Compositional PAC-Bayes: Generalization of GNNs with persistence and beyond

Authors: kirill  brilliantov
amauri h souza
vikas  garg
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-10 17:58:27

Abstract: Heterogeneity, e.g., due to different types of layers or multiple sub-models, poses key challenges in analyzing the generalization behavior of several modern architectures. For instance, descriptors based on Persistent Homology (PH) are being increasingly integrated into Graph Neural Networks (GNNs) to augment them with rich topological features; however, the generalization of such PH schemes remains unexplored. We introduce a novel _compositional_ PAC-Bayes framework that provides a general recipe to analyze a broad spectrum of models including those with heterogeneous layers. Specifically, we provide the first data-dependent generalization bounds for a widely adopted PH vectorization scheme (that subsumes persistence landscapes, images, and silhouettes) as well as PH-augmented GNNs. Using our framework, we also obtain bounds for GNNs and neural nets with ease. Our bounds also inform the design of novel regularizers. Empirical evaluations on several standard real-world datasets demonstrate that our theoretical bounds highly correlate with empirical generalization performance, leading to improved classifier design via our regularizers. Overall, this work bridges a crucial gap in the theoretical understanding of PH methods and general heterogeneous models, paving the way for the design of better models for (graph) representation learning.

Main paper: 
 introduction: 
 Topological data analysis (TDA) harnesses tools from algebraic topology to unveil the underlying shape and structure of data. TDA has recently gained significant traction within machine learning mainly due to its flagship method: persistent homology (PH) [11] , which allows for capturing topological invariants (like connected components and loops) of the input domain at multiple scales. In particular, PH has recently been leveraged as a tool to augment the representational capabilities of graph neural networks (GNNs) [20, 53, 58] , with expressivity gains formally established [22, 44] . Intuitively, PH furnishes global structural signatures that complement the local nature of GNNs [5, 18, 20, 57] .Understanding the generalization behavior of these models is crucial as it plays a pivotal role in ensuring their reliability and applicability [41] . In this context, there are two fundamental approaches to achieving generalization bounds: data-independent and data-dependent [47] , each offering unique insights into the generalization problem. Both these approaches have been investigated to analyze the generalization ability of GNNs [12, 14, 25, 33, 36, 46, 48, 52, 59] . Data-dependent generalization bounds evoke particular interest since they are typically much tighter than the agnostic bounds afforded by, e.g., VC dimension. However, no such bounds have been unearthed for PH methods (i.e., learnable vectorization schemes) and, consequently, for GNNs enhanced with PH-based descriptors.We approach this gap with the first data-dependent generalization bound for classifiers based on a versatile and widely used vectorization framework for persistence diagrams, namely, PersLay [5] . PersLay leverages extended persistence to effectively represent detailed topological features, and subsumes commonly used methods such as persistence landscapes [4] , images [1] , and silhouettes [6] .Central to our analysis is a novel PAC-Bayes framework (Lemma 2) that provides a general recipe to analyze the generalization of a broad spectrum of models, including those with heterogeneous layers and those comprising multiple sub-models. To achieve this, we introduce general conditions (Equations 6-9) that are satisfied by commonly used learning architectures and, surprisingly, their compositions (Section 4). Leveraging Lemma 2, we show how to obtain bounds for heterogeneous MLPs and GNNs in a straightforward manner (Table 2 ). Notably, we also establish the first generalization bounds for GNNs augmented with persistence layers (PersLay). Our exposition focuses on graphs; however, i) our Lemma 2 can be used in any domain and ii) our bound for PersLay considers persistence diagrams obtained from any non-learnable filtration function, and therefore extends more generally to input domains beyond graphs. From a technical perspective, our approach hinges on contrasting previous analyses within the PAC-Bayes framework [9, 37, 38] to extract the common structure encoded in the general conditions of Lemma 2. This allows us to overcome challenges arising from the heterogeneity of the models we consider.Our experiments on several standard real-world datasets confirm strong correlation between the empirical performance and our theoretical bounds. We reinforce the merits of our analysis via regularized PH-based models informed by our bounds with demonstrable empirical benefits.Our main contributions are: (Theoretical, see Table 1 ) We develop a general recipe for obtaining PAC-Bayes bounds for a broad class of (possibly heterogeneous) models and their compositions. We also provide the first data-dependent bounds for PH-based classifiers and combinations of GNNs and PersLay;(Empirical) We show that the dependence on parameters depicted in our bounds strongly correlates with the observed performance. We also show that novel regularization schemes based on our bounds can reduce the generalization gap of PH-augmented GNNs on multiple datasets. 
preliminaries: 
 This section overviews GNNs, persistent homology on graphs, and their combination. We also provide basic notions and results in PAC-Bayes learning, which serve as a background for this work.Notation. We consider attributed graphs denoted as a tuple G = (V, E, z), where V = {1, 2, ..., n} is the vertex set, E ⊆ V × V is the edge set, and z : V → R dz assigns to each vertex v ∈ V an attribute (or color) z(v). For convenience, hereafter, we denote the feature vector of v by z v . We consider classification tasks with input and label spaces X and Y = {1, . . . , K} (K is the number of classes) and the γ-margin loss l γ : Y × R K → {0, 1} where l γ (y, ŷ) = 1(ŷ y ≤ γ + max j̸ =y ŷj ) and γ ≥ 0 is the margin parameter. Let S = {(x i , y i )} m i=1 ⊆ X × Y denote a collection of m input/label pairs sampled i.i.d. from some unknown distribution D. Then, the empirical error of a hypothesis g w : X → R K with parameters w is defined as LS,γ (g w ) = 1 /m m i=1 l γ (y i , g w (x i )). Accordingly, we can define the generalization error as L D,γ (g w ) = E (x,y)∼D [l γ (y, g w (x))]. We use ∥ • ∥ 2 to refer the ℓ 2 norm (vectors) and the spectral norm (matrices), and ∥ • ∥ F to refer to the Frobenius norm. Also, we denote the set {1, ..., n} by [n] . We provide a notation table in the Appendix (Table 5 ).Graph neural networks (GNNs). Message-passing GNNs [15, 55] employ a sequence of messagepassing steps, where each node v aggregates messages from its neighbors N (v) = {u : (v, u) ∈ E} and use the resulting vector to update its own embedding. Starting from z (0) v = z v , GNNs recursively applyEQUATIONwhere { {•} } denotes a multiset, Agg ℓ is an order-invariant function and Upd ℓ is an arbitrary update function -often a multilayer perceptron (MLP).Persistence homology (PH) on graphs aims to extract detailed (multiscale) topological features from graphs. A filtration of a graph G is a finite nested sequence of subgraphs of G, i.e., ∅ = G 0 ⊂ G 1 ⊂ ... ⊂ G -alternatively, clique complexes can also be built at each step (see [2] ). While filtrations can be obtained in different ways [2, 18] , a typical choice consists of leveraging a real-valued filtering function f on the vertices of G (or their features) to compute the vertex level set V α = {v : f (v) ≤ α} at scale α ∈ R. Let G α be the subgraph of G induced by V α . By increasing α from -∞ to ∞, we obtain a nested sequence of subgraphs called the sub-level filtration of G induced by f . The idea of PH is to keep track of the appearance and disappearance of topological features (e.g., connected components, loops) in a filtration. If a topological feature first appears in G α b and disappears in G α d , then we encode its persistence as a pair (α b , α d ); if a feature does not disappear, then its persistence is (α b , ∞). The collection of all pairs forms a multiset that we call persistence diagram. We use Dg i (G) to denote the persistence diagram for i-dim topological features of graph G.For details on PH, we refer to Edelsbrunner and Harer [10] , Hensel et al. [17] , and Hofer et al. [19] .Persistence layers (PersLay). Carrière et al. [5] introduced a general way to vectorize persistence diagrams. Given a persistence diagram Dg(G) for an arbitrary dimension, PERSLAY computesEQUATION)where Agg is any permutation invariant operation (e.g., minimum, maximum, sum, or kth largest value), ω : R 2 → R is a weight function for the elements in Dg(G), and φ : R 2 → R h is the so-called point transformation. More specifically, given a persistence pair p = [p 1 , p 2 ] ⊤ ∈ R 2 , PersLay introduces the triangle point transformation (Λ):EQUATIONthe Gaussian point transformation (Γ):EQUATIONand the line point transformation (Ψ):EQUATIONIn all of these transformations t 1 , . . . , t h are learnable parameters. Notably, the architectural design of PERSLAY is quite versatile and accommodates a wide range of traditional persistence diagram vectorizations, extending DeepSets [56] and including persistence landscapes [4] , persistence silhouette [6] , persistence images [1] , and other Gaussian-based kernel approaches [23, 29, 31] .To obtain class predictions, the output of PersLay is typically fed to an MLP with Lipschitz activations. We refer to this joint model (PersLay followed by MLP) as PersLay Classifier (PC). GNNs with persistence. Recently, PH has been used to boost the expressive power of GNNs. For instance, Horn et al. [20] and Immonen et al. [22] leverage node embeddings at each layer of a GNN to obtain persistence descriptors. This topological information can be added