On Error Bounds for Rank-Constrained Affine Matrix Sets

Abstract: 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.