or the complexity of input data Konuk & Smith (2019).

Original: Furthermore, recent works have provided topological analysis of the decision boundaries ofclassiﬁers based on PH and Betti numbers [24, 22].
Revised: Furthermore, recent works have provided topological analysis of the decision boundaries of classiﬁers based on PH and Betti numbers Ramamurthy et al.

Revised: ; Naitzat et al.

Original: Graph and topological representations of neural networks Gebhart et al.
Revised: Graph and topological representations of neural networks Gebhart et al.

Original: [12] suggest a methodfor computing the PH over the graphical activation neural networks, while Watanabe and Yamana[29] propose representing neural networks via simplicial complexes based on Taylor decomposition,from which one can compute the PH.
Revised: (2019) suggest a method for computing the PH over the graphical activation neural networks, while Watanabe & Yamana (2020a) propose representing neural networks via simplicial complexes based on Taylor decomposition, from which one can compute the PH.

Original: Chowdhury et al.
Revised: Chowdhury et al.

Original: [7] show that directed homology can be usedto represent MLPs.
Revised: (2019) show that directed homology can be used to represent MLPs.

Original: Anonymous [2] concurrently show neural networks, when represented as directed,acyclic graphs, can be associated to an Algebraic Topology object.
Revised: Anonymous (2021) concurrently show neural networks, when represented as directed, acyclic graphs, can be associated to an Algebraic Topology object, speciﬁcally to a directed ﬂag complex.

Original: By computing the PH diagram,one can effectively characterize neural networks, and even compute distances between two givenneural networks, which can be used to measure their similarity.
Revised: By computing the PH diagram, one can effectively characterize neural networks, and even compute distances between two given neural networks, which can be used to measure their similarity.

Original: This is unlike other works [11, 13]approximating neural networks representations with regard to the input space. 
Revised: This is unlike other works Corneanu et al. ; Guss & Salakhutdinov (2018b) approximating neural networks representations with regard to the input space. Relevant to our work, in Rieck et al. (2019b)

Revised: authors propose a complexity measure of neural networks based on persistent homology.

Revised: However, we will see that their representation does not fulﬁll our requirements in Section 4.

Original: Estimating the generalization and studying the learning process We are, though, speciﬁcallyinterested in the use of PH for analyzing the learning process, especially with the goal of estimatinggeneralization.
Revised: Estimating the generalization and studying the learning process We are, though, speciﬁcally interested in the use of PH for analyzing the learning process, especially with the goal of estimating generalization.

Original: In this regard, the literature is perhaps more limited.
Revised: In this regard, the literature is perhaps more limited.

Original: Jiang et al.
Revised: Jiang et al.

Original: [16] work onunderstanding what drives generalization in deep networks from a Bayesian of view.
Revised: (2019) work on understanding what drives generalization in deep networks from a Bayesian of view.

Original: Neyshabur et al.[23] study the generalization gap prediction from the training data and network parameters using amargin distribution, which are the distances of training points to the decision boundary.
Revised: Neyshabur et al. (2017) study the generalization gap prediction from the training data and network parameters using a margin distribution, which are the distances of training points to the decision boundary.

Original: In Li et al.[21] , authors propose an alternative to cross-validation for model selection based on training once onthe whole train set, without any data split, deriving a validation set with data augmentation.
Revised: In Li et al. (2020), authors propose an alternative to cross-validation for model selection based on training once on the whole train set, without any data split, deriving a validation set with data augmentation.

Original: Corneanu et al.
Revised: Corneanu et al.

Original: [10] try to estimate the performance gap between training and testing using PHmeasures.
Revised: (2020) try to estimate the performance gap between training and testing using PH measures.

Original: They claim.
Revised: They claim.

Original: However, one can observe some caveats.
Revised: However, one can observe some caveats.

Original: The ﬁrst one is that their regressionﬁtted to predict the test error has a considerably high error, making it not usable in practice.
Revised: The ﬁrst one is that their regression ﬁtted to predict the test error has a considerably high error, making it not usable in practice.

Original: Thesecond caveat is that for ﬁtting the regression one needs at least part of the sequestered testing set.
Revised: The second caveat is that for ﬁtting the regression one needs at least part of the sequestered testing set.

Original: In this work, motivated by the interest of having a better understanding of whether it would bepossible to estimate the generalization of neural networks without a holdout set, we suggest using thetopological characterization and distances concurrently proposed in Anonymous [2] but, crucially,measured between consecutive weight updates.
Revised: In this work, motivated by the interest of having a better understanding of whether it would be possible to estimate the generalization of neural networks without a holdout set, we suggest using the topological characterization and distances concurrently proposed in Anonymous (2021) but, crucially, measured between consecutive weight updates.