to GNN's node embeddings (as in [20] ) or concatenated with GNN's graph-level representation (as in [22] ) -which we refer to the parallel mode of integrating PH into GNNs. Figure 1 illustrates the latter (parallel mode) with persistence diagrams vectorized using PersLay.PAC-Bayesian analysis adopts a Bayesian approach to the PAC learning framework [30, 37, 38, 50] .The idea consists of placing a prior distribution P over our hypothesis class and then use the training data to obtain a posterior Q, i.e., the learning process induces a posterior distribution over the hypothesis class. Importantly, we can leverage the Kullback-Leibler (KL) divergence between Q and P to bound the difference between the generalization and empirical errors [37] . To compute PAC-Bayes bounds for models like neural networks, we can i) choose a prior, ii) apply a learning algorithm; and iii) add random perturbations (from some known distribution) to the learned parameters such that we ensure tractability of the KL divergence. Following this recipe, Neyshabur et al. [40] introduced the important result in Lemma 1. Notably, Lemma 1 tells us that if we have prior and posterior distributions and guarantees that the change of the model's output due to perturbations over the learned parameters is small with high probability, we can obtain a generalization bound.Lemma 1 (Neyshabur et al. [40] ). Let g w (x) : X → R K be any model with parameters w, and let P be any distribution on the parameters that is independent of the training data. For any w, we construct a posterior Q(w + u) by adding any random perturbation u to w, s.t.,P u (max x∈X |g w+u (x) -g w (x)| ∞ < γ 4 ) > 1 2 .Then, for any γ, δ > 0, with probability at least 1δ over an i.i.d. size-m training set S drawn according to D, for any w, we have:L D,0 (g w ) ≤ LS,γ (g w ) + 4 D KL (Q(w + u)||P) + log 6m δ m -1 . 
generalized pac-bayes: 
 This section first presents a general procedure for obtaining generalization bounds for heterogeneous models, i.e., going beyond spectrally-normalized layers and architecture-specific models (as in [33, 40] ). Then, we show how to leverage such a procedure to extend existing bounds in the literature and to obtain the first generalization bound for PersLay.Our next result (Lemma 2) applied perturbation-based PAC-Bayes bounds to arbitrary models with (possibly) non-homogeneous layers. To achieve this generality, we carefully contrasted results in [33, 40, 48] to identify the conditions (Equations 6, 7, 8, and 9) that are sufficient to subsume the considered models as well as to extend to a broader class of models. In Section 4, we will also exploit Lemma 2 in the analysis of different combinations of neural models (e.g., GNNs and PersLay).General recipe for PAC-Bayesian bounds for heterogeneous models Lemma 2. Let f w : X → R K be a model with parameters w = vec{W 1 , ..., W n }. If there exists T ∈ R + and a sequence (S i ) i∈ [n] with S i ∈ R + both of which may depend on w, and parameter-independentEQUATION• the output change can be bounded under a small perturbation of the parameters, i.e., for all u = vec{U 1 , ..., U n } with ∥U i ∥ 2 ≤ η i S i :EQUATION• the following auxiliary conditions hold:EQUATIONη := minEQUATION)Then, for any γ, δ > 0 with probability at least 1δ over the choice of training sets S with m i.i.d. samples drawn according to some distribution D, we have:EQUATION)where h is the maximum dimension across the matrices(W i ) i∈[n] .Proof sketch. We build on the main result in [40] by extending it to a broader context. The general idea involves employing Lemma 1. Following [40] , we define the prior distribution P as an isotropic Gaussian with variance σ 2 and the posterior distribution Q as a shifted isotropic Gaussian with the same variance. To achieve a tighter bound, it is essential to maximize σ (since the KL-divergence scales as O( 1 /σ 2 )); consequently, σ should be determined based on the parameter β. However, since P must remain independent of the learned weights, we set σ according to an approximation of the learned weights. Specifically, we define β = T i 1 /Si and at first consider only w such that β fall within the range |β -β| ≤ 1 /2β for some arbitrary β, an approximation. We then select σ based on this approximation, β. At this point we can apply Lemma 1 for all w such that β falls into the defined earlier interval. To account for other values of β, we establish a finite grid across the relevant β values and choose an appropriate β for each interval on the grid. Finally, a union-bound argument across all β values provides the final result. Although Equation 7and Lemma 1 have their own constraints on the random perturbation, the above steps outline a method to set the variance σ that satisfies these constraints and maintains independence from the learned weights. = Lip ϕ Lip g Lip ρ ∥W 2 ∥, λ = ∥W 1 ∥ 2 ∥W 3 ∥ 2 andξ = ((dC) n-1 -1) /(dC-1). Comparing to [33] we do not add Lip ϕ to ξ and instead of W l we have W 3 .Model (reference) T S i η i C 1 C 2MLP (Neyshabur et al. [40] )n i=1 ∥W i ∥ 2 ∥W i ∥ 2 1 6nB Lip eB Lip GCN (Liao et al. [33] )n i=1 ∥W i ∥ 2 ∥W i ∥ 2 1 6n d n-1 2 B Lip ed n-1 2 B LipGCN (Sun and Lin [48] )n i=1 ∥W i ∥ 2 ∥W i ∥ 2 1 6nB Lip eB Lip MPGNN, dC ̸ = 1 (Liao et al. [33] )λξ ∥W i ∥ 2 * 1 6nB Lip ϕ eBn Lip ϕ MPGNN, dC = 1 (Liao et al. [33] )λ ∥W i ∥ 2 * 1 6n Bn Lip ϕ eBn 2 Lip ϕ *S 2 = min{dC, ∥W 2 ∥ 2 }Discussion. We note that Lemma 2 requires choosing values for the variables (S i ) i∈[n] and T . In this regard, one might set S i as the spectral norm of the weight matrix W i , i.e., S i = ∥W i ∥ 2 , and make T equal to i S i . In this case, the condition in Equation 8 is satisfied -the geometric mean is always smaller than or equal to the arithmetic mean. Regarding the variables (η i ) i , a typical choice is to setη i = O( 1 /n).By doing so, our bound implicitly depends on n also through η i .The role of Equation 6 is to constraint w to non-trivial parameter spaces. In particular, if T is too small, the magnitude of the model output might not be sufficient to distinguish different inputs up to a margin γ. In this case, the model would have large empirical loss. In turn, Equation 7 is directly associated with the condition in Lemma 1, enabling us to use it.As discussed, Neyshabur et al. [40] assume spectrally-normalized weight matrices. To avoid this assumption, we introduce the conditions in Equation 8 and Equation 9, which allow us to pick perturbations that meet the condition in Equation 7, again justifying the application of Lemma 1.The bound also includes a somewhat unconventional term, max{1, ∥w∥ 2 2 }. While this technical term allows for a more concise proof, we note that it does not impose suboptimality. More specifically, in most real-world cases, the squared norm of w is greater than 1. See Appendix I for a discussion.Applying Lemma 2 to MLPs and GNNs. As previously mentioned, using Lemma 2 involves defining the variables T, (S i ) i , (η i ) i , C 1 , C 2 to meet all conditions in Lemma 2's statement. Typically, this definition comes naturally from the perturbation analysis of the model. To illustrate the power of Lemma 2, Table 2 shows how we can apply it using the perturbation analysis for MLPs and GNNs provided in [33, 40, 48] . Detailed proofs are given in Appendix E. Importantly, we note that we do not make any additional assumption beyond those in the original papers -we state all assumptions before each proof in the Appendix for clarity. Notably, our approach generalizes results by Neyshabur et al. [40] and Liao et al. [33] to MLPs/GCNs with different activation functions -note that the original works only consider ReLU activations. For inherently non-homogeneous models like MPGNNs, our method leads to tighter bounds in several settings. We provide a comparison between our bounds and previous ones in Appendices F and G.Applying Lemma 2 to PersLay. Next, Corollary 1 provides a PAC-Bayes bound for PersLay under fixed filtration functions -see Appendix H for a discussion about filtration functions. To the best of our knowledge, this is the first generalization result for vectorization schemes of persistence diagrams. Again, the proof consists of verifying if the requirements of Lemma 2 are met. For readability, we omit the proof and provide only the constant weighting function case in the main text, proof as well as the arbitrary weighting function case can be found in Appendix E. The perturbation analysis of PersLay is also given in the supplementary material (Lemma 10).Corollary 1 (PersLay with constant weighting function). Let f w : G → R k with w = vec{W (φ) } be a PersLay where W (φ) denotes the parameters of the point-transformation function, φ. Let Dg be a mapping from graphs to (extended) persistence diagrams with a fixed filtration function and B such that max G∈G max p∈Dg(G) ∥p∥ 2 ≤ B, then f w satisfies the requirements of Lemma 2 with: T = T (φ) and S = ∥W (φ) ∥ 2 and η = 1,C 1 = 2C 2 and C 2 = 2A 2 max{Lip (φ) , C (φ) }, where A 1 = A 2 = max G∈G card(Dg(G))if Agg = "sum"A 1 = 1, A 2 = 3 if Agg = "k-max" or "mean"andT φ C φ Lip φ if φ = Λ max{1, ∥W (φ) ∥ 2 } B √ h 1 if φ = Γ max{1, ∥W (φ) ∥ 2 } √ h 1 τ √ e if φ = Ψ ∥W (φ) ∥ 2 √ 3 max{1, B} max{1, B}PersLay's special cases. Carrière et al. [5] designed PersLay with flexibility in mind to subsume commonly used persistence vectorization schemes in the literature. Consequently, we can obtain bounds for these schemes -we provide the values of C 2 (divided by 2) which is enough to compare the schemes since everything else is the same in the constant weighting function case:Persistence k-landscapes [4] Persistence Images [1] Persistence Silhouettes [6]3 max{B √ h, 1} card • max{ √ h, 1 τ √ e } card • max{B √ h, 1}where card is the max G∈G Dg(G). If "images" and "silhouettes" use "sum" as an aggregating function, then our generalization analysis suggests that "k-landscapes" would have stronger guarantees. If these schemes use "mean" as an aggregating function, then C 2 for "k-landscapes" would be at most C 2 for "silhouettes", and the result of comparison of "k-landscapes" and "images" could be in favor of both "landscapes" and "images" depending on chosen parameters τ and B. 
