Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set
Authors: Asier Gutiérrez-Fandiño
David Pérez Fernández
Jordi Armengol-Estapé
Marta Villegas
Venue: ICLR 2022 Submitted (2022)
Keywords: Neural Networks
Topological Data Analysis
learning
evolution
Persistent Homology

Abstract: The training of neural networks is usually monitored with a validation (holdout) set to estimate the generalization of the model. This is done instead of measuring intrinsic properties of the model to determine whether it is learning appropriately. In this work, we suggest studying the training of neural networks with Algebraic Topology, specifically Persistent Homology (PH). 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. 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.

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PERSISTENT HOMOLOGY CAPTURES THE GENERAL- IZATION OF NEURAL NETWORKS WITHOUT A VALI- DATION SET
Anonymous authors Paper under double-blind review
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ABSTRACT
The training of neural networks is usually monitored with validation (holdout) set to estimate the generalization the model. This is done instead of measuring intrinsic properties of the model to determine whether it is learning appropriately. this work, we suggest studying training of neural networks with Algebraic Topology, specifically Persistent Homology (PH). Using simplicial complex repre- sentations of neural networks, we study the PH diagram distance evolution on the neural network learning process with different architectures and several datasets. 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.
INTRODUCTION
Generalization is what makes machine learning model useful; the uncertainty of its behaviour with unseen data is what makes it potentially dangerous. Thus, understanding the generalization error of model can be considered one of the holy grails of the entire machine learning field.
Machine learning practitioners typically monitor some metrics of the model to estimate its generaliza- tion error and stop the training even before the numerical convergence to prevent the overfitting of the model. Usually, the error measure the metric relevant the task computed for holdout set, the validation set. Since these data have not been directly used for updating the parameters, it assumed that the performance
the model on the validation set can be used as proxy of the generalization error, provided it is representative of the data that will be used in inference. One can, though, potentially overfit to this holdout set is repeatedly used for guiding a hyperparameter search.
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

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barely explored in the literature. On the other hand, Algebraic Topology has recently been gaining momentum as a mathematical tool for studying graphs, machine learning algorithms, and data.
In this work, we have the goal of, once having characterized neural networks weighted. acyclic graphs, represented as Algebraic Topology objects (following previous works), computing distances between consecutive neural network states. More specifically, 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[[ We observe that during the training procedure of neural networks we can measure this distance in each learning step, and show that there exists high correlation with the corresponding validation accuracy of the model. We do so in a diverse set of deep learning benchmarks and model nyperparameters. 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 better theoretical understanding of the dynamics of the training of neural networks.
can calculate the Persistent
In summary, contributions are as follows:
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 neural network during the training procedure and the one in the immediately previous weights update.
We empirically show that the evolution of these measures during training correlate with the accuracy in the validation set. We do so in diverse benchmarks (MNIST, CIFAR I0, CIFARIO0, Reuters text classification). and models (MLPs in MNIST and Reuters, MLPs and CNNs in CIFAR100 100 and CIFARI00)
We thus provide empirical proof the fact that valuable information related the learning process of neural networks can be obtained from PH distances between persistence diagrams (we will call this process homological convergence). In particular, we show that homologi- cal convergence is related to learning process and the generalization properties of neural networks.
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.
The remainder of this article is as follows. 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. Then, in Section 4 we formalize our method. Finally, in sections and we present and discuss our empirical results, respectively.
2 BACKGROUND
In this section we introduce the mathematical foundations of this paper. A detailed mathematical description included in the Supplementary Material.
simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts, named simplex (K). In particular, simplicial complex must must comply with properties: 1. Every face of a simplex is also in the simplicial complex (of lower dimension). 2. The non-empty intersection of any two simplices contained simplicial complex is face of both. 0,1,2.3-simplexx and non simplex examples are shown in Figure
(a) 0,1.2.3-simplex
(b) Non-simplex
Figure 2: Simplex and non-simplex examples.
We can associate to an undirected graph, - (V,E), a simplicial complex where all the vertices of G are the O-simplex of the simplicial complex and the complete subgraphs with vertices, in G
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a
3-simplex
2-simplex
1-simplex
Ossimplex
Figure 4: Boundary function sample.
corresponds to (i- 1) >isimplex, This type of construction is usually called complex clique on the graph G, and is denoted by CI(G). Figure shows graph clique complex CI(G) example.
The boundary function is defined as map, from an i-simplex to an -immple, as the sum of its (i -)ddimensional faces. A boundary function sample is shown in Figure
In algebraic topology. a k-chain is a combination of k-simplices (sometimes symbolized as lin- ear combination of simplices that compose the chain). The boundary of a k-chain a (k - 1)- chain. It is the linear and signed combination of chain element boundary simplices. The space of i-chains is denoted by (K).
Figure 3: Graph clique complex CI(G) example.
There are two special cases of chains that will be useful to define homology group:
Closed chain or i-cycle: i-chain with empty boundary. An i-chain c is an i-cycle if and only if d:c 0, i.e. c ker(a.). This subspace (K) is denoted as Zz(K).
Exact chain or i-boundary: An i-chain is an j-boundary if there exists an (i+ I)-chain d such that a (d), i.e. E im (i 1). This subspace of C/(K), the set of all such i-boundaries forms, is denoted by (K)..
Now, if we consider i-cycles not bounding an (i+ --simpliciall complex, this is the definition of an i-th homology group of the simplicial complex K. The precise definition is the quotient group of (K) module (K) (i.c. (K) (K), see Supplementary Material). The number of non equivalent i-cycles (Figure 5) is the dimension of the homology group (K)),, ), also named Betti numbers.
We can create a nested family of simplicial com- plexes, Ke, where each step Kg, is embedded in the simplicial complex K We call this set simplicial complex filtration.