Original: We will show that the evolution of this distanceis similar to the one of the validation accuracy.
Revised: We will show that the evolution of this distance is similar to the one of the validation accuracy.

Original: Unlike Li et al.
Revised: Unlike Li et al.

Original: [21], we do not use any data atall.
Revised: (2020), we do not use any data at all.

Original: Unlike [10], we do not build a statistical or machine learning model (linear regression) forpredicting the testing error.
Revised: Unlike Corneanu et al. (2020), we do not build a statistical or machine learning model (linear regression) for predicting the testing error.

Original: Instead, we propose a new measure, and we empirically show that ithighly correlates with the validation accuracy.
Revised: Instead, we propose a new measure, and we empirically show that it highly correlates with the validation accuracy.

Original: Note that in this work we do not work with any inputdata and activations, but with the parameters of the neural network themselves.
Revised: Note that in this work we do not work with any input data and activations, but with the parameters of the neural network themselves.

Revised: The code and outputs are fully available in the Supplementary Material under a MIT License.

Original:  Approach
Revised:  A PPROACH

Original: Representation For representing neural networks as graphs, we follow the approach proposedconcurrently in Anonymous [2].
Revised: Representation For representing neural networks as graphs, we follow the approach proposed concurrently in Anonymous (2021).

Original: We associate to the neural network, at each learning state (deﬁnedby its weights), a weighted directed graph that is analyzed as an abstract simplicial complex.
Revised: We associate to the neural network, at each learning state (deﬁned by its weights), a weighted directed graph that is analyzed as an abstract simplicial complex.

Original: It isimportant to note that abstract simplicial complex are used in opposition to geometric simplicialcomplex.
Revised: It is important to note that abstract simplicial complex are used in opposition to geometric simplicial complex (e.g. Vietoris-Rips complex).

Original: For every training state, neural network connections are considered as directed and weighted edgesbetween neurons, represented by graph nodes.
Revised: For every training state, neural network connections are considered as directed and weighted edges between neurons, represented by graph nodes.

Original: Biases are considered as new edges that join to isolatevertices.
Revised: Biases are considered as new edges that join to isolate vertices.

Original: In this representation, activation functions are lost.
Revised: In this representation, activation functions are lost.

Original: Bias information could also have been ignored because, as we will see, it is not very informative in terms of homology, but we decided topreserve it.
Revised: Bias information could also have been ignored because, as we will see, it is not very informative in terms of homology, but we decided to preserve it.

Original: Negative edge weights are represented with reverse edges with the same weight absolute value.
Revised: Negative edge weights are represented with reverse edges with the same weight absolute value.

Original: Wediscard the use of weight absolute value  as neural networks are not invariant under weight signtransformations.
Revised: We discard the use of the absolute value of weights as neural networks are not invariant to weight sign transformations.

Original: This representation is consistent with the fact that every neuron can be replaced by aneuron from which two edges with opposite weights emerge and converge again on another neuronwith opposite weights.
Revised: This representation is consistent with the fact that every neuron can be replaced by a neuron from which two edges with opposite weights emerge and converge again on another neuron with opposite weights.

Original: From an homological point of view, this would be represented as a closedcycle.
Revised: From an homological point of view, this would be represented as a closed cycle.

Original: Weights are normalized following the Equation 1.
Revised: Weights are normalized following the Equation 1.

Original: ζ is an smoothing parameter that we set to1e-6.
Revised: ζ is an smoothing parameter that we set to 1e-6.

Original: This smoothing parameter is necessary as we want to avoid normalized weights of edges to be0 (in our representation 0 implies a lack of connection):
Revised: This smoothing parameter is necessary as we want to avoid normalized weights of edges to be 0 (in our representation 0 implies a lack of connection):

Original: [Equation]
Revised: [Equation]

Revised: Unlike Rieck et al.

Revised: (2019b)

Revised: we do not represent the neural network as a multipartite graph with a persistent homology ﬁltration that contains at most 1-simplices (edges), which only capturrd zero-dimensional topological information, i.e. connectivity information.

Revised: We do not believe the strong assumption that the NNs encode the learned information layer pairwise exclusively since there are trivial global transformations of a NN that are not captured by analyzing pairs of layers, more speciﬁcally 1.

Original:  Algebraic Topology object For each weighted directed graph associated with the state of a neuralnetwork, we link a directed ﬂag complex to it.
Revised: Superﬂuous cycle insertions: for instance, add two neurons and connect their input to a single neuron in a lower layer and their two outputs to a single output neuron in an upper layer with opposite weights; 2. Identity layer insertions: insert an intermediate identity layer with neurons and trivially connect to the next layer; and 3. Non-planar neural networks analysis: the analysis of neural networks that use multiple connections between non-consecutive layers require higher order topological analysis. Algebraic Topology object For each weighted directed graph associated with the state of a neural network, we link a directed ﬂag complex to it.