compositional pac-bayes: 
 In this section, we present two lemmas (Lemmas 3 & 4) that allow us to compose models satisfying Lemma 2 requirements. At the end of the section (Corollary 2), we showcase our framework by getting generalization bounds for combinations of GNNs and PersLay. For readability, here we provide informal statements and defer the formal ones to the Appendix (Lemma 5, Lemma 6).In particular, Lemma 3 establishes that the composition of MLPs with models that satisfy Lemma 2 also satisfy it. As a result, we can derive PAC-Bayes bounds for heterogeneous models that leverage MLPs using our framework in a straightforward way. This is particularly relevant since deep learning models often employ learnable feature extractors followed by MLPs as classification heads.Lemma 3 (Informal; Composition with MLP). Let f be an MLP and g be a model satisfying Lemma 2 requirements, then f • g also satisfies Lemma 2 requirements.In addition, we show in Lemma 5 (Appendix) that this result also extends to an arbitrary number of models beyond MLPs. In particular, the result holds whenever we can upper bound output deviations due to perturbations on parameters and inputs, i.e., sup x∈X ∥f w+u (x + ∆x)f w (x)∥ 2 is bounded.Our next lemma suggests that models satisfying Lemma 2 requirements are closed under parallel concatenation. We note that combining two (or more) models in parallel is also a common design choice in deep learning. For instance, this encompasses persistence-augmented GNNs [22] and ensemble methods [13] . Lemma 4 (Informal; Models in parallel). Let f 1 , f 2 be two models satisfying Lemma 2 requirements and g be an aggregating Lipschtiz function. Then, g(f1 (•), f 2 (•)) also satisfies Lemma 2 requirements.We also provide a generalization of this lemma in the Appendix (Lemma 6) for n > 2 models in parallel. It leads to tighter bounds than a naive 2-by-2 sequential application of Lemma 4.Corollary 2 (Informal). By combining the results for MLPs, GCNs, MPGNNs (Table 2 ) and that for PersLay (Corollary 1) with Lemma 3 and Lemma 4, we can get generalization bounds on various compositions of these models. In particular, for GNNs with persistence (see Figure 1 ), we have• T for the overall model scales with the product of PersLay's and GNN's T variables;• C 1 and C 2 of the overall model scale linearly with C 1 , C 2 of PersLay and GNN.Despite the generality of our results, Corollary 2 demonstrates the benefits of our framework in the domain of graph representational learning. To the best of our knowledge, this the first result providing generalization guarantees for graph neural networks combined with persistence vectorization schemes. Furthermore, our findings can aid practitioners in making informed architectural decisions to enhance the generalizability of their models. Specifically, in the case of combining PersLay with GNNs, a tighter bound can be achieved by selecting a PersLay dimension that is considerably smaller than the GNN dimension. Failing to do so may result in a bound dependency on the width of the form O(h √ ln h) rather than O( √ h ln h). Additionally, we recommend using aggregation functions such as "mean" or "k-max" instead of "sum" as the latter introduces a term max G∈G card(G) to the bound, which may be large in practical scenarios. 
related works: 
 Expressivity and generalization of GNNs. GNNs have achieved state-of-the-art performance across various applications [16, 28, 45, 51, 54] , and have garnered significant attention. Maron et al. [35] , Xu et al. [55] analyzed the representational power of GNNs in terms of the 1-WL test, revealing theoretical limits on their expressivity. This has motivated a surge of works aiming to go beyond 1-WL with GNNs [e.g., 32]. Regarding generalization, Scarselli et al. [46] provided upper bounds on the order of growth of VC-dimension for GNNs. Garg et al. [14] presented the first data-dependent generalization bounds for GNNs via Rademacher complexity. Recently, Morris et al. [39] employed the WL test alongside VC-dimension to gain insights about the generalization performance of GNNs. For details about the expressivity and learning of GNNs, we refer to Jegelka [24] .Learning theory and PH. Birdal et al. [3] , Dupuis et al. [8] and Chen et al. [7] investigate connections between learning theory and topological data analysis. In particular, Birdal et al. [3] , Dupuis et al. [8] explored the concept of PH dimension as a complexity measure to analyze generalization. Chen et al. [7] proposed a topological regularizer to simplify decision boundaries by penalizing non-essential topological features. In contrast, we apply learning theory to derive data-dependent generalization bounds for arbitrary heterogeneous layers, specifically targetting persistence-aware GNNs, and introduce a regularizer informed by these bounds to guide the design of robust and generalizable models. PAC-Bayes. The PAC-Bayes framework [37, 38] allows us to leverage knowledge about learning algorithms and distributions over the hypothesis set to achieve tighter generalization bounds. Remarkably, Neyshabur et al. [40] presented a generalization bound for feedforward networks in terms of the product of the spectral norm of weights using a PAC-Bayes analysis. Liao et al. [33] provided PAC-Bayes bounds for GNNs, and Sun and Lin [48] enhanced their analysis considering the adversarial case as well. In a seminal work, Dziugaite and Roy [9] optimized PAC-Bayes bounds directly to obtain non-vacuous generalization bounds for deep stochastic neural networks. 
experiments: 
 To illustrate the practical relevance of our analysis, we now consider the generalization of persistenceaware models on real-world datasets, and report results for regularized models based on our bounds. In particular, we conduct two main experiments. The first one aims to analyze how well our bounds capture generalization gaps as a function of model variables. The second assesses to which extent a structural risk minimization algorithm that uses our bound on the weights spectral norm improve generalization compared to empirical risk minimizers. We implemented all experiments using PyTorch [42] , and implementation details are given in the Appendix J. Our code is available at https://github.com/Aalto-QuML/Compositional-PAC-Bayes.Datasets and evaluation setup. We use six popular benchmarks for graph classification: DHFR, MUTAG, PROTEINS, NCI1, IMDB-BINARY, MOLHIV, which are available as part of TUDatasets [26] and OGB [21] . We use a 80/10/10% (train/val/test) split for all datasets when we perform model selection. Here, we consider both PersLay Classifiers and GNNs with persistence models with constant weight functions and Gaussian point transformations. For the experiments with GNNs, we kept only the larger datasets (and added results for the NCI109 dataset). Regarding filtration functions, we closely follow [5] and use Heat kernels with parameter values equal to 0.1 and 10.Dependence on model components. Figure 2 and Figure 4 show the generalization gap (measured as L D,0 -LS,γ=1 ) and the bound on the weights spectral norm (T from Lemma 2) over the training epochs for PersLay Classifier and GNNs with persistence, respectively. To evaluate how well our bound captures the trend observed in the empirical gap, we compute correlation coefficients between the two sequences across different seeds and report their mean and standard deviation for each dataset. Overall, the coefficients are greater than 0.7 in 7 out of 9 experiments, indicating a good correlation.Figure 3 shows the empirical gap and our estimated bound as a function of the model's width for the PersLay classifier. Again, we compute correlation coefficients between the two curves and find they are highly correlated (with an average correlation above 0.91 on 4 out of 5 datasets). Also, we note that these curves are obtained at the final training epoch. We report additional results across different epochs and hyper-parameters in the supplementary material. Again, these results validate that our theoretical bounds can capture the trend observed in the empirical generalization gap. Regularizing PersLay. We compare variants of PersLay trained via ERM (empirical risk minimization) and a regularized version with loss given by LS,1 +α r n 2 ln n∥w∥ 2 2 β 2 ; α r is a hyper-parameter that balances the influence of the two terms and β = T i 1 /Si -see proof sketch of Lemma 2 in Section 3. This is similar to a weight-decay regularization approach, with the spectral norm of weights appearing in β. Here, we consider models with n = 1 or 2, selected via hold-out validation. We note that PersLay classifier does not use node features, it only exploits graph structures.Table 3 : Comparison of PersLay with and without spectral norm regularization. We report accuracy statistics (except for MOLHIV, which uses AUROC) computed over five independent runs. In 5 out of 6 cases, the models using SpecNorm achieve better test results.Method DRFH MUTAG PROTEINS IMDB-B NCI1 MOLHIV PersLay (ERM)0.71 ± 0.04 0.88 ± 0.02 0.65 ± 0.03 0.65 ± 0.01 0.68 ± 0.01 0.7005 ± 0.0177 PersLay (w/ SpecNorm) 0.72 ± 0.02 0.94 ± 0.01 0.72 ± 0.01 0.70 ± 0.01 0.68 ± 0.01 0.7185 ± 0.0106 Table 3 reports accuracy results (mean and standard deviations) computed over five runs. Overall, the regularized approach significantly outperforms the ERM variant despite the use of small-sized networks. On 5/6 datasets, PersLay with spectral norm regularization is the best-performing model.Regularizing GNNs with persistence. We now evaluate the impact of using our bound to regularize different GNNs combined with persistent homology (PersLay) in parallel mode. We consider GCN [28] , GraphSage [16] , and GIN [55] architectures. Table 4 reports the test classification error and the generalization gap on the NCI, NCI109, and PROTEINS datasets -mean and standard deviation obtained over five independent runs. For our regularizer, we select the optimal penalization factor α r ∈ {1e-5, 1e-6, 1e-7, 1e-8} using the validation