{"url":"/method/kaf","slug":"kaf","name":"KAF","full_name":"Kernel Activation Function","full_name_withheld":false,"description_markdown":"A **Kernel Activation Function** is a non-parametric activation function defined as a one-dimensional kernel approximator:\r\n\r\n$$ f(s) = \\sum_{i=1}^D \\alpha_i \\kappa( s, d_i) $$\r\n\r\nwhere:\r\n\r\n1. The dictionary of the kernel elements $d_0, \\ldots, d_D$ is fixed by sampling the $x$-axis with a uniform step around 0.\r\n2. The user selects the kernel function (e.g., Gaussian, [ReLU](https://paperswithcode.com/method/relu), [Softplus](https://paperswithcode.com/method/softplus)) and the number of kernel elements $D$ as a hyper-parameter. A larger dictionary leads to more expressive activation functions and a larger number of trainable parameters.\r\n3. The linear coefficients are adapted independently at every neuron via standard back-propagation.\r\n\r\nIn addition, the linear coefficients can be initialized using kernel ridge regression to behave similarly to a known function in the beginning of the optimization process.","description_state":"present","introduced_year":null,"introduced_by":{"title":"Kafnets: kernel-based non-parametric activation functions for neural networks","paper":"/paper/kafnets-kernel-based-non-parametric","first_author":"Simone Scardapane","n_authors":4,"url_abs":null,"archive_paper_url":"https://paperswithcode.com/paper/kafnets-kernel-based-non-parametric"},"source":{"url":"http://arxiv.org/abs/1707.04035v2","title":"Kafnets: kernel-based non-parametric activation functions for neural networks","url_on_a_paper_host":true},"code_snippet_url":"https://github.com/ispamm/kernel-activation-functions/blob/e525f7e82b54508ac262f82c9557ad66be2732f3/pytorch/kafnets.py#L11","code_snippet_url_on_a_code_host":true,"categories":[{"area":"General","area_id":"general","collection":"Activation Functions","url":"/methods/category/activation-functions","pwc_aliases":[]}],"n_papers_tagged":9,"archive_num_papers":9,"papers_newest_first":[{"paper":null,"title":"Knowledge Augmentation in Federation: Rethinking What Collaborative Learning Can Bring Back to Decentralized Data","date":"2025-03-05","arxiv_id":"2503.03140","n_code_links":0,"syntology":null},{"paper":null,"title":"Kolmogorov-Arnold Fourier Networks","date":"2025-02-09","arxiv_id":"2502.06018","n_code_links":0,"syntology":null},{"paper":null,"title":"An Analytic Solution for Kernel Adaptive Filtering","date":"2024-02-05","arxiv_id":"2402.03497","n_code_links":0,"syntology":null},{"paper":null,"title":"Association of stroke lesion distributions with atrial fibrillation detected after stroke","date":"2023-05-24","arxiv_id":"2305.14609","n_code_links":0,"syntology":null},{"paper":null,"title":"The Functional Wiener Filter","date":"2022-12-31","arxiv_id":"2301.00291","n_code_links":0,"syntology":null},{"paper":"/paper/learning-to-forecast-dynamical-systems-from","title":"Learning to Forecast Dynamical Systems from Streaming Data","date":"2021-09-20","arxiv_id":"2109.09703","n_code_links":1,"syntology":null},{"paper":null,"title":"Multikernel activation functions: formulation and a case study","date":"2019-01-29","arxiv_id":"1901.10232","n_code_links":0,"syntology":null},{"paper":null,"title":"Improving Sparsity in Kernel Adaptive Filters Using a Unit-Norm Dictionary","date":"2017-07-13","arxiv_id":"1707.04236","n_code_links":0,"syntology":null},{"paper":"/paper/kafnets-kernel-based-non-parametric","title":"Kafnets: kernel-based non-parametric activation functions for neural networks","date":"2017-07-13","arxiv_id":"1707.04035","n_code_links":2,"syntology":null}],"papers_shown":9,"tasks":[{"task":"/task/time-series-1","name":"Time Series","papers":3},{"task":"/task/time-series","name":"Time Series Analysis","papers":3},{"task":"/task/computational-efficiency","name":"Computational Efficiency","papers":1},{"task":"/task/fairness","name":"Fairness","papers":1},{"task":"/task/federated-learning","name":"Federated Learning","papers":1},{"task":"/task/optical-character-recognition","name":"Optical Character Recognition (OCR)","papers":1},{"task":"/task/privacy-preserving","name":"Privacy Preserving","papers":1},{"task":"/task/regression-1","name":"regression","papers":1}],"tasks_shown":8,"n_tasks":8,"usage_by_year":[{"year":"2017","papers":2},{"year":"2019","papers":1},{"year":"2021","papers":1},{"year":"2022","papers":1},{"year":"2023","papers":1},{"year":"2024","papers":1},{"year":"2025","papers":2}],"row_source":"methods_table","archive":{"source":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","archive_url":"https://paperswithcode.com/method/kaf"},"syntology_read_at":"2026-09-24T18:15:14+00:00"}