Related papers: Universal Approximation and the Topological Neural…
Deep Neural Networks (DNNs) are universal function approximators providing state-of- the-art solutions on wide range of applications. Common perceptual tasks such as speech recognition, image classification, and object tracking are now…
We study feedforward neural networks with inputs from a topological space (TFNNs). We prove a universal approximation theorem for shallow TFNNs, which demonstrates their capacity to approximate any continuous function defined on this…
We study feedforward neural networks with inputs from a topological vector space (TVS-FNNs). Unlike traditional feedforward neural networks, TVS-FNNs can process a broader range of inputs, including sequences, matrices, functions and more.…
We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We…
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 standard cosmological model with cold dark matter posits a hierarchical formation of structures. We introduce topological neural networks (TNNs), implemented as message-passing neural networks on higher-order structures, to effectively…
Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function…
Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine…
We propose tensorial neural networks (TNNs), a generalization of existing neural networks by extending tensor operations on low order operands to those on high order ones. The problem of parameter learning is challenging, as it corresponds…
Is there any theoretical guarantee for the approximation ability of neural networks? The answer to this question is the "Universal Approximation Theorem for Neural Networks". This theorem states that a neural network is dense in a certain…
In this paper, a universal approximation theorem (UAT) for shallow neural networks whose inputs belong to a topological vector space (TVS) and whose outputs take values in a Hausdorff locally convex TVS is established. The networks are…
How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use…
Deep neural networks (DNN) are black box algorithms. They are trained using a gradient descent back propagation technique which trains weights in each layer for the sole goal of minimizing training error. Hence, the resulting weights cannot…
The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications,…
The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the…
In this paper, we aim at establishing an approximation theory and a learning theory of distribution regression via a fully connected neural network (FNN). In contrast to the classical regression methods, the input variables of distribution…
Neural networks (NNs) are known for their high predictive accuracy in complex learning problems. Beside practical advantages, NNs also indicate favourable theoretical properties such as universal approximation (UA) theorems. Binarized…
Topological Neural Networks (TNNs) incorporate higher-order relational information beyond pairwise interactions, enabling richer representations than Graph Neural Networks (GNNs). Concurrently, topological descriptors based on persistent…
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear…
This paper introduces a tensor neural network (TNN) to address nonparametric regression problems, leveraging its distinct sub-network structure to effectively facilitate variable separation and enhance the approximation of complex,…