set. Overall, the results show that the regularized methods achieve smaller generalization gaps and slightly lower classification errors. In particular, our spectral regularizer leads to a significant drop in generalization gap in all experiments. Table 4 : Test classification error (0-1 loss) and generalization gap (L D,0 -LS,γ ) for PH-augmented GNNs. ERM means empirical risk minimizer (no regularization). We denote the best-performing methods in bold. In almost all cases, employing the method derived from our theoretical analysis leads to the smallest test errors and generalization gaps. We derive the first generalization bounds for neural networks that appeal to persistent homology for graph learning. Notably, the analyzed framework (PersLay) offers a flexible and general way to extract vector representations from persistence diagrams. Due to this generality, our analysis covers several methods available in the literature. The developed framework also allows to analyze composite models like, GNNs combined with PersLay. Our constructions involve a perturbation and generalization behavior analysis of non-homogeneous networks in rather general setting, which poses specific technical challenges.While we provide valuable insights and methodologies, we would like to underscore the need for future investigations to delve into PH schemes that encompass parametrized filtration functions. Nonetheless, while some works showed gains using learnable filtrations [20] , others have reported no benefits and advocated fixed functions instead [5, 34] . Moreover, the tightness of our bounds can further be improved since there is still considerable gap between empirical results and the theoretical one. By shedding new light on the generalization of machine learning models based on persistent homology, we hope to contribute to the community by providing key insights about the limits and power of these methods, paving the path to further theoretical developments on PH-based neural networks for graph representation learning.Table 5 summarizes the main mathematical symbols and abbreviations used in this work. EQUATIONwhere H k ∈ R h k are the vectors computed after layer k, and GCN.W k ∈ R h k-1 ×h k , k ∈ [n]EQUATIONwhereH k ∈ R |V |×h k , k ∈ [n -1] are the node representations in each layer, H n ∈ R 1×K is the readout, and W k ∈ R h k-1 ×h k , k ∈ [n]are the parameters in the k-th layer. And we let H 0 = Z, where Z is the matrix constructed from node features, z v for v ∈ V . The matrixP G ∈ R |V |×|V | is related to the graph structure and ψ k (•), k ∈ [n -1]are some element-wise non-linear Lip k -Lipschitz mappings. Practically, for GCN, we take P G as the Laplacian of the graph, defined as D-1/2 Ã D-1/2 , where Ã is the adjacency matrix with +1 on the diagonal and D = diag|V | j=1 Ãij , i ∈ [|V |]is the degree matrix of Ã. Note that we sightly changed (comparing to Liao et al. [33] , Sun and Lin [48] ) the notation to be consistent: instead of l, number of layers, we have n, instead of n, number of nodes, we have |V | and instead of σ i we have ψ i as activation functions.EQUATIONwhere k ∈ [n -1], H k ∈ R |V |×h k are node representations/states and H n ∈ R 1×K is the output representation. Here we initialize H 0 = 0. WLOG, we assume ∀k ∈[n -1], H k ∈ R |V |×h and M k ∈ R |V |×h since h is the maximum hidden dimension. C in ∈ R |V |×|E| and C out ∈ R |V |×|E|are the incidence matrices corresponding to incoming and outgoing nodes1 respectively. Specifically, rows and columns of C in and C out correspond to nodes and edges respectively. C in [i, j] = 1 indicates that the incoming node of the j-th edge is the i-th node. Similarly, C out [i, j] = 1 indicates that the outgoing node of the j-th edge is the i-th node. g, ϕ, ρ are nonlinear mappings, e.g., ReLU and Tanh. Technically speaking, g : R h → R h , ϕ : R h → R h , and ρ : R h → R h operate on vector-states of individual node/edge. However, since we share these functions across nodes/edges, we can naturally generalize them to matrix-states, e.g., φ : R |V |×h → R |V |×h where φ(X)[i, :] = ϕ(X[i, :]). By doing so, the same function could be applied to matrices with varying size of the first dimension.For simplicity, we use g, ϕ, ρ to denote such generalization to matrices. We denote the Lipschitz constants of g, ϕ, ρ under the vector 2-norm as C g , C ϕ , C ρ respectively. We also assume g(0) = 0, ϕ(0) = 0, and ρ(0) = 0 and define the percolation complexity as C = C g C ϕ C ρ ∥W 2 ∥ 2 following Garg et al. [14] . Note that we sightly changed (comparing to Liao et al. [33] , Sun and Lin [48] ) the notation to be consistent: instead of l, number of layers, we have n, instead of n, number of nodes, we have |V | and instead of W l , parameter matrix, we have W 3 .Lemma 2. Let f w : X → R K be a model with parameters w = vec{W 1 , ..., W n }. If there exists T ∈ R andS i ∈ R for i ∈ [n] depending on W 1 , ..., W n ; C 1 , C 2 > 0; η i ∈ (0, 1] for i ∈ [n] such that:maximum norm of the output is bounded by T and C 1 :EQUATIONmaximum change of the output with perturbed weights, u = vec{U 1 , ..., U n }, is bounded if the perturbation is small, ∥U i ∥ 2 ≤ η i S i :EQUATIONarithmetic mean of S i inverses is at least the nth root of T inverse:EQUATION)minimum η i is at most the ratio of C 1 and 2C 2 :EQUATIONthen for any γ, δ > 0 with probability at least 1δ over the choice of training set S of size m sampled accordingly to some data distribution D we have the bound on the generalization gap:EQUATION)where h is the upper bound for dimensions of W i .Proof. First, we note that if w such that C 1 T ≤ γ 2 , then this imply for any i ∈[K] max x∈X |f w (x) i | ≤ max x∈X ∥f w (x)∥ 2 ≤ C 1 T ≤ γ 2 ,where we used Equation 6 and the fact that every coordinate of the vector is at most ℓ 2 -norm of the vector. In this case the empirical loss would be equal to one, since no index i can satisfyf w (x) i ≤ γ + max i̸ =j f w (x)and the inequality in Equation 10 becomes trivial since generalization error can not be greater than 1.So from now on we consider w such that C 1 T ≥ γ 2 . Moreover we note that:EQUATIONWe denoteEQUATION)Following Neyshabur et al. [40] and Liao et al. [33] , we consider the prior P = N (0, σ 2 I) and random perturbation u ∼ N (0, σ 2 I). Note that the σ of the prior and the perturbation are the same and will be set according to β. More precisely, we will set the σ based on some approximation β of β since the prior cannot depend on any learned weights directly. The approximation β is chosen to be a cover set which covers the meaningful range of β. For now, let us assume that we have a fixed β and consider β which satisfies |β -β| ≤ εβ for some ε > 0. Note that this also implies:EQUATIONThis setup is very like the setup in Neyshabur et al. [40] and Liao et al. [33] Choosing σ. From Tropp [49] we know that if U ∈ R h×h ∼ N (0, σ 2 I), thenP (∥U ∥ 2 ≥ t) ≤ 2h exp - t 2 2hσ 2So applying it to U 1 , ..., U n (all of them are smaller than h × h) and t = σ √ 2h ln 4nh we have:EQUATIONFor simplicity of notation we denote C t = √ 2h ln 4nh . Using defined perturbation u = vec{U 1 , ..., U n } and Equation 23 combined with Equation 7 we get with probability at least 1 2 :EQUATIONIfσ ≤ γ(1 -ε) 4C 2 C t β ,then the perturbation is bounded by γ 4 . However, we need to satisfy the condition of Equation 7, i.e. ∀i ∈ [n] : ∥U i ∥ ≤ η i S i . Let us find C σ ≥ 1 such that:EQUATIONand that ∀i ∈ [n] :EQUATION(i) comes from Equation 21. (ii) we just multiplied and divided on η i S i . (iii) we plugged in the expression for β, Equation 20. (iv) 1 + 19We choosek̸ =i Si S k ≥ 1. (v) comes from EquationEQUATIONIn this case C σ ≥ 1 because of Equation 9and continuing Equation 26:σC t ≤ η i S i C 1 2C 2 C σ η i = η i S i η η i (i) ≤ η i S i ,where (i) comes from the fact that η is the minimum among η i s, gives us that ∀i ∈ [n] :∥U i ∥ 2 ≤ η i S i .Plugging this into expression for σ, Equation 25, we get:EQUATIONGetting the bound for weights |β -β| ≤ 1/2β. Now, we can compute the KL-divergence and get the bound using Lemma 1 for case when β is around β, i.e. w such that |β -β| ≤ εβ.EQUATIONwhere (i) comes from σ definition, Equation 28, (ii) comes from the upper-bound on β, Equation 22.We are ready to put this into Lemma 1:EQUATIONwhere we upper-bounded ∥w∥ 2 2 as max{1, ∥w∥ 2 2 } to simplify further derivations.Union bound. Finally, we need to consider multiple choices of β so that for any β, we can bound the generalization error like Equation 30. In order to do this we (i) find interval of reasonable βs, (ii) define a covering and upper-bound the number of balls in the covering, (iii) combine bounds for every β together and get the final result.If β is too large, the KL-divergence would be too large and generalization gap would be greater than 1 which thereby trivialize the bound since the generalization error cannot be greater than 1.EQUATIONwhere (i) holds since we remove multipliers that are greater than 1 and ln m δ which is ≥ 0 and appears in sum, (ii) comes from the fact that η ≤ 1.So, if β ≥ γ √ mC1 , then the bound becomes trivial. On the other hand, if the β is too small, then the empirical loss would be too large since the model would not be able to classify with margin, γ, as it was shown in Equation 19. Using this equation and Arithmetic versus Geometric mean we get the following inequality:EQUATIONwhere (i) comes from Equation 8and (ii) comes from Equation 19.Combining lower (Equation 32) and upper (Equation 31) bound on β we get:EQUATIONTo satisfy the condition |β -β| ≤ εβ we can take εn γ EQUATIONLet us consider cases:     1 n 1 C1 1/n ≤ 1 C1 , C 1 < 1 1 n 1 C1 1/n ≤ 1, C 1 ≥ 1First inequality holds because function a 1/x x is monotonously decreasing on the interval (0, +∞). So, we can combine:EQUATIONIt is left to apply the union bound argument for the events of Equation 30 happening with β taking value from ith ball. Let us denote such event E i :EQUATIONHence with probability at least 1δ for all w we have:L D,0 (f w ) ≤ LS,γ (f w )+ + O       max{1, ∥w∥ 2 2 }T 2 n i=1 1 Si 2 C 2 t C 2 1 η-2 + log m δ max 1, 1 C1 γ 2 m      Note that we substituted ε with 1 2 and now max{1, 1 C1 } appearing under logarithm comparing to Equation 30.Lemma 5 (Generalization of Lemma 3). Let f (1) w1 : X 1 → R h1 , ..., f (k) w k : X k → R h k for k ∈ N with w i = vec{W (i) 1 , ..., W (i)ni } and h 1 = K be some models that we can compose, i.e. f = f (1) (f (2) (...