Time-frequency analysis in deep learning framework: theory, computation and applications

异常的大脑活动或结构振动突然发作并以不同的频率范围发生，可能提供异常情况或结构故障的早期预警。时频分析（TFA）由描述频率内容如何随时间演变的各种数学变换组成。局部特征通常首先从时变信号的时频表示中提取，然后提供给机器学习算法进行分类和模式识别。因此，将TFA集成到实时监测和诊断系统中，我们可以及早发现，防止昂贵和致命的损害。尽管TFA很有用，但两个主要问题严重阻碍了基于TFA的监测和诊断的有效性。首先，TFA的计算工作量通常很大。这在处理超大型数据集和/或实时数据时尤其成问题。其次，许多标准的TFA技术使用一组固定的基/帧来分解信号。在实际应用中，许多重要的应用都会产生高度随机和时变的信号。使用同一组基/帧来分析整个信号可能效果不佳。近年来，随着强大的计算资源的唾手可得，深度学习神经网络在图像识别、制造、疾病诊断、语音处理等方面可以获得更好的性能。深度学习网络的主要优点是它不需要专业知识，但通常比经典方法产生更高的准确性。但对深度学习网络的数学理解仍不清楚。本研究计划的主要目的是从理论上研究深度学习框架下的时频分析，并探讨这种组合是否可以克服现有经典时频分析的局限性。我们的目标是回答以下问题：1)时频分析的深度学习框架是否会改变原始信号？2)它是否有助于动态确定信号的最佳时频表示？3)如果是这样，使用深度学习架构的动态时频分析的基本数学性质是什么？ 4)我们能否实现快速算法来计算这种表示？这个研究计划是跨学科的和实用的。它集成了数学、深度学习、计算和应用程序开发。该计划的学员将与数据科学、医疗保健和工业领域的科学家密切合作，并对现实世界的重要问题做出重大的原创性贡献。该项目的成功将推进时频分析领域，为特定类型的信号创建复杂的时频分析，并导致各种应用的计算机辅助监测和检测软件的研发发展。

Abnormal brain activities or structural vibrations that have a sudden onset and occur at a distinct frequency range, may provide early warnings of abnormal conditions or structural failure. Time-frequency analysis (TFA) is comprised of various mathematical transforms that describe how frequency content evolves with time. Local features are often first extracted from the time-frequency representation of time-varying signals and then feed to the machine learning algorithms for classification and pattern recognition. Thus, integrating TFA into a realtime monitoring and diagnosis system, we can make early detection to prevent expensive and fatal damages. Despite of its usefulness, two main issues severely impede the effectiveness of the TFA based monitoring and diagnoses. First, the computational effort of TFA is often substantial. This is particularly problematic for processing ultra large data set and/or realtime data. Second, many standard TFA techniques use a fixed set of bases/frames to decompose a signal. In practice, many of important applications will generate highly random and timevarying signals. Using the same set of bases/frames to analyze the entire signal may not be effective. In recent years, with the easy access to powerful computing resources, better performance can be obtained using deep learning neural networks in image recognition, manufacturing], disease diagnosis, and speech processing. The main advantages of deep learning network is that it does not require expert knowledge but often produces higher accuracy than the classic methods. But mathematical understanding of deep learning networks is still not yet clear. The main objective of this research program is to theoretically investigate the time-frequency analysis in the deep learning framework and explore whether such a combination can overcome the limitations of existing classic time-frequency analysis. We aim to answer the following questions: 1) Does the deep learning framework of a time-frequency analysis alter the original signal? 2) Can it help to determine an optimal time-frequency representation of a signal dynamically? 3) If so, what are the essential mathematical properties of dynamic timefrequency analysis using deep learning architecture 4) Can we implement fast algorithms to compute such a representation? This research proposal is interdisciplinary and practical. It integrates mathematics, deep learning, computing, and application development. The trainees in the program will work closely with scientists in data science, health care and industries and make significant original contributions relevant to important real world problems. The success of this program will advance time-frequency analysis field, create sophisticated  time-frequency analysis tailored for specific types of signals and lead to the R&D development of computer-assisted monitoring and detection software for various applications.