摘要翻译：
在本文中，我们提出并奠定了一个表示信号的功能框架的基础。在经典集合论测度理论的基础上，引入了额外的范畴论相对和生成的观点，以一种新的真实箭论的方式描述了冗余的基本概念，压缩。将信号表示为适当线性空间向量的现有经典框架作为该框架的特例。其次，在信号空间作为范畴的背景下，我们利用可测或测度空间范畴及其对立面，包括布尔代数和测度代数以及部分扩张，研究了$L^0$和$L^2$函子的各种协变和逆变形式。最后，利用同构箭头的一般概念，给出了信号内冗余的一个新定义，并将平移情况和其他情况作为特例。通过范畴理论，我们对著名的无损差分编码标准的启发式给出了一个简单而精确的解释，从而在线条图、图标图像、文本等图像类型中产生更好的压缩；与JPEG等经典表示技术相比，它在全局希尔伯特空间中选择基或帧。
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英文标题：
《Unified Functorial Signal Representation III: Foundations, Redundancy,
  $L^0$ and $L^2$ functors》
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作者：
Salil Samant and Shiv Dutt Joshi
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最新提交年份：
2017
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分类信息：

一级分类：Electrical Engineering and Systems Science        电气工程与系统科学
二级分类：Signal Processing        信号处理
分类描述：Theory, algorithms, performance analysis and applications of signal and data analysis, including physical modeling, processing, detection and parameter estimation, learning, mining, retrieval, and information extraction. The term "signal" includes speech, audio, sonar, radar, geophysical, physiological, (bio-) medical, image, video, and multimodal natural and man-made signals, including communication signals and data. Topics of interest include: statistical signal processing, spectral estimation and system identification; filter design, adaptive filtering / stochastic learning; (compressive) sampling, sensing, and transform-domain methods including fast algorithms; signal processing for machine learning and machine learning for signal processing applications; in-network and graph signal processing; convex and nonconvex optimization methods for signal processing applications; radar, sonar, and sensor array beamforming and direction finding; communications signal processing; low power, multi-core and system-on-chip signal processing; sensing, communication, analysis and optimization for cyber-physical systems such as power grids and the Internet of Things.
信号和数据分析的理论、算法、性能分析和应用，包括物理建模、处理、检测和参数估计、学习、挖掘、检索和信息提取。“信号”一词包括语音、音频、声纳、雷达、地球物理、生理、（生物）医学、图像、视频和多模态自然和人为信号，包括通信信号和数据。感兴趣的主题包括：统计信号处理、谱估计和系统辨识；滤波器设计；自适应滤波/随机学习；（压缩）采样、传感和变换域方法，包括快速算法；用于机器学习的信号处理和用于信号处理应用的机器学习；网络与图形信号处理；信号处理中的凸和非凸优化方法；雷达、声纳和传感器阵列波束形成和测向；通信信号处理；低功耗、多核、片上系统信号处理；信息物理系统的传感、通信、分析和优化，如电网和物联网。
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一级分类：Computer Science        计算机科学
二级分类：Information Theory        信息论
分类描述：Covers theoretical and experimental aspects of information theory and coding. Includes material in ACM Subject Class E.4 and intersects with H.1.1.
涵盖信息论和编码的理论和实验方面。包括ACM学科类E.4中的材料，并与H.1.1有交集。
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一级分类：Mathematics        数学
二级分类：Category Theory        范畴理论
分类描述：Enriched categories, topoi, abelian categories, monoidal categories, homological algebra
丰富范畴，topoi，abelian范畴，monoidal范畴，同调代数
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一级分类：Mathematics        数学
二级分类：Information Theory        信息论
分类描述：math.IT is an alias for cs.IT. Covers theoretical and experimental aspects of information theory and coding.
它是cs.it的别名。涵盖信息论和编码的理论和实验方面。
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英文摘要：
  In this paper we propose and lay the foundations of a functorial framework for representing signals. By incorporating additional category-theoretic relative and generative perspective alongside the classic set-theoretic measure theory the fundamental concepts of redundancy, compression are formulated in a novel authentic arrow-theoretic way. The existing classic framework representing a signal as a vector of appropriate linear space is shown as a special case of the proposed framework.   Next in the context of signal-spaces as a categories we study the various covariant and contravariant forms of $L^0$ and $L^2$ functors using categories of measurable or measure spaces and their opposites involving Boolean and measure algebras along with partial extension. Finally we contribute a novel definition of intra-signal redundancy using general concept of isomorphism arrow in a category covering the translation case and others as special cases. Through category-theory we provide a simple yet precise explanation for the well-known heuristic of lossless differential encoding standards yielding better compressions in image types such as line drawings, iconic image, text etc; as compared to classic representation techniques such as JPEG which choose bases or frames in a global Hilbert space.
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PDF链接：
https://arxiv.org/pdf/1710.10227