Piecewise Linear Neural Networks Research Articles

Achieving Sharp Upper Bounds on the Expressive Power of Neural Networks via Tropical Polynomials.

The expressive power of neural networks describes the ability to represent or approximate complex functions. The number of linear regions is the standard and most natural measure of expressive power. However, a major challenge in utilizing the number of linear regions as a measure of expressive power is the exponential gap between the theoretical upper and lower bounds, which becomes more pronounced as the neural network capacity increases. In this article, we aim to derive a sharp upper bound on piecewise linear neural networks (PLNNs) to bridge this gap. Specifically, we first establish the relationship between tropical polynomials and PLNNs. In the unexpanded tropical polynomials form, we make the proposition that hyperplanes are not all in the general positions, thereby reducing the number of intersecting hyperplanes. We propose a rank-based approach and present the empirical analysis that this approach outperforms previous Zaslavsky's theorem-based methods. In the expanded tropical polynomials form, accounting for limitations in weight initialization and model computational precision, we raise the concept that the values range of each term is bounded. We propose a precision-based approach that transforms the approximate exponential growth of the number of linear regions into polynomial growth with width, which is effective at larger layer widths. Finally, we compare the number of linear regions that can be represented by each hidden layer in both forms and derive a sharp upper bound for PLNNs. Empirical analysis and experimental results provide compelling evidence for the efficacy and feasibility of this sharp upper bound on both simulated experiments and real datasets.

Activation Control of Multiple Piecewise Linear Neural Networks

Activation Control of Multiple Piecewise Linear Neural Networks

On the number of regions of piecewise linear neural networks

Many feedforward neural networks (NNs) generate continuous and piecewise-linear (CPWL) mappings. Specifically, they partition the input domain into regions on which the mapping is affine. The number of these so-called linear regions offers a natural metric to characterize the expressiveness of CPWL NNs. The precise determination of this quantity is often out of reach in practice, and bounds have been proposed for specific architectures, including for ReLU and Maxout NNs. In this work, we generalize these bounds to NNs with arbitrary and possibly multivariate CPWL activation functions. We first provide upper and lower bounds on the maximal number of linear regions of a CPWL NN given its depth, width, and the number of linear regions of its activation functions. Our results rely on the combinatorial structure of convex partitions and confirm the distinctive role of depth which, on its own, is able to exponentially increase the number of regions. We then introduce a complementary stochastic framework to estimate the average number of linear regions produced by a CPWL NN. Under reasonable assumptions, the expected density of linear regions along any 1D path is bounded by the product of depth, width, and a measure of activation complexity (up to a scaling factor). This yields an identical role to the three sources of expressiveness: no exponential growth with depth is observed anymore.

Towards rigorous understanding of neural networks via semantics-preserving transformations

AbstractIn this paper, we present an algebraic approach to the precise and global verification and explanation of Rectifier Neural Networks, a subclass of Piece-wise Linear Neural Networks (PLNNs), i.e., networks that semantically represent piece-wise affine functions. Key to our approach is the symbolic execution of these networks that allows the construction of semantically equivalent Typed Affine Decision Structures (TADS). Due to their deterministic and sequential nature, TADS can, similarly to decision trees, be considered as white-box models and therefore as precise solutions to the model and outcome explanation problem. TADS are linear algebras, which allows one to elegantly compare Rectifier Networks for equivalence or similarity, both with precise diagnostic information in case of failure, and to characterize their classification potential by precisely characterizing the set of inputs that are specifically classified, or the set of inputs where two network-based classifiers differ. All phenomena are illustrated along a detailed discussion of a minimal, illustrative example: the continuous XOR function.

