Function Approximation in Numerically Rank-Deficient Bases

We study linear function approximation in a finite basis under finite-precision arithmetic. In a highly non-orthogonal basis, certain directions are only weakly represented, so that rounding errors can significantly distort the effectively spanned space. In the first part of the paper, we formalize this phenomenon through the notion of a numerical span. Using a novel model for the rounding errors involved, we prove that approximation in the numerical span behaves like approximation in exact arithmetic subject to an additional penalty proportional to the size of the expansion coefficients and the unit roundoff. A key implication is that straightforward numerical orthogonalization cannot mitigate the effects induced by finite-precision arithmetic. The framework also provides a theoretical justification for $\ell^2$-regularized approximation. Moreover, regularization controls the amplification of rounding errors in the computation of expansion coefficients. In the second part of the paper, we address sampling for function approximation in the presence of numerical rank-deficiency. We demonstrate that regularization has another fundamental benefit: it relaxes the conditions required for accurate least squares approximation from sampled data. This effect is made concrete through an analysis of randomized sampling based on a regularized variant of the Christoffel function. The resulting sample complexity bounds depend on an effective dimension that measures the number of directions that remain useful after finite-precision rounding. We also show that regularization renders the Christoffel function computable in contrast to the standard Christoffel function, whose numerical evaluation may require arbitrarily high precision in the presence of numerical rank-deficiency. We apply the derived theory to obtain new results for the discretization of univariate Fourier extension frames.

💡 Research Summary

The paper investigates linear function approximation in a finite basis when computations are performed in finite‑precision floating‑point arithmetic. When the basis is highly non‑orthogonal, some directions are only weakly represented (i.e., they correspond to small singular values of the synthesis operator). In such directions the coefficients required for an accurate representation become large, and rounding errors of magnitude proportional to the unit round‑off u are amplified, leading to a “numerical rank‑deficiency” of the basis. The authors formalize this phenomenon by introducing the numerical span, defined as the span of the rounded basis functions f(ϕ_i). They show that if the condition number κ(T)=σ₁/σ_{ĥn} exceeds 1/u, at least one direction is substantially perturbed by rounding.

A central result is the error decomposition
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