Group Invariant Scattering

Stéphane Mallat

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Group Invariant Scattering

Stéphane Mallat

arXiv:1101.2286v3Jan 12, 2011math.FAcs.CV

TL;DR

The paper addresses the difficulty of constructing translation-invariant representations that remain stable to diffeomorphisms and preserve high-frequency information. It builds scattering transforms from cascaded wavelet-modulus operators, extends them to stationary processes and compact Lie groups, and establishes translation, deformation, and rotation-invariance results within the stated settings.

Problem
Existing translation-invariant operators are not Lipschitz continuous to diffeomorphisms, while high-frequency information is important for discriminating signals.
Method
The paper constructs scattering propagators from path-ordered wavelet transforms and modulus operators, uses windowed local integration, and extends the construction to compact Lie groups.
Results
The resulting representations are translation invariant and Lipschitz continuous to diffeomorphisms; scattering coefficients can discriminate stationary processes with identical second-order moments, and group constructions yield translation- and rotation-invariant scattering.
Takeaways & Limitations
Scattering provides a deformation-stable representation that retains high-frequency information and captures higher-order statistical structure beyond the Fourier power spectrum.
Takeaways & Limitations
The paper leaves open a conjecture concerning conditions for strong convergence in L2(R^d), and the path-space subdivision map is discontinuous at subdivision boundaries.

Abstract

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This paper constructs translation invariant operators on L2(R^d), which are Lipschitz continuous to the action of diffeomorphisms. A scattering propagator is a path ordered product of non-linear and non-commuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz continuous to the action of diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform which is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L2 (G), where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on L2(R^d) and on Ld (SO(d)) defines a translation and rotation invariant scattering on L2(R^d).

1 Introduction

The paper develops translation-invariant representations that retain high-frequency information while remaining Lipschitz stable to diffeomorphisms. Its scattering construction extends to stationary processes and compact Lie-group actions, including joint translation and rotation invariance.

Motivation: Existing translation-invariant operators are not Lipschitz continuous to diffeomorphisms, especially at high frequencies, motivating representations that preserve high-frequency information while remaining deformation-stable.The paper frames maintaining Lipschitz continuity over high frequencies as the central difficulty.
Scattering construction: A scattering propagator cascades wavelet transforms and modulus operators along multiple paths, while windowed local integration yields a nonexpansive transform that preserves the L2 norm and is Lipschitz continuous to C2 diffeomorphisms.The construction uses nonlinear, non-commuting operators and locally integrates their outputs.
Translation invariance: As the window size increases, windowed scattering transforms converge to a translation-invariant scattering transform defined on a path set.The limiting transform is associated with a measure and metric on the path space, and its outputs belong to the corresponding L2 space.
Stationary processes: Scattering coefficients depend on high-order moments, allowing discrimination between stationary processes with identical second-order moments, while expected scattering transforms map stationary processes into l2.For broad classes of ergodic processes, a single realization numerically provides a mean-square consistent estimator of the expected transform.
Group invariance: The paper extends scattering operators to compact Lie groups and combines scattering on L2(R^d) and L2(SO(d)) to obtain translation- and rotation-invariant scattering on L2(R^d).Group invariance is defined through invariance under the left action of each group element.

2 Finite Path Scattering

Finite-path scattering iteratively applies wavelet modulus operators and low-pass averaging to produce translation-invariant, deformation-stable representations while preserving signal energy. Its energy concentrates on frequency-decreasing paths, enabling efficient computation and supporting convergence, stability, and displacement estimation results.

Path-ordered scattering: A wavelet transform groups frequencies into dyadic packets, while scattering iteratively applies wavelet transforms and modulus operators along ordered paths.Each U[λ]f captures energy in a wavelet frequency band and propagates it toward lower frequencies.
Norm preservation: Scattering is nonexpansive and preserves the L2(R^d) norm under the admissibility condition for the scattering wavelet.The propagator preserves norms, and the full scattering energy ultimately reaches the low-pass output.
Efficient path computation: More than 99.5% of the energy for a cubic spline wavelet in d = 1 is concentrated on frequency-decreasing paths, allowing O(N log N) computation for signals of size N.These paths satisfy |λk+1| < |λk|, and practical implementations restrict computation to them.
Translation invariance: As J increases, the scattering distance is non-increasing and converges to a translation-invariant limit, with lim J→∞∥SJ[PJ]f −SJ[PJ]Lcf∥ = 0.For admissible wavelets, the finite-window transform retains the signal norm while its translation discrepancy vanishes in the limit.
Diffeomorphism stability: A windowed scattering is Lipschitz continuous to sufficiently small diffeomorphisms, with translation error proportional to 2^-J∥τ∥∞ and deformation error proportional to ∥∇τ∥∞.The result applies under conditions including ∥∇τ∥∞ ≤ 1/2 and finite mixed scattering norm.

3 Normalized Scattering Transform

The normalized scattering transform is built on finite and infinite path spaces equipped with cylinder sets and a Dirac scattering measure. Windowed transforms converge toward the limiting transform, whose behavior is compared with the Fourier modulus through frequency mapping and numerical examples.