(f (k) )...)). If there exists for j ∈ [k] : T (j) depending on W(j) 1 , ..., W (j) nj ; for j ∈ [j] : S (j) 1 , ..., S (1) nj ∈ R depending on W (j) 1 , ..., W (j) nj ; η (1) 1 , ..., η (1) n1 , ..., η (k) 1 , ..., η (k) n k ∈ (0, 1] and C (1) 1 , ..., C (k) 1 ∈ R, C (1) 2 , ..., C (k) 2∈ R such that: maximum output for every model is bounded as follows:EQUATIONand for j = k we can have weaker condition -the same as in Lemma 2, maximum perturbation of the output during small perturbation of weights, ∥U(j) i ∥ 2 ≤ η (j) i S (j)i , for every model is bounded as follows:EQUATION)and for j = k we can have weaker condition -the same as in Lemma 2, arithmetic mean of inverses of S (j) i is greater than n j -root of T (j) :EQUATIONEQUATION)and η(j) := mini∈[nj ] η (j)i is upper-bounded by the ration C (j)EQUATIONwe denote n = n k η 1:n = η (1)1 , ..., η (1) n1 , ..., η(k) 1 , ..., η (k) n k C 1 = max x∈X k ∥x∥ 2 max    k j=1 C (j) 1 , C • C (ind) 1 C (ind) 2    C 2 = max x∈X k ∥x∥ 2 C, where ind = arg min j∈[k] η(j) and C = max j∈[k] j i=1 C (i) 2 k i=j+1 C (i) 1Proof. We denote as f k:j composition of models from k to j First we test Equation 6 by plugging in Equation 37for every model:max x∈X k ∥f k:1 (x)∥ 2 ≤ max x∈X k C (1) 1 T (1) ∥f k:2 (x)∥ 2 ≤ ≤ max x∈X k ∥x∥ 2 k j=1 C (i) 1 T (i) ≤ C 1 T (41)Next, we test Equation 7by plugging in Equation 38and Equation 37 for every model:EQUATIONTo test Equation 8we use arithmetic versus geometric mean inequality:EQUATION)It is left to verify Equation 9. As one can noticeC1 2C2 ≥ C (ind) 1 C (ind) 2 ≥ η.Lemma 6 (Generalization of Lemma 4). Let f(1)w1 : X → R h1 , ..., f (k) w k : X → R h k with w 1 = vec{W (1) 1 , ..., W (1) n1 }, ..., w k = vec{W (k) 1 , ..., W (k)n k } be some models satisfying Lemma 2 conditions. If Agg : R h1 × ... × R h k → R K such that:∀x 1 , y 1 ∈ R h1 , ..., x k , y k ∈ R h k : ∥Agg(x 1 , ..., x k ) -Agg(y 1 , ..., y k )∥ 2 ≤ A k j=1 ∥x j -y j ∥ 2for some A > 0 and Agg(0, ..., 0) = 0. Also we denote n = k j=1 n j , then Agg(f(1) w1 (•), ..., f (k) w k (•))satisfies Lemma 2 conditions with either:T = max    T (1) , ..., T (k) , k j=1 T (j)    and S 1:n = S (1)1 , ..., S (1) n1 , ..., S1 , ..., S (k)n k η 1:n = η (1)1 , ..., η (1) n1 , ..., η(k) 1 , ..., η (k) n k C 1 = A max    k j=1 C (j) 1 , max j∈[k] C (j) 2 • C (ind) 1 C (ind) 2    C 2 = A max j∈[k] C (j) 2 ,or withT = max    k j=1 T (j) , k j=1 T (j)    and S 1:n = S (1)1 , ..., S (1) n1 , ..., S1 , ..., S (k)n k η 1:n = η (1)1 , ..., η (1) n1 , ..., η(k) 1 , ..., η (k) n k C 1 = A max max j∈[k] C (j) 1 , max j∈[k] C (j) 2 • C (ind) 1 C (ind) 2 C 2 = A max j∈[k] C (j) 2 ,where ind = arg minj∈[k] η(j) .Proof. First, we test Equation 6for composite model by applying assumption about Agg and its Equation 6 for every f (j) :EQUATIONTo test Equation 7for composite model we again apply assumption about aggregation function and its Equation 7version for every f (j) :EQUATIONw k +u k (x)) -Agg(f (1) w1 (x), ..., f (k) w k (w))∥ 2 ≤ ≤ A k j=1 ∥f (j) wj +uj (x) -f (j) wj (x)∥ 2 ≤ A k j=1 C (j) 2 T (j) nj i=1 ∥U (j) i ∥ 2 S (j) i ≤ ≤ AT max j∈[k] C (j) 2 k j=1 ni i=1 ∥U (j) i ∥ S (j) i (45)To test Equation 8for composite model we employ arithmetic vs geometric mean together with Equation 8 for every model: (46) where the last inequality comes from the fact that k j=1 T k j=1 nj i=1 1 S (j) i = k j=1 T T (j) nj i=1 T (j) S (j) i ≥ k j=1 n j T (T (j) ) 1/nj ≥ n T k j=1 T (j) 1/n ≥ nT n-1 n ,EQUATION)EQUATION)where we denote Lip n = 1.Proof. The proof follows the Neyshabur et al. [40] with some modifications concerning the ∆x and Lip i and η i .We denote truncated f after i + 1th layer as f (i+1) . First, we provide the bound on the norm of the output after i + 1th layer:EQUATIONwhere (i) is the definition of MLP, (ii) comes from the definition of operator norm, (iii) comes from Lipschitzness. Unrolling the recursion of Equation 49 will get us (we denote fw (x) to be x):EQUATIONNow let us provide the bound for the perturbation after i + 1th layer. We denote this perturbation as ∆ i+1 :EQUATIONwhere (i) comes from the definition of ∆ i+1 , (ii) comes from the defintion of an MLP, (iii) add and substract U i+1 ψ i (f(i)w (x)), (iv) comes from triangle inequality combined with operator norm definition, (v) comes from the definition of η i+1 , Lipschitzness and Equation 50, (vi) comes from multiplying and dividing by ∥W i+1 ∥ 2 the second term. Now we unroll the recursion of Equation 51∆ i+1EQUATIONwhere (i) comes from Equation 51, (ii) comes from unrolling the recursion ∆ i , (iii) comes from the fact that 1 + η k ≥ 1.Lemma 8 (GCN perturbation analysis). Let f w : X → R K with w = vec{W 1 , ..., W n } be a n-layer GCN model. Let ψ i be Lip i activation functions after layer i ∈[n -1]. If u = vec{U 1 , ..., U n }, B ∈ R are such that: ∀i ∈ [n] : ∥U i ∥ 2 ≤ 1 n ∥W i ∥ 2 and ∀G ∈ X , ∀v ∈ G : |z v | 2 ≤ B,and ∀G ∈ X , G is simple and have maximum degree of d, then for anyEQUATIONwhere Z ∈ R |V |×dz is the matrix consisting of z v for v ∈ V , and P G is a Laplacian of the graph G which is equal to D-1/2 Ã D-1/2 where Ã is an adjacency matrix with +1 on the diagonal andD = diag |V | j=1 Ãij , i ∈ [|V |] and d is maximum degree of a graph in X .Proof. The proof follows [48] except for the fact that we add Lip i to the bound. This change is straightforward, however, for the completeness of the picture we provide this proof here. The detailed description of the architectures can be found in Appendix B.First we prove two helpful propositions Proposition 1. For any matrix A ∈ R n×m , B ∈ R m×p , we have,∥AB∥ F ≤ ∥A∥ F ∥B∥ 2Proof. Let x ⊤ i , a ⊤ i be ith row of AB and A respectively, then we have: where∥AB∥ 2 F ≤ i=1 ∥x ⊤ i | 2 2 = n i=1 ∥a ⊤ i B∥ 2 2 ≤ n i=1 ∥a ⊤ i ∥ 2 2 ∥B∥ 2 2 = ∥A∥ 2 F ∥B∥ 2D i = |V | j=1 A ij , i ∈ [|V |]. Then we have, (i) ∥A∥ 2 ≤ d; (ii) D-1 2 Ã D-1 2 2 ≤ 1, where Ã = A + I and D = diag( D1 , . . . , D|V | ) = diag   |V | j=1 Ãij , i ∈ [|V |]   .Proof. For (i), by the definition of spectral norm, we have∥A∥ 2 = max ∥x∥2=1 x ⊤ Ax = max ∥x∥2=1 (i,j)∈E x i x j ≤ max ∥x∥2=1 (i,j)∈E x 2 i + x 2 j 2 ≤ max ∥x∥2=1 d i∈V x 2 i = d.For (ii), let Ẽ = E ∪ {(i, i)|i ∈ V } be the edge set associated with the adjacency matrix Ã. By the definition of spectral norm, we haveD -1 2 ÃD -1 2 2 = max ∥x∥2=1 x ⊤ D -1 2 ÃD -1 2 x = max ∥x∥2=1 (i,j)∈ Ẽ x i x j Di Dj ≤ ≤ max ∥x∥2=1 (i,j)∈ Ẽ x 2 i 2 Di + x 2 j 2 Dj ≤ max ∥x∥2=1 i∈V x 2 i = 1.Before proving the first inequality we note that 1√ |V | ∥X∥ F ≤ B and ∥P G ∥ n-1 2≤ 1, so the second inequality is rather straightforward. The last inequality comes from the fact that d ≥ 1.We denote the node representation in the j-th (j ≤ l) layer asf j w (G) = H j = ψ j (P G H j-1 W j ), j ∈ [n -1], f n w (G) = H n = 1 |V | 1 |V | H n-1 W n .Adding perturbation u to the parameter w, that is, for the j-th (j ≤ n) layer, the perturbed parameters are W j + U j and denoteH ′ j = f j w+u (G), j ∈ [n].Upper Bound on the Node Representation. For any j < n,∥H j ∥ F = ∥ψ j (P G H j-1 W j )∥ F ≤ ≤ Lip j ∥P G H j-1 W j ∥ F ≤ ≤ Lip j ∥P G H j-1 ∥ F ∥W j ∥ 2 ≤ ≤ Lip j ∥P G ∥ 2 ∥H j-1 ∥ F ∥W j ∥ 2 ,where the first inequality holds since ψ j is a Lipschitz and ψ j (0) = 0, and the second and the last ones hold by Proposition 1. Then, unrolling the recursion and setting H 0 = X, we haveEQUATION)Upper Bound on the Change of Node Representation. For any j < |V |,EQUATIONUsing the triangle inequality, we have∥H ′ j -H j ∥ F ≤ Lip j ∥P G (W j + U j )(H ′ j-1 -H j-1 )∥ F + ∥P G H j-1 U j ∥ F .The first term can be bounded as∥P G (H ′ j-1 -H j-1 )(W j + U j )∥ F = ∥P G ∥ 2 ∥H ′ j-1 -H j-1 ∥ F ∥W j + U j ∥ 2, and the second term can be bounded as∥P G H j-1 U j ∥ F = ∥P G ∥ 2 ∥H j-1 ∥ F ∥U j ∥ 2 .Therefore,EQUATION)Unrolling the recursion while simplifying notation as:∥H j -H ′ j ∥ F ≤ a j-1 ∥H ′ j-1 -H j-1 ∥ F + b j-1 we get: ∥H ′ j -H j ∥ F ≤ j-1 k=0 b k j-1 i=k+1 a i = = j-1 k=0 Lip k+1 ∥P G ∥∥H k ∥ F ∥U k+1 ∥ 2 j-1 i=k+1 Lip i+1 ∥P G ∥ 2 ∥W i+1 + U i+1 ∥ 2 = = j-1 k=0 ∥P G ∥ j-k 2 ∥H k ∥ F ∥U k+1 ∥ 2 j-1 i=k Lip i+1 j-1 i=k+1 ∥W i+1 + U i+1 ∥ 2 .Plugging in Equation 54, we have:EQUATIONFinal Bound on the Readout Layer.∥H ′ n -H n ∥ 2 = 1 |V | 1 |V | H ′ n-1 (W n + U n ) - 1 |V | 1 |V | H n-1 W n 2 = = 1 |V | 1 |V | (H ′ n-1 -H n-1 )(W n + U n ) + 1 |V | 1 |V | H n-1 U n 2 ≤ ≤ 1 |V | 1 |V | (H ′ n-1 -H n-1 )(W n + U n ) 2 + 1 |V | 1 |V | H n-1 U n 2 ≤ ≤ 1 |V | 1 |V | 2 ∥(H ′ n-1 -H n-1 )(W n + U n )∥ 2 + 1 |V | 1 |V | 2 ∥H n-1 U n ∥ 2 ≤ ≤ 1 |V | ∥H ′ n-1 -H n-1 ∥ F ∥W n + U n ∥ 2 + 1 |V | ∥H n-1 ∥ F ∥U n ∥ 2 ,where in a last inequality we first apply that ∥A∥ 2 ≤ ∥A∥ F for any matrix A and then use Proposition 1.Using Equation 54and Equation 57, we have:X sorted by f :. . . m(k) . . . n(k) . . . n(i) . . . X sorted by g: . . . n(i) . . . m(k) . . . n(k) . . . X sorted by f : . . . m(k) . . . n(k) . . . X sorted by g: . . . n(k) . . . m(k) . . .Figure 6: Case g(n(k)) > g(m(k)), f (n(k)) ≤ f (m(k)) ∥H ′ n -H n ∥ 2 ≤ 1 |V | ∥W n + U n ∥ 2 ∥X∥ F ∥P G ∥ n-1 n-1 i=1 Lip i ∥W i ∥ 2 n-1 k=1 ∥U k ∥ 2 ∥W k ∥ 2 1 + 1 n n-k-1 + + 1 |V | ∥U n ∥ 2 ∥P G ∥ n-1 2 ∥X∥ F n-1 i=1 Lip i ∥W i ∥ 2 ≤ ≤ 1 |V | ∥Z∥ F ∥P G ∥ n-1 ∥W n + U n ∥ 2 n-1 i=1 Lip i ∥W i ∥ 2 n-1 k=1 ∥U k ∥ 2 ∥W k ∥ 2 1 + 1 n n-k-1 + ∥U n ∥ 2 n-1 i=1 Lip i ∥W i ∥ 2 = 1 |V | ∥Z∥ F ∥P G ∥ n-1 n i=1 Lip i ∥W i ∥ 2 ∥W n + U n ∥ 2 ∥W n ∥ 2 n-1 k=1 ∥U k ∥ 2 ∥W k ∥ 2 1 + 1 n n-k-1 + ∥U n ∥ 2 ∥W n ∥ 2 ≤ ≤ 1 |V | ∥Z∥ F ∥P G ∥ n-1 2 n i=1 Lip i ∥W i ∥ 2 1 + 1 n n-1 k=1 ∥U k ∥ 2 ∥W k ∥ 2 1 + 1 n n-k-1 + ∥U n ∥ 2 ∥W n ∥ 2 ≤ ≤ e |V | ∥Z∥ F ∥P G ∥ n-1 2 n i=1 Lip i ∥W i ∥ 2 n k=1 ∥U k ∥ 2 ∥W k ∥ 2 ,where we set Lip n = 1 for simplicity of notation and last inequality holds since 1≤ (1 + 1 n ) n ≤ e.Our next lemma helps us to upper bound the PERSLAY's perturbation, when Agg = k-max.Lemma 9. Let X be an arbitrary finite set and f, g : X → R. Then we can say that:|k-max x∈X f (x) -k-max x∈X g(x)| ≤ 3 max x∈X |f (x) -g(x)|Proof. Denote n : N → X by a function that maps natural number k to an element of X that would be on kth position in order sorted by f . Denote m as an analogous function but for g. Then, we are interested in the following expression: |f (n(k))g(m(k))|. Let us rewrite it:|f (n(k)) -g(m(k))| = |f (n(k)) -g(n(k)) + g(n(k)) -g(m(k))| ≤ ≤ |f (n(k)) -g(n(k))| + |g(n(k)) -g(m(k))| ≤ ≤ max x∈X |f (x) -g(x)| + |g(n(k)) -g(m(k))| Now, the task is to prove that |g(n(k)) -g(m(k))| ≤ 2 max x∈X |f (x) -g(x)|Let us consider four cases:• g(n(k)) > g(m(k)) and f (n(k)) ≥ f (m(k)) (Fig. 5 ). In this case ∃i ∈ N such that f (n(i)) > f (n(k)) and g(n(i)) < g(m(k)). Indeed, if none of the elements "to the right" of n(k) moved "to the left" of m(k), then "to the right" of m(k), there are at least nk + 1 elements; however, there are must be exactly nk elements.