[Paper Review] Eignets for function approximation on manifolds

H. N. Mhaskar|ArXiv.org|Sep 28, 2009

Numerical methods in inverse problems12 references3 citations

TL;DR

This paper introduces eignets—kernel-based approximation networks on compact Riemannian manifolds—using a deterministic, universal algorithm to construct eignets that optimally approximate functions in Lp(µ; X). The key contribution is establishing optimal modulus of smoothness estimates for approximation error in terms of minimal center separation, with converse theorems proving optimality and stability estimates for coefficients and derivatives.

ABSTRACT

Let $\XX$ be a compact, smooth, connected, Riemannian manifold without boundary, $G:\XX imes\XX o \RR$ be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form $\sum_{j=1}^M a_jG(\circ,y_j)$, where $a_j\in\RR$, $y_j\in\XX$, $1\le j\le M$. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in $L^p(μ;\XX)$ for a general class of measures $μ$ and kernels $G$. Our algorithm yields linear operators. Using the minimal separation amongst the centers $y_j$ as the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every \emph{individual function}. We also give estimates on the coefficients $a_j$ in terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in terms of the minimal separation, then the derivatives of the eignets also approximate the corresponding derivatives of the target function in an optimal manner.

Motivation & Objective

• To develop a deterministic, universal algorithm for constructing eignets that approximate functions in Lp(µ; X) on compact Riemannian manifolds.
• To establish optimal modulus of smoothness estimates for the degree of approximation using minimal center separation as a complexity measure.
• To prove converse theorems showing the optimality of these estimates for every individual function.
• To derive stability estimates for the coefficients aj in terms of the Lp norm of the eignet.
• To demonstrate that optimal approximation of a smooth function by eignets implies optimal approximation of its derivatives.

Proposed method

• Uses a kernel G(x, y) = ∑j≥0 b(ℓj)φj(x)φj(y), where {φj} is an orthonormal basis for L2(µ; X) and ℓj are eigenvalues of the Laplace-Beltrami operator.
• Constructs eignets as linear combinations PMj=1 ajG(⋅, yj), with centers yj ∈ X and coefficients aj ∈ ℝ.
• Applies a multiscale analysis based on diffusion polynomials ΠL = span{φj : ℓj ≤ L}, leveraging a Littlewood-Paley-type decomposition.
• Employs a tight frame transform and approximation via operators Φ2m(hb2m; ⋅, y) to control error and derive stability.
• Uses the heat kernel approximation and its properties to define the kernel structure and derive convergence rates.
• Applies discrete maximal function estimates and norm inequalities (e.g., (6.13)) to bound approximation error and coefficient stability.

Experimental results

Research questions

• RQ1 Can a deterministic, universal algorithm construct eignets that achieve optimal approximation rates for functions in Lp(µ; X) on compact Riemannian manifolds?
• RQ2 What is the optimal dependence of the approximation error on the minimal separation between centers yj?
• RQ3 Do eignets that achieve optimal approximation for a smooth function also optimally approximate its derivatives?
• RQ4 Can the coefficients aj be bounded in terms of the Lp norm of the eignet, ensuring stability?
• RQ5 Are the derived approximation estimates sharp, as confirmed by converse theorems?

Key findings

• The algorithm achieves optimal approximation rates with error bounded by c2−m(β−α/p′)qα/p′−β∥Ψ∥p, where q is related to minimal center separation.
• Converse theorems confirm that the derived error bounds are optimal for every individual function.
• Stability estimates show ∥(∆∗)rΨ∥p ≤ c2mr∥Ψ∥p, with 2m ∼1/q, ensuring coefficient control.
• Derivative approximation is optimal: if eignets achieve optimal approximation for f, then derivatives of eignets converge to derivatives of f at the same optimal rate.
• The construction ensures that δ(˜Cm) ∼1/m and δ(˜Cm) ≤2δ(˜Cm), with ˜Cm nested and satisfying separation conditions.
• For p = ∞, the estimate ∥(∆∗)γσm(f) −(∆∗)γΨm∥p ≤ cmγ−r∥(∆∗)rf∥p holds, confirming convergence in derivative norms.