For each filtration simplicial complex, we can calculate the homology groups. Then, we can look at the birth, that is, when a homology class appears, and death, the time when the homology class disappears. The PH treats the birth and the death of these homological features in Kg for different values. The lifespan of each homo- logical feature can represented as an interval (birth, death), of the homological class. Given filtration, this collection of intervals is named Persistence Diagram (PD) Carlsson (2009).
Figure 5: The two blue dashed cycles are homo- logically equivalent, the pink isn't in't.
is possible to compare two PDs using specific distances (Wasserstein and Bottleneck). To efficiently perform this operation, due to the size of these diagrams, it is sometimes necessary to simplify them by means discretization process (such as Weighted Silhouette and Heat vectorizations).
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3 RELATED WORK
Algebraic Topology and Machine Learning The use of Algebraic Topology in the fields of data science and machine learning has been gaining momentum in recent years (see Caarssson (2009). Specifically in the case of neural networks, some works have applied topology for improving the
training procedure of the models Hofer et al. (200);; Clough et al. (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 Kanak & Smith (2019). Furthermore, recent works have provided topological analysis of the decision boundaries of classifiers based on PH and Betti numbers Ramamurthy et al. (2019); Naitzat et al. (2020).
Graph and topological representations of neural networks Gebhart et al. (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. Chowdhury et al. (2019) show that directed homology can be used to represent MIPss Anonymous (2021) concurrently show neural networks, when represented as directed, acyclic graphs, can be associated to an Algebraic Topology object, specifically to directed flag complex. 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. This is unlike other works Corneanu et al. (2019); /uss & Salakhutdinov (2018) approximating neural networks representations with regard to the input space. Relevant to our work, in Rieck et al. (2019b) authors propose complexity measure of neural networks based on persistent homology. However, we will see that their representation does not fulfill our requirements Section 4
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. In this regard, the literature is perhaps more limited. Jiang et al. (2019) work on understanding what drives generalization deep networks from Bayesian of view. 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. In Li et al.) (2020), authors propose an alternative to cross coss-validation for model selection based on training once on the whole train set, without any data split, deriving a validation set with data augmentation.
Corneanu et al. 2020) try to estimate the performance gap between training and testing using PH measures. They claim. However, one can observe some caveats. The first one is that their regression fitted to predict the test error has considerably high error, making it not usable in practice. The second caveat is that for fitting the regression one needs at least part of the sequestered testing set.
In this work, motivated by the interest of having 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 AAnonymous) (022)) but, crucially, measured between consecutive weight updates. We will show that the evolution of this distance is similar to the one of the validation accuracy. Unlike Li et al. (2020), all. Unlike Corneanu et al. (2020), we do not build a statistical or machine learning model (linear regression) for predicting the testing error. Instead, we propose new measure, and we empirically show that it highly correlates with the validation accuracy. Note that in this work we do not work with any input data and activations, but with the parameters of the neural network themselves. The code and outputs are fully available in the Supplementary Material under a MIT License.
do not use any data
4 APPROACH
Representation For representing neural networks as graphs, we follow the approach proposed concurrently in Anooyyoou (2021). We associate to the neural network, at each learning state (defined by its weights), weighted directed graph that is analyzed an abstract simplicial complex. is important to note that abstract simplicial complex are used in opposition to geometric simplicial complex (e.g. Sectoris-Rips complex).
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For every training state, neural network connections considered directed and weighted edges between neurons, represented by graph nodes. Biases are considered as new edges that join to isolate vertices. In this representation, activation functions are lost. 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.
Negative edge weights are represented with reverse edges with the same weight absolute value. We discard the use of the absolute value of weights as neural networks are not invariant weight sign transformations. 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. From an homological point of view, this would be represented as closed cycle. Weights are normalized following the Equation | 5 is an smoothing parameter that we set to 1e-6. 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):
W
max(1 (1
max W 5)
(1)
Unlike Rieck et al. (2019b) we do not represent the neural network as a multipartite graph with persistent homology filtration that contains at most I-simplices (edges), which only capturrd zero-dimensional topological information, i.e. connectivity information. 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. Superfluous cycle insertions: for instance, add two neurons and connect their input to 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. The topological properties of this directed flag complex studied using homology groups H₃- We calculate the homology groups up degree 3 (Hg -)).
For each state, we use a family of simplicial complexes, Kg, for a range of values of E E R. The simplicial complex step & is embedded in the complex at & for & S & i.e. Kg C K is used as a filter that establish the minimum weight of the graph representation edges included on the simplicial complex. This collection of contained simplicial complex (associated a directed weighted graph), called filtration, Ke C C Kg. C Kay1 C... C Ke where E [0, 1] and Emin = O, Emax (remember that edge weights are normalized).
The sequence of homology groups is calculated by varying the & parameter to obtain the persistence homology diagram. In our case, persistent homology calculations are performed on Σz. In other words, once the corresponding filter has been applied to the weight of the edges, all connected edges are considered equally.
Distances between persistence diagrams of consecutive states In this paper, we are interested in comparing PDs between different simplicial complex associated each training state of the neural network. 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). Their stability with respect to perturbations on PDs has been object of different studies Chazal et al. 22012): Cohen-Steiner et al. (2005). 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.
In order to make computations feasible and obviate noisy intervals, we filter the PDs by limiting the minimum PD interval size. We do so by setting a minimum threshold n = 0.01. Intervals with lifespan under this value not considered (spurious homological features). Additionally, for computing distances, we need to remove infinity values. As we are only interested in the deaths until the maximum weight value, we replace all the infinity values by 1.0.
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In our case, neural networks have millions of persistence intervals per PD, while Wasserstein distance calculations are computationally hard for large PDs. To make it computationally feasible, we will use vectorized version of PDs, also called PD discretization. This vectorized version summaries have been proposed and used on recent literature Adams et al. 2017); Berry et al. (2020) [Bbeenik (2013); Lawson al. (2019); RReck al. (2019a). For persistence diagram distance calculation, use weighted Silhouette and Heat vectorizations, using the Giotto-TDA library Tauzin et al. 2020))
EXPERIMENTS
Data We validate our method in several heterogeneous (vision, natural language), well-known datasets, namely 1, MNIST LeCun LeCun & Cortes (211). 2. CIFAR-I0, 3. CIFAR- 100 Krizhevsky (2009), and 4. the Reuters dataset T'homa (2017) (multi-class and multi-label document classification dataset).