Original: The topological properties of this directed ﬂag complexare studied using homology groups H n .
Revised: The topological properties of this directed ﬂag complex are studied using homology groups H n .

Original: We calculate the homology groups up to degree 3 ( H 0 - H 3 ).
Revised: We calculate the homology groups up to degree 3 ( H 0 - H 3 ).

Original: For each state, we use a family of simplicial complexes, K ε , for a range of values of ε ∈ R .
Revised: For each state, we use a family of simplicial complexes, K ε , for a range of values of ε ∈ R .

Original: Thesimplicial complex at step ε t is embedded in the complex at ε t + 1 , for ε t ≤ ε t + 1 , i.e. K ε ⊆
Revised: The simplicial complex at step ε t is embedded in the complex at ε t + 1 , for ε t ≤ ε t + 1 , i.e. K ε ⊆

Original: K ε t
Revised: K ε t

Original: + 1 .
Revised: + 1 .

Original: εis used as a ﬁlter that establish the minimum weight of the graph representation edges included onthe simplicial complex.
Revised: ε is used as a ﬁlter that establish the minimum weight of the graph representation edges included on the simplicial complex.

Original: This collection of contained simplicial complex (associated to a directedweighted graph), called ﬁltration, K ε min ⊆ ... ⊆
Revised: This collection of contained simplicial complex (associated to a directed weighted graph), called ﬁltration, K ε min ⊆ ... ⊆

Original: K ε t ⊆
Revised: K ε t ⊆

Original: K ε t + 1 ⊆ ... ⊆
Revised: K ε t + 1 ⊆ ... ⊆

Original: K ε max , where t ∈ [ 0 , 1 ] and ε min = 0,ε max = 1 (remember that edge weights are normalized).
Revised: K ε max , where t ∈ [ 0 , 1 ] and ε min = 0, ε max = 1 (remember that edge weights are normalized).

Original: The sequence of homology groups is calculated by varying the ε parameter to obtain the persistencehomology diagram.
Revised: The sequence of homology groups is calculated by varying the ε parameter to obtain the persistence homology diagram.

Original: In our case, persistent homology calculations are performed on Z 2 .
Revised: In our case, persistent homology calculations are performed on Z 2 .

Original: In otherwords, once the corresponding ﬁlter has been applied to the weight of the edges, all connected edgesare considered equally.
Revised: In other words, once the corresponding ﬁlter has been applied to the weight of the edges, all connected edges are considered equally.

Original: Distances between persistence diagrams of consecutive states In this paper, we are interested incomparing PDs between different simplicial complex associated to each training state of the neuralnetwork.
Revised: Distances between persistence diagrams of consecutive states In this paper, we are interested in comparing PDs between different simplicial complex associated to each training state of the neural network.

Original: There are two distances traditionally used to compare PDs, Bottleneck distance (the lengthof the longest edge) and Wasserstein distance (using the sum of all edges lengths, instead of themaximum).
Revised: There are two distances traditionally used to compare PDs, Bottleneck distance (the length of the longest edge) and Wasserstein distance (using the sum of all edges lengths, instead of the maximum).

Original: Their stability with respect to perturbations on PDs has been object of different studies[6, 9].
Revised: Their stability with respect to perturbations on PDs has been object of different studies Chazal et al.

Revised: ; Cohen-Steiner et al.

Revised: As shown in comparative studies such as in ? , different distances and different ways of vectorizing persistence diagrams have results with different levels of stability and quality.

Original: In order to make computations feasible and obviate noisy intervals, we ﬁlter the PDs by limitingthe minimum PD interval size.
Revised: In order to make computations feasible and obviate noisy intervals, we ﬁlter the PDs by limiting the minimum PD interval size.

Original: We do so by setting a minimum threshold η = 0 .
Revised: We do so by setting a minimum threshold η = 0 .

Original: Intervals witha lifespan under this value are not considered (spurious homological features).
Revised: Intervals with a lifespan under this value are not considered (spurious homological features).

Original: Additionally, forcomputing distances, we need to remove inﬁnity values.
Revised: Additionally, for computing distances, we need to remove inﬁnity values.

Original: As we are only interested in the deaths untilthe maximum weight value, we replace all the inﬁnity values by 1 . 0.
Revised: As we are only interested in the deaths until the maximum weight value, we replace all the inﬁnity values by 1 . 0.

Original: In our case,our neural networks have millions of persistence intervals per Persistence Diagram,while Wasserstein distance calculations are computationally hard for large PDs.
Revised: In our case, neural networks have millions of persistence intervals per PD, while Wasserstein distance calculations are computationally hard for large PDs.