Physics Informed Piecewise Linear Neural Networks for Process Optimization

Constructing first-principles models is usually a challenging and time-consuming task due to the complexity of real-life processes. On the other hand, data-driven modeling, particularly a neural network model, often suffers from overfitting and lack of useful and high-quality data. At the same time, embedding trained machine learning models directly into the optimization problems has become an effective and state-of-the-art approach for surrogate optimization, whose performance can be improved by physics-informed machine learning. This study proposes using piecewise linear neural network models with physics-informed knowledge for optimization problems with neural network models embedded. In addition to using widely accepted and naturally piecewise linear rectified linear unit (ReLU) activation functions, this study also suggests piecewise linear approximations for the hyperbolic tangent activation function to widen the domain. Optimization of three case studies, a blending process, an industrial distillation column, and a crude oil column are investigated. Physics-informed trained neural network-based optimal results are closer to global optimality for all cases. Finally, associated CPU times for the optimization problems are much shorter than the standard optimization results.

Piecewise linear neural networks and deep learning

As a powerful modelling method, PieceWise Linear Neural Networks (PWLNNs) have proven successful in various fields, most recently in deep learning. To apply PWLNN methods, both the representation and the learning have long been studied. In 1977, the canonical representation pioneered the works of shallow PWLNNs learned by incremental designs, but the applications to large-scale data were prohibited. In 2010, the Rectified Linear Unit (ReLU) advocated the prevalence of PWLNNs in deep learning. Ever since, PWLNNs have been successfully applied to extensive tasks and achieved advantageous performances. In this Primer, we systematically introduce the methodology of PWLNNs by grouping the works into shallow and deep networks. Firstly, different PWLNN representation models are constructed with elaborated examples. With PWLNNs, the evolution of learning algorithms for data is presented and fundamental theoretical analysis follows up for in-depth understandings. Then, representative applications are introduced together with discussions and outlooks.

Piecewise linear neural networks and deep learning

Piecewise linear neural networks and deep learning

Analysis on the Number of Linear Regions of Piecewise Linear Neural Networks.

Deep neural networks (DNNs) are shown to be excellent solutions to staggering and sophisticated problems in machine learning. A key reason for their success is due to the strong expressive power of function representation. For piecewise linear neural networks (PLNNs), the number of linear regions is a natural measure of their expressive power since it characterizes the number of linear pieces available to model complex patterns. In this article, we theoretically analyze the expressive power of PLNNs by counting and bounding the number of linear regions. We first refine the existing upper and lower bounds on the number of linear regions of PLNNs with rectified linear units (ReLU PLNNs). Next, we extend the analysis to PLNNs with general piecewise linear (PWL) activation functions and derive the exact maximum number of linear regions of single-layer PLNNs. Moreover, the upper and lower bounds on the number of linear regions of multilayer PLNNs are obtained, both of which scale polynomially with the number of neurons at each layer and pieces of PWL activation function but exponentially with the number of layers. This key property enables deep PLNNs with complex activation functions to outperform their shallow counterparts when computing highly complex and structured functions, which, to some extent, explains the performance improvement of deep PLNNs in classification and function fitting.

High-precision linearized interpretation for fully connected neural network

Despite the widespread application of deep neural networks in finance, medical treatment, and autonomous driving, these networks face multiple security threats, such as maliciously constructed adversarial samples that can easily mislead deep neural network model classification, causing errors. Therefore, creating an interpretable model or designing an interpretation method is necessary to improve its security. This paper presents an interpretation scheme, named Convergent Interpretation for Deep Neural Networks (CIDNN), to obtain a provably convergent and consistent interpretation for deep neural networks. The main idea of CIDNN is to first convert the deep neural networks into a set of mathematically convergent Piecewise Linear Neural Networks (PLNN), then convert the PLNN into a set of equivalent linear classifiers. In this way, each linear classifier can be interpreted by its decision features. By analyzing the convergence of the local approximation interpretation scheme, we prove that this interpretable model can be sufficiently close to the deep neural network with certain conditions. Experiments show the convergence of CIDNN’s interpretation, and the interpretation conforms with similar samples in the synthetic dataset. Besides, we demonstrate the semantical meaning of CIDNN in the Fashion-MNIST dataset.

