Related papers: An approach to reachability analysis for feed-forw…

Over-approximating the reachable sets of dynamical systems is a fundamental problem in safety verification and robust control synthesis. The representation of these sets is a key factor that affects the computational complexity and the…

In this paper, we develop a framework for showing that neural networks can overcome the curse of dimensionality in different high-dimensional approximation problems. Our approach is based on the notion of a catalog network, which is a…

Deep learning models have achieved state-of-the-art performance in many classification tasks. However, most of them cannot provide an interpretation for their classification results. Machine learning models that are interpretable are…

We consider approximations of general continuous functions on finite-dimensional cubes by general deep ReLU neural networks and study the approximation rates with respect to the modulus of continuity of the function and the total number of…

Neural networks are playing a crucial role in everyday life, with the most modern generative models able to achieve impressive results. Nonetheless, their functioning is still not very clear, and several strategies have been adopted to…

This paper presents an investigation of the approximation property of neural networks with unbounded activation functions, such as the rectified linear unit (ReLU), which is the new de-facto standard of deep learning. The ReLU network can…

The Rectified Power Unit (RePU) activation function, a differentiable generalization of the Rectified Linear Unit (ReLU), has shown promise in constructing neural networks due to its smoothness properties. However, deep RePU networks often…

We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a…

In this article we study high-dimensional approximation capacities of shallow and deep artificial neural networks (ANNs) with the rectified linear unit (ReLU) activation. In particular, it is a key contribution of this work to reveal that…

Nonlinear activation functions are widely recognized for enhancing the expressivity of neural networks, which is the primary reason for their widespread implementation. In this work, we focus on ReLU activation and reveal a novel and…

Complementarity problems, a class of mathematical optimization problems with orthogonality constraints, are widely used in many robotics tasks, such as locomotion and manipulation, due to their ability to model non-smooth phenomena (e.g.,…

Neural networks are powerful functions with widespread use, but the theoretical behaviour of these functions is not fully understood. Creating deep neural networks by stacking many layers has achieved exceptional performance in many…

The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth…

We study the complexity of the problem of training neural networks defined via various activation functions. The training problem is known to be existsR-complete with respect to linear activation functions and the ReLU activation function.…

Deep neural networks paved the way for significant improvements in image visual categorization during the last years. However, even though the tasks are highly varying, differing in complexity and difficulty, existing solutions mostly build…

Robust control problems have significant practical implications since external disturbances can significantly impact the performance of control methods. Existing robust control methods excel at control-affine systems but fail at neural…

In this paper, a robust optimization framework is developed to train shallow neural networks based on reachability analysis of neural networks. To characterize noises of input data, the input training data is disturbed in the description of…

In the first part of this paper we introduced an algorithm that uses reachable set approximation to approximate the minimum time function of linear control problems. To illustrate the error estimates and to demonstrate differences to other…

Would it be possible to explain the emergence of new computational ideas using the computation itself? Would it be feasible to describe the discovery process of new algorithmic solutions using only mathematics? This study is the first…

An established measure of the expressive power of a given ReLU neural network is the number of linear regions into which it partitions the input space. There exist many different, non-equivalent definitions of what a linear region actually…