![]() ![]() The connection with maxout in PWL-DNNs can be referred to. This paper presents the idea of inserting multiple linear functions in the hinge, and formal proofs are given for the universal representation ability for continuous PWL functions. A compact f–f model of high-dimensional piecewise-linear function over a degenerate intersection. The complete canonical piecewise-linear representation: functional form for minimal degenerate intersections. This work presents formal proofs on the universal representation ability of the lattice representation and summarizes different locally linear subregion realizations. Region configurations for realizability of lattice piecewise-linear models. The connection with ReLU in PWL-DNNs can be referred to. This paper introduces the hinging hyperplanes representation model and its hinge-finding learning algorithm. Hinging hyperplanes for regression, classification, and function approximation. Canonical representation: from piecewise-linear function to piecewise-smooth functions. A global representation of multidimensional piecewise-linear functions with linear partitions. This paper initiates the prevalence and state-of-the-art performance of PWL-DNNs, and establishes the most popular ReLU. This paper proposes the pioneering compact expression for PWL functions and formally introduces it for circuit systems, and analytical analysis for PWL functions has since become viable. Section-wise piecewise-linear functions: canonical representation, properties, and applications. This paper presents a systematic analysis of CPLR, including some crucial properties of PWLNNs.Ĭhua, L. Canonical piecewise-linear representation. Solving nonlinear resistive networks using piecewise-linear analysis and simplicial subdivision. Real Analysis: Modern Techniques and Their Applications (Wiley Interscience, 1999).Ĭhien, M.-J. Piecewise Linear Modeling and Analysis (Springer Science & Business Media, 2013).įolland, G. Then, representative applications are introduced together with discussions and outlooks. With PWLNNs, the evolution of learning algorithms for data is presented and fundamental theoretical analysis follows up for in-depth understandings. First, different PWLNN representation models are constructed with elaborated examples. In this Primer, we systematically introduce the methodology of PWLNNs by grouping the works into shallow and deep networks. Ever since, PWLNNs have been successfully applied to many tasks and achieved excellent performance. In 2010, rectified linear units (ReLU) advocated the prevalence of PWLNNs in deep learning. In 1977, the canonical representation pioneered the works of shallow PWLNNs learned by incremental designs, but the applications to large-scale data were prohibited. To apply PWLNN methods, both the representation and the learning have long been studied. As a powerful modelling method, piecewise linear neural networks (PWLNNs) have proven successful in various fields, most recently in deep learning. ![]()
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