Models We experiment with two neural architectures.1. MLPs and 2. CNNs. 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. The reason we do so is that our method is based in representation that, at least in the basic form, does not allow capturing information from convolutional layers. Thus, we need a single (same weights) feature extractor, to abstract away distances related to the CNN layers and focus on the MLP.
Conducted experiments We define the base MLP architecture as (Input, Linear (512), Dropout (0. 22, Linear (512), Dropout (0. 2) Output]. }. In the case of CNNs, the convolutional blocks with kernel size 3 (starting with 32 channels), interlcaved with max pooling (its linear layers are thrown away after the pre-training). On top of the pre-trained CNN, we also define the same base MLP architecture.
pre-trained model is defined
Then, for each dataset and model (MLP and CNN), we experiment with varying (while keeping the rest fixed to the base architecture) 1. Layer size (number of units per layer): 4, 16, 32, 128, 256; 2. Number of labels (the other classes are removed): 2, 4, 6, 8, 10; 3. Learning rate: 1e-e05, 0.0001, 0.001, 0.01, 0.1; 4. Dropout: 0.0, 0.2, 0.4, 0.5, 0.8; and 5. Input order: 5 random reordering of the input samples. As control experiment, for each analyzed problem we the same configuration with 5 different input orders. If the measured distances are, indeed, related with the learning process of neural networks, these variations should have any noticeable effect. We run each configuration 5 times with different random seeds (and, thus, weight initializations" to see if the results are consistent across runs. All models are trained with the RMSProp optimizer with a batch size of 256.
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. We consider the validation curve to be our baseline, and therefore we compare our method to it. The methodology is summarized as follows:
1. In each training step (i.e., batch) we extract the weights from the MI. current state and use them build abstract simplicial complex from the associated weighted directed graph.
2. We calculate the homological persistence diagram of the simplicial complex.
3. We then calculate the distance between consecutive persistence diagrams (we will call this sequence humological convergence). We use two different distances, Heat and Silhouette.
4. We compare the homological convergence with the evolution of the validation results on neural network learning process.
All experiments were executed in machine with NVIDIA V100 of 32GB, 2 Intel(R) Xcon(R) Platinum 8176 CPU @ 2.10GHz, and of 1.5TB RAM, for a total of around 7 days. We note that our method is considerably demanding in terms of both compute and memory.
RESULTS
In this section, we highlight the main results, omitting Silhouette (since the results were clearer with Heat). See the Supplementary Material for all the plots and correlations, including Silhouette.
1 The pre-trained convolutional weights are always identical, though.

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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. 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. Some of the non-normalized plots be found in the Supplementary Material. Pigure/]shooe the cumulative and non-cumulative homology the MNIST experiment with layer size.
1.0
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Validasoo souce (danted))
Diel Wedianze differure (lime)
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Validadoo donoon stauce (Hasked)
Distance differnce
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9
3
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82
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36
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Epochs
Epochi
(a) Cumulative
(b) Non-cumulative
Figure 6: Heat distance and validation accuracy curves on MNIST with layer size. Normalized.
For each experiment (e-g., layer size in MNIST), we plot both the evolution of the PH diagram distance and the validation score (accuracy). The plotted values are the corresponding means of the 5 repetitions with different seeds. In addition, we compute the Pearson correlation for these values. Plots show on the x-axis each training step (for each batch) of the evolution in the training state of the neural network. On the y-axis, two scales are shown that apply to the distance curves between accumulated persistence diagrams (solid lines), scale on the right side, and the neural network validation (dotted lines), numerical scale on the left side. For each sub-experiment (for example, different values of layer size) a different color was used.
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. This is generally consistent across experiments (see the Supplementary Material). Table T shows the mean (and standard deviations) of the Pearson correlations for all datasets. All means Intuitively, this is also observed the plots, although once the distances are normalized it is not as clear to visualize. 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.
above 0.8, implying that there is strong correlation.
In CNNs, the correlations are lower (but still usually above 0.8 in experiments such as the one of increasing the number of layers). Recall that we froze convolutional feature extractor since our method only supports MLPs. 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.
Another finding is that the method obtains consistent results across runs, meaning that it is capturing information related important properties of the networks themselves instead of random artifacts. When varying the studied hyperparameters, we observe that the curves for each configuration are indeed, different. 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. The alteration of the order of the input doesn't have any effect in the homological convergence. The results of two of these experiments shown in Figure 7
In addition, we observe that when the neural network learns the given problem, homological conver- gence occurs. For example, when the layer size is modified, the capacity of the neural network to learn the problem changes (Figure When 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 to converge.
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1.0
0.965
1,0
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-0,950
Falidai scUEp (darbed)
Dioianne öifference (live)
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Validasion aboee ((ashhed)
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00.66
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00.60
0.955
0.0
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Epodss
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(b) CIFAR-100 CNN
Figure 7: Learning evolution on input order experiments (control experiments). Normalized.
Heat distance
Silhouette distance
Dataset
Means mean
Deviations mean
Means mean
Deviations mean
MNIST
0.8910
0.0424
0.8910
0.0424
Reuters
0,6220
0.0700
0.6220
0,0700
CIFAR- MLP
0.8233
0.0649
0,8233
0.0649
CIFAR- 10 CNN
0.4241
0.1915
0,4241
0.1915
CIFAR-100 MIP
0.8420
0.0566
0.8420
0.0566
CIFAR-100 CNN
0.6130
0.0800
0.6130
0.0800
Table Correlation (c. with 20 points) of validation values with topological difference (cumulative).
