Generalized moduli of continuity under irregular or random deformations via multiscale analysis

TL;DR

The paper develops a scale-aware theory for the stability of signal processing mappings under irregular input deformations, introducing a generalized modulus of continuity built on multiresolution spaces $U_s$ and the Wiener amalgam spaces $X^{p,q}_r$. It establishes sharp, regime-dependent bounds for $\|F_\tau f - f\|_{L^2}$ in terms of the distortion size $\|\tau\|_{\infty}$ and the scale $s$, extends results to Besov spaces $\dot B^{d/2}_{2,1}$ and to frequency-modulated deformations, and proves the optimality of the exponents via band-limited tests. The framework also handles random deformations, giving mean-square stability bounds in terms of moments of $|\tau|$, thereby connecting uncertainty principles, multiscale analysis, and robustness under both deterministic and stochastic distortions. Overall, the work provides a rigorous, scalable foundation for robustness of multiscale signal representations in the presence of irregular or random deformations, with direct implications for stable feature extraction in neural networks and related systems.

Abstract

Motivated by the problem of robustness to deformations of the input for deep convolutional neural networks, we identify signal classes which are inherently stable to irregular deformations induced by distortion fields $τ\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$, to be characterized in terms of a generalized modulus of continuity associated with the deformation operator. Resorting to ideas of harmonic and multiscale analysis, we prove that for signals in multiresolution approximation spaces $U_s$ at scale $s$, stability in $L^2$ holds in the regime $\|τ\|_{L^\infty}/s\ll 1$ - essentially as an effect of the uncertainty principle. Instability occurs when $\|τ\|_{L^\infty}/s\gg 1$, and we provide a sharp upper bound for the asymptotic growth rate. The stability results are then extended to signals in the Besov space $B^{d/2}_{2,1}$ tailored to the given multiresolution approximation. We also consider the case of more general time-frequency deformations. Finally, we provide stochastic versions of the aforementioned results, namely we study the issue of stability in mean when $τ(x)$ is modeled as a random field (not bounded, in general) with identically distributed variables $|τ(x)|$, $x\in\mathbb{R}^d$.

Figures (1)

• Figure 1: (A) A signal $f$ supported on $[-s,s]$ and its deformation $F_\tau f$, where $\tau(x)=x$ for $|x|<K$, with $K>s$. The plateau level corresponds to the value $f(0)$. The operator $F_\tau$ (with the choice of $\tau$ specified above) performs a single-point sampling of $f$, hence it does not make sense on discontinuous signals. (B) A signal $f$ localized in frequency where $|\omega|\approx s^{-1}$. With the choice of the deformation $\tau=s\mathbbm{1}_{\{f=-1\}}$, the signal $F_\tau f$ is low-pass (a similar example with $f$ continuous is easily obtained by smoothing the steps).

Theorems & Definitions (31)

• Definition 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• Proposition 3.5
• Proposition 4.1
• Remark 4.2
• proof : Proof of Proposition \ref{['maint xopt']}