Decomposition of the Kostlan--Shub--Smale model for random polynomials

TL;DR

This paper studies the decomposition of random homogeneous polynomials under the Kostlan--Shub--Smale model, showing they can be approximated by lower-degree polynomials with high probability in Sobolev spaces.

Contribution

It provides a new decomposition approach for these polynomials and establishes probabilistic bounds for their approximation by lower-degree polynomials.

Findings

High-probability approximation of random polynomials by lower-degree polynomials.

Explicit bounds on approximation error depending on polynomial degree and parameters.

Exponential decay of error and probability deviation when degree thresholds are exceeded.

Abstract

Let $\cP_n$ be the space of homogeneous polynomials of degree $n$ on $\bbR^{m+1}$. We consider the asymptotic behavior of some coefficients relating to the decomposition of $\cP_n$ into the sum of $\SO(m+1)$-irreducible components. Using the results, we prove that a random Kostlan--Shub--Smale polynomial $u\in\cP_n$ can be approximated by polynomials of lower degree in the Sobolev spaces $H^k(S^m)$ on the unit sphere $S^m$ with small error and probability close to $1$. For example, if $l_n>\sqrt{(m+2k+8\ep)n\ln n}$, then the inequality $\dist(u,\cP_{l_n})<An^{-\ep}\|u\|$ holds for any sufficiently large $n$ with probability greater than $1-Bn^{-2\ep}$, where $\dist$ and $\|\ \|$ are the distance and norm in $H^k(S^m)$, respectively, $\ep\in(0,1)$, and $A,B$ depend only on $m$ and $k$. If $l_n>\ep n$, then both the approximation error and the deviation of probability from $1$ decay exponentially.