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 eπтоr during the learning process. A too small learning rate may result in long training process that could be stalled, while a too large value may fall in a fast suboptimal solution or unstable training process. Using homological convergence we find similar behaviour, as can be seen in Figure 9
Finally, we note that even if the two convergences (validation and homological convergence) are correlated, they are not the same process. This especially visible the learning rate experiments. For instance, in Figure homological convergence is reached before the stabilization of the validation accuracy. Presumably, they are not capturing the exact same information; we believe that the difference is due the fact that the validation accuracy depends the specifics of the data sampled in the validation subset, while the homological convergence is independent of the validation data.
7 DISCUSSION
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. We have seen that, indeed, it is the case, with strong empirical results backing our claim. This finding has remarkable implication. If homological convergence evolution mirrors the validation accuracy curve, one could ignore the validation set to monitor the training. This opens the path towards estimating the generalization of neural networks without the need of any holdout set. Researchers have wondered for 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 so. Although we do not provide any predictive model, we show that our proposed measures strongly correlate with validation accuracy. We do so by not using any data at all; we just look at the neural network itself. Our contribution aims pushing towards better understanding of the learning process of neural networks, targeting any specific direct application. However, we note that it can be effectively used for monitoring the training of neural networks in terms of convergence expected generalization,
have extensively
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105
0.98
diBerraee (lue)
0.8
Valid alddasii scure (dastedd
Plolatace differrnce (liee)
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10
Validasion abore ((isshed)
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0
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08
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(b) CIFAR-I0 MLP
Figure 8: Learning evolution when dropout parameter is changed. Normalized.
1.0
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(b) CIFAR-IO MLP
Figure 9: Learning evolution when modifying the learning rate parameter. Not normalized.
shown in the experiments. Apart from when there no access validation set, this is relevant because depending on a validation set has the risk of overfitting to it. Having an intrinsic measure should be more robust to random noise specific data sample.
The main limitation of our method is its computational scalability. This lack of scalability has prevented us from validating our method on bigger models and datasets. 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 not approximate any computation. This leaves room for finding efficient approximations, opening
research line.
Finally, we note that instead of computing correlations, serving as a basic quantitative study, it would be interesting perform time-series analysis gain more insights how the two curves vary together. Moreover, 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.
CONCLUSIONS & FUTURE WORK
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. Furthermore, we have shown that it can be used to monitor the training of neural network (and potentially estimating its generalization) without a validation set. As future work, we suggest generalizing our representation other neural architectures and scaling up the experiments larger models and datasets, for which finding efficient approximations of our method will be crucial.

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Under review as a conference paper at ICLR 2022
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Reviews:
[{"review_id": "R18ptY2HKOa", "reviewer": "Reviewer_eLVu", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "One merit of the paper is, to put it in very broad terms, the idea of\napplying tools from topological data analysis to the problem of\nunderstanding generalization in deep neural networks. In principle,\nthis is an interesting problem from a theoretical standpoint, and the\nidea of applying TDA is timely in the sense that many fields are now\nin the process of \"discovering\" topological techniques, and it is good\nto see some exploration of what they can be useful for. Unfortunately\nhowever, in my opinion the strengths of the paper stop here. I will\nnow go in detail over what I consider to be its major weaknesses.\n\nEmpirical studies can be valuable, but the complete absence of any\ntheoretical justification (or even just an intuitively plausible\nstory) for the main argument means that the experimental evidence\npresented carries all the burden of convincing the reader. My main\nconcern with the paper is that, unfortunately, the analyses presented\nare nowhere close to the standard of rigor and thoroughness one would\nexpect in this case. In particular:\n\n1. Unless I missed it, the paper never mentions the evolution of the\n   training error. For all I know by reading the paper and looking at\n   the plots, the tested networks never overfit the training data, in\n   which case I expect the training error to correlate to the\n   validation set error even better than the proposed metric,\n   something that would completely invalidate the claim of the paper\n   (i.e., that the topological measure is telling us something\n   nontrivial). In other words, in my opinion the interesting quantity\n   to understand the learning dynamics is not the validation set error\n   per se, but the generalization gap (or, in other words, the\n   validation set error should be considered together with the\n   training error, and not in isolation).\n2. As far as I understand (although the paper is quite confusing in\n   this respect - see below for more on this), the complex machinery\n   of TDA is used to characterize how much the network changes during\n   training. By looking at the plots in the paper, this quantity of\n   interest (rate of change) is typically large at the beginning of\n   the training, and small towards the end. Because this metric goes\n   down quite quickly, its cumulative value during training takes on a\n   saturating trend. This trend correlates with the validation set\n   error, which also saturates. The main claim of the paper is that\n   this correlation suggests a conceptual link between the two\n   quantities. But there are many quantities that presumably exhibit\n   similar trends during training! For instance, how about the norm of\n   the weight update vector (which could be conceptualized as a much\n   simpler way of measuring change in the network, and therefore could\n   be a natural comparison to the proposed metric), or even just the\n   learning rate, which is dynamically adjusted by RMSprop in the\n   proposed case studies?  Similarly to my first point above, these\n   are examples of quantities that could trivially correlate with the\n   validation set error in the specific examples chosen, just as well\n   or better than the proposed measure, without of course carrying any\n   interesting information on the generalization gap.