|g(n(k)) -g(m(k))| = g(n(k)) -g(m(k)) ≤ g(n(k)) -g(n(i)) ≤ ≤ f (n(k)) + (max x∈X |f (x) -g(x)|) -g(n(i)) < < f (n(i)) + (max x∈X |f (x) -g(x)|) -g(n(i)) < 2(max x∈X |f (x) -g(x)|)• g(n(k)) > g(m(k)) and f (n(k)) ≤ f (m(k)) (Fig. 6 )|g(n(k)) -g(m(k))| = g(n(k)) -g(m(k)) ≤ ≤ f (n(k)) + max x∈X |f (x) -g(x)| -g(m(k)) ≤ ≤ f (m(k)) + max x∈X |f (x) -g(x)| -g(m(k)) ≤ ≤ 2 max x∈X |f (x) -g(x)|• The rest of the cases can be handled analogously.Lemma 10 (Perturbation analysis of PersLay). Let f w : G → R k with w = {W (ω) , W (φ) } be a PersLay where W (ω) is a parameter vector (matrix) of weight function, ω, and W (φ) is a parameter vector (matrix) of point-transformation function, φ. Let Dg be a mapping from graphs to (extended) persistence diagrams with a fixed filtration function and B such that maxG∈G max p∈Dg(G)∥p∥ 2 ≤ B, then:EQUATIONand for η (ω) and u = {U (ω) , U (φ) } such that ∥U (ω) ∥ 2 ≤ η (ω) T (ω) , we have:EQUATIONwhere for any w and u.EQUATIONProof. First we prove the inequaliry about maximum output norm.Maximum output norm.EQUATIONwhereA 1 is max G∈G card(Dg(G)) if Agg is sum and A 1 = 1 if Agg is k-max or mean.From the Lemma statement we can upper bound ω w (p): maxG∈G,p∈Dg(G) ω w (p) ≤ C (ω) T (ω)Maximum norm of φ:EQUATIONHence:max G∈G,p∈Dg(G) ∥φ w (p)∥ 2 = h i=1 |φ w (p) i | 2 1/2 ≤ h i=1 1 1/2 = √ h φ = Ψ. max G∈G,p∈Dg(G) |φ w (p) i | = max G∈G,p∈Dg(G) |p 1 t i [1] + p 2 t i [2] + t i [3]| ≤ ≤ max{B, 1}|t i,1 + t i,2 + t i [3]| Hence: max G∈G,p∈Dg(G) ∥φ w (p)∥ 2 = h i=1 |(φ w ) i | 2 1/2 ≤ h i=1 max{B, 1} 2 |t i,1 + t i,2 + t i [3]| 2 1/2 ≤ ≤ max{b, 1} h i=1 3(t 2 i,1 + t 2 i,2 + t i [3] 2 ) ≤ ≤ √ 3 max{B, 1}∥vec(W (φ) )∥ 2Combining all together we get the Equation 58.Maximum perturbation of the PersLay output. max G∈G ∥f w+u (Dg(G)) -f w (Dg(G))∥ 2 ≤ A 2 max G∈G,p∈Dg(G)∥ω w+u ∥ 2 ≤ C ω (1 + η (ω) )T (ω) ,and maxG∈G,p∈Dg(G) |ω w+u (p) -ω w (p)| ≤ Lip (ω) ∥U (ω) ∥ 2 .Moreover, from Paragraph about max norm of φ:max G∈G,p∈Dg(G) ∥φ w (p)∥ 2 ≤ C (φ) T (φ)Maximum perturbation of φ:φ = Λ. Since g(x) = |x| is 1-Lipschitz we have that: max G∈G,p∈Dg(G) ∥(φ w+u (p) -φ w (p)) i ∥ ≤ U (φ) i Hence, max G∈G,p∈Dg(G) ∥φ w+u (p) -φ w (p)∥ 2 ≤ h i=1 |(φ w+u (p) -φ w (p)) i | 2 1/2 ≤ ≤ h i=1 (U (φ) i ) 2 1/2 = ∥U (φ) ∥ 2 φ = Γ. Suppose g(x, y) = exp -x 2 +y 2 2τ 2 . Then ∥g(x, y) -g(x + ∆x, y + ∆y)∥ 2 = |∇g(x ′ , y ′ )| 2 ∆x 2 + ∆y 2 ≤ ≤ max x ′ ,y ′ |∇g(x ′ , y ′ )| 2 ∆x 2 +∆y 2 by mean-value theorem for some x ′ , y ′ between x and x + ∆x and y and y + ∆y.Let us find maximum of the gradient by every coordinate. Since the function is symmetric we need to do it only for one of the coordinates.The maximum of the norm of the first coordinate of the gradient is achieving at the point t i,1 = p 1 ± τ , and the gradient value at these points is at most 1 τ e 1/2 . So we have:max G∈G,p∈Dg(G) ∥φ w+u (p) -φ w (p)∥ 2 ≤ h i=1 ∥U (φ) i ∥ 2 2 τ 2 e 1/2 ≤ ∥vec(U (φ) )∥ 2 τ e 1/2 φ = Ψ. In this case g(x, y, z) = p 1 x + p 2 y + z is max{B, 1}-Lipschitz, so max G∈G,p∈Dg(G) ∥φ w+u (p)-φ w (p)∥ 2 ≤ h i=1 max{B, 1} 2 ∥U (φ) i ∥ 2 2 1/2 = max{B, 1}∥vec(U (φ) )∥ 2 2Combining all together we have: max G∈G ∥f w+u (G) -f w (G)∥ 2 ≤ A 2 (1 + η (ω) )C (ω) T (ω) Lip (φ) ∥U (φ) ∥ 2 + C (φ) T (φ) Lip (ω) ∥U (ω) ∥ 2 ≤ ≤ A 2 T (φ) T (ω) (1 + η (φ) ) max{Lip (φ) C (ω) , Lip (ω) C (φ) } ∥U (φ) ∥ 2 ∥T (φ) ∥ 2 + ∥U (ω) ∥ 2 ∥T (ω) ∥ 2T = n i=1 ∥W i ∥ 2 and ∀i ∈ [n] : S i = ∥W i ∥ 2 , η i = 1 6n C 1 = B n-1 i=1 Lip i and C 2 = eB n-1 i=1 Lip iProof. To test Equation 6and Equation 7we use Lemma 7.To check Equation 8we apply arithmetic-geometric mean inequality:EQUATIONIt is left to show that Equation 9 holds: EQUATIONT = n i=1 ∥W i ∥ 2 and ∀i ∈ n : S i = ∥W i ∥ 2 , η i = 1 6nif using perturbation analysis from Liao et al. [33] C1 = d n-1 2 B n-1 i=1 Lip i and C 2 = eBd n-1 2 n-1 i=1Lip i or if using perturbation analysis from Sun and Lin [48] C1 = B n-1 i=1 Lip i and C 2 = eB n-1 i=1 Lip iProof. To test Equation 6, Equation 7we use Lemma 8. We can apply Lemma 8 because η = 1 6n ≤ 1 n . To test Equation 8 we use arithmetic vs geometric mean inequality:n i=1 ∥W i ∥ 2 n i=1 1 ∥W i ∥ 2 = n i=1 j̸ =i ∥W j ∥ 2 ≥ ≥ n   n i=1 j̸ =i ∥W j ∥ 2   = n n i=1 T i n-1 n = nT n-1 nIt is left to test Equation 9 C 1 2C 2 1 6n ≥ 3n e ≥ 1 Corollary 5. Let f w : X → R k with w = {W 1 , W 2 , W 3 } be n-layer MPGNN (n > 2). Let g,T = ∥W 1 ∥ 2 ∥W 3 ∥ 2 (dC) n-1 -1 dC -1 , S 1 = ∥W 1 ∥, S 2 = min{dC, ∥W 2 ∥ 2 }, S 3 = ∥W 3 ∥ 2 , η 1 = η 2 = η 3 = 1 6n C 1 = BC ϕ and C 2 = eBC ϕ n if dC = 1: T = ∥W 1 ∥ 2 ∥W 3 ∥ 2 S 1 = ∥W 1 ∥ 2 , S 2 = min{1, ∥W 2 ∥ 2 }, S 3 = ∥W 3 ∥ 2 η 1 = η 2 = η 3 = 1 6(n + 1) C 1 = BC ϕ (n + 1) and C 2 = eBC ϕ (n + 1) 2Proof. From [33] (Lemma 3.3) we know that:EQUATIONwhere η = max ∥U1∥2 ∥W1∥2 , ∥U2∥2 ∥W2∥2 , ∥U3∥2 ∥W3∥2 ≤ 1 n . In our case η ≤ 1 6n , so we can apply these inequalities. Note that in our notation we have n-layer MPGNN and instead of W l we have W 3 (and instead of U l we have U 3 ).EQUATIONEq. ( 76) and Eq. ( 68) respectively.Equation 6 follows from Equation 64.To test Equation 7we note that:EQUATIONNow Equation 7follows from Equation 63 and Equation 65.To test Equation 8 we employ arithmetic vs geometric mean inequality:T 1 S 1 + 1 S 2 + 1 S 3 ≥ 3T (S 1 S 2 S 3 ) 1/3 ≥    3 ∥W1∥2∥W3∥2 (∥W1∥2∥W3∥2) 1/3 ≥ 3T 2/3 , dC = 1 3 ∥W1∥2∥W3∥2 (dC) n-1 -1 dC-1 (∥W1∥2dC∥W3∥2) 1/3 ≥ 3T 2/3 , dC ̸ = 1, where in case dC = 1, S 2 ≤ 1 and in case dC ̸ = 1, S 2 ≤ dC and (dC) n-1 -1 dC-1≥ dC for n > 2.To test Equation 9we provide the following derivation: EQUATIONT = T (φ) T (ω) and (S (φ) , S (ω) ) = (∥W (φ) ∥ 2 , ∥W (ω) ∥ 2 ) and (η (φ) , η (ω) ) = (1, 1) C 1 = 2 max A 1 C (ω) C (φ) , 2A 2 max{C (ω) Lip (φ) , C (φ) Lip (ω) } C 2 = 2A 2 max{C (ω) Lip (φ) , C (φ) Lip (ω) },where ω) are such that:EQUATIONmax G∈G max p∈Dg(G) |ω w (p)| 2 ≤ C (ω) T (ω) and max G∈G max p∈Dg(G) |ω w+u (p) -ω w (p)| ≤ Lip (ω) ∥U (ω) ∥ 2for any w and u.Proof. To test Equation 6 and Equation 7 we apply Lemma 10. η (ω) in Lemma 10 is arbitrary, so we can use this lemma with η (ω) = 1 and in this case 1 + η (ω) = 2, so our choice of C 2 works.To test Equation 8 we use arithmetic vs geometric mean inequality:T (φ) T (ω) 1 S (φ) + 1 S (ω) ≥ 2 T (φ) T (ω) (∥W (ω) ∥ 2 ∥W (φ) ∥ 2 ) 1/3 ≥ 2 T (ω) T (φ) (T (ω) T (φ) ) 1/3 = 2T 2/3 It is left to test Equation 9 C 1 2C 2 ≥ 2A 1 C (ω) C (φ) max 1, 2A2 max{C (ω) Lip (φ) ,C (φ) Lip (ω) } A1C (ω) C (φ)2A 2 max{C (ω) Lip (φ) , C (φ) Lip (ω) ≥ 1MLP & GCN The result is not as tight as in [40] because of the homogenity assumption that they do. Specificaly, matching our result to theirs, we have B 2 (h ln nh)η -2 the same as B 2 n 2 (h ln nh) (instead of n they have d in their notation). We have log m δ max{1, 1 B } which is in most cases better than log m•n δ . However, the suboptimality of our bound comes from:n i=1 ∥W i ∥ 2 F ∥W i ∥ 2 2 ≤ n i=1 ∥W i ∥ 2 F n i=1 1 ∥W i ∥ 2 2 ≤ max 1, n i=1 ∥W i ∥ 2 F n i=1 1 ∥W i ∥ 2 2Table 6 : The dependency of the bound on the model parameters. All models have maximum width of weight matrices h. We consider n-layer multilayer perceptron (MLP) with weights W 1 , ..., W n . We consider n-layer GCN with weights W 1 , ..., W n . We consider n-layer (n > 2) MPGNN with weights W 1 , W 2 , W 3 . We consider simple graphs with maximum degree d. We denote C = Lip ϕ Lip g Lip ρ ∥W 2 ∥, λ = ∥W 1 ∥ 2 ∥W 3 ∥ 2 and ξ = ((dC) n-1 -1) /(dC-1) and |w| 2 is the norm of all parameters in the model. Comparing to [33] we do not add Lip ϕ to ξ and instead of W l we have W 3 . We consider PersLay with one of the classical point-transformation (described in [5] ) and weight function with weights W (φ) , W (ω) and ∥w∥ 2 is the norm of all parameters in the model. We consider abstract model that satisfy Lemma filtration function parameters. This would additionally require an analysis of Wasserstein distances between persistence diagrams obtained with different parameters. We believe that for a specific class of graphs we can get modified upper bounds for perturbation with respect to filtration function parameters that would depend on Wasserstein distance of the same order. This additional analysis could be readily integrated into our framework to get generalization bounds for learnable filtrations.I Discussion on the max over 1 and parameters norm.From the first glance it may seem that introducing max{1, ∥w∥ 2 2 } is suboptimal; however, ∥w∥ 2 2 is greater than 1 in many real-world scenarios. For instance, suppose we have parameters within the range of O(ε); then the squared norm would be greater than 1 if the number of parameters exceeds 1 ε 2 . Considering a typical choice in the Deep Learning literature, ε = 0.01, we need more than 10, 000 parameters. For example, a two-layer MLP with a hidden dimension of 64 (also a typical choice) has more than 8, 000 parameters, so in practice, ∥w∥ 2 2 ≥ 1 is usually true.Table 7 reports summary statistics of the datasets used in this paper. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: We do not foresee any direct risk stemming from our work. Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [NA] Justification: All code was made by the authors Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.For undirected graphs, we convert each edge into two directed edges. 
 