Original: In order to makecalculations computationally feasible, we will use a vectorized version of PDs, also called PDdiscretization.
Revised: To make it computationally feasible, we will use a vectorized version of PDs, also called PD discretization.

Original: This vectorized version summaries have been proposed and used on recent literature[1, 3, 4, 19, 25].
Revised: This vectorized version summaries have been proposed and used on recent literature Adams et al. ; Berry et al. (2020); Bubenik (2015); Lawson et al. ; Rieck et al. (2019a).

Original: For persistence diagram distance calculation, we use weighted Silhouette and Heatvectorizations, using the Giotto-TDA library [27].
Revised: For persistence diagram distance calculation, we use weighted Silhouette and Heat vectorizations, using the Giotto-TDA library Tauzin et al.

Original:  Experiments
Revised:  E XPERIMENTS

Original: Data We validate our method in several heterogeneous (vision, natural language), well-knowndatasets, namely 1.
Revised: Data We validate our method in several heterogeneous (vision, natural language), well-known datasets, namely 1.

Original: MNIST [20], 2.
Revised: MNIST LeCun & Cortes (2010), 2.

Original: CIFAR-10, 3.
Revised: CIFAR-10, 3.

Original: CIFAR-100 [18], and 4.
Revised: CIFAR-100 Krizhevsky (2009), and 4.

Original: the Reuters dataset [28](multi-class and multi-label document classiﬁcation dataset).
Revised: the Reuters dataset Thoma (2017) (multi-class and multi-label document classiﬁcation dataset).

Original: Models We experiment with two neural architectures,1.
Revised: Models We experiment with two neural architectures,1.

Original: MLPs and 2.
Revised: MLPs and 2.

Original: CNNs.
Revised: CNNs.

Original: In the latter case, weuse the convolutional layers as a pre-trained model with frozen weights, and we learn an MLP on topof it.
Revised: In the latter case, we use the convolutional layers as a pre-trained model with frozen weights, and we learn an MLP on top of it.

Original: The reason we do so is that our method is based in a representation that, at least in the basicform, does not allow capturing information from convolutional layers.
Revised: The reason we do so is that our method is based in a representation that, at least in the basic form, does not allow capturing information from convolutional layers.

Original: Thus, we need a single (exact same weights) feature extractor, to abstract away distances related to the CNN layers and focus onthe MLP.
Revised: Thus, we need a single (same weights) feature extractor, to abstract away distances related to the CNN layers and focus on the MLP.

Original: Conducted experiments We deﬁne the base MLP architecture as {Input, Linear(512),
Revised: Conducted experiments We deﬁne the base MLP architecture as {Input, Linear(512),

Original: Dropout(0.2), Linear(512), Dropout(0.2), Output} .
Revised: Dropout(0.2), Linear(512), Dropout(0.2), Output} .

Original: In the case of CNNs, the pre-trainedmodel is deﬁned as 3 convolutional blocks with kernel size 3 (starting with 32 channels), interleavedwith max pooling (its linear layers are thrown away after the pre-training).
Revised: In the case of CNNs, the pre-trained model is deﬁned as 3 convolutional blocks with kernel size 3 (starting with 32 channels), interleaved with max pooling (its linear layers are thrown away after the pre-training).

Original: On top of the pre-trained CNN, we also deﬁne the same base MLP architecture.
Revised: On top of the pre-trained CNN, we also deﬁne the same base MLP architecture.

Original: Then, for eachdataset andmodel (MLP and CNN) , we experiment with varying (while keeping therest ﬁxed tothe base architecture)
Revised: Then, for each and and CNN), we experiment with varying (while keeping the to

Original: Layer size (number of units per layer): 4, 16, 32, 128, 256.

Original: Number of labels (the other classes are removed): 2, 4, 6, 8, 10.

Original: Learning rate: 1e-e05, 0.0001, 0.001, 0.01, 0.

Original: Dropout: 0.0, 0.2, 0.4, 0.5, 0.8.

Original: Input order: 5 random input orders. As a control experiment, for each analyzed problem we run the same conﬁguration 
Revised: As a control experiment, for each analyzed problem we run the same conﬁguration with 5 different input orders.

Original: with

Original: 5 different input orders.

Original: If the measured

Original: distances

Original: are,

Original: indeed

Original: , related with the learning process of neural networks , these variations should not 
Revised: If the measured distances are, indeed, related with the learning process of neural networks, these variations should not have any noticeable effect.

Original: have any noticeable effect. We run each conﬁguration 5 times with different random seeds (and, thus, weight initializations 1 ) tosee if the results are consistent across runs.
Revised: We run each conﬁguration 5 times with different random seeds (and, thus, weight initializations 1 ) to see if the results are consistent across runs.

