Geometry-Driven Conditioning of Multivariate Vandermonde Matrices in High-Degree Regimes
Omer Friedland, Yosef Yomdin
TL;DR
The paper addresses stability of multivariate monomial Vandermonde matrices in the high-degree regime without assuming node separation. It introduces the projection-based geometry parameter $\kappa(Z)$ and constructs coefficient-bounded Lagrange polynomials $Q_j$ to establish explicit row-to-span separation and a right inverse for $V_N(Z)$. The main results provide dimension-explicit lower bounds on $\sigma_{\min}$ and upper bounds on $\sigma_{\max}$, leading to a concrete condition-number bound $\kappa(V_N(Z)) \lesssim s^2\nu(4n/\kappa(Z))^{s-1}$, valid for all distinct node sets $Z\subset B_2^n$ when $N\ge s-1$. These findings enhance understanding of interpolation stability in high-dimensional, high-degree settings and offer explicit geometric control via $\kappa(Z)$ without relying on global separation assumptions.
Abstract
We study multivariate monomial Vandermonde matrices $V_N(Z)$ with arbitrary distinct nodes $Z=\{z_1,\dots,z_s\}\subset B_2^n$ in the high-degree regime $N\ge s-1$. Introducing a projection-based geometric statistic -- the \emph{max-min projection separation} $ρ(Z,j)$ and its minimum $κ(Z)=\min_jρ(Z,j)$ -- we construct Lagrange polynomials $Q_j\in\mathcal P_N^n$ with explicit coefficient bounds $$ \|Q_j\|_\infty \lesssim s\Bigl(\frac{4n}{ρ(Z,j)}\Bigr)^{s-1}. $$ These polynomials yield quantitative distance-to-span estimates for the rows of $V_N(Z)$ and, as consequences, $$ σ_{\min}(V_N(Z)) \gtrsim \frac{κ(Z)^{s-1}}{(4n)^{s-1} s\sqrt{s ν(n,N)}}, \quad ν(n,N)={N+n\choose N}, $$ and an explicit right inverse $V_N(Z)^+$ with operator-norm control $$ \|V_N(Z)^+\| \lesssim s^{3/2}\sqrt{ν(n,N)}\Bigl(\frac{4n}{κ(Z)}\Bigr)^{s-1}. $$ Our estimates are dimension-explicit and expressed directly in terms of the local geometry parameter $κ(Z)$; they apply to \emph{every} distinct node set $Z\subset B_2^n$ without any \emph{a priori} separation assumptions. In particular, $V_N(Z)$ has full row rank whenever $N\ge s-1$. The results complement the Fourier-type theory (on the complex unit circle/torus), where lower bounds for $σ_{\min}$ hinge on uniform separation or cluster structure; here stability is quantified instead via high polynomial degree and the projection geometry of $Z$.