 Appendix:  


Decision: This paper uses a PAC-Bayes framework to recover known generalization bounds for MLPs and GNNs, and derive new generalization bounds for GNNs with additional topological features. Experiments show that these bounds correlate well with empirical generalisation performance, and a regularization technique based on the analysis in this paper was shown to yield good results.

The reviewers have voiced some concerns regarding the nichenss of the particular topological GNNs used. Nonetheless, most reviewers found the technical contribution of generalization using PAC-Bayes important, and recommended acceptance.

Reviews: review: paper_summary: paper_summary: This paper presents a compositional PAC learning framework for bounding the generalization gap in deep graph networks that are augmented by PersLay-vectorization of persistent homology features. Topological features can be complementary to deep features. Empirically, this combination can boost the empirical test performance. This paper investigates whether this observation reflects to the theory, and concludes that it does, up to some assumptions, for example in filtrations.
strength_weakness: strength_weakness: - One obvious drawback is that the architectures of concern in this paper are rather limited and not the ones used in practice. While this limits the applicability, the paper's theoretical contributions make it less of a concern. According to me, speaking of generalization for PH-augmented GNNs through the lens of compositional pac-bayes is interesting in its own right.

- Most importantly, does the proposed bound, similar to the KL-terms in Neyshabur's bound, depend on the number of parameters in the network? (I was suspecting this due to the use of the norm.) If so, it is definitely worth discussing, as the recent theory of deep learning states that the intrinsic dimension matters rather than the ambient dimension. And if not, please clarify.

