Senwei Liang, Liyao Lyu, Chunmei Wang, Haizhao Yang
{"title":"Reproducing activation function for deep learning","authors":"Senwei Liang, Liyao Lyu, Chunmei Wang, Haizhao Yang","doi":"10.4310/cms.2024.v22.n2.a1","DOIUrl":null,"url":null,"abstract":"We propose reproducing activation functions (RAFs) motivated by applied and computational harmonic analysis to improve deep learning accuracy for various applications ranging from computer vision to scientific computing. The idea is to employ several basic functions and their learnable linear combination to construct neuron-wise data-driven activation functions for each neuron. Armed with RAFs, neural networks (NNs) can reproduce traditional approximation tools and, therefore, approximate target functions with a smaller number of parameters than traditional NNs. As demonstrated by extensive numerical tests, the proposed RAFs can facilitate the convergence of deep learning optimization for a solution with higher accuracy than existing deep learning solvers for audio/image/video reconstruction, PDEs, and eigenvalue problems. With RAFs, the errors of audio/video reconstruction, PDEs, and eigenvalue problems are decreased by over 14%, 73%, 99%, respectively, compared with baseline, while the performance of image reconstruction increases by 58%. Numerically, in the NN training, RAFs can generate neural tangent kernels with better condition numbers than traditional activation functions, which provides a prospective for understanding the improved optimization convergence using the theory of neural tangent kernel. The code is available $\\href{https://github.com/LeungSamWai/Reproducing-Activation-Function}{https://github.com/LeungSamWai/Reproducing-Activation-Function}$.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"12 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/cms.2024.v22.n2.a1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We propose reproducing activation functions (RAFs) motivated by applied and computational harmonic analysis to improve deep learning accuracy for various applications ranging from computer vision to scientific computing. The idea is to employ several basic functions and their learnable linear combination to construct neuron-wise data-driven activation functions for each neuron. Armed with RAFs, neural networks (NNs) can reproduce traditional approximation tools and, therefore, approximate target functions with a smaller number of parameters than traditional NNs. As demonstrated by extensive numerical tests, the proposed RAFs can facilitate the convergence of deep learning optimization for a solution with higher accuracy than existing deep learning solvers for audio/image/video reconstruction, PDEs, and eigenvalue problems. With RAFs, the errors of audio/video reconstruction, PDEs, and eigenvalue problems are decreased by over 14%, 73%, 99%, respectively, compared with baseline, while the performance of image reconstruction increases by 58%. Numerically, in the NN training, RAFs can generate neural tangent kernels with better condition numbers than traditional activation functions, which provides a prospective for understanding the improved optimization convergence using the theory of neural tangent kernel. The code is available $\href{https://github.com/LeungSamWai/Reproducing-Activation-Function}{https://github.com/LeungSamWai/Reproducing-Activation-Function}$.
期刊介绍:
Covers modern applied mathematics in the fields of modeling, applied and stochastic analyses and numerical computations—on problems that arise in physical, biological, engineering, and financial applications. The journal publishes high-quality, original research articles, reviews, and expository papers.