Extracting low-dimensional signals from high-dimensional data

我的工作的主要推力是发展与本质上低维信号的高维推理的理论基础。有一些美丽的开放性问题，它们的答案可能对数据科学社区非常有用。理解这些问题的关键是理解高维空间中结构化信号的几何形状，以及它们在随机投影下的行为。我的大部分工作都涉及信号估计，尽管数据缺乏确定性。这可以表现为高噪声、极端量化或信号的环境维度超过数据本身的大小。特别是，压缩感知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，问题是需要多少样本来重建。在矩阵补全中，假设给定一个矩阵项的子采样，并要求填充缺失的项。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。以下具体项目特别值得关注。

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.