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: NeurIPS 2021 Submitted (2021)
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.

Generalization is what makes machine learning model useful; the uncertainty of its behaviour with 13 unseen data is what makes potentially dangerous. Thus, understanding the generalization error of Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set
Anonymous Author(s)
Affiliation Address email
Input
Training
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Figure 1: Our proposal.
Abstract
The training of neural networks is usually monitored with a validation (holdout) set estimate the generalization of the model. This is done instead of measuring intrinsic properties the model to determine whether is learning appropriately. In this work, we suggest studying the 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 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 neural network could be intrinsically estimated without any holdout set.
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Introduction
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14 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- 16 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
18 set, the validation set. Since these data have not been directly used for updating the parameters, it assumed that the performance of the model the validation set can be used proxy of the
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Submitted to 35th Conference on Neural Information Processing Systems (NeurIPS 2021). Do not distribute.

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20 generalization error, provided it is representative of the data that will be used in inference. One
21 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 he 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. On the other hand, Algebraic Topology has recently been gaining momentum as a mathematical tool for studying graphs, machine learning algorithms, and data.
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27 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,
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can calculate the Persistent 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 training step) (see 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 hyperparameters. 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.
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In summary, our contributions are as follows:
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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.
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We empirically show that the evolution of these measures during training correlate with the accuracy in the validation set. We do in diverse benchmarks (MNIST, CIFARIO, 10, CIFARI00 100, Reuters text classification), and models (MLPs in MNIST and Reuters, MLPs and CNNs in CIFAR100 and CIFAR100).
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We thus provide empirical proof of the fact that valuable information related the learning process of neural networks can be obtained from PH distances between persistence diagrams (homological convergence). In particular, we show that homological convergence is related to learning process and the generalization properties of neural networks.
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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.
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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 [ we formalize our method. Finally, in sections 6 and 7 we present and discuss our empirical results, respectively.
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2
Background
58 In this section we introduce the mathematical foundations of this paper. detailed mathematical description is included in the Supplementary Material.
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80 simplicial complex is set composed of points, line segments, triangles, and their n-dimensional
61 counterparts, named simplex (K). In particular, a simplicial complex must comply with two properties: Every face of a simplex is also in the simplicial complex (of lower dimension). 2. The non-empty intersection of any two simplices contained on simplicial complex is face of both. 0,1,2,3 0,1,2,3-simplex
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84 and non simplex examples are shown in Figure [
65 We can associate to an undirected graph, G= (V,E), a simplicial complex where all the vertices of G are the O-simplex of the simplicial complex and the complete subgraphs with i vertices, in G
67 corresponds to a (i- 1) -immple. This type of construction is usually called a complex clique on the denoted by CI) Figure shows graph clique complex CI(G) example.
88 graph G, and is

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(a) 0, 1.1.i..xs
(b) Non-simplex
Figure 2: Simplex and non-simplex examples.
a
a
3-simplex
2-simplex
1-simplex
O-simplex
Figure 4: Boundary function sample.
69 The boundary function is defined as a map, from
an i-simplex to an (i- -immppe,,, the of its (i ( di-dimensio faces. boundary function sample is shown in Figure 4
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algebraic topology, a k-chain a combination of k-simplices (sometimes symbolized as lin-
75 car combination of simplices that compose the chain). The boundary of a k-chain is a (k I)- chain. It is the linear and signed combination of chain element boundary simplices. The space of
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Figure 3: Graph clique complex CI(G) example.
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i-chains denoted by (K).
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There are two special cases of chains that will be useful to define homology:
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Closed chain or i-cycle: i-chain with empty boundary. An i-chain e an i-cycle and only if 0;c = i.e. ker(a)). This subspace of C,(K) is denoted as (K).
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Exact chain j-boundary: An i-chain is i-boundary if there exists (i+ 1)-chain d such that c 1 i.c. E im (m(a++1) This subspace of C)KK), the set of all such i-boundaries forms, is denoted by (K).
