Summary and Contributions: The paper proposes a Dirichlet graph variational autoencoder, an instance of a variational autoencoder in which the input graph is encoded into Dirichlet-distributed latent variables. As a consequence, they can be interpreted as cluster memberships, similar to topic model VAEs for text generation. The paper also establishes a connection between a term in the ELBO and an instance of graph cuts, which motivates a novel graph encoding strategy based on a Taylor approximation of heat kernels in spectral graph convolutions.
Strengths: - The paper addresses an important problem and its results, specifically regarding the semantic understanding of latent variables, could be relevant in a variety of contexts. - The proposed Heatts encoder is well motivated and analyzed. I particularly liked the theoretical comparison to GCNs in terms of their distance to the ideal low pass filter. - I enjoyed reading about the interesting relationship between elements of the ELBO and a version of graph cuts. [see below for questions about the proof, though] - The experiments show performance improvements over popular baselines in a number of relevant scenarios. Furthermore, an ablation study shows the effectiveness of the proposed Heatts encoder compared to traditional GCN-based encoding. - Visualizations of graph clusters based on latent cluster memberships serve as empirical evidence of the paper’s technical contributions.
Weaknesses: - The established relationship between spectrally relaxed graph cuts and the reconstruction term of the ELBO (claim 4.1) feels fragile; please see below for additional details. I am also confused about the implications of claim 4.1 in a more general sense: does the proof rely on a specific distribution of the latent variables? If not, does that mean the statement is also true for VGAEs, which use a very similar generative model? - Section 2 provides a good summary about graph cuts, but more intuition about the ratio cut objective and additional details about spectral clustering would have been helpful; maybe they can be added to the supplementary material. Especially the role of Eq.(4) does not become immediately clear. - Section 3 heavily builds upon results in , but in many cases the necessary context is missing. For example, it is not possible to understand why Eq.(8) is Dirichlet-distributed or why Eqs.(9-10) describe a logistic normal distribution without consulting external references. I encourage the authors to update this section with additional details to make it more self-contained. - The quality of the Laplace approximation proposed in section 3 is unclear. Is it possible to make a theoretical statement in terms of an error bound? - The experimental evaluation could be improved in two ways: (1) The description of the datasets is very brief, simply referring to  is not enough; (2) The evaluation distinguishes x-AE and x-VAE, but it is nowhere mentioned what the difference is.
Correctness: - Eq.(12) seems to differ from the decoding strategy proposed in . Were these differences introduced on purpose and, if so, what is the motivation behind them? Why does Z not appear on the RHS of the equation? Is there an implicit assumption that the latent variables correspond to cluster memberships? - Claim 4.1 establishes a connection between the ELBO reconstruction term and spectrally relaxed graph cuts. However, it only considers the decoder p(A|Z) and ignores the effects of the approximate posterior in the reconstruction term. Furthermore, it is not clear for which distance metrics f claim 4.1 holds. I encourage the authors to comment on that.
Clarity: - The paper is well written and for the most part easy to follow. The structure is clear and the goals and objectives are well motivated. The notation and terminology are consistent.
Relation to Prior Work: - The introduction puts the proposed model in context with relevant prior works on graph variational autoencoders, graph cuts, and topic model VAEs for text generation. The related work section expands this discussion to Dirichlet VAEs and literature on the low pass properties of GNNs. - Since the proposed model is heavily inspired by topic model VAEs for text generation, it is not always clear which ideas are simple adaptions of earlier works and which ones are truly novel contributions. A more detailed discussion of these earlier works feels necessary.
Additional Feedback: I'm willing to raise my score if my questions/concerns under 'Correctness' are addressed. Post-rebuttal comment: I thank the authors for addressing my concerns about some of the technical aspects of the paper and increase my score from (6) to (7).
Summary and Contributions: Authors propose to extend the variational graph autoencoders by replacing Gaussian distributed latent variables with the Dirichlet distributed variables (approximated by logistic normal) such that the latent variables can be directly used to describe graph cluster memberships. The model is trained by optimising the evidence lower bound, and authors show that maximizing the reconstruction term is equivalent to minimizing the spectral graph cut and that the regularization term (KL) promotes balanced cluster sizes in the latent space. Authors report competitive results on graph generation and graph clustering, when compared to existing methods. Based on authors' ablation study much of the improved performance can be attributed to a new GNN encoder that embeds a given input graph into (latent) cluster membership. The proposed GNN uses graph convolutional neural network with a so-called heat kernel (together with a tailor approximation).