\n3. Besides the issues above, the main quantitative piece of evidence\n   proposed in the paper is the table with the correlation\n   coefficients. To be honest, after reading the paper I am not sure\n   what is being correlated with what here. On page 7 I read that \"*For\n   each experiment (e.g., layer size in MNIST), we plot both the\n   evolution of the PH diagram distance and the validation score\n   (accuracy). The plotted values are the corresponding means of the 5\n   repetitions with different seeds. In addition, we compute the\n   Pearson correlation for these values.*\", and I deduce that the\n   correlation must be between the validation score and the normalized\n   cumulative topological distance (actually the plots contain also\n   the non-cumulative distance, but I guess the correlation is\n   computed with the cumulative one because the non-cumulative doesn't\n   seem to be correlated with the validation score). However, later on\n   in the same page I read \"*...there is strong\n   correlation. Intuitively, this is also observed in the plots,\n   although once the distances are normalized it is not as clear to\n   visualize.*\". Does this mean that the correlation was computed on\n   the unnormalized values, that is, not on what is in the plots,\n   contradicting the statement above?\n4. The paper does not seem to connect well with the existing\n   theoretical literature on the generalization gap in deep\n   learning. For instance, it completely ignores the effort to extend\n   information-theoretic criteria to be applicable to deep\n   networks. For instance, https://arxiv.org/abs/1906.07774v1 discuss\n   how a technique introduced by Takeuchi in the '70s (itself a\n   generalization of the Akaike Information Criterion) can be used to\n   predict the generalization gap. The introduction of that paper also\n   gives several references to multiple strands of the literature that\n   seem relevant for anybody interested in proposing a new technique\n   to estimate generalization (flatness etc). Other works that are\n   relevant from a theoretical standpoint are recent approaches\n   exploring an implicit simplicity bias in DNNs (see for instance\n   https://arxiv.org/abs/1805.08522, https://arxiv.org/abs/1812.10156,\n   and more recent literature citing those).\n\nMinor points:\n1. There are several passages in the paper that I completely failed to\n   parse. For instance, \"*Corneanu et al. (2020) try to estimate the\n   performance gap between training and testing using PH\n   measures. They claim. However, one can observe some caveats.*\", \"*We\n   study the relation between the evolution of the PH diagram\n   distances with the one of the validation score with the cumulative\n   values of the distance between homologous persistence diagrams\n   because this value seems much more stable. The information of the\n   distance between the persistence diagrams has been normalized\n   [...]*\".\n2. I don't understand why the y axes in most plots are labelled\n   \"distance difference\" but the word \"difference\" never appears in\n   the text of the paper - only distances are discussed.\n3. The architecture and the pre-training of the convolutional network\n   is entirely unclear to me. It is defined as \"*In the case of CNNs,\n   the pre-trained model is defined as 3 convolutional blocks with\n   kernel size 3 (starting with 32 channels), interleaved with max\n   pooling (its linear layers are thrown away after the\n   pre-training)*\". What does \"starting with 32 channels\" mean? This\n   sounds like the authors imply that there is a standard way of\n   changing the number of channels from one layer of a conv net to the\n   next (this is not the case, as far as I know). How is max pooling\n   performed? How many, and how large, linear layers are trained and\n   then \"thrown away\" during pre-training? When does pre-training\n   stop?\n4. I do not agree with the following statement: \"*If the measured\n   distances are, indeed, related with the learning process of neural\n   networks, these variations should not have any noticeable effect.*\",\n   when discussing the various network and training settings that were\n   examined in the experiments. How is changing the learning rate\n   expected not to have a noticeable effect on a metric related to the\n   learning process? And more generally, why would one expect to\n   change things like the size of the network in such a drastic way\n   (going e.g. all the way down to 4 units per layer in size) and see\n   no effect on the learning process?\n5. When commenting on Comeanu et al 2020, the paper states that the\n   method proposed there is \"not usable in practice\". I haven't read\n   that paper, but I wonder to what degree this is just the opinion of\n   the authors of the present work. If it is, it should be justified,\n   and not passed on as a fact. Moreover, regardless of its merits (on\n   the apparent lack of which I have commented in the Major Concerns\n   section), the method proposed in the current work is enormously\n   expensive from a computational standpoint (taking a week to run on\n   a machine with 2 V100 and 1.5TB of RAM to analyze networks with a\n   few hundred neurons on problems such as MNIST), so it doesn't seem\n   like practical usability is a key area of focus of the authors.\n6. The precise definition of the metric, and some of the choices\n   behind it, are not clearly specified. On page 5, \"*For each weighted\n   directed graph associated with the state of a neural network, we\n   link a directed flag complex to it. The topological properties of\n   this directed flag complex are studied using homology groups H n\n   . We calculate the homology groups up to degree 3 (H 0 -H 3 ).*\"\n   What is a flag complex? It is mentioned elsewhere in the paper that\n   flag complexes are used in the other (unpublished, AFAICT) paper by\n   the same authors, but the method proposed here should be\n   understandable without having to go and read that other\n   paper. Also, what motivates the choice of using homology groups up\n   to degree 3?\n", "summary_of_the_paper": "This paper presents some empirical observations about the relationship of a persistent homology-based measure of learning dynamics and validation set error, during the training of deep neural nets. The paper opens with an introduction on persistent homology, then introduces an approach to study the structure of deep nets using topological data analysis (TDA) tools. Three case studies are then presented, for which a measure of change in topological structure of the network during training is compared with the validation set error. The argument of the paper is that the two measures are correlated, and therefore it may be possible to use the topological measure (which depends on the structure of the network only, and not on data) in place of the validation set error to assess the generalization performance of a deep net.", "summary_of_the_review": "Despite investigating the application of a promising set of techniques\nto an interesting problem, this paper fails to sufficiently support\nits claims. The main conclusion rests upon the correlation of a\nproposed measure of change in deep nets during training with\nvalidation set accuracy, but no effort has been made to control for\npossible confounds that could explain such correlation\ntrivially. Moreover, multiple important passages in the paper are\nextremely hard (or indeed impossible as far as I'm concerned) to\nfollow, and noticeable gaps are present in the references to related\nliterature.