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Now, if think in the i-cycles that do not bound (i+ 1i-simplicial complex, this is the definition i-th homology of the simplicial complex K. The precise definition the quotient space of IB, (K) subspace of (K) (see Supplementary Material). The number of non equivalent i-cycles (Figure 5) the dimension of the homology group H, (K), also named Betti numbers.
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We can create nested family of simplicial com- plexes, Kg, where each step Kg, embedded in the simplicial complex K We call this set simplicial complex filtration.
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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) (5).
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Figure 5: The two blue dashed cycles are homo- logically equivalent, the pink isn't 't.
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It is possible to compare two PDs using specific distances (Wasserstein and Bottleneck). To efficiently perform this operation, due the size of these diagrams, it sometimes necessary to simplify them by means of a discretization process (such as Weighted Silhouette and Heat vectorizations).
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Related Work
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Algebraic Topology and Machine Learning The use of Algebraic Topology in the fields of
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data science and machine learning has been gaining momentum in recent years (see Carlsson [5]) Specifically in the case of neural networks, some works have applied topology for improving the
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training procedure of the models [15] [] or pruning the model afterwards [30]. Other works have
focused on analyzing the capacity of neural networks [1, 26 17 or the complexity of input data
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Furthermore, recent works have provided topological analysis of the decision boundaries of
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classifiers based on PH and Betti numbers 24 22
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Graph and topological representations of neural networks Gebhart et al. 12] suggest a method for 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. Chowdhury al. 77) show that directed homology can be used to represent MLPs. Anonymous 2 concurrently show neural networks, when represented as directed, acyclic graphs, can be associated an Algebraic Topology object. 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 [[] approximating neural networks representations with regard to the input space.
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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. [16] work on understanding what drives generalization in deep networks from Bayesian of view. Neyshabur et al.
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23 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.
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II. authors propose an alternative to cross-validation for model selection based on training once on the whole train set, without any data split, deriving validation set with data augmentation.
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Corneanu et al. [O] try to estimate the performance gap between training and testing using PH measures. They claim. However,
can observe some caveats. The first one is that their regression fitted to predict the test error has a 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.
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137 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 topologiaal characterization and distances concurrently proposed in Anonymous ( 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 et al. (21) we do not use any data at all. Unlike (TO). we do not build a statistical or machine learning model (linear regression) for predicting the testing error. Instead, we propose a new measure, and we empirically show that 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.
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4 Approach
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Representation For representing neural networks as graphs, we follow the approach proposed concurrently in Anonymous (2). 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. It is important to note that abstract simplicial complex used in opposition to geometric simplicial
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complex.
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For every training state, neural network connections are 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
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ignored because, as we will see, it is not very informative in terms of homology, but we decided to preserve
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Negative edge weights are represented with reverse edges with the same weight absolute value. We discard the use of weight absolute value as neural networks not invariant under 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 [ S is an smoothing parameter that we set to
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le-6. This smoothing parameter is necessary as we want to avoid normalized weights of edges to be
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(in our representation 0 implies a lack of connection):
w
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Algebraic Topology object For each weighted directed graph associated with the state of a neural network, we link a directed flag complex it. The topological properties of this directed flag complex are studied using homology groups Hn. We calculate the homology groups up to degree 3 (Ho-Hs)
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For each state, we use family of simplicial complexes, Ke, for a range of values of E R. The simplicial complex at step & is embedded in the complex at 6r+1, for & S 62+1, i.e. K₂ C K E used filter that establish the minimum weight of the graph representation edges included on the simplicial complex. This collection of contained simplicial complex (associated to a directed weighted graph), called filtration, Ke C ... Kee S Ke+1 C... Ke where E [0, and Emin
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Emax- = (remember that edge weights are normalized).
