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Register a new userLearning Sub-Patterns in Piecewise Continuous Functions
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Added by
Anastasis Kratsios
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters, which requires their activation function to exhibit a degree of continuity. However, this continuity constraint on the activation function prevents these neural models from uniformly approximating discontinuous functions. This paper focuses on the case where the discontinuities arise from distinct sub-patterns, each defined on different parts of the input space. We propose a new discontinuous deep neural network model trainable via a decoupled two-step procedure that avoids passing gradient updates through the networks non-differentiable unit. We provide universal approximation guarantees for our architecture in the space of bounded continuous functions and in the space of piecewise continuous functions, which we introduced herein. We present a novel semi-supervised two-step training procedure for our discontinuous deep learning model, and we provide theoretical support for its effectiveness. The performance of our architecture is evaluated experimentally on two real-world datasets and one synthetic dataset.
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This paper presents a Gaussian process (GP) model for estimating piecewise continuous regression functions. In scientific and engineering applications of regression analysis, the underlying regression functions are piecewise continuous in that data follow different continuous regression models for different regions of the data with possible discontinuities between the regions. However, many conventional GP regression approaches are not designed for piecewise regression analysis. We propose a new GP modeling approach for estimating an unknown piecewise continuous regression function. The new GP model seeks for a local GP estimate of an unknown regression function at each test location, using local data neighboring to the test location. To accommodate the possibilities of the local data from different regions, the local data is partitioned into two sides by a local linear boundary, and only the local data belonging to the same side as the test location is used for the regression estimate. This local split works very well when the input regions are bounded by smooth boundaries, so the local linear approximation of the smooth boundaries works well. We estimate the local linear boundary jointly with the other hyperparameters of the GP model, using the maximum likelihood approach. Its computation time is as low as the local GPs time. The superior numerical performance of the proposed approach over the conventional GP modeling approaches is shown using various simulated piecewise regression functions.
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This paper introduces Associative Compression Networks (ACNs), a new framework for variational autoencoding with neural networks. The system differs from existing variational autoencoders (VAEs) in that the prior distribution used to model each code is conditioned on a similar code from the dataset. In compression terms this equates to sequentially transmitting the dataset using an ordering determined by proximity in latent space. Since the prior need only account for local, rather than global variations in the latent space, the coding cost is greatly reduced, leading to rich, informative codes. Crucially, the codes remain informative when powerful, autoregressive decoders are used, which we argue is fundamentally difficult with normal VAEs. Experimental results on MNIST, CIFAR-10, ImageNet and CelebA show that ACNs discover high-level latent features such as object class, writing style, pose and facial expression, which can be used to cluster and classify the data, as well as to generate diverse and convincing samples. We conclude that ACNs are a promising new direction for representation learning: one that steps away from IID modelling, and towards learning a structured description of the dataset as a whole.
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