相关论文: Any Target Function Exists in a Neighborhood of An…

We study agents playing a pure coordination game on a large social network. Agents are restricted to coordinate locally, without access to a global communication device, and so different regions of the network will converge to different…

In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure…

We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of…

Complex systems, ranging from soft materials to wireless communication, are often organised as random geometric networks in which nodes and edges evenly fill up the volume of some space. Studying such networks is difficult because they…

Generative networks have shown remarkable success in learning complex data distributions, particularly in generating high-dimensional data from lower-dimensional inputs. While this capability is well-documented empirically, its theoretical…

Training neural networks means solving a high-dimensional optimization problem. Normally the goal is to minimize a loss function that depends on what is called the network function, or in other words the function that gives the network…

Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation…

The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer…

Neural network width and depth are fundamental aspects of network topology. Universal approximation theorems provide that with increasing width or depth, there exists a neural network that approximates a function arbitrarily well. These…

The functionality of a graph $G$ is the minimum number $k$ such that in every induced subgraph of $G$ there exists a vertex whose neighbourhood is uniquely determined by the neighborhoods of at most $k$ other vertices in the subgraph. The…

In this work, we address the question whether a sufficiently deep quantum neural network can approximate a target function as accurate as possible. We start with simple but typical physical situations that the target functions are physical…

We consider the ability of deep neural networks to represent data that lies near a low-dimensional manifold in a high-dimensional space. We show that deep networks can efficiently extract the intrinsic, low-dimensional coordinates of such…

The success of Neural networks in providing miraculous results when applied to a wide variety of tasks is astonishing. Insight in the working can be obtained by studying the universal approximation property of neural networks. It is proved…

Network models with latent geometry have been used successfully in many applications in network science and other disciplines, yet it is usually impossible to tell if a given real network is geometric, meaning if it is a typical element in…

The stunning empirical successes of neural networks currently lack rigorous theoretical explanation. What form would such an explanation take, in the face of existing complexity-theoretic lower bounds? A first step might be to show that…

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a…

This paper investigates the connection between neural networks and sufficient dimension reduction (SDR), demonstrating that neural networks inherently perform SDR in regression tasks under appropriate rank regularizations. Specifically, the…

We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global…

Training neural networks to be certifiably robust is critical to ensure their safety against adversarial attacks. However, it is currently very difficult to train a neural network that is both accurate and certifiably robust. In this work…

Random projections are able to perform dimension reduction efficiently for datasets with nonlinear low-dimensional structures. One well-known example is that random matrices embed sparse vectors into a low-dimensional subspace nearly…