The sequence of homology groups is calculated by varying the e parameter obtain the persistence homology diagram. In our case, persistent homology calculations are performed on Zz- In other words, once the corresponding filter has been applied to the weight of the edges, all connected edges are considered equally.
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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. 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
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In order to make computations feasible and obviate noisy intervals. 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, our neural networks have millions of persistence intervals per Persistence Diagram, while Wasserstein distance calculations are computationally hard for large PDs. In order to make calculations computationally feasible, we will use a vectorized version of PDs, also called PD discretization. This vectorized version summaries have been proposed and used on recent literature
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3 19 (5). For persistence diagram distance calculation, we use weighted Silhouctte and Heat 193
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vectorizations, using the Giotto-TDA library (27
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Experiments
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We validate our method in several heterogeneous (vision, natural language), well-known datasets, namely 1. MNIST 20). 2. CIFAR-10, 3. CIFAR-100 [88. and the Reuters dataset 28 (multi-class and multi-label document classification dataset).
Data
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Models
We experiment with two neural architectures, MLPs and 2. CNNs. the latter case, we use the convolutional layers as pre-trained model with frozen weights, and we learn an MIP on top of it. The reason we do so is that our method based in a representation that, least in the basic form, does not allow capturing information from convolutional layers. Thus, need single (exact
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same weights) feature extractor, to abstract away distances related to the CNN layers and focus on the MLP.
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Conducted experiments We define the base MLP architecture as (Input, Linear (512), Dropout (0.2), Linear (512), Dropout 2), Output}. 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-traine pre-training). On top of the pre-traine trained
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CNN, we also define the same base MLP architecture. Then, for each dataset and model (MLP and
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CNN), we experiment with varying (while keeping the rest fixed the base architecture)
1. Layer size (number of units per layer): 4, 16, 32, 128, 256.
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2. Number of labels (the other classes are removed): 2, 4, 6, 8, 10.
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3. Learning rate: le-c05, 0.0001, 0.001, 0.01, 0.1
4. Dropout: 0.0, 0.2, 0.4, 0.5, 0.8.
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5. Input order: random input orders. As a 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 not have any noticeable effect.
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We run each configuration 5 times with different random seeds (and, thus, weight initializations to if the results are consistent across runs. All models are trained with the RMSProp optimizer with a batch size of 256.
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Distances and validation accuracy computation Note that homological distances obtained at the end of each batch, while validation metrics are only computed on each epoch. The methodology follow to analyze the learning process each different problem be summarized with the following steps:
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1. In each training step (i.e., for each batch) extract the weights from the MLP current state and use them to build an abstract simplicial complex from the associated weighted directed graph.
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2. We calculate the homological persistence diagram of the simplicial complex.
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3. We then calculate the distance between consecutive persistence diagrams (we will call this sequence homological convergence). We use two different distances, namely, Heat and Silhouctte.
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4. We compare the homological convergence with the evolution of the validation results on neural network learning process.
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All experiments were executed in a machine with 2 NVIDIA V100 of 32GB, 2 Intel(R) Xeon(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.
Hardware
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The code and outputs are fully available in the Supplementary Material under MIT License.
Results
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In this section, we highlight the main results, omitting the ones with Silhouette (since the obtained results were clearer with Heat). See the Supplementary Material for the full results (plots and correlations), including the ones with Silhouette distance.
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243 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 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 can be found in the Supplementary Material. Figure(BJshow the cumulative and non-cumulative homology the MNIST experiment with layer size.
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The traitra convolutional weights always identical, though.

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Figure 6: Heat distance and validation accuracy curves on the MNIST experiment with layer size. Normalized.
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For each experiment (e.g., layer size MNIST), we plot both the evolution of the diagram distance and the validation score (accuracy). The plotted values the corresponding means of the 5 repetitions with different sceds. 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 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.
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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 D shows the mean (and standard deviations) of the Pearson correlations for all datasets. All means
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above 0.8, implying that there is strong correlation. 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 't end up learning properly.
