Likelihood-based, or explicit, deep generative models use neural networks to construct flexible high-dimensional densities. This formulation directly contradicts the manifold hypothesis, which states that observed data lies on a low-dimensional manifold embedded in high-dimensional ambient space. In this paper we investigate the pathologies of maximum-likelihood training in the presence of this dimensionality mismatch. We formally prove that degenerate optima are achieved wherein the manifold itself is learned but not the distribution on it, a phenomenon we call manifold overfitting. We propose a class of two-step procedures consisting of a dimensionality reduction step followed by maximum-likelihood density estimation, and prove that they recover the data-generating distribution in the nonparametric regime, thus avoiding manifold overfitting. We also show that these procedures enable density estimation on the manifolds learned by implicit models, such as generative adversarial networks, hence addressing a major shortcoming of these models. Several recently proposed methods are instances of our two-step procedures; we thus unify, extend, and theoretically justify a large class of models.
Normalizing flows are generative models that provide tractable density estimation by transforming a simple base distribution into a complex target distribution. However, this technique cannot directly model data supported on an unknown low-dimensional manifold, a common occurrence in real-world domains such as image data. Recent attempts to remedy this limitation have introduced geometric complications that defeat a central benefit of normalizing flows: exact density estimation. We recover this benefit with Conformal Embedding Flows, a framework for designing flows that learn manifolds with tractable densities. We argue that composing a standard flow with a trainable conformal embedding is the most natural way to model manifold-supported data. To this end, we present a series of conformal building blocks and apply them in experiments with real-world and synthetic data to demonstrate that flows can model manifold-supported distributions without sacrificing tractable likelihoods.Preprint. Under review.
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In the traditional federated learning setting, a central server coordinates a network of clients to train one global model. However, the global model may serve many clients poorly due to data heterogeneity. Moreover, there may not exist a trusted central party that can coordinate the clients to ensure that each of them can benefit from others. To address these concerns, we present a novel decentralized framework, FedeRiCo, where each client can learn as much or as little from other clients as is optimal for its local data distribution. Based on expectationmaximization, FedeRiCo estimates the utilities of other participants' models on each client's data so that everyone can select the right collaborators for learning. As a result, our algorithm outperforms other federated, personalized, and/or decentralized approaches on several benchmark datasets, being the only approach that consistently performs better than training with local data only.
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