Out-of-sample guarantees for data-driven frequency and value-space truncation

Establish rigorous out-of-sample guarantees showing that frequency sets and finite-dimensional value subspaces estimated from finitely many samples control the approximation error uniformly over the entire family K of Hilbert-space-valued functions, either under explicit Fourier-decay assumptions or through a learning-theoretic covering-number or Rademacher-complexity argument.

Background

The paper’s main compactness theorems establish the existence of a finite frequency set S and, under uniform tightness, a finite-dimensional subspace V that provide uniform approximation over the full family K. However, the algorithmic discussion considers estimating S and V from a finite sample f_1,...,f_n drawn from K, using empirical Fourier magnitudes and an empirical covariance operator. The paper explicitly notes that these procedures do not by themselves guarantee control over unsampled members of K.

The unresolved issue is to convert the in-sample estimates into a uniform bound for all of K. The authors identify two possible routes: impose an explicit, verifiable Fourier-decay rate on K, or develop a learning-theoretic argument based on covering numbers or Rademacher complexity. This problem is distinct from the existence results proved earlier, which provide no quantitative sampling or cardinality guarantees.

References

If both steps above are performed, the triangle inequality gives $$ |f - P_{S_\varepsilon}(Q_{\widehat V} f)|{L2(G,\mathcal{H})} \;\le\; |f - Q{\widehat V}f|{L2(G,\mathcal{H})} + |Q{\widehat V}f - P_{S_\varepsilon}(Q_{\widehat V}f)|{L2(G,\mathcal{H})}, $$ mirroring the proof of Theorem~\ref{thm:main-tight}; but turning this into a numerical bound requires the two heuristic steps above to actually control their respective terms on the full set $K$, not just on the $n$ samples used to construct $S\varepsilon$ and $\widehat{V}$, a gap between in-sample and out-of-sample guarantees that is a genuine open point, not a detail we are glossing over.

— Pego theorem for Hilbert space-valued functions on compact groups  (2608.13142 - Lakmon et al., 13 Aug 2026) in Section “Toward Applications,” subsection “Algorithmic and numerical aspects,” subsubsection “(b) Choosing S and V from data,” item “Joint error accounting”