Low-dimensional structures in high-dimensional data

自然图像往往是可压缩的，也就是说，编码图像所需的信息量很小。例如，大脑的核磁共振图像（MRI）可以通过最大的5- 10%的小波系数很好地近似。这种信息的简洁性——换句话说，信号的低维度——在从MRI到雷达到量子态断层扫描的大量应用中都可以找到。人们自然会问：确定一个信号所需的测量次数能否与信息内容相比较？测量可以采取多种形式，这种性质的问题在数据科学中尤其普遍。的确，对大数据的分析常常归结为考虑什么低维模型，以及如何利用这个模型来综合数据。此外，随着环境维度的增大，使用低维度结构的好处变得更加明显。近年来，压缩感知MRI利用图像的固有结构，形成了一种新的测量范式，并最终对医疗技术产生了重大影响。2017年，美国食品和药物管理局批准了西门子和通用电气的核磁共振设备，这两家公司利用压缩传感将程序速度提高了8-16倍。例如，心脏成像从4分钟减少到16秒。我对压缩感知的基础理解做出了重大贡献，并继续在其推广方面工作。最近，使用固定信号结构的想法已经被用深度神经网络学习信号结构的想法所取代。神经网络在学习信号结构和分类方面都取得了巨大的成功。然而，从基础的角度来看，它们并没有得到很好的理解。我正在朝着这个基本的理解努力。与此同时，部分是为了脚踏实地，我正在研究将深度学习用于视网膜图像分类，最终目标是帮助医生预测眼疾。

Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. For example, a magnetic resonance image (MRI) of the brain can be well-approximated by just the largest 5--10% of its wavelet coefficients. This conciseness of information---in other words, low dimensionality of the signal---is found throughout a plethora of applications ranging from MRI to radar to quantum state tomography.  It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content?   Measurements can take many forms, and questions of this nature are especially prevalent in data science. Indeed, the analysis of Big Data often reduces to the question of what low-dimensional model to consider and how to use this model to synthesize the data. Furthermore, as the ambient dimension grows bigger, the gains from using lower dimensional structures become much more apparent. In recent years, compressed sensing MRI has taken advantage of the inherent structure of images leading to a new measurement paradigm which has ultimately had a large impact on medical technology. In 2017, the USA Food and Drug Administration approved of MRI devices from Siemens and GE which have leveraged compressed sensing to speed up the procedure by a factor of 8-16. For example, cardiac imaging is reduced from four minutes to 16 seconds.   I have contributed significantly to the foundational understanding of compressed sensing and continue to work in its generalizations. Recently, the idea of using a fixed signal structure has been replaced with the idea of learning the signal structure with a deep neural net.  Neural nets have had great success for both learning signal structures and classification.  However, they are not well understood from a foundational perspective.  I am working towards this foundational understanding.  At the same time, and in part to stay grounded, I am working to use deep learning for retinal image classification, with the ultimate goal of helping doctors to predict eye diseases.