[Paper Review] Kolmogorov and Linear Widths of Balls in Sobolev and Besov Norms on Compact Manifolds

Daryl Geller, Isaac Z. Pesenson|arXiv (Cornell University)|Apr 4, 2011

Advanced Harmonic Analysis Research33 references4 citations

TL;DR

This paper establishes asymptotically exact upper and lower bounds for Kolmogorov, linear, and Gelfand $n$-widths of the unit ball in Sobolev spaces on compact Riemannian manifolds, particularly for compact homogeneous manifolds. Using spectral theory, kernel localization, and cubature formulas, it derives sharp asymptotic estimates of the form $ n^{-r/s + \max(0, 1/p - 1/q)} $, where $ s $ is the manifold dimension and $ r $ the smoothness index, under natural parameter ranges.

ABSTRACT

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact Riemannian manifolds. For compact homogeneous manifolds, we establish estimates which are asymptotically exact, for the natural ranges of indices. The proofs heavily rely on our previous results such as: estimates for the near-diagonal localization of the kernels of elliptic operators, Plancherel-Polya inequalities on manifolds of bounded geometry, cubature formulas with positive coefficients and uniform estimates on Clebsch-Gordon coefficients on general compact homogeneous manifolds.

Motivation & Objective

• To determine asymptotic estimates of Kolmogorov and linear $n$-widths of the unit ball in Sobolev spaces on smooth compact Riemannian manifolds.
• To establish asymptotically exact bounds for these widths on compact homogeneous manifolds, particularly for natural ranges of indices $p$, $q$, and $r$.
• To derive lower bounds for Gelfand widths on compact homogeneous manifolds using spectral and harmonic analysis tools.
• To unify and extend prior results on $n$-widths by leveraging Plancherel-Polya inequalities, positive cubature formulas, and uniform estimates on Clebsch-Gordon coefficients.
• To provide a comprehensive framework for function approximation on manifolds using $n$-widths as a measure of best-approximation efficiency.

Proposed method

• The authors use the spectral theory of elliptic operators, particularly the Laplace-Beltrami and Casimir operators, to define Sobolev norms on compact manifolds.
• They apply near-diagonal localization estimates for kernels of elliptic operators to control the behavior of singular integrals and projections.
• Plancherel-Polya inequalities on manifolds are used to relate $L_p$-norms of functions to their coefficients in spectral expansions.
• Positive cubature formulas with uniform error bounds are constructed to discretize $L_p$-norms and approximate $n$-widths via finite-dimensional projections.
• Uniform estimates on Clebsch-Gordon coefficients on compact homogeneous manifolds are used to control the growth of matrix coefficients in tensor product decompositions.
• The proofs rely on comparison techniques using $\ell_p$-ball widths in finite dimensions, via embeddings and duality, to derive lower bounds for the manifold case.

Experimental results

Research questions

• RQ1 What are the asymptotic behaviors of Kolmogorov and linear $n$-widths of the unit ball in Sobolev spaces on compact Riemannian manifolds?
• RQ2 How do these widths scale with $n$, the dimension of the approximating subspace, and the smoothness $r$ and integrability $p$, $q$ parameters?
• RQ3 Can the upper bounds for $n$-widths be made asymptotically exact on compact homogeneous manifolds?
• RQ4 What are the sharp lower bounds for Gelfand $n$-widths on compact homogeneous manifolds, and how do they compare to Kolmogorov and linear widths?
• RQ5 To what extent do the results depend on the choice of elliptic operator $L$, and when is the Casimir operator optimal?

Key findings

• For compact homogeneous manifolds, the Kolmogorov $n$-width of the unit ball in $W_p^r(M)$ satisfies $ d_n(B_p^r, L_q) \asymp n^{-r/s + \max(0, 1/p - 1/q)} $, with $ s = \dim M $, for natural ranges of $p$, $q$, and $r$.
• The linear $n$-width $\delta_n(B_p^r, L_q) \asymp n^{-r/s + \max(0, 1/p - 1/q)} $, matching the Kolmogorov width up to constants.
• For Gelfand widths, the lower bound $ d^n(B_p^r, L_q) \gg n^{-r/s + \max(0, 1/p - 1/q)} $ holds, and the asymptotic order is sharp in many cases.
• The asymptotic estimates are exact when $p \leq q$ and $p \geq 1$, with the exponent $ -r/s + \max(0, 1/p - 1/q) $, and the bounds are independent of the specific choice of elliptic operator $L$ for homogeneous spaces.
• The results are derived via comparison with finite-dimensional $\ell_p$-ball widths, using the embedding $ b_p^M \subset M^{1/p - 1/q} b_q^M $ and duality techniques.
• The proof relies on constructing a projection $Q_N$ onto a spectral subspace of dimension $P_N \asymp n$, and using the fact that $ \|Q_N h\|_q \asymp N^{-1/q} \|a\|_q $ for coefficients $a$ of $h$.