Many different methods to train deep generative models have been introduced in the past. In this paper, we propose to extend the variational auto-encoder (VAE) framework with a new type of prior which we call "Variational Mixture of Posteriors" prior, or VampPrior for short. The VampPrior consists of a mixture distribution (e.g., a mixture of Gaussians) with components given by variational posteriors conditioned on learnable pseudo-inputs. We further extend this prior to a two layer hierarchical model and show that this architecture with a coupled prior and posterior, learns significantly better models. The model also avoids the usual local optima issues related to useless latent dimensions that plague VAEs. We provide empirical studies on six datasets, namely, static and binary MNIST, OMNIGLOT, Caltech 101 Silhouettes, Frey Faces and Histopathology patches, and show that applying the hierarchical VampPrior delivers state-of-the-art results on all datasets in the unsupervised permutation invariant setting and the best results or comparable to SOTA methods for the approach with convolutional networks.
Very recently, the VampPrior was shown to be a perfect match for the information-theoretic approach to learn latent representation @cite_27 . Additionally, the authors of @cite_27 proposed to use a weighted version of the VampPrior: where @math are trainable parameters such that @math and @math . This allows the VampPrior to learn which components (, pseudo-inputs) are meaningful and may prevent from potential overfitting.
@cite_27
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Many different methods to train deep generative models have been introduced in the past. In this paper, we propose to extend the variational auto-encoder (VAE) framework with a new type of prior which we call "Variational Mixture of Posteriors" prior, or VampPrior for short. The VampPrior consists of a mixture distribution (e.g., a mixture of Gaussians) with components given by variational posteriors conditioned on learnable pseudo-inputs. We further extend this prior to a two layer hierarchical model and show that this architecture with a coupled prior and posterior, learns significantly better models. The model also avoids the usual local optima issues related to useless latent dimensions that plague VAEs. We provide empirical studies on six datasets, namely, static and binary MNIST, OMNIGLOT, Caltech 101 Silhouettes, Frey Faces and Histopathology patches, and show that applying the hierarchical VampPrior delivers state-of-the-art results on all datasets in the unsupervised permutation invariant setting and the best results or comparable to SOTA methods for the approach with convolutional networks. An arithmetic read-once formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are + ,× and such that every input variable labels at most one leaf. In this paper we study the problems of giving deterministic identity testing and reconstruction algorithms for ROFs. Our main result is an @math time deterministic algorithm for checking whether a black box holding the sum of k n -variate ROFs computes the zero polynomial. In other words, we provide a hitting set of size @math for the sum of k ROFs. This result greatly improves [27] where an @math algorithm was given for the problem. Using our new results we obtain a deterministic reconstruction algorithms for read-once formulas that runs in time @math . In fact, our results also hold for the more general model of preprocessed read-once formulas that we define in this paper. In this model we are allowed to replace each variable x i with a polynomial T i (x i ). Our techniques are very close to the techniques in [27]. The main difference is that we obtain several tighter versions of the tools first used there. In particular we obtain a better version of the hardness of representation approach which was first used in [27]. This technique can be thought of as a very explicit way of transforming (mild) hardness of a very structured polynomial to an identity testing algorithm. We study the flow of information and the evolution of internal representations during deep neural network (DNN) training, aiming to demystify the compression aspect of the information bottleneck theory. The theory suggests that DNN training comprises a rapid fitting phase followed by a slower compression phase, in which the mutual information @math between the input @math and internal representations @math decreases. Several papers observe compression of estimated mutual information on different DNN models, but the true @math over these networks is provably either constant (discrete @math ) or infinite (continuous @math ). This work explains the discrepancy between theory and experiments, and clarifies what was actually measured by these past works. To this end, we introduce an auxiliary (noisy) DNN framework for which @math is a meaningful quantity that depends on the network's parameters. This noisy framework is shown to be a good proxy for the original (deterministic) DNN both in terms of performance and the learned representations. We then develop a rigorous estimator for @math in noisy DNNs and observe compression in various models. By relating @math in the noisy DNN to an information-theoretic communication problem, we show that compression is driven by the progressive clustering of hidden representations of inputs from the same class. Several methods to directly monitor clustering of hidden representations, both in noisy and deterministic DNNs, are used to show that meaningful clusters form in the @math space. Finally, we return to the estimator of @math employed in past works, and demonstrate that while it fails to capture the true (vacuous) mutual information, it does serve as a measure for clustering. This clarifies the past observations of compression and isolates the geometric clustering of hidden representations as the true phenomenon of interest. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math .