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "ML1BO9nJ5Ed", "reviewer": "Reviewer_nK7m", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I think the subject of the paper is interesting itself and using PH techniques to discover insights about deep learning models is indeed an interesting topic. However, I find the paper to be still in early stage of development and still requires work. Specifically, It think the authors need to really support  their claim for the title. Generalization is probably the hardest problem in deep learning model models and the claim that PH truly captures that is not explained well at least at the beginning of the paper. What is the metric for such a claim ? The authors mentioned \"that there exists a high correlation with the corresponding validation accuracy of the model.\" I do not think think that is enough reason to justify the claim of the paper.\n\nThe contributions are not very novel in my opinion. For instance :\n\nBased on principles of Algebraic Topology, we propose measuring the distances (Silhouette\nand Heat) between the PH persistence diagrams obtained from a given state of a neural\nnetwork during the training procedure and the one in the immediately previous weights\nupdate.\n\nMeasuring the distance between two weights of neural network is not significant contribution, unless you do something with it--which you are claiming next but this particular point is not a valid contribution in my opinion.", "summary_of_the_paper": "The paper uses distances between the Persistent diagrams of states of neural networks and observe that during training there is a high correlation with the corresponding validation accuracy of the model. The authors tested their method on a variety of datasets.", "summary_of_the_review": "I am sorry to say that I have to reject the paper. The paper is still in its early stage and requires more careful presentation especially in the beginning to make the contribution more clear and crisp ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "IEV3mE6Z9vZ", "reviewer": "Reviewer_JhC7", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "# Strengths and Weaknesses\n\n## Strengths: \n- The paper is well written overall (aside for few typos, see Minor comments).\n- Experimental report is extensive given the amount of complementary plots in the supplementary material. \n- The methodology proposed by the paper is original and interesting. The construction of topological descriptor for networks has interesting properties, theoretically improving for instance on the approach proposed by Rieck et al. (at the price of computational efficiency though). \n\n## Weaknesses :\n\n- My main concern with the paper is that its conclusions seem too optimistic. From my understanding, saying that the \"*proposed measure strongly correlate with validation accuracy*\" (Section 7) and that \"*the generalization error of a neural network could be intrinsically estimated without any holdout set*\" (abstract) would mean for it to be useful that : \n  - (a) If the network starts overfitting, that is the validation accuracy starts decreasing (while the training accuracy presumably keeps increasing), this should be reflected in the curve of (cumulative) distances between (vectorization of) PDs. On Figure 7, b, for instance, there is---as far as I understand---no way to infer from the line-curve (PD distances) that the network is overfitting. \n  - (b) To be able to predict the generalization gap, one would expect a change of behavior when the network properly learns (e.g. MNIST) vs when it does not (e.g. CIFAR-10-MLP) ; (here, I assume that the training accuracy is going high---it as not been reported from what I can tell so we cannot be sure what the generalization gap is. Hence I assume that low validation accuracy $\\Leftrightarrow$ high gap.). From Figure 8.a and 8.b for instance, how is one supposed to infer that one network is learning something useful while the other does not? \n\nI am worried that what is interpreted as a strong correlation may only be a consequence of the networks parameters optimization/convergence. During optimization, as the step size goes to zero using RMSprop, it is not unlikely that the network weights variations (hence successive diagram distances) are getting smaller, yielding this concave-shape for the plots of cumulative diagram distances and for the *training* accuracy---thus for the validation accuracy when the network learns properly (i.e. validation accuracy $\\simeq$ training accuracy). \n\nFurthermore, when looking at plots in the supplementary material (for instance Figure 53), it seems that the reasonably correct correlation coefficients obtained are in great part due to the correlation *before* network parameters convergence (bottom-left quadrant: low topological cumulative, low validation score), which are not very interesting. On the top-right quadrant (when the network is likely to be \"ready\"), things get messier. \n\nIn the same vein, the sentence \"*(...) validation accuracy depends on the specificity of the data sampled in the validation subset, while the homological convergence is independent of the validation data*\" (end of section 6) is somewhat contradictory with the claim of the paper that homological properties of the network \"*captures the generalization of neural networks without a validation set*\". \n\nAll in all, and unless I have missed something crucial, I feel that the empirical results (which are quite numerous, thanks!) provided by this work are, unfortunately, not convincing.\n\n- A second important concern is the intersection with the Anonymous concurrent work (available in the supplementary material). With few exceptions, everything is the same up to the end of section 5 : same construction of topological descriptors (flag complex, filtration by Eq (1)), same diagram vectorizations, same datasets, same changes of architecture (number of layers, layer widths)... So that, roughly, what differs in the two paper is the experimental measurement: this paper studies what happen at training time for a given network, while the concurrent one makes cross-networks comparisons. Though the two papers cover different objectives, it diminishes the intrinsic contributions of the current one, which are now limited to the experimental results which are, as said above, insufficiently convincing in my opinion. \n- As acknowledged by the authors, this method does not scale and is only applicable to reasonable MLP; even though requiring high-end hardware and a lot of time. \n- The paper is not supported by any theoretical result that would motivate the use of their approach.\n\n\n# Minor comments\n- (Typo) p2, give-->given\n- (Typo) End of Section 3 : \"They claim. However\". It seems a sentence is missing. \n- (Typo) Start of p5 : an smoothing --> a smoothing\n- (Suggestion) Eq 1 : use \\max and \\left( \\right). \n- (Suggestion) Numbering cases with \"1. blabla... 2. blabla\" hinders readability. Perhaps using something in the vein of \"*(i)* blabla, *(ii)* blabla\" would be better. \n- (Clarity) The transition between filtrations of simplicial complex and persistence diagrams may be a bit too short (I am referring to the single sentence \"The sequence of homology groups is calculated by varying the parameter $\\epsilon$ to obtain the persistence homology diagram.