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the case of CNNs, the correlations are lower (although still almost always above 0.8 in experiments such as the one of increasing the number of layers). Recall that in the case of CNN we froze single convolutional feature extractor, since our method only supports MIPs. We believe these lower correlations can be explained because an important part of the learning process happened in the convolutional layers (in the pre-training, which we do not capture.
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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 random artifacts.
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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 are shown Figure 7
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In addition, 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 6). When it can't learn the problem, because the network does not have sufficient capacity (the layer size too small, 4 units), the homology does not seem to converge.
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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 estimated error during 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 an unstable training process. Using homological convergence we find similar behaviour, can be seen in Figure 9
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0.955
1.0
-0.950
Valida 6on 50070 (dantred))
Ditaarr ditference (line)
Valilation foo moore (dasheel)
0.8
-0.975
didfererre
0.6
0.970
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0.965
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0.960
0.955
0.42
Ejochs
Epodks
(a) MNIST
(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
Deviations mean
MNIST
0.8910
0.0424
0.8910
0.0424
Reuters
0,6220
0.0700
0.6220
0,0700
CIFAR-IO MLP
0.8233
0.0649
0.8233
0.0649
CIFAR-I0 CNN
0.4241
0.1915
0,4241
0.1915
CIFAR-100 MLP
0.8420
0.0566
0.8420
0.0566
100 CNN
0.6130
0.0800
0.6130
0.0800
Table 1: Correlation of validation values with topological difference cumulative. Correlation is computed with 20 points.
Finally, we note that even if the two convergences (validation and homological convergence) are correlated, they are not the same process. This is especially visible in the case of the learning rate experiments. For instance, in Figure 9 homological convergence is reached before the stabilization of the validation accuracy. Presumably, they are not capturing the exact same information; specifically, believe that the difference is due to the fact that the validation accuracy depends the specifics of the data sampled in the validation subset, while the homological convergence is independent the validation data.
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Discussion
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We posed the of question whether homological convergence (in terms of distances between PH diagrams in consecutive neural network states) related to the learning process of neural networks. We have seen that, indeed, it is the case, with strong empirical results backing claim.
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This finding has remarkable implication. If the homological convergence evolution mirrors 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 a long time whether generalization could be predicted from intrinsic properties of the model or training data alone (i.c., without a holdout set), and in fact other works have claimed to do so. Although we do not provide any predictive model, we show that our proposed measures strongly correlate with the validation accuracy. In addition, we do so by not using any data all; just look at the neural network itself.
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Our contribution aims pushing towards having a better understanding
learning process neural networks, not 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 shown in the experiments. Apart from the cases without access validation
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set, this is relevant because depending validation has the risk of overfitting it. Having intrinsic, well-principled measure should be more robust to random noise in a specific data sample.
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The main limitation of our method is its computational scalability. As we said in Section 5 our method took more than 7 days of compute in a HPC machine, even if we restricted the experiments to small datasets and parameter count. However, we note that our approach computes the exact persistence diagram distances, that is, we do not simplify the graph representation of the neural networks (we keep every single neuron and connections) and do not approximate any computation. This leaves room for finding efficient approximations, opening new research line. addition, this lack of scalability has prevented us from validating method on bigger models and datasets.
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Finally, we note that instead of computing correlations, serving as basic quantitative study, it would be interesting to perform a time-series analysis to gain more insights on how the two curves vary together. Moreover, would have been interesting investigate how build predictive model of the validation accuracy from the PH distances, but it is was of the scope of this work.
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8 Conclusions & Future Work
323 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, have shown that it can be used to monitor the training of a neural network (and potentially estimating its generalization) without validation set. As future work, suggest generalizing 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.
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Checklist
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1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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contributions and scope? [Yes]
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(b) Did you descrihe the limitations of your work? [Yes] See the Discussion Section.
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(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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2. If you are including theoretical results...
