Distributional Algebro-Geometric Persistence (dAGP): Wasserstein-Stable Polynomial-Relation Signatures

A circle and a rounded square are indistinguishable to homology: both are simple closed curves. They are not indistinguishable algebraically, since their defining equations have degree $2$ and degree $4$. We study a representation of point-cloud distributions that targets this second kind of structure. For a probability measure $\mu$ on $\Omega = [-1,1]^d$ and an orthonormal polynomial feature map $\phi_m$ of degree $m$, the moment matrix $G_m(\mu) = E_{X\sim\mu}[\phi_m(X)\phi_m(X)^\top]$ satisfies $c^\top G_m(\mu)c = E_\mu[f_c(X)^2]$, so coefficient directions associated with its small eigenvalues represent low-degree polynomials that nearly vanish on $\mu$. Counting them as the degree $m$ and the tolerance $\varepsilon$ vary gives a two-parameter integer signature. We call it distributional algebro-geometric persistence (dAGP), because it is the population counterpart of algebro-geometric persistence (AGP), a point-cloud construction we are developing separately and restate in full here. We show that the signature is the sample-normalised counterpart of that construction, and that $\mu \mapsto G_m(\mu)$ is Lipschitz in the $1$-Wasserstein distance, so its ordered eigenvalues vary Lipschitz-continuously under transport of mass, while its threshold counts remain unchanged under perturbations satisfying a spectral-margin condition. We give the Lipschitz constant explicitly, in a form localised to the supports, and we are precise about the regime in which the resulting guarantee is informative. Experimentally, the population noise-floor prediction agrees with the finite-sample mean to $0.1\%$, within sampling error; relation onset orders a conic, a quartic and a non-algebraic closed curve, while their full count profiles distinguish them from a two-dimensional control; calibration identifies a usable tolerance window; and a $100$-repetition sensitivity study quantifies finite-sample onset recovery and finds that the count grid and two continuous spectral baselines all nearly saturate a simple four-family discrimination task.