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

突然发作并以不同频率范围发生的异常大脑活动或结构振动可以提供异常状况或结构故障的早期警告。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，然后馈送到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，我们可以进行早期检测，以防止昂贵和致命的损害。 尽管其有用性，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的计算工作量通常是巨大的。这对于处理超大数据集和/或实时数据尤其成问题。其次，许多标准TFA技术使用固定的基/帧集合来分解信号。在实际应用中，许多重要的应用将产生高度随机和时变的信号。使用相同的基/帧集合来分析整个信号可能不是有效的。近年来，随着强大的计算资源的容易获取，使用深度学习神经网络可以在图像识别、制造、疾病诊断和语音处理中获得更好的性能。深度学习网络的主要优点是它不需要专家知识，但往往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然不清楚。该研究计划的主要目标是从理论上研究深度学习框架中的时频分析，并探索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)它能帮助动态地确定信号的最佳时频表示吗？3)如果是这样的话，使用深度学习架构的动态时频分析的基本数学属性是什么4）我们可以实现快速算法来计算这样的表示吗？本研究具有跨学科性和实用性。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，并为重要的真实的世界问题做出重要的原创性贡献。该计划的成功将推动时频分析领域的发展，为特定类型的信号量身定制复杂的时频分析，并导致用于各种应用的计算机辅助监测和检测软件的研发。

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.