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(a) Did you state the full set of assumptions of all theoretical results? [N/A]
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(b) Did you include complete proofs of all theoretical results? [N/A]
3. If you experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either the supplemental material or as a URL)? [Yes] Both code and outputs.
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Check the Experiments Section and the code.
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(c) Did you report error bars (e.g., with respect to the random seed after running ex- periments multiple times)? [Yes] We include means, standard deviations and raw outputs.
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(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Check Experiments Section.
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4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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(a) If your work uses existing assets, did you cite the creators? [Yes] In the case of the datasets. We do not use any other additional asset.
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(b) Did you mention the license of the assets? [No]
(c) Did you include any new assets either in the supplemental material URL? [Yes] Code, results and pictures we have made for explanations.
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Reviews:
[{"review_id": "sqBofKizg9e", "reviewer": "Reviewer_YPfU", "summary": "Summary:\nThe paper proposes to monitor neural networks during training by considering a momentary state of a network (MLP or CNN) as an abstract simplicial complex as defined by considering the network as a weighted directed graph. Then persistent homology features are computed resulting in persistence diagrams which are vectorized to compute distances between subsequent network states during training. The paper proposes to monitor this measure they call homological convergence to identify training stages where generalization error is low -- without the need to estimate performance by means of a validation set.", "questions": "", "limitations": "", "rating": 4, "confidence": 3, "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": "Major points:\n-I find it relevant and interesting to assess generalization capabilities of neural networks by means of TDA without requiring a validation set, and I am curious to see where this line of research is developing.\n- The paper is clearly written, and the non-familiar reader also receives a clear introduction to TDA.\n- My biggest concern is that the main finding, i.e., that PH of the network architecture allows to monitor performance without a validation set, is not novel. For instance, in Reference [26], Rieck et al employed PH of the network architecture (albeit using a weighted undirected graph) to distinguish good from poor performing networks and even proposed an early stopping scheme with a PH-based patience approach for guiding training and monitoring of neural nets without a validation set, amongst others also using MNIST and CIFAR-10. That being said, even though the empirical observation that PH of the network correlates with generalization has been known before, to my knowledge it is still an open question *why* this is the case, i.e., the theoretical workup of this phenomenon is still pending and would be quite relevant to have addressed - which I encourage the authors of this paper to investigate, as this would considerably strengthen the paper.\n\nOne way I could imagine that potential explanations of this phenomen could be identified could be via dedicated comparison studies:\ne.g. do simple statistical moments of the weight distribution also correlate with generalisation, or could there be a way to identify relevant cliques in the network which are topologically relevant and also coinciding with a \"lottery winning\" subnetwork, etc?\n\n- The experiments could be strenghened by the addition of certain monitoring baselines to showcase the gained value of employing PH here.\n    For instance, in [26] the PH-based measure was compared to patience with validation loss (what is the difference in performance and epochs after when early stopping was initiated). Also, it would be interesting to add simple baselines measures to check that they are not already helpful alone (like non-topological information about the weight distribution).\n\n\nMinor points:\n- Figure 6: it took me some time to understand the legend / caption. I would suggest to repeat \"# layers\" in the legend, such that is easier to understand the legend without reading the caption first.\n- Line 294, typo: \"We posed the of question whether\"", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Computation scalability was considered the main limitation in the paper. Otherwise, I could not find any discussion of potentially negative societal impact. For instance, when significant runtime due to scalability issues are in the focus, the paper could mention that this could affect CO2 expenditures of models employing these techniques.\n", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "~ 5 hours"}, "extra_text": {}}, {"review_id": "Z9aLrNxNkUL", "reviewer": "Reviewer_Kand", "summary": "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.\n", "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": "## Summary of the review\n\nI enjoyed the ideas presented in this paper; the analysis of\ngeneralisation performance is a highly relevant and timely.