Original: All models are trained with the RMSProp optimizer witha batch size of 256.
Revised: All models are trained with the RMSProp optimizer with a batch size of 256.

Original: Distances and validation accuracy computation Note that homological distances are obtained atthe end of each batch, while validation metrics are only computed on each epoch.
Revised: Distances and validation accuracy computation Note that homological distances are obtained at the end of each batch, while validation metrics are only computed on each epoch.

Original: The methodologywe follow to analyze the learning process on each different problem can be  summarized with thefollowing steps:
Revised: We consider the validation curve to be our baseline, and therefore we compare our method to it. The methodology is summarized as follows:

Original: In each training step (i.e.,for each batch) we extract the weights from the MLP current stateand use them to build an abstract simplicial complex from the associated weighted directedgraph.
Revised: In each training step (i.e., batch) we extract the weights from the MLP current state and use them to build an abstract simplicial complex from the associated weighted directed graph.

Original: We calculate the homological persistence diagram of the simplicial complex.
Revised: We calculate the homological persistence diagram of the simplicial complex.

Original: We then calculate the distance between consecutive persistence diagrams (we will call thissequence homological convergence )
Revised: We then calculate the distance between consecutive persistence diagrams (we will call this sequence homological convergence ).

Original: We use two different distances,namely, Heat and Silhouette
Revised: We use two different distances, Heat and Silhouette.

Original: We compare the homological convergence with the evolution of the validation results onneural network learning process.
Revised: We compare the homological convergence with the evolution of the validation results on neural network learning process.

Original: Hardware All experiments were executed in a machine with 2 NVIDIA V100 of 32GB, 2 Intel(R)
Revised: All experiments were executed in a machine with 2 NVIDIA V100 of 32GB, 2 Intel(R)

Original: Xeon(R)
Revised: Xeon(R)

Original: Platinum 8176 CPU @ 2.10GHz, and of 1.5TB RAM, for a total of around 7 days.
Revised: Platinum 8176 CPU @ 2.10GHz, and of 1.5TB RAM, for a total of around 7 days.

Original: We notethat our method is considerably demanding in terms of both compute and memory.
Revised: We note that our method is considerably demanding in terms of both compute and memory.

Original: The code and outputs are fully available in the Supplementary Material under MIT License.

Original:  Results
Revised:  R ESULTS

Original: In this section, we highlight the main results, omittingthe ones with Silhouette (since the obtainedresults were clearer with Heat).
Revised: In this section, we highlight the main results, omitting Silhouette (since the results were clearer with Heat).

Original: See the Supplementary Material for  the full results (plots andcorrelations), including the ones with Silhouette distance.
Revised: See the Supplementary Material for all the plots and correlations, including Silhouette.

Original: We study the relation between the evolution of the PH diagram distances with the one of the validationscore with the cumulative values of the distance between homologous persistence diagrams becausethis value seems much more stable.
Revised: We study the relation between the evolution of the PH diagram distances with the one of the validation score with the cumulative values of the distance between homologous persistence diagrams because this value seems much more stable.

Original: The information of the distance between the persistence diagramshas been normalized to visualize clearly the type of evolution of each curve on the same scale.
Revised: The information of the distance between the persistence diagrams has been normalized to visualize clearly the type of evolution of each curve on the same scale.

Original: Some of the non-normalized plots can be found in the Supplementary Material.
Revised: Some of the non-normalized plots can be found in the Supplementary Material.

Original: Figure 6 shows thecumulative and non-cumulative homology the MNIST experiment with layer size.Epochs 0.
Revised: Figure 6 shows the cumulative and non-cumulative homology the MNIST experiment with layer size.

Original: [-D i s t a n ce d i ff e r e n ce ( li n e )     0. V a li d a t i o n s c o r e ( d a s h e d ) (a) Cumulative 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 
 Epochs 0.-]
Revised: [-D i s t a n ce d i ff e r e n ce ( li n e )     0. V a li d a t i o n s c o r e ( d a s h e d ) (a) Cumulative 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 
 Epochs 0.-][Figure]

Original: D i s t a n ce d i ff e r e n ce ( li n e )     0. V a li d a t i o n s c o r e ( d a s h e d ) (b) Non-cumulative For each experiment (e.g., layer size in MNIST), we plot both the evolution of the PH diagramdistance and the validation score (accuracy).
Revised: For each experiment (e.g., layer size in MNIST), we plot both the evolution of the PH diagram distance and the validation score (accuracy).

Original: The plotted values are the corresponding means ofthe 5 repetitions with different seeds.
Revised: The plotted values are the corresponding means of the 5 repetitions with different seeds.

Original: In addition, we compute the Pearson correlation for thesevalues.
Revised: In addition, we compute the Pearson correlation for these values.