Limit Set Dichotomy and Convergence of Cooperative Piecewise Linear Neural Networks

This paper considers a class of nonsymmetric cooperative neural networks (NNs) where the neurons are fully interconnected and the neuron activations are modeled by piecewise linear (PL) functions. The solution semiflow generated by cooperative PLNNs is monotone but, due to the horizontal segments in the neuron activations, is not eventually strongly monotone (ESM). The main result in this paper is that it is possible to prove a peculiar form of the Limit Set Dichotomy for this class of cooperative PLNNs. Such a form is slightly weaker than the standard form valid for ESM semiflows, but this notwithstanding it permits to establish a result on convergence analogous to that valid for ESM semiflows. Namely, for almost every choice of the initial conditions, each solution of a fully interconnected cooperative PLNN converges toward an equilibrium point, depending on the initial conditions, as t → +∞. From a methodological viewpoint, this paper extends some basic techniques and tools valid for ESM semiflows, in order that they can be applied to the monotone semiflows generated by the considered class of cooperative PLNNs.

Configuration of Continuous Piecewise-Linear Neural Networks

The problem of constructing a general continuous piecewise-linear neural network is considered in this paper. It is shown that every projection domain of an arbitrary continuous piecewise-linear function can be partitioned into convex polyhedra by using difference functions of its local linear functions. Based on these convex polyhedra, a group of continuous piecewise-linear basis functions are formulated. It is proven that a linear combination of these basis functions plus a constant, which we call a standard continuous piecewise-linear neural network, can represent all continuous piecewise-linear functions. In addition, the proposed standard continuous piecewise-linear neural network is applied to solve some function approximation problems. A number of numerical experiments are presented to illustrate that the standard continuous piecewise-linear neural network can be a promising tool for function approximation.

Piecewise-Linear Neural Networks and Their Relationship to Rule Extraction from Data

This article addresses the topic of extracting logical rules from data by means of artificial neural networks. The approach based on piecewise linear neural networks is revisited, which has already been used for the extraction of Boolean rules in the past, and it is shown that this approach can be important also for the extraction of fuzzy rules. Two important theoretical properties of piecewise-linear neural networks are proved, allowing an elaboration of the basic ideas of the approach into several variants of an algorithm for the extraction of Boolean rules. That algorithm has already been used in two real-world applications. Finally, a connection to the extraction of rules of the Łukasiewicz logic is established, relying on recent results about rational McNaughton functions. Based on one of the constructive proofs of the McNaughton theorem, an algorithm is formulated that in principle allows extracting a particular kind of formulas of the Łukasiewicz predicate logic from piecewise-linear neural networks trained with rational data.

Convergent design of piecewise linear neural networks

Piecewise linear networks (PLNs) are attractive because they can be trained quickly and provide good performance in many nonlinear approximation problems. Most existing design algorithms for piecewise linear networks are not convergent, non-optimal, or are not designed to handle noisy data. In this paper, four algorithms are presented which attack this problem. They are: (1) a convergent design algorithm which builds the PLN one module at a time using a branch and bound technique; (2) two pruning algorithms which eliminate less useful modules from the network; and (3) a sifting algorithm which picks the best networks out of the many designed. The performance of the PLN is compared with that of the multilayer perceptron (MLP) using several benchmark data sets. Numerical results demonstrate that piecewise linear networks are adequate for many approximation problems.

A new algorithm for learning in piecewise-linear neural networks

Piecewise-linear (PWL) neural networks are widely known for their amenability to digital implementation. This paper presents a new algorithm for learning in PWL networks consisting of a single hidden layer. The approach adopted is based upon constructing a continuous PWL error function and developing an efficient algorithm to minimize it. The algorithm consists of two basic stages in searching the weight space. The first stage of the optimization algorithm is used to locate a point in the weight space representing the intersection of N linearly independent hyperplanes, with N being the number of weights in the network. The second stage is then called to use this point as a starting point in order to continue searching by moving along the single-dimension boundaries between the different linear regions of the error function, hopping from one point (representing the intersection of N hyperplanes) to another. The proposed algorithm exhibits significantly accelerated convergence, as compared to standard algorithms such as back-propagation and improved versions of it, such as the conjugate gradient algorithm. In addition, it has the distinct advantage that there are no parameters to adjust, and therefore there is no time-consuming parameters tuning step. The new algorithm is expected to find applications in function approximation, time series prediction and binary classification problems.