元素唯一性、先验知识和数据依赖性分辨率
Element-wise uniqueness, prior knowledge, and data-dependent resolution
AI总结：
本文研究了在欠定线性系统中通过正则化求解的技术，提出了一种基于优化方法的系统表征方法，用于确定信息可恢复性，并通过凸优化问题分析解的唯一性，从而改进了传统闭式解方法的局限性。
摘要：
用于寻找欠定线性系统正则化解的技术可以被视为在未知向量上施加先验知识。现代技术的成功在于优化算法的进步，使得能够施加如稀疏性和非负性等先验条件。然而，用于表征和分析系统以确定信息可恢复性的技术仍通常依赖于闭式解方法，如奇异值分解或滤波截止。本文提出了一种优化方法，以扩展系统表征方法。我们首先推导出在广义先验知识下，系统每个未知元素何时可以唯一解的条件。通过这种方法，可以提出一个凸优化问题来确定每个解元素的“唯一性”，这可以视为对传统分辨率的泛化，以纳入先验知识。我们发现结果取决于未知向量本身，即数据依赖性，例如当解的稀疏性提高其唯一重建的可能性。该方法可用于对系统进行个案分析，估计数据中重要信息的含量，并定量理解正则化解的可信任程度。
英文摘要：
Techniques for finding regularized solutions to underdetermined linear systems can be viewed as imposing prior knowledge on the unknown vector. The success of modern techniques, which can impose priors such as sparsity and non-negativity, is the result of advances in optimization algorithms to solve problems which lack closed-form solutions. Techniques for characterization and analysis of the system to determined when information is recoverable, however, still typically rely on closed-form solution techniques such as singular value decomposition or a filter cutoff, for example. In this letter we pose optimization approaches to broaden the approach to system characterization. We start by deriving conditions for when each unknown element of a system admits a unique solution, subject to a broad class of types of prior knowledge. With this approach we can pose a convex optimization problem to find "how unique" each element of the solution is, which may be viewed as a generalization of resolution to incorporate prior knowledge. We find that the result varies with the unknown vector itself, i.e. is data-dependent, such as when the sparsity of the solution improves the chance it can be uniquely reconstructed. The approach can be used to analyze systems on a case-by-case basis, estimate the amount of important information present in the data, and quantitatively understand the degree to which the regularized solution may be trusted.
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