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Register a new userKafnets: kernel-based non-parametric activation functions for neural networks
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Neural networks are generally built by interleaving (adaptable) linear layers with (fixed) nonlinear activation functions. To increase their flexibility, several authors have proposed methods for adapting the activation functions themselves, endowing them with varying degrees of flexibility. None of these approaches, however, have gained wide acceptance in practice, and research in this topic remains open. In this paper, we introduce a novel family of flexible activation functions that are based on an inexpensive kernel expansion at every neuron. Leveraging over several properties of kernel-based models, we propose multiple variations for designing and initializing these kernel activation functions (KAFs), including a multidimensional scheme allowing to nonlinearly combine information from different paths in the network. The resulting KAFs can approximate any mapping defined over a subset of the real line, either convex or nonconvex. Furthermore, they are smooth over their entire domain, linear in their parameters, and they can be regularized using any known scheme, including the use of $ell_1$ penalties to enforce sparseness. To the best of our knowledge, no other known model satisfies all these properties simultaneously. In addition, we provide a relatively complete overview on alternative techniques for adapting the activation functions, which is currently lacking in the literature. A large set of experiments validates our proposal.
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Although neural networks are capable of reaching astonishing performances on a wide variety of contexts, properly training networks on complicated tasks requires expertise and can be expensive from a computational perspective. In industrial applications, data coming from an open-world setting might widely differ from the benchmark datasets on which a network was trained. Being able to monitor the presence of such variations without retraining the network is of crucial importance. In this article, we develop a method to monitor trained neural networks based on the topological properties of their activation graphs. To each new observation, we assign a Topological Uncertainty, a score that aims to assess the reliability of the predictions by investigating the whole network instead of its final layer only, as typically done by practitioners. Our approach entirely works at a post-training level and does not require any assumption on the network architecture, optimization scheme, nor the use of data augmentation or auxiliary datasets; and can be faithfully applied on a large range of network architectures and data types. We showcase experimentally the potential of Topological Uncertainty in the context of trained network selection, Out-Of-Distribution detection, and shift-detection, both on synthetic and real datasets of images and graphs.
In this paper, we propose a novel adaptive kernel for the radial basis function (RBF) neural networks. The proposed kernel adaptively fuses the Euclidean and cosine distance measures to exploit the reciprocating properties of the two. The proposed framework dynamically adapts the weights of the participating kernels using the gradient descent method thereby alleviating the need for predetermined weights. The proposed method is shown to outperform the manual fusion of the kernels on three major problems of estimation namely nonlinear system identification, pattern classification and function approximation.
Achieving transparency in black-box deep learning algorithms is still an open challenge. High dimensional features and decisions given by deep neural networks (NN) require new algorithms and methods to expose its mechanisms. Current state-of-the-art NN interpretation methods (e.g. Saliency maps, DeepLIFT, LIME, etc.) focus more on the direct relationship between NN outputs and inputs rather than the NN structure and operations itself. In current deep NN operations, there is uncertainty over the exact role played by neurons with fixed activation functions. In this paper, we achieve partially explainable learning model by symbolically explaining the role of activation functions (AF) under a scalable topology. This is carried out by modeling the AFs as adaptive Gaussian Processes (GP), which sit within a novel scalable NN topology, based on the Kolmogorov-Arnold Superposition Theorem (KST). In this scalable NN architecture, the AFs are generated by GP interpolation between control points and can thus be tuned during the back-propagation procedure via gradient descent. The control points act as the core enabler to both local and global adjustability of AF, where the GP interpolation constrains the intrinsic autocorrelation to avoid over-fitting. We show that there exists a trade-off between the NNs expressive power and interpretation complexity, under linear KST topology scaling. To demonstrate this, we perform a case study on a binary classification dataset of banknote authentication. By quantitatively and qualitatively investigating the mapping relationship between inputs and output, our explainable model can provide interpretation over each of the one-dimensional attributes. These early results suggest that our model has the potential to act as the final interpretation layer for deep neural networks.
Standard convolutional neural networks assume a grid structured input is available and exploit discrete convolutions as their fundamental building blocks. This limits their applicability to many real-world applications. In this paper we propose Parametric Continuous Convolution, a new learnable operator that operates over non-grid structured data. The key idea is to exploit parameterized kernel functions that span the full continuous vector space. This generalization allows us to learn over arbitrary data structures as long as their support relationship is computable. Our experiments show significant improvement over the state-of-the-art in point cloud segmentation of indoor and outdoor scenes, and lidar motion estimation of driving scenes.
Linear models have shown great effectiveness and flexibility in many fields such as machine learning, signal processing and statistics. They can represent rich spaces of functions while preserving the convexity of the optimization problems where they are used, and are simple to evaluate, differentiate and integrate. However, for modeling non-negative functions, which are crucial for unsupervised learning, density estimation, or non-parametric Bayesian methods, linear models are not applicable directly. Moreover, current state-of-the-art models like generalized linear models either lead to non-convex optimization problems, or cannot be easily integrated. In this paper we provide the first model for non-negative functions which benefits from the same good properties of linear models. In particular, we prove that it admits a representer theorem and provide an efficient dual formulation for convex problems. We study its representation power, showing that the resulting space of functions is strictly richer than that of generalized linear models. Finally we extend the model and the theoretical results to functions with outputs in convex cones. The paper is complemented by an experimental evaluation of the model showing its effectiveness in terms of formulation, algorithmic derivation and practical results on the problems of density estimation, regression with heteroscedastic errors, and multiple quantile regression.
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