Extracting low-dimensional signals from high-dimensional data

我的工作的主旨是发展用本质上的低维信号进行高维推理的理论基础。有一些美丽的开放问题，它们的答案可能会对数据科学界有很大帮助。理解这些问题的一个关键是理解高维空间中结构化信号的几何形状，以及这些信号在随机投影下是如何作用的。尽管数据缺乏确定性，但我的大部分工作都与信号估计有关。这可能以高噪声、极端量化或其环境维度超过数据本身大小的信号为特征。特别是，压缩传感和矩阵补全属于后一类。在压缩感知中，假设信号是随机采样的，问题是重建所需的样本有多少。在矩阵补全中，假设给出了矩阵条目的子采样，并要求填充缺失的条目。在这两种情况下，在信号重建中利用底层低维信号结构是很重要的。*1.了解矩阵补全和压缩感知中的测量模式：矩阵补全理论主要假设对条目进行随机抽样，压缩感知中也有类似的假设。虽然简化概率模型以支持强大的理论是至关重要的，但这种方法有两个明显的缺陷：1)在实际数据中，通常很明显，抽样中存在不反映随机模型的模式；2)抽样模式本身提供信息；基于统一随机模型的方法没有利用这些信息。我的目标是在我未来的研究中解决这两点。*2.测量的离散化：大数据分析的一般思想是，非常粗略的测量可以结合在一起做出强有力的推断，前提是这些测量的数量足够多，并且信号的维度足够小。这一想法在一位压缩传感中变得相当精确，这是一种极端量化的模型，在这种模型中，只保留每个测量的符号。*采样的不平稳性导致了新的理论和计算挑战。*3.维度的有用描述：被二次采样的信号的维度或复杂性的适当概念是什么？在给定任意信号集的情况下，能否以计算效率的方式估计该维度？这是统计信号处理中最重要的问题之一，有着悠久的方法历史。我的目标是通过将勒卡姆和科尔莫戈洛夫的经典思想与源自压缩传感文献的现代模型结合起来，为这一挑战赋予现代色彩。**

The main thrust of my work is to develop the theoretical underpinnings of high-dimensional inference with intrinsically low-dimensional signals.  There are beautiful open questions whose answers could be of much use to the data science community. A key to understanding these questions is to understand the geometry of structured signals in high dimensional space, and how these act under random projections. Much of my work concerns signal estimation despite a lack of certainty in the data. This can be characterized by high-noise, extreme quantization, or a signal whose ambient dimension exceeds the size of the data itself. In particular, compressive sensing and matrix completion fall into this latter category. In compressive sensing, one assumes that the signal is randomly sampled and the question is how few samples are necessary to reconstruct. In matrix completion one assumes that one is given a subsampling of entries of a matrix and asked to fill in the missing entries. In both cases, it is important to utilize an underlying low-dimensional signal structure in signal reconstruction.  The following specific projects are of special interest.*** 1. Understanding the measurement pattern in matrix completion and compressive sensing: The theory of matrix completion by largely assumes random sampling of entries and similar assumptions are made in compressive sensing. While it is vital to make simplifying probabilistic models to allow powerful theory, there are two apparent drawbacks of this approach: 1) In real data, it is often clear that there is a pattern in the sampling which does not reflect the random model and 2) the sampling pattern itself gives information; this information is not taken advantage of by methods based on a uniform random model. I aim to address these two points with my future research. ***2. Discretization in the measurements: A general idea in the analysis of Big Data is that very rough measurements can be leveraged together to make powerful inferences, provided there are a sufficient number of them, and the signal has sufficiently small dimension. This idea is made quite precise in one-bit compressive sensing, a model of extreme quantization, in which only the sign of each measurement is retained.  The lack of smoothness of the sampling leads to new theoretical and computational challenges.***3. Useful characterizations of dimension: What is the appropriate notion of the dimension or complexity of a signal which is being subsampled? And given an arbitrary signal set, can this dimension be estimated in a computationally efficient manner? This is one of the most important questions of statistical signal processing with a long history of approaches. I aim to give a modern twist on this challenge by combining classical ideas of Le Cam and Kolmogorov with modern models stemming from the compressed sensing literature.**