Variational Diffusion Autoencoders with Random Walk Sampling
Variational inference (VI) methods and especially variational autoencoders (VAEs) specify scalable generative models that enjoy an intuitive connection to manifold learning --- with many default priors the posterior/likelihood pair / can be viewed as an approximate homeomorphism (and its inverse) between the data manifold and a latent Euclidean space. However, these approximations are well-documented to become degenerate in training. Unless the subjective prior is carefully chosen, the topologies of the prior and data distributions often will not match. Conversely, diffusion maps (DM) automatically 	extit{infer} the data topology and enjoy a rigorous connection to manifold learning, but do not scale easily or provide the inverse homeomorphism. In this paper, we propose 	extbf{a)} a principled measure for recognizing the mismatch between data and latent distributions and 	extbf{b)} a method that combines the advantages of variational inference and diffusion maps to learn a homeomorphic generative model. The measure, the 	extit{locally bi-Lipschitz property}, is a sufficient condition for a homeomorphism and easy to compute and interpret. The method, the 	extit{variational diffusion autoencoder} (VDAE), is a novel generative algorithm that first infers the topology of the data distribution, then models a diffusion random walk over the data. To achieve efficient computation in VDAEs, we use stochastic versions of both variational inference and manifold learning optimization. We prove approximation theoretic results for the dimension dependence of VDAEs, and that locally isotropic sampling in the latent space results in a random walk over the reconstructed manifold. Finally, we demonstrate the utility of our method on various real and synthetic datasets, and show that it exhibits performance superior to other generative models.
** Summary The paper studies the problem of density estimation and learning accurate generative models. The authors start from the observation that this problem has been approaches either using variational inference models, that scale very well but whose approximations may lead to degenerate results in practice, and diffusion maps, that scale poorly but are very effective in capturing the underlying data manifold. From here, the authors propose integrating the notion of random walk from diffusion maps into VAEs to avoid degenerate conditions. The proposed method is first defined in its generality, a practical implementation is presented, theoretical guarantees are provided, and empirical evidence of its effectiveness is reported. ** Evaluation The starting point of the paper seems very solid: diffusion maps are capturing the geometry of data very effectively and bringing some of those characteristics into the more scalable approach of VAE is an interesting approach. In the proposed method, this translates into introducing a learned diffusion map from manifold to an Euclidean space into the inference part. As a result, the lower bound optimized by the method now contains local information about the accuracy of the one-step random walk. How this can be translated into a practical implementation is also convincing. My main concern is that the overall method is now approximating many different elements in the original formulation, such as the diffusion map, its inverse, and the covariance of the random walk. Although theory seems to support that as these approximation become more accurate the overall result is reliable, in practice I wonder how they could combine and deteriorate the final result. The empirical validation is relatively simple but in my opinion it provides enough insights about the advantages of the proposed method compared to VAEs and GANs. More solid and extensive evaluation is definitely needed in the future to have a more thorough comparison and a more careful assessment of the limits of the proposed method, but at this stage, I think the evaluation is sufficient.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Variational Diffusion Autoencoders with Random Walk Sampling
Variational inference (VI) methods and especially variational autoencoders (VAEs) specify scalable generative models that enjoy an intuitive connection to manifold learning --- with many default priors the posterior/likelihood pair / can be viewed as an approximate homeomorphism (and its inverse) between the data manifold and a latent Euclidean space. However, these approximations are well-documented to become degenerate in training. Unless the subjective prior is carefully chosen, the topologies of the prior and data distributions often will not match. Conversely, diffusion maps (DM) automatically 	extit{infer} the data topology and enjoy a rigorous connection to manifold learning, but do not scale easily or provide the inverse homeomorphism. In this paper, we propose 	extbf{a)} a principled measure for recognizing the mismatch between data and latent distributions and 	extbf{b)} a method that combines the advantages of variational inference and diffusion maps to learn a homeomorphic generative model. The measure, the 	extit{locally bi-Lipschitz property}, is a sufficient condition for a homeomorphism and easy to compute and interpret. The method, the 	extit{variational diffusion autoencoder} (VDAE), is a novel generative algorithm that first infers the topology of the data distribution, then models a diffusion random walk over the data. To achieve efficient computation in VDAEs, we use stochastic versions of both variational inference and manifold learning optimization. We prove approximation theoretic results for the dimension dependence of VDAEs, and that locally isotropic sampling in the latent space results in a random walk over the reconstructed manifold. Finally, we demonstrate the utility of our method on various real and synthetic datasets, and show that it exhibits performance superior to other generative models.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



** Summary The paper studies the problem of density estimation and learning accurate generative models. The authors start from the observation that this problem has been approaches either using variational inference models, that scale very well but whose approximations may lead to degenerate results in practice, and diffusion maps, that scale poorly but are very effective in capturing the underlying data manifold. From here, the authors propose integrating the notion of random walk from diffusion maps into VAEs to avoid degenerate conditions. The proposed method is first defined in its generality, a practical implementation is presented, theoretical guarantees are provided, and empirical evidence of its effectiveness is reported. ** Evaluation The starting point of the paper seems very solid: diffusion maps are capturing the geometry of data very effectively and bringing some of those characteristics into the more scalable approach of VAE is an interesting approach. In the proposed method, this translates into introducing a learned diffusion map from manifold to an Euclidean space into the inference part. As a result, the lower bound optimized by the method now contains local information about the accuracy of the one-step random walk. How this can be translated into a practical implementation is also convincing. My main concern is that the overall method is now approximating many different elements in the original formulation, such as the diffusion map, its inverse, and the covariance of the random walk. Although theory seems to support that as these approximation become more accurate the overall result is reliable, in practice I wonder how they could combine and deteriorate the final result. The empirical validation is relatively simple but in my opinion it provides enough insights about the advantages of the proposed method compared to VAEs and GANs. More solid and extensive evaluation is definitely needed in the future to have a more thorough comparison and a more careful assessment of the limits of the proposed method, but at this stage, I think the evaluation is sufficient.
assistant
** Summary The paper studies the problem of density estimation and learning accurate generative models. The authors start from the observation that this problem has been approaches either using variational inference models, that scale very well but whose approximations may lead to degenerate results in practice, and diffusion maps, that scale poorly but are very effective in capturing the underlying data manifold. From here, the authors propose integrating the notion of random walk from diffusion maps into VAEs to avoid degenerate conditions. The proposed method is first defined in its generality, a practical implementation is presented, theoretical guarantees are provided, and empirical evidence of its effectiveness is reported. ** Evaluation The starting point of the paper seems very solid: diffusion maps are capturing the geometry of data very effectively and bringing some of those characteristics into the more scalable approach of VAE is an interesting approach. In the proposed method, this translates into introducing a learned diffusion map from manifold to an Euclidean space into the inference part. As a result, the lower bound optimized by the method now contains local information about the accuracy of the one-step random walk. How this can be translated into a practical implementation is also convincing. My main concern is that the overall method is now approximating many different elements in the original formulation, such as the diffusion map, its inverse, and the covariance of the random walk. Although theory seems to support that as these approximation become more accurate the overall result is reliable, in practice I wonder how they could combine and deteriorate the final result. The empirical validation is relatively simple but in my opinion it provides enough insights about the advantages of the proposed method compared to VAEs and GANs. More solid and extensive evaluation is definitely needed in the future to have a more thorough comparison and a more careful assessment of the limits of the proposed method, but at this stage, I think the evaluation is sufficient.