Extracting low-dimensional signals from high-dimensional data

我的工作的主旨是发展的理论基础，高维推理与内在的低维信号。 有一些美丽的开放式问题，其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结构，以及这些信号在随机投影下的作用。我的大部分工作都涉及信号估计，尽管数据缺乏确定性。这可以通过高噪声、极端量化或其环境维度超过数据本身大小的信号来表征。特别地，压缩感测和矩阵补全属于后一类别。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，假设给定一个矩阵的条目的子采样，并要求填充缺失的条目。在这两种情况下，重要的是在信号重构中利用底层低维信号结构。 以下具体项目具有特殊意义。

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.