Original: Plots show on the x-axis each training step (for each batch) of the evolution in the trainingstate of the neural network.
Revised: Plots show on batch evolution training the neural network.

Original: On the y-axis, two scalesare shown thatapply to the distancecurvesbetween accumulated persistence diagrams (solid lines), scale on the right side, and the neuralnetwork validation (dotted lines), numerical scale on the left side. For each sub-experiment (forexample, different values of layer size) a different color was used.
Revised: On the y-axis two scales shown that distance right neural

Original: The general result is that the evolution of the homological convergence of the MLPs seems to bevery similar to the one of the validation score.
Revised: The general result is that the evolution of the homological convergence of the MLPs seems to be very similar to the one of the validation score.

Original: This is generally consistent across experiments (seethe Supplementary Material).
Revised: This is generally consistent across experiments (see the Supplementary Material).

Original: Table 1 shows the mean (and standard deviations) of the Pearsoncorrelations for all datasets.
Revised: Table 1 shows the mean (and standard deviations) of the Pearson correlations for all datasets.

Original: All means are above 0.8, implying that there is strong correlation.
Revised: All means are above 0.8, implying that there is strong correlation.

Original: Intuitively, this is also observed in the plots, although once the distances are normalized it is not asclear to visualize.
Revised: Intuitively, this is also observed in the plots, although once the distances are normalized it is not as clear to visualize.

Original: Interestingly, we ﬁnd that the very few exceptions in which the correlation is lowcorresponds to extreme values (very small number of neurons per layer, very high learning rate, veryhigh dropout), in which the neural network doesn’t end up learning properly.
Revised: Interestingly, we ﬁnd that the very few exceptions in which the correlation is low corresponds to extreme values (very small number of neurons per layer, very high learning rate, very high dropout), in which the neural network doesn’t end up learning properly.

Original: Inthe case of CNNs, the correlations are lower (although still almost always above 0.8 in experimentssuch as the one of increasing the number of layers).
Revised: In CNNs, the correlations are lower (but still usually above 0.8 in experiments such as the one of increasing the number of layers).

Original: Recall thatin the case of CNN we froze asingle convolutional feature extractor, since our method only supports MLPs.
Revised: Recall that we froze a convolutional feature extractor since our method only supports MLPs.

Original: We believe these lowercorrelations can be explained because an important part of the learning process happened in theconvolutional layers (in the pre-training),which we do not capture.
Revised: We believe these lower correlations occur because an important part of the learning process happened in the convolutional layers (in the pre-training), not captured.

Original: Another ﬁnding is that the method obtains consistent results across runs, meaning that it is capturinginformation related to important properties of the networks themselves instead of random artifacts.
Revised: Another ﬁnding is that the method obtains consistent results across runs, meaning that it is capturing information related to important properties of the networks themselves instead of random artifacts.

Original: When varying the studied hyperparameters, we observe that the curves for each conﬁguration areindeed, different.
Revised: When varying the studied hyperparameters, we observe that the curves for each conﬁguration are indeed, different.

Original: Remarkably, in the control experiments, this is not the case; results show that thehomological convergence during the learning of the same problem with the same model but withdifferent input order is very similar.
Revised: Remarkably, in the control experiments, this is not the case; results show that the homological convergence during the learning of the same problem with the same model but with different input order is very similar.

Original: The alteration of the order of the input doesn’t have any effect inthe homological convergence.
Revised: The alteration of the order of the input doesn’t have any effect in the homological convergence.

Original: The results of two of these experiments are shown in Figure 7.
Revised: The results of two of these experiments are shown in Figure 7.

Original: In addition, we observe that when the neural network learns the given problem, homological convergence occurs.
Revised: In addition, we observe that when the neural network learns the given problem, homological convergence occurs.

Original: For example, when the layer size is modiﬁed, the capacity of the neural network tolearn the problem changes (Figure 6).
Revised: For example, when the layer size is modiﬁed, the capacity of the neural network to learn the problem changes (Figure 6).

Original: When it can’t learn the problem, because the network does nothave sufﬁcient capacity (the layer size is too small, 4 units), the homology does not seem to converge.
Revised: When it can’t learn the problem, because the network does not have sufﬁcient capacity (the layer size is too small, 4 units), the homology does not seem to converge.

Original: Regarding the learning rate, the results are coherent with the intuition that it is a fundamentalparameter that controls how much to change the model in response to the estimated error during thelearning process.

Original: A too small learning rate may result in a long training process that could be stalled,while a too large value may fall in a fast suboptimal solution or an unstable training process.

Original: Usinghomological convergence we ﬁnd similar behaviour, as can be seen in Figure 9.

Original: Epochs 0.
Revised: Epochs 0.

