标题： 关于秩约束仿射矩阵集的误差界 显示英文标题

标题： On Error Bounds for Rank-Constrained Affine Matrix Sets

摘要： 秩约束矩阵问题在科学和工程中经常出现。 用于这些问题的迭代算法的收敛性分析通常依赖于局部误差界，这些误差界将到可行集的距离与约束违反程度的度量相关联。 半代数几何中的基础结果保证了此类界限的存在，但相关的指数通常没有明确确定。 本文为规范的秩约束仿射可行性集建立了具有显式指数的局部Hölder误差界。 本文证明了，在任何紧集上，到可行集的距离被一个自然残差函数的幂所限制，该函数捕捉了秩和仿射约束中的违反情况。 此界限中的指数以问题的维度显式给出。 这提供了关于解集几何学的基本定量结果，为一大类数值方法的收敛性分析铺平了道路。 显示英文摘要

摘要： Rank-constrained matrix problems appear frequently across science and engineering. The convergence analysis of iterative algorithms developed for these problems often hinges on local error bounds, which correlate the distance to the feasible set with a measure of how much the constraints are violated. Foundational results in semi-algebraic geometry guarantee that such bounds exist, yet the associated exponents are generally not explicitly determined. This paper establishes a local H\"olderian error bound with an explicit exponent for the canonical rank-constrained affine feasibility set. This paper proves that, on any compact set, the distance to the feasible set is bounded by a power of a natural residual function capturing violations in both the rank and affine constraints. The exponent in this bound is given explicitly in terms of the problem's dimensions. This provides a fundamental quantitative result on the geometry of the solution set, paving the way for the convergence analysis of a broad class of numerical methods.