学术报告报告主题: Approximation Theory of Deep Neural Network: Spherical Analysis with Deep Neural Networks by Encoder-Decoder Framework
报 告 人: 杨珍瑜
报告时间：2025年6月27日（星期五）下午14:30-15:30
报告地点：清清文理楼3A02
邀 请 人: 包学莲副教授
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数学学院
2025年6月20日
报告摘要：
Despite the phenomenal achievements of deep learning in areas such as speech recognition and computer vision, its mathematical theory is still in its early stages, particularly when it comes to functionals defined in non-Euclidean spaces, like spheres. Deep architectures utilizing ReLU neural networks provide a theoretical foundation for addressing these challenges and demonstrate tractability, as they effectively overcome the vanishing gradient problem and enable complex function approximation with fewer parameters. In this talk, we propose an encoder-decoder framework for continuous functionals on Sobolev spaces. The encoder projects functions from infinite-dimensional spherical domains into finite-dimensional spaces via linear operators and isometric isomorphism, while the decoder employs deep ReLU networks to approximate the target functionals. We construct three types of encoders to handle: (i) continuous inputs in Sobolev spaces, (ii) discretely sampled inputs, and (iii) discrete inputs corrupted by noise. Our theoretical analysis establishes approximation rates of logarithmic fractional form, parameterized by the total number of network parameters N, with the exponent scaling proportionally to the smoothness of the input over the ambient dimension. Furthermore, we address practical considerations arising from real-world data acquisition, including discretization and noise. By introducing a discrete inner product via cubature rules, we show that the approximation guarantees remain valid under these conditions.
报告人介绍：
杨珍瑜，香港城市大学数学系博士研究生。2019年本科毕业于武汉大学数学基地班，2022年硕士毕业于北京师范大学。研究方向为统计学习的数学理论与逼近论的应用。