Strengths: Competitive results on several data sets when compared to previous methods. Theoretical results showing that the proposed method encourages balanced graph cuts and balanced cluster sizes. For the most part, the manuscript is well-written, and the introductory/preliminaries part is written in an educative way.
Weaknesses: While results on simulation studies seem competitive, the best results seem to be achieved with DGAE: I understood that DGAE is a non-variationally trained alternative of the proposed method, and authors need to further clarify whether/to what extend the theoretical results apply to DGAE variant of the method. The fact that the method promotes balanced cluster sizes may perhaps also be seen as a limitation, as not all data sets necessarily have equally sized clusters.
Correctness: I did not check all derivations thoroughly but they seem correct.
Clarity: For the most part, yes. I would like to see a more detailed description of Heats kernel based encode at the level of connecting it to the parameters of the variational approximation q().
Relation to Prior Work: Yes.
Additional Feedback: I would like to see a more detailed description of Heats kernel based encode at the level of connecting it to the parameters of the variational approximation q(). Eq. 5: Expectation with respect to q_\phi(Z \mid G). UPDATE: upon reading authors‘ response letter, I changed my overall score = 7
Summary and Contributions: The paper deals with learning a variational autoencoder for graph structure generation. The authors draw connections to spherical clustering and the corresponding balanced graph cut. In addition, they introduce a novel graph neural network based on heat kernels in conjunction with a Taylor series for fast computation. The methods has been evaluated on graph generation and graph clustering tasks on both artificial and real world datasets.
Strengths: * The authors introduce a three-fold contribution for learning graph-structures in a VAE setting. Especially, they propose a new GNN based on heat kernels that learns better latent representations for graph structure generation tasks and draw connections to spherical clustering. * The proposed method was extensively tested on multiple datasets and compared to multiple baseline approaches.
Weaknesses: * The authors formulate the problem that the explanation of latent factors remain unclear in the context GGNs and VAEs. To overcome these limitations, they proposed DGVAE however it is unclear to me in what sense the Dirichlet approach leads to a better explanation compared to other clustering approaches. * It would be interesting to compare your approach to more recent state-of-the-art clustering approach instead of plain k-means such as  * The three proposed contributions seems to be only loosely connected. For example, in the experimental evaluation one gets the impression that only Heatts is required to obtain the good results (e.g. l. 214). Therefore, I sometimes got the impression that the other contributions are nice to have however not necessary for the whole approach. It would be nice if the authors could try embed their contributions better in their story such that they are also reflected in the experiments.
Correctness: The claims of the paper seem to be correct and the empirical methodology seems to be correct.
Clarity: The paper is tightly written but the methodology is sufficiently explained. Sometimes I miss a bit the common thread throughout the paper.
Relation to Prior Work: The authors clearly discussed their method and the differences to prior work. However, I would have appreciated a discussion about archetypal analysis as it seems to be highly-related to Dirichlet-based autoencoders, e.g. [1,2,3]
Additional Feedback: * How do you make sure that data points lie at the corners of your simplex. Is it is only imposed by the prior assumption and by optimizing the reconstruction term with more updates? It would be interesting to know how stable this approach is? * As a follow up question, how do you sample from our latent space and are there big holes? Because if you try to cluster the data points in the corners most parts of the remaining space should be empty? * In line 108: The authors write that the Dirichlet assumption makes the latent factors more interpretable. In what sense and how does it differ from a normal clustering? * I am wondering if it is fair to compare cluster-based methods such as k-means that look for typical observations (cluster centers) with archetypal-like constructions that seek for extreme points where all other data points are convex-combinations of such extreme points (corners of the Dirichlet simplex). =============================================================== After rebuttal: Dear authors, thank you for answering and clarifying my questions. After reading your response and the discussion with the other reviewers, I will adjust my overall score to 7.