Approximation Theory

Approximation theory focuses on finding simpler, computationally tractable representations of complex functions or data, aiming to minimize the error introduced by this simplification. Current research emphasizes optimal sampling strategies for least-squares approximation, analyzing the approximation capabilities of deep neural networks (including convolutional and ReLU networks) and their application to diverse problems like image processing and stochastic process modeling, and developing theoretical frameworks for understanding approximation errors in models such as CycleGANs and Graph Convolutional Networks. These advancements have significant implications for machine learning, scientific computing, and various engineering applications by enabling efficient and accurate modeling of high-dimensional data and complex systems.