代数问题的几何建模与正则化
Geometric modeling and regularization of algebraic problems
摘要：
数据扰动的不连续性在代数计算中很常见，其中解通常是高度敏感的。这些问题可以建模为在给定的数据参数下求解方程组。通过附加辅助方程，模型可以被公式化以满足四个容易验证的条件，使得数据形成复解析流形，在该流形上解保持其结构和Lipschitz连续性。当这样一个问题是与经验数据，解决系统成为一个最小二乘问题，其解决方案唯一存在，并享有Lipschitz连续性，只要数据点是在一个管状邻域的流形。其结果是，奇异问题是正则化的适定计算问题。
英文摘要：
Discontinuity with respect to data perturbations is common in algebraic computation where solutions are often highly sensitive. Such problems can be modeled as solving systems of equations at given data parameters. By appending auxiliary equations, the models can be formulated to satisfy four easily verifiable conditions so that the data form complex analytic manifolds on which the solutions maintain their structures and the Lipschitz continuity. When such a problem is given with empirical data, solving the system becomes a least squares problem whose solution uniquely exists and enjoys Lipschitz continuity as long as the data point is in a tubular neighborhood of the manifold. As a result, the singular problem is regularized as a well-posed computational problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