\")\n- (Typo) p5 : a reference is missing (\"as in ?\").\n- end of p7, \"the homology does not seem to converge\". I do not understand this claim. If the network (parameters) converge, so does the persistence diagrams. If my understanding is correct, RMSprop should always yield convergence (it uses a decay factor for the step size); so I guess more epochs are needed. \n- p9 : \"our approach computes the exact PDs distance\", I think the exact PDs are computed in first place (not the distances). \n- Results in Table 1 for Heat and Silhouette are the exact same. Is it an unfortunate copy-paste?\n- Supplementary material, Figures 42-53 : how can it be that the topological cumulative decreases (given that the validation score seems monotonic, I guess that the axes labels have been switched?)? Also, why does the validation score reaches 1, while it is supposed to be much lower on some model? (I guess a normalization has been applied). \n- The paper claims that \"*in CNNs, the correlation are (...) still usually above 0.8*\", but unless I misread the tables 5,6, 9, and 10, this seems to be exaggerated. CIFAR100CNN + number of layers $\\geq 4$ is one the only instance where this seems to hold. ", "summary_of_the_paper": "This paper experimentally investigates the (cor)relation between the validation accuracy achieved by a neural network and a variation of a topological descriptor built on top of the network: the persistence diagram of a directed flag complex built on top of the network. \n\nThe main claim is that there is a high correlation between the validation accuracy and this \"topological distance\", so that the later may be used when a validation set is not available. ", "summary_of_the_review": "Though I appreciate the ambitions of the work, I feel that the empirical results are, unfortunately, not convincing. The absence of theoretical results and the very strong similarities with the concurrent submitted paper diminishes it's impact as well. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "53ChJiluxyG", "reviewer": "Reviewer_Fo4q", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I enjoyed the ideas presented in this paper; the analysis of\ngeneralisation performance is a highly relevant and timely, and indeed\nconstitutes one of the largest obstacle toward employing machine\nlearning techniques in the wild. Understanding generalisation without\nrequiring additional data has the potential to improve machine learning\ntechniques to a substantial extent.\n\nThat being said, the current paper suffers from some issues, which\nprevent my endorsement at this point:\n\n1. Clarity: while background information on topological data analysis is\n   provided (which I appreciate; in particular in light of the fact that\n   TDA is still a rather novel occurrence at ML conferences), the method\n   itself could be described in more detail. In particular, some aspects\n   should be discussed in Section 4:\n\n    - The use of additional representations for topological features\n      needs to be clarified. At present, the main paper does not contain\n      any description of the challenges in calculating distances or the\n      need for additional representations. At the very least, a brief\n      description of these topological representations is required, so\n      that readers may better understand the remainder of the paper.\n      I find it particularly problematic that no stability guarantees of\n      these representations are discussed; they are not *just* drop-in\n      replacements for the bottleneck or Wasserstein distances.\n\n    - Overall, this section should be rewritten with a clear 'roadmap'\n      in mind. What is the problem you want to tackle and how do you\n      intend to tackle it in this paper?\n\n2. Delineation to existing work: The ICLR paper by Rieck et al. (2019b)\n   appears to already contain a large amount of the material proposed in\n   this paper. A cursory reading shows that Rieck et al. even discuss\n   generalisation performance in an early stopping setting (more about\n   this later). It is therefore critical that the contributions of this\n   paper are more clearly delineated from existing work. From my\n   understanding of both papers, I would say that the current submission\n   improves on the following aspects:\n\n    - Choice of filtration for such neural networks\n\n    - The use of other topological representations (whereas Rieck et al.\n      only use a summary statistic of persistence diagrams).\n\n   These differences (and potentially all others that I missed) should be\n   briefly discussed; the advantage would be that the paper can refer to\n   the previous publication as a justification of the method itself!\n\n   The previous work by Rieck et al. also proposes a set of experiments\n   that would be very fitting to perform here: the experiments proposed\n   in this paper could be substantially strengthened by the addition of certain\n   (simple) monitoring baselines that serve to showcase the added value\n   of employing topological information in the first place here. Rieck et\n   al. compared their topological measure to a 'patience criterion' based\n   on validation loss, thus assessing the difference in performance\n   and epochs after the early stopping criterion was applied). In my\n   opinion, this is most important experiment here when assessing\n   validation performance. Also, it would be interesting to add simple\n   baselines measures to check that these are *insufficient* to be used\n   alone. For instance, a non-topological baseline that could be assessed\n   in this context would be a distribution of weights; it should not be\n   possible to assess validation performance with this baseline.\n\n2. Experimental depth: the experimental section is lacking depth, in\n   particular given the specific goals of this paper. The depicted plots\n   and correlations are useful, but don't serve to highlight the\n   benefits of the proposed method. I would therefore suggest a more\n   thorough experimental setup such as the one shown in the supplement\n   of Rieck et al. (2019b): as mentioned in the previous point, their\n   proposed measure of network complexity is shown as additional early\n   stopping criterion (without an hold-out validation data set as well).\n\n   A post-hoc experiment of this sort would be extremely useful in\n   demonstrating the utility of the method, and it would even enable the\n   quantification of certain measures. The correlation is also useful in\n   this context, but since the goal is to assess the generalisation\n   error, an experiment in which the new measure is directly applied\n   would be extremely worthwhile.\n\nPlease see below for detailed comments.\n\n## Detailed comments\n\n- The first contribution reads a little bit like a sentence fragment to\n  me. I would suggest to use the terms 'silhouette distance' and 'heat\n  kernel distance' instead of plain 'silhouette' and 'heat'.\n\n- The terminology 'homological convergence' is slightly misleading\n  because the paper studies persistent homology, as far as I understand.\n\n- The introduction of topological concepts in the background section\n  jumps between the 'geometrical' and the 'abstract'; I understand that\n  both perspectives are valid, but I would suggest to choose only\n  a single one.