- I'm curious about the limitations of the compositional framework. Can we leverage the nature of PH and the integration to do something more specific to tighten the bound? The compositional framework seems to adapt a rather late fusion. 

-  Ln. 81: The generalization error is defined to be the population risk and not as the difference between them. I believe some sort of a difference between test/train would be more appropriate. Why is this chosen? Is this common convention? 

- I see PH as a hand-crafted way of extracting topological features, which goes a bit orthogonal to the current trends. Can the same analysis be extended to hybrid classical & topological deep learning (like the ones presented in [*,**]) which also operates on complexes? If so, do the authors see a straightforward way? 

- There are multiple ways to combine PH(-vectorization) and GNNs. A two-branch architecture + PersLay is definitely a good way. Yet, I would expect that the paper investigates other forms of combination. For example, what about learning a filtration?

- I'm noticing that only a single family of GNNs is used in the paper. Is it possible to show results with more modern architectures at least like GraphSage?

- There is also a rather recent literature of leveraging PH to bound generalization [***] (other way around), which this paper does not seem to discuss. It would be good to have both sides of this picture to stress on the impact of PH in the generalization theory.

[*] Hajij, Mustafa et al. "Topological deep learning: Going beyond graph data." arXiv preprint arXiv:2206.00606 (2022).

[**] Papamarkou, Theodore, Tolga Birdal, Michael M. Bronstein, Gunnar E. Carlsson, Justin Curry, Yue Gao, Mustafa Hajij et al. "Position: Topological Deep Learning is the New Frontier for Relational Learning." In Forty-first International Conference on Machine Learning. 2024.

[***] Birdal, Tolga, Aaron Lou, Leonidas J. Guibas, and Umut Simsekli. "Intrinsic dimension, persistent homology and generalization in neural networks." Advances in Neural Information Processing Systems 34 (2021): 6776-6789.
questions: questions: I have appended my questions after each relevant weakness. I would be happy if the authors could address them. In addition: 
- Why would the bounds worsen with the increase number of epochs (where I believe training gets better)?
- There is also growth in the width disproportionately to the empirical gap. Why would this happen?

In general, some more light on the empirical findings would be useful.
limitations: limitations: The paper appropriately discusses the limitations and remains to be largely theoretical in a very niche domain of machine learning. As such, I don't see any issues with broader impact.
paper_summary: paper_summary: This paper introduces a novel compositional PAC-Bayes framework for analyzing the generalization of heterogeneous machine learning models, with a particular focus on graph neural networks (GNNs) augmented with persistent homology (PH) features. The work develops a general PAC-Bayes lemma for heterogeneous models that not only recovers existing bounds for neural networks and GNNs but also extends them to more complex architectures. Notably, it provides the first data-dependent generalization bounds for PH-based models.
The key innovation lies in the compositional approach, presenting lemmas that allow for combining bounds from different model components. This enables the analysis of complex architectures like GNNs augmented with PH features, bridging an important gap in the theoretical understanding of topology-based graph representation learning methods. The framework's versatility is demonstrated by recovering existing bounds for various models and deriving new bounds for PersLay and its variants. The empirical evaluations across multiple datasets validate the theoretical results, showing correlation between the derived bounds and observed generalization performance.
strength_weakness: strength_weakness: - Empirical evaluation focuses mostly on graph classification tasks; additional experiments on node classification or link prediction could strengthen the results
questions: questions: - The theoretical analysis assumes fixed filtration functions for PH. How limiting is this in practice, and could the framework be extended to learnable filtrations?
limitations: limitations: - While the experiments cover several datasets, they focus primarily on graph classification tasks. The paper could benefit from a broader range of experiments, including node classification or link prediction tasks, to demonstrate the generality of the approach.
- While the paper derives a regularization scheme from the bounds, it doesn't fully explore how the theoretical results could guide the design of better GNN+PH architectures in general. Some discussion on how the bounds suggest optimal ways to combine GNNs and PersLay could enhance the practical impact of the work. 

These points did not diminish the overall contribution of the paper but addressing them could significantly strengthen its impact and applicability.
paper_summary: paper_summary: The paper introduces a novel Compositional PAC-Bayes framework that addresses challenges related to the heterogeneity of Graph Neural Network (GNN) layers and persistent vectorization components. It provides data-dependent generalization bounds for PH vectorization schemes and persistence-augmented GNNs, offering insights into improved classifier design and generalization performance predictions.
strength_weakness: strength_weakness: Clarity: Some sections of the paper may require further clarification to enhance readability and understanding for a broader audience. From experiments, we can observe that there exists a considerable gap between the theoretical results and empirical results.
questions: questions: 1. This paper mentioned the expressivity of PH and GNNs for many times. Does the expressivity have anything to do with the generalization?
2. What is the influence of using different filtration functions in the proposed framework?
3. Does the proposed framework have any concrete applications? Can you provide a case study?
limitations: limitations: yes
paper_summary: paper_summary: This paper derives novel generalization bounds for a special class of GNNs augmented with persistent homology descriptors, specifically PersLay. They empirically test the bound and use the bound to propose a regularization, which works well in practice.
strength_weakness: strength_weakness: 1. It is hard to parse the significance of this work as the bound applies only to a restricted family of models augmented with PersLay. This issue could be mitigated by demonstrating that a model derived from this analysis is comparable with more advanced GNN models (not necessarily with PH), which the paper fails to do.
2. The paper would benefit from listing all assumptions used in the derivation.
3. The experiments could be strengthened by considering different GNN models augmented by different PersLay variants.
questions: questions: How does the bound for GNNs with PH compare with their counterparts without PH? Does the result provide any insights into why PH is beneficial for enhancing GNN performance?
limitations: limitations: Please refer to the weaknesses section.

Comments: title: to reviewers: please respond to rebuttal
comment: Thank you for the feedback and clarification. I will keep my score.
Dear reviewer,

Thank you again for your constructive feedback.

As detailed above, we've acted on all your comments and suggestions (including clarifying how our main result applied broadly significantly beyond PersLay, listing all assumptions, and providing results of additional experiments that demonstrate improved generalization with regularized PH-augmented GraphSAGE, GCN, GIN). We will include all these in the updated version. Please also see our global response, where we summarize the key steps taken to address the concerns of all the reviewers.

We believe acting on your feedback has helped us consolidate our contributions, and reinforced the strengths of this work. Since only a few hours are left until the end of the discussion period,  we would greatly appreciate if you could update your score to reflect the same. Many thanks!

Best regards.
I thank the authors for the good work and their explanations. I will maintain my recommendation of acceptance. Note that, in the newly provided plots, the gap between the empirical error and the proposed bound grows with epochs. This might be seen as a little concerning and stresses the importance of tightening the bounds in future works, for example through some of the directions I suggested.
Dear reviewer,

Thank you for your reviews. The authors have tried to address your concerns in the rebuttal. If you have not already done so, please carefully read their rebuttal, and let them know what you think, and whether there is any more clarifications you require. Note that author-reviewer discussion ends on August 13th.
Thanks!
the AC
We are glad that our answers addressed your concerns and that you are positive about acceptance. Thank you again for your review and for acknowledging our rebuttal.
Thanks for all the authors' efforts to address my other concerns.  I have no further questions. I am positive to an acceptance.