\nUnderstanding generalisation without requiring additional data has the\npotential to improve machine learning techniques to a significant\nextent.\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    - How does the calculation work in practice? The paper only provides\n      a description of simplicial homology in the undirected setting but\n      Section 4 appears to discuss directed graphs as well. I understand\n      that the directed graph is turned into an undirected one, but this\n      needs to be clarified.\n\n    - Would it be possible to use a directed flag complex or a Dokwer\n      complex here instead? Maybe this is more of an expert discussion\n      topic, but if directionality matters, that could be a good option.\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 paper by Rieck et al. [26] appears\n   to already contain a large amount of the material proposed in this\n   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\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. [26]: here, their proposed measure of network\n   complexity is shown as additional early stopping criterion (without an\n   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- 'complex clique' --> 'clique complex'\n\n- The discussion of chains and boundaries requires more explanations (or\n  could be partially relegated to the supplementary materials). For\n  instance, what is as 'signed combination' in this context? If\n  I understand it correctly, the paper uses the usual $\\mathbb{Z}_2$\n  coefficients, so there are no signs?\n\n- I would suggest not to use $\\mathbb{B}$ and $\\mathbb{Z}$ to denote the\n  boundary and cycle subgroups, respectively, as the later one can be\n  easily confused with the set of integers or certain characteristic-$p$\n  fields. The more common terminology seems to be to just use $B$ and\n  $Z$ here.\n\n- There is a distinction between the 'homology' and the 'homology\n  group'. The way the paper defines it, I assume that 'homology group'\n  is meant in l. 87.\n\n- The definition of homology group needs some refinement. The sentence\n  'is the quotient space of $\\mathbb{B}$ a subspace of $\\mathbb{Z}$' is\n  confusing to me. Please rewrite this more precisely.\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. [26].\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.\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 don't 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' somewhat 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 `sort` as a `natbib` option in order to ensure that\n  citations are sorted; this makes the manuscript easier to read.\n\n- 'contained simplicial complex' --> 'nested simplicial complexes'\n\n- can't --> cannot\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- '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. Paper #24, for instance, has\n  already been published as an ICML paper.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Limitations of the method could be more prominently discussed. As in\nprevious work, the current method does not work well with convolutional\narchitectures; an additional subsection on limitations would therefore\nbe appreciated, in particular by readers that are not yet familiar with\nTDA. This could be integrated into Section 7, which already contains\na discussion of the computational requirements.\n\nBeing mostly a theoretical analysis of generalisation performance, no\nnegative societal impact is bound to arise specifically from this work.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "3"}, "extra_text": {}}, {"review_id": "OtfbXylJZuG", "reviewer": "Reviewer_RrH9", "summary": "This paper proposes a method to evaluate the generalization of a treined neural network using persistent homology. They measure the distance between Persistent Diagrams corresponding to Neural Networks in the training process, and show that the accumulation correlates with the Validation Curve. They also discuss which method(difinition) is appropriate for the distance between PDs.", "questions": "", "limitations": "", "rating": 4, "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": "In training NNs, the capture of generalization is very important, so we often divide the data into training set and validation set. On the other hand, there are some problems with this method, such as the dependency on the partitioning method and the small amount of training data. From these perspectives, it would be a great advantage if generality could be evaluated without a validation set. This paper represents the NN of the learning process as a weighted graph and evaluates generality by calculating persistent homology. Evaluating the network using persistent homology is a very interesting approach because it may allow us to ignore the randomness of the order of the neuron layers. On the other hand, there are some concerns about the evaluation method of Generality:\n1. Since the score of the proposed method is obtained only from the training data, we cannot evaluate whether the method can be used for data whose domain is different from the training data. I think it is used to make sure that the model does not overfit the training data, but can the detection of overfit be proven by simply checking the correlation with the validation curve? In particular, it is more difficult to understand because of the normalization. I think you need more discussion.