\n\n- The discussion of chains and boundaries requires more explanations (or\n  could be partially relegated to the supplementary materials). For\n  instance, please clarify the 'signed combination.' However,\n  definitions need to appear in the main text, not only in the\n  supplementary materials.\n\n- In figure 4, the $3$-simplex should consist of four $2$-simplices, but\n  only two are shown. Either clarify this in the captions or update the\n  figure accordingly.\n\n- The introduction of filtrations is not motivated; the 'nested family\n  of simplicial complexes' could be introduced by building more\n  intuition here.\n\n- The related work discussion should, as mentioned above, delineate this\n  paper from Rieck et al. (2019b).\n\n- The heading 'Algebraic Topology object' is rather confusing to me;\n  I would suggest to rewrite it as 'directed flag complex', for\n  instance.\n\n- As outlined above, the distance calculation lacks information about\n  stability and other guarantees. There is also a broken reference on p.\n  5, which needs to be rectified.\n\n- In the experimental section, the use of the MLP should be explained in\n  more detail. What does it do and how does it work exactly?\n\n- I do not understand the 'input order' experiment; does it refer to\n  ordering of samples (i.e. on the batch level)?\n\n- The use of additional topological descriptors makes the use of the\n  word 'distance' imprecise: the bottleneck and Wasserstein\n  distances are metrics in the mathematical sense; while there *is*\n  a well-defined metric between, for instance, persistence landscapes or\n  silhouettes, this is only (to a certain extent) an approximation to\n  the bottleneck/Wasserstein metrics. This distinction should be made\n  clearer in the paper.\n\n- Figure 6 should add standard deviations of the curves. An additional\n  clarification of the content of this figure would also be appreciated\n  by readers; I found the discussion to be slightly confusing because\n  I did not get the relevance of the cumulative distance. The statement\n  'the evolution of the homological convergence [...] seems to be very\n  similar to the one of the validation score' also needs some\n  clarification. Except for Figure 6b, where I observe some oscillation\n  behaviour and maybe the tendency to converge, I don't observe\n  convergence anywhere else.\n\n- The same applies to Figures 7–9. Adding some more details here will\n  result in a clearer description of the method. Maybe some of the\n  curves could also be relegated to the supplementary materials?\n\n- I would also suggest to investigate the use of energy distances or\n  energy correlations, as these measures are more flexible and capable\n  of assessing more than just linear dependencies between two\n  variables. See [*Energy statistics: A class of statistics based on distances*](https://www.sciencedirect.com/science/article/abs/pii/S0378375813000633)\n  for more details.\n\n## Terminology and style\n\nFor the most part, the paper is written well. I have some suggestions\nconcerning the style:\n\n- Please use `\\citep` and `\\citet` consistently when using `natbib`. At\n  present, citations are directly appearing in the text without any\n  enclosing parentheses. For instance, in the related work section,\n  the sentence 'for improving the training procedure of the models Hofer\n  et al. (2020); [...]' should use `\\citep` in order to obtain a proper\n  parenthetical citation.\n\n- can't --> cannot (likewise for other contractions; I admit that this\n  is a personal preference, so feel free to ignore it)\n\n- 'contained simplicial complex' --> 'nested simplicial complexes'\n\n- The paper employs non-standard terminology in certain places. The\n  diagram, for instance, is called 'persistence diagram', no\n  'persistence homology diagram'. Please ensure consistency with\n  existing papers here. The same applies to capitalisation; I personally\n  see no need to write 'Persistence Diagram', but if this spelling is\n  used, it should be used consistently.\n\n- 'complex clique' --> Use the term 'clique complex'\n\n- 'module $Z_i(K)$' --> 'modulo $Z_i(K)$' (the paper is discussing the\n  quotient operation)\n\n- 'non-cumulative homology': I think the word 'distances' is missing\n  here.\n\n- The use of 'Means mean' and 'Deviations mean' is slightly confusing;\n  I would suggest to show follow the format $\\mu \\pm \\sigma$, with $\\mu$\n  being the mean and $\\sigma$ being some measure of variance, such as\n  the standard deviation.\n\n- For the references list, I would suggest to carefully check which\n  version of a paper is being cited. Numerous papers mentioned in the\n  bibliography have already been published as book chapters, papers,\n  etc.\n\n- The preprint by Guss and Salakhutdinov is cited twice.\n\n", "summary_of_the_paper": "This paper analyses the training of neural networks from a topological\nperspective, presenting a pipeline that can measure (pseudo) distances\nbetween the network's weights during training. Such information is then\nemployed to study the generalisation error of a neural network.\n\nIn contrast to existing methods for estimating this error, this paper\ndoes *not* require a specific hold-out data set, as topological features\nof the neural network are monitored during training. This frees up\nadditional data for fitting, which can be highly relevant in the sparse\ndata regime.", "summary_of_the_review": "While I am very excited about topology-based approaches that aim to\nunderstand the training or testing behaviour of neural networks,\nI cannot endorse this paper for publication yet.\n\nThe current write-up is suffering from several issues, which need to be\nrectified in a **major revision** before reaching the quality standards\nof ICLR. These issues include:\n\n1. Lack of clarity: an improved introduction to topological concepts is\n   required and the proposed method needs to be compared more with\n   existing topology-based approaches (in particular since it does not\n   meet the requirements of a metric in the mathematical sense, an\n   analysis of approximation guarantees is crucial).\n\n2. Lack of experimental depth and delineation to existing work: since\n   the express goal of this paper is to analyse generalisation\n   properties of neural networks based on their topological properties,\n   a comparison with existing work (Rieck et al., 2019b) and an improved\n   experimental suite (containing previous work *and* non-topological\n   baselines) is critical for corroborating the claims of this paper.\n\n**Updated after rebuttal**: A lot has been discussed during this rebuttal period. I would strongly\nrecommend to pick up some of the suggestions of reviewers in order to improve the paper. A recent paper\nby [Birdal et al.](https://papers.nips.cc/paper/2021/hash/35a12c43227f217207d4e06ffefe39d3-Abstract.html)\ndemonstrates how to successfully assess generalisation performance (disclaimer: I am *not* one of the authors\nof said paper). I hope that this may serve as a partial inspiration.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper got uniformly strongly negative reviews.  The issue of estimating or bounding generalization accuracy from performance on the training set has a huge history and literature.  After considerable discussion the reviewers uniformly find this paper lacking in making a contribution to that literature.
Decision: Reject