集合上的次高斯矩阵：最优尾依赖性及其应用
Sub-Gaussian Matrices on Sets: Optimal Tail Dependence and Applications
摘要：
随机线性映射在现代信号处理、压缩感知和机器学习中有着广泛的应用。这些映射可以用于将数据嵌入到显著更低的维度中，同时保留有用的信息。这是通过近似保留数据点之间的距离来完成的，这些数据点被假定属于$\mathbb{R}^n$。因此，这些映射的性能通常通过它们与数据上的等距的接近程度来捕获。高斯线性映射一直是许多研究的对象，而亚高斯设置尚未完全理解。在后一种情况下，性能取决于行的次高斯范数。在许多应用中，例如，在压缩感测中，该范数可能很大，或者甚至随着维度而增长，并且因此表征该依赖性是重要的。我们研究了次高斯矩阵何时可以成为集合上的近等距，证明了以前最好的依赖于次高斯范数是次优的，并提出了最优的依赖。我们的结果不仅回答了Liaw，Mehrabian，Plan和Vershynin在2017年提出的一个遗留问题，而且还推广了他们的工作。我们还开发了一个新的Bernstein型不等式的次指数随机变量，和一个新的Hanson-Wright不等式的二次型的次高斯随机变量，在这两种情况下，改善了在矩约束下的次高斯制度的界限。最后，我们举例说明流行的应用，如约翰逊-林登施特劳斯嵌入，零空间性质的0-1矩阵，随机草图和盲解调，其理论保证可以提高我们的结果（在亚高斯的情况下）。
英文摘要：
Random linear mappings are widely used in modern signal processing, compressed sensing and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful information. This is done by approximately preserving the distances between data points, which are assumed to belong to $\mathbb{R}^n$. Thus, the performance of these mappings is usually captured by how close they are to an isometry on the data. Gaussian linear mappings have been the object of much study, while the sub-Gaussian settings is not yet fully understood. In the latter case, the performance depends on the sub-Gaussian norm of the rows. In many applications, e.g., compressed sensing, this norm may be large, or even growing with dimension, and thus it is important to characterize this dependence. We study when a sub-Gaussian matrix can become a near isometry on a set, show that previous best known dependence on the sub-Gaussian norm was sub-optimal, and present the optimal dependence. Our result not only answers a remaining question posed by Liaw, Mehrabian, Plan and Vershynin in 2017, but also generalizes their work. We also develop a new Bernstein type inequality for sub-exponential random variables, and a new Hanson-Wright inequality for quadratic forms of sub-Gaussian random variables, in both cases improving the bounds in the sub-Gaussian regime under moment constraints. Finally, we illustrate popular applications such as Johnson-Lindenstrauss embeddings, null space property for 0-1 matrices, randomized sketches and blind demodulation, whose theoretical guarantees can be improved by our results (in the sub-Gaussian case).
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