Continuous Nonlinear Widths in Approximation

• Continuous nonlinear widths are quantitative invariants that generalize classical linear widths by measuring optimal performance in nonlinear approximations under regularity constraints.
• They employ methods such as Lipschitz parameterizations, manifold embeddings, and sphere-embedding arguments to derive tight, often dimension-independent bounds.
• Applications span geometric analysis, PDE control, and deep learning, offering rigorous insights into compressibility and the intrinsic complexity of functional sets.

Continuous nonlinear widths are a class of quantitative invariants designed to measure the optimal performance of nonlinear approximation and encoding schemes under regularity constraints, generalizing classical linear widths. They play a pivotal role in geometric analysis, approximation theory, and the theory of parametric representations in functional spaces, enabling rigorous quantification of the compressibility and geometric complexity of sets in metric, Banach, and function spaces.

1. Foundational Definitions and Key Notions

Several related, but distinct, definitions for continuous nonlinear widths exist, each appropriate for a different analytic or geometric context.

General Metric Setting. For a compact metric space $(X,d)$ and integer $k \ge 0$, the (continuous) nonlinear $k$-width is

$$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$

where the infimum is over compact $k$-dimensional polyhedra and all continuous maps, capturing how finely $X$ can be contracted to a $k$-complex via continuous collapse (Avvakumov et al., 2024).

Manifold Widths. For a compact $K$ in a Banach space $X$, the manifold $n$-width is

$k \ge 0$0

where $k \ge 0$1 and $k \ge 0$2 are both continuous, encoding and decoding the elements of $k \ge 0$3 via $k \ge 0$4-parameters (Siegel, 2024).

Lipschitz Widths. For a bounded $k \ge 0$5 and $k \ge 0$6, the $k \ge 0$7th Lipschitz width is

$k \ge 0$8

where $k \ge 0$9 is the unit ball in $k$0 under a suitable norm and $k$1 is $k$2-Lipschitz (Petrova et al., 2021, Petrova et al., 2022). These invariants quantify approximation under controlled parameter dependence.

$k$3-Widths in Geometric Analysis. In the min-max theory, for a compact domain $k$4,

$k$5

where $k$6 is a $k$7-sweepout, $k$8 is the mass norm, and the sweepout has nontrivial cohomology (Chodosh et al., 5 May 2025). These form a nonlinear analogue of Laplacian eigenvalues.

2. Contrasts with Classical Linear Widths

Classic Kolmogorov $k$9-widths $$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$0 use linear $$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$1-dimensional subspaces for approximating $$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$2,

$$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$3

measuring optimal performance for linear encoders/decoders. Nonlinear widths allow nonlinear parameterizations, markedly improving compressibility for classes of functions and sets where nonlinear structures are natural.

The Urysohn width $$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$4 is closely related, capturing the minimal fiber diameter over continuous maps into $$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$5-complexes. Both Kolmogorov and Urysohn widths are generally larger than their continuous nonlinear analogues, since the latter admit non-linear collapsing schemes (Avvakumov et al., 2024).

3. Invariant Properties, Continuity, and Scaling Laws

Continuous nonlinear widths exhibit key stability and monotonicity properties:

• Monotonicity: They decrease with increasing parameter dimension and increased regularity of the parameterization (e.g., larger Lipschitz constant).
• Continuity: Nonlinear widths vary continuously under domain deformations (e.g., $$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$6/Hausdorff topology for domains in $$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$7), an essential feature for geometric invariants (Chodosh et al., 5 May 2025).
• Asymptotics: In the Gromov–Guth $$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$8-width theory,

$$w_k(X) = \inf_{\substack{K^k \subset X \ \dim K^k \le k}} \enspace \inf_{\phi:X \rightarrow K^k} \sup_{x \in X} d(x, \phi(x)),$$9

for planar domains, interpolating between geometric width and spectral invariants (Chodosh et al., 5 May 2025).

For $k$0-widths of $k$1-manifolds in $k$2,

$k$3

with an explicit bound of $k$4 in codimension one, reflecting dimension-independent compressibility (Avvakumov et al., 2024).

4. Computational Examples and Explicit Bounds

Low-parameter continuous nonlinear widths have been computed explicitly in model domains:

• For the equilateral triangle of side $k$5,
  • $k$6
  • $k$7
  • $k$8
• For the square of side $k$9,
  • $X$0
  • $X$1
  • $X$2
  • (Chodosh et al., 5 May 2025)

For Sobolev and Besov balls with unit radius in $X$3-norm, sharp lower and upper bounds for manifold widths are

$X$4

where $X$5 is the smoothness and $X$6 is the spatial dimension, provided $X$7. Notably, the decay of manifold widths can strictly exceed that of corresponding Bernstein widths, showing that the restriction to continuous parameterizations fundamentally alters approximation rates (Siegel, 2024).

5. Geometric and Analytical Applications

Continuous nonlinear widths provide tight control over performance in nonlinear parametric methods and have been employed in distinctive analytical and geometric contexts:

• Geometric Min-Max Theory: The realization of $X$8-widths as sums of billiard trajectory lengths in polygons provides a direct connection between geometric optimization and nonlinear width theory (Chodosh et al., 5 May 2025).
• Systolic Geometry: Dimension-independent upper bounds for $X$9-widths yield universal estimates for systolic volume and the existence of noncontractible curves lying in small cubes (Avvakumov et al., 2024).
• Control Theory: Quantitative upper bounds for the reachable set of nonlinear distributed-parameter systems (e.g., Euler–Bernoulli beams, controlled Schrödinger systems) are given using affine and nonlinear estimates of the widths of image sets under nonlinear mappings (Zuyev et al., 2024).

6. Links to Entropy Numbers and Approximation Theory

There exist explicit quantitative relationships between continuous nonlinear widths and entropy numbers:

• For compact $k$0,

$k$1

where $k$2 is the $k$3-th entropy number. Conversely, lower bounds on entropy numbers furnish limits for Lipschitz widths (Petrova et al., 2021, Petrova et al., 2022).

These invariants also control the performance ceiling for deep neural networks and $k$4-term approximation schemes. For example, the Lipschitz widths yield fundamental lower bounds for the approximation error achievable by ReLU networks with fixed architecture and uniform parameter bounds (Petrova et al., 2021).

7. Techniques for Bounding and Constructing Nonlinear Widths

Key techniques for deriving bounds include:

• Nonlinear Federer–Fleming Push-Out: Replacing linear projections onto grid skeletons with collapse onto low-volume separators (e.g., optimal foams) (Avvakumov et al., 2024).
• Sphere-Embedding Arguments: Proving lower bounds for manifold widths via topological constructions using Borsuk–Ulam-type theorems (Siegel, 2024).
• Kinematic Averaging and Group Actions: Using averaging over isometries (e.g., signed permutations for $k$5) to optimize separators and control mass intersections (Avvakumov et al., 2024).
• Explicit Lipschitz Constructions: Employing parameterizations with controlled Lipschitz constants mapped into target function or geometric spaces (Petrova et al., 2021).

These analytical and combinatorial tools allow stronger, often dimension-independent, and sometimes matching upper and lower bounds in both geometric and analytic settings.

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Continuous nonlinear widths provide the modern framework for understanding the intrinsic complexity and optimal parameterization of compact sets under continuous or Lipschitz maps, unifying themes in geometry, PDE, nonlinear approximation, learning theory, and control (Chodosh et al., 5 May 2025, Avvakumov et al., 2024, Siegel, 2024, Zuyev et al., 2024, Petrova et al., 2021, Petrova et al., 2022).