RKHS代数的卷积滤波

IF 4.6 2区 工程技术 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC
Alejandro Parada-Mayorga;Leopoldo Agorio;Alejandro Ribeiro;Juan Bazerque
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引用次数: 0

摘要

在本文中,我们发展了卷积信号处理和神经网络的广义理论,用于再现核希尔伯特空间(RKHS)。利用代数信号处理(ASP)理论,我们证明了任何RKHS都允许对多个代数卷积模型进行形式化定义。我们证明了任何RKHS诱导代数,其元素决定作用于RKHS元素的卷积算子。这种方法允许我们实现可扩展的过滤和学习作为卷积模型的副产品,同时利用在RKHS中处理信息的众所周知的好处。为了强调我们方法的通用性和实用性,我们展示了如何使用代数RKHS来定义群、图元和传统欧几里得信号空间上的卷积信号模型。此外,使用代数RKHS模型,我们建立了卷积网络,正式定义了点非线性的概念,并推导了训练的显式表达式。这样的推导是根据RKHS的代数表示得到的。我们提出了一组在实际数据上的数值实验,其中无线覆盖是由无人机捕获的测量预测的。这个特殊的现实场景强调了卷积RKHS模型在神经网络中与完全连接和标准卷积算子相比的优势。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Convolutional Filtering With RKHS Algebras
In this paper, we develop a generalized theory of convolutional signal processing and neural networks for Reproducing Kernel Hilbert Spaces (RKHS). Leveraging the theory of algebraic signal processing (ASP), we show that any RKHS allows the formal definition of multiple algebraic convolutional models. We show that any RKHS induces algebras whose elements determine convolutional operators acting on RKHS elements. This approach allows us to achieve scalable filtering and learning as a byproduct of the convolutional model, and simultaneously take advantage of the well-known benefits of processing information in an RKHS. To emphasize the generality and usefulness of our approach, we show how algebraic RKHS can be used to define convolutional signal models on groups, graphons, and traditional Euclidean signal spaces. Furthermore, using algebraic RKHS models, we build convolutional networks, formally defining the notion of pointwise nonlinearities and deriving explicit expressions for the training. Such derivations are obtained in terms of the algebraic representation of the RKHS. We present a set of numerical experiments on real data in which wireless coverage is predicted from measurements captured by unmaned aerial vehicles. This particular real-life scenario emphasizes the benefits of the convolutional RKHS models in neural networks compared to fully connected and standard convolutional operators.
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来源期刊
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing 工程技术-工程:电子与电气
CiteScore
11.20
自引率
9.30%
发文量
310
审稿时长
3.0 months
期刊介绍: The IEEE Transactions on Signal Processing covers novel theory, algorithms, performance analyses and applications of techniques for the processing, understanding, learning, retrieval, mining, and extraction of information from signals. The term “signal” includes, among others, audio, video, speech, image, communication, geophysical, sonar, radar, medical and musical signals. Examples of topics of interest include, but are not limited to, information processing and the theory and application of filtering, coding, transmitting, estimating, detecting, analyzing, recognizing, synthesizing, recording, and reproducing signals.
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