\n2. In the proposed method, each score is obtained from successive NNs in the training process. Intuitively, I think this captures the fact that NNs change less with the trainning process. Since the training curve and the variation of NN are strongly correlated, isn't the proposed method looking at the convergence of the training curve? If the training set and validation set are from the same domain and there is no model overfit, the training curve and validation curve will be correlated. You may want to make sure that your experiment is not just correlating with the training curve.\n3. For practical use, the proposed score has a different range of values depending on the problem. If the value of the score changes every time, it may be difficult for users to judge the generality of the score by looking at the value.\n\nAnother comment:\nFor example, it would be more interesting to discuss how to measure the stability of NNs based on the variation of each batch in each epoch.\n\nMinor comments:\n- The figure appears to have a solid line and a dotted line, but this is very unclear and we suggest you make it clearer.\n- I think you should write what W,w is in definition of normalized weight(3) to make it easier to read.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "I do not believe that the content of this paper will have any negative social impact. The ability to evaluate generarization without validation set is useful for practitioners, considering the situation where it is practically difficult to collect data. Due to the above concerns, it is still unclear whether it can be used in practice.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "6 hours"}, "extra_text": {}}, {"review_id": "GNvIBuUhtWc", "reviewer": "Reviewer_MNbS", "summary": "This paper proposes a topological data analysis approach to investigate the inner workings of a deep network. In particular, the generalization performance of a deep network is measured without a holdout set by analyzing the topological change of the network during training. Multiple experiments conducted over various datasets demonstrate that the method performs reasonably for various learning rates, batch sizes, and dropout probabilities. ", "questions": "", "limitations": "", "rating": 5, "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": "PROS\nThe approach is really sensible. In fact, I strongly believe that the topology of training dynamics can reveal why and how a neural network generalizes. In fact, in the future, by encouraging certain topological properties during training, it may be possible discover networks that generalize better. The paper delivers a clear message and is easy to understand.\n\nCONS\n- I am not entirely convinced by the experimental evaluation given in this paper. On one hand, because of the convergence properties of the (stochastic) gradient descent algorithm,  the magnitude of the 'updates' to the weights of a neural network will decrease as training progresses. On the other hand, any successful training is expected to decrease the validation error (stochastically). Hence, it is not surprising that these quantities correlate - even if not causally related. How would this be an indicator of the generalization performance? Is there any theoretical line of thought which would justify this? From an empirical standpoint, I suggest that the authors compare different architectures. Can the PD distance identify which of these networks would generalize better? \n\n- The proposed method seems to be agnostic to the construction of the graph. One could also construct the graph as in Corneanu [10], or potentially in many other different ways. Why do we pick the proposed method of graph construction? The impracticality claim for [10] also does not seem to be justified in this paper. Unfortunately, [2], which shares the exact similar graph construction, does not explain this either. \n\n- This paper seems to contain extreme resemblance to [2]. If [2] (attached in supp.) is also a submission, I would doubt the contributions of this paper. I invite the area chairs to look into this issue.\n\n- Maybe we can have the definition of Heat vectorization in the paper?\n\n- In general it would be good to improve the captions of the figures to be more descriptive. \n\n- Why are silhouette and heat distances the same in Table 1?", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "I do not see issues related to the societal impact of this work. However, this paper is not applicable to CNNs, which are crucial for modern deep learning. This could be added to the limitations section of the paper. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper proposes a novel method for evaluating the generalization of NN using persistent homology.   While the paper introduces an interesting and novel idea, there are some weak points.  The empirical validation of the proposed idea is not strong enough. As the method is not supported by theory, more careful empirical justification is needed.  Also, there are previous works that have some overlap with this submission in the direction, such as Rieck et al. [26].  More comparison is necessary to highlight the advantage of the proposed method. 
 Based on these observations, we need to judge the current paper is unfortunately not above the acceptance threshold.   However, we encourage the authors to improve the work and submit it to another conference or journal.  
Decision: Reject