Original: D i s t a n ce d i ff e r e n ce ( li n e )     0.
Revised: D i s t a n ce d i ff e r e n ce ( li n e )     0.

Original: V a li d a t i o n s c o r e
Revised: V a li d a t i o n s c o r e

Original: ( d a s h e d ) (a) MNIST 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 

Revised: ( d a s h e d ) (a) MNIST 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Original: Epochs 0.
Revised: Epochs 0.

Original: D i s t a n ce d i ff e r e n ce ( li n e )     0.
Revised: D i s t a n ce d i ff e r e n ce ( li n e )     0.

Original: V a li d a t i o n s c o r e
Revised: V a li d a t i o n s c o r e

Original: ( d a s h e d ) (b) CIFAR-100 CNN
Revised: ( d a s h e d ) (b) CIFAR-100 CNN

Revised: Heat distance

Revised: Silhouette distance

Original: [Table]
Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Original: Finally, we note that even if the two convergences (validation and homological convergence) arecorrelated, they are not the same process.
Revised: [Table] [Table] [Table] Regarding the learning rate, the results are coherent with the intuition that it is a fundamental parameter that controls how much to change the model in response to the estimated error during the learning process.

Revised: A too small learning rate may result in a long training process that could be stalled, while a too large value may fall in a fast suboptimal solution or an unstable training process.

Revised: Using homological convergence we ﬁnd similar behaviour, as can be seen in Figure 9.

Revised: Finally, we note that even if the two convergences (validation and homological convergence) are correlated, they are not the same process.

Original: This is especially visible in thecase of the learning rateexperiments.
Revised: This is especially visible in the learning rate experiments.

Original: For instance, in Figure 9, homological convergence is reached before the stabilization ofthe validation accuracy.
Revised: For instance, in Figure 9, homological convergence is reached before the stabilization of the validation accuracy.

Original: Presumably, they are not capturing the exact same information; speciﬁcally,we believe that the difference is due to the fact that the validation accuracy depends on the speciﬁcsof the data sampled in the validation subset, while the homological convergence is independent of thevalidation data.
Revised: Presumably, they are not capturing the exact same information; we believe that the difference is due to the fact that the validation accuracy depends on the speciﬁcs of the data sampled in the validation subset, while the homological convergence is independent of the validation data.

Original:  Discussion
Revised:  D ISCUSSION

Original: We posed theof question whether homological convergence (in terms of distances between PHdiagrams in consecutive neural network states) is related to the learning process of neural networks.
Revised: We posed the question whether homological convergence (in terms of distances between PH diagrams in consecutive neural network states) is related to the learning process of neural networks.

Original: We have seen that, indeed, it is the case, with strong empirical results backing our claim.
Revised: We have seen that, indeed, it is the case, with strong empirical results backing our claim.

Original: This ﬁnding has a remarkable implication.
Revised: This ﬁnding has a remarkable implication.

Original: Ifthe homological convergence evolution mirrors thevalidation accuracy curve, one could ignore the validation set to monitor the training.
Revised: If homological convergence evolution mirrors the validation accuracy curve, one could ignore the validation set to monitor the training.

Original: This opens thepath towards estimating the generalization of neural networks without the need of any holdout set.
Revised: This opens the path towards estimating the generalization of neural networks without the need of any holdout set.

Original: Researchers have wondered for a long time whether generalization could be predicted from intrinsicproperties of the model or training data alone (i.e., without a holdout set), and in fact other workshave claimed to do so.
Revised: Researchers have wondered for a long time whether generalization could be predicted from intrinsic properties of the model or training data alone, and in fact other works have claimed to do so.

Original: Although we do not provide any predictive model, we show that our proposedmeasures strongly correlate withthe validation accuracy.
Revised: Although we do not provide any predictive model, we show that our proposed measures strongly correlate with validation accuracy.

Original: In addition, we do so by not using any dataat all; we just look at the neural network itself.
Revised: We do so by not using any data at all; we just look at the neural network itself.

Original: Our contribution aims pushing towardshaving a better understanding of the learning process of neuralnetworks, not targeting any speciﬁc direct application.
Revised: Our contribution aims pushing towards a better understanding of the learning process of neural networks, not targeting any speciﬁc direct application.

Original: However, we note that it can be effectivelyused for monitoring the training of neural networks in terms of convergence expected generalization,as we have extensivelyshown in the experiments. Apart from the cases without access to a validation
Revised: However, we note that it can be effectively used for monitoring the training of neural networks in terms of convergence expected generalization, as we have extensively

Original: [Figure]
Revised: [Figure]

Original: D i s t a n ce d i ff e r e n ce ( li n e )

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: [Figure]

Original: V a li d a t i o n s c o r e

Original: ( d a s h e d

Original: [Figure]

Original: V a li

Original: [Figure]

Original: a t

Original: [Figure]
Revised: [Figure]

Original: o

Original: [Figure]

Original: c o

Original: [Figure]

Original: e ( d

Original: [Figure] e d ) [Figure] [Figure]
Revised: shown in the experiments.

