On Error Bounds for Rank-Constrained Affine Matrix Sets
Ruoning Chen, Defeng Sun, Liping Zhang
TL;DR
We address the geometry of the rank-constrained affine feasibility set ${\mathcal S} = \{X \in \mathbb{R}^{m \times n} \mid \mathcal{A}(X)=b, \mathrm{rank}(X)\le r\}$ and derive a local Hölderian error bound linking ${\sf dist}(X, {\mathcal S})$ to a residual that combines rank violations and affine violations. The approach introduces a polynomial lifting $g(X,V)$ of degree 4 and uses the Łojasiewicz inequality to obtain an explicit exponent $\tau = \frac{1}{R(n(m+n-r),4)}$ with $R(l,d) = d(3d-3)^{l-1}$, yielding $c\, {\sf dist}(X, {\mathcal S}) \le f(X)^{\tau}$ for $f(X) = \sum_{i=n-r+1}^{n} \sigma_i^2(X) + \tfrac{1}{2}\|\mathcal{A}(X)-b\|^2$. This provides a quantitative, geometry-based tool to analyze convergence of iterative methods addressing rank-constrained affine feasibility problems. The bound is explicit but may be conservative due to the polynomial exponent, and the paper notes directions for tightening and extending the results.
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 Hölderian 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.