Original: D i s t a n ce d i ff e r e n ce ( li n e )

Original: [Figure] D i s t a n ce d i ff e r e n ce ( li n e ) 1e 1e0. V a li d a t i o n s c o r e ( d a s h e d ) (b) CIFAR-10 MLP set, this is relevant because depending on a validation set has the risk of overﬁtting to it.
Revised: Apart from when there is no access to a validation set, this is relevant because depending on a validation set has the risk of overﬁtting to it.

Original: Having anintrinsic, well-principled measure should be more robust to random noise in a speciﬁc data sample.
Revised: Having an intrinsic measure should be more robust to random noise in a speciﬁc data sample.

Original: The main limitation of our method is its computational scalability.
Revised: The main limitation of our method is its computational scalability.

Original: As we said in Section 5, ourmethod took more than 7 days of compute in a HPC machine, even if we restricted the experimentsto small datasets and parameter count.
Revised: This lack of scalability has prevented us from validating our method on bigger models and datasets.

Original: However, we note that our approach computes the exactpersistence diagram distances, that is, we do not simplify the graph representation of the neuralnetworks (we keep every single neuron and connections) and we do not approximate any computation.
Revised: However, we note that our approach computes the exact PD distances, that is, we do not simplify the graph representation of the neural networks (we keep every single neuron and connections) and we do not approximate any computation.

Original: This leaves room for ﬁnding efﬁcient approximations, opening a new research line.
Revised: This leaves room for ﬁnding efﬁcient approximations, opening a new research line.

Original: In addition, thislack of scalability has prevented us from validating our method on bigger models and datasets.

Original: Finally, we note that instead of computing correlations, serving as a basic quantitative study, it wouldbe interesting to perform a time-series analysis to gain more insights on how the two curves varytogether.
Revised: Finally, we note that instead of computing correlations, serving as a basic quantitative study, it would be interesting to perform a time-series analysis to gain more insights on how the two curves vary together.

Original: Moreover, it would have been interesting to investigate how to build a predictive model ofthe validation accuracy from the PH distances, but it is was of the scope of this work.
Revised: Moreover, it would have been interesting to investigate how to build a predictive model of the validation accuracy from the PH distances, but it is was of the scope of this work.

Original:  Conclusions & Future Work
Revised:  C ONCLUSIONS & F UTURE W ORK

Original: In this work, we have provided an empirical proof of the fact that homological convergence is relatedto the learning process and generalization properties of neural networks.
Revised: In this work, we have provided an empirical proof of the fact that homological convergence is related to the learning process and generalization properties of neural networks.

Original: Furthermore, we haveshown that it can be used to monitor the training of a neural network (and potentially estimating itsgeneralization) without a validation set.
Revised: Furthermore, we have shown that it can be used to monitor the training of a neural network (and potentially estimating its generalization) without a validation set.

Original: As future work, we suggest generalizing our representationto other neural architectures and scaling up the experiments to larger models and datasets, for whichﬁnding efﬁcient approximations of our method will be crucial.Checklist For all authors...
Revised: As future work, we suggest generalizing our representation to other neural architectures and scaling up the experiments to larger models and datasets, for which ﬁnding efﬁcient approximations of our method will be crucial.

Original: (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’scontributions and scope?

Original: [Yes]

Original: (b) Did you describe the limitations of your work?

Original: [Yes] See the Discussion Section.

Original: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A]

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?

Original: [Yes]

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Original: [N/A]

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Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: mental results (either in the supplemental material or as a URL)?

Original: [Yes] Both code and

Original: outputs.

Original: (b) Did you specify all the training details (e

Original: g., data splits, hyperparameters,

Original: how they

Original: were chosen)?

Original: [Yes] Check the Experiments Section and the code.

Original: (c) Did you report error bars (e

Original: g., with respect to the random seed after running ex-

Original: periments multiple times)?

Original: [Yes] We include means, standard deviations and raw

Original: outputs.

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type

Original: of GPUs, internal cluster, or cloud provider)?

Original: [Yes] Check Experiments Section.4.

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Original: [Yes]

Original: In the case of thedatasets.

Original: We do not use any other additional asset.

Original: (b) Did you mention the license of the assets?

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Original: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [Yes]

Original: Code, results and pictures we have made for explanations.

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Original: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable

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Original: (a) Did you include the full text of instructions given to participants and screenshots, ifapplicable?

Original: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?

Original: [N/A]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?

Original: [N/A]