Path spaces: Finite paths are embedded in an extended path space, while increasing resolution produces higher-resolution path extensions converging toward infinite paths.The finite path set includes the empty path, and each path can be extended by longer paths as J increases.
Path spaces: Cylinder sets organize the path space, generate its sigma algebra, and support a unique σ-finite Dirac scattering measure defined by the norm of scattering coefficients of a Dirac.The measure satisfies a scale relation and is nonzero under a nonvanishing condition on the wavelet Fourier modulus.
Frequency mapping: A surjective frequency-to-path map preserves the relevant measure structure but is discontinuous at subdivision boundaries.Nearby frequencies on opposite sides of a boundary can map to paths that remain separated in the path metric.
Frequency comparisons: The limiting scattering transform has Fourier-like frequency behavior: its coefficients and the Fourier modulus have equivalent decay over dyadic frequency bands.The construction also yields a frequency partition whose interval widths decrease with increasing resolution.
Numerical comparisons with Fourier: For a scaled Gabor function, the Fourier-modulus deformation constant is C = 13.5, whereas the scattering constant is C = 1.5 and does not grow with center frequency.For two high-frequency Gabor components, second-order scattering captures low-frequency interference information that the wavelet transform does not resolve directly.

4 Scattering Stationary Processes

The expected scattering transform represents stationary processes in l2(P∞), with coefficients sensitive to normalized high-order moments and stable to random deformations up to a logarithmic term. Windowed scattering estimates expected coefficients as the averaging scale grows, with numerical mean-square convergence for broad ergodic classes.

Expected scattering maps stationary processes to l2(P∞) and is Lipschitz continuous to random deformations up to a log term.
Scattering coefficients at path length m depend on normalized moments of order 2m, successively filtered by wavelets.
The expected scattering coefficient is SX(p) = E(U[p]X), obtained as the window scale tends to infinity; windowed scattering provides its estimator.
For a large class of ergodic processes, including Gaussian processes, mean-square convergence of windowed scattering is observed numerically.
For Gaussian white noise and moving-average Gaussian processes, the estimation error decays linearly with J along frequency-decreasing paths.
Unlike the Fourier power spectrum, scattering power spectra can discriminate stationary processes with identical second-order moments and Fourier power spectra.

5 Invariance to Actions of Compact Lie Groups

The paper extends scattering to compact Lie groups, where group-invariant operators are built from cascaded wavelet-modulus operators. Combining translation scattering on L2(R^d) with rotation scattering on L2(SO(d)) yields translation- and rotation-invariant representations.

Scattering on L2(G) is constructed by cascading modulus operators of wavelet transforms defined with group convolutions.
The group wavelet-modulus propagator is nonexpansive, and the wavelet transform preserves the L2(G) norm.
For general compact Lie groups, the norm-preservation extension requires additional verification; validity is established for the rotation group SO(2).
At the integration scale, the compact-group scattering transform is invariant to the left action L_g f(r) = f(g^-1r).
A combined scattering applies rotation-group scattering to the rotation variable of translation-scattering paths.
With admissible wavelets, the combined scattering preserves the norm and is invariant to translations and rotations.

A Proof of Lemma 2.8

The proof shows that scattering energy moves toward lower frequencies as path length increases. This frequency propagation yields bounds controlling the scattering norm across path lengths.

The average arrival log frequency of scattering energy increases with path length, so energy propagates toward lower frequencies.
The arrival log frequency of a path is the log-frequency index of its final wavelet element.
Lemma A.1 states that increasing path length by one decreases the average arrival frequency by nearly α/2 under condition (30).
Summing the recurrence bounds the cumulative scattering energy across path lengths by a finite expression involving e0 and (a1 + J)e1.

B Proof of Lemma 2.11

The proof of Lemma 2.11 bounds the deformation commutator using Schur’s lemma. A Taylor expansion and Jacobian control yield the required operator-norm estimate.

Schur’s lemma provides operator-norm bounds for integral operators from their kernels.
The proof applies Schur’s lemma to the kernel of kJ = L_τA_J − A_J.
A first-order Taylor expansion bounds the deformation-induced difference between the translated and undeformed averaging operators.
When ||∇τ||∞ ≤ 1/2, the change-of-variable Jacobian is bounded below by 2^-d, enabling the final operator-norm estimate.

C Proof of (69)

The proof bounds the kernel generated by the discrepancy between translated and locally displaced averaging operators. Taylor expansion, Hessian decay, and a Jacobian estimate yield the desired operator bound.

Taylor expansion expresses the kernel discrepancy through second derivatives of the averaging function.The Hessian matrix controls the remainder after subtracting the zeroth- and first-order displacement terms.
The scaled averaging function inherits Hessian decay from the mother function under the change of variables.The proof uses φ2J(x) = 2^-dJφ(2^-Jx) and the corresponding Hessian scaling.
The Jacobian of the displacement change of variables is bounded below by (1 − ∥∇τ∥∞)^d.This lower bound controls the change-of-variable factor when ∥∇τ∥∞ is small.
Schur’s lemma combines the kernel bounds to prove the target operator inequality.The upper bounds obtained for the kernel and its integral estimates establish the result.

D Proof of Lemma 2.13

This proof establishes commutator identities for the scattering propagator by iterating through finite path lengths. Nonexpansiveness then allows the finite-length result to extend to all paths.

The wavelet modulus propagator applies modulus wavelet transforms indexed by scales and orientations without averaging.VJf collects |f ⋆ ψλ| over λ ∈ ΛJ, and UJ combines this propagator with averaging.
The proof restricts the path set to paths shorter than m and establishes the commutator relation inductively.The induction introduces the remainder operator Km and repeatedly substitutes the same identity.
Letting m tend to infinity completes the proof of the all-path commutator property.The finite-path identity is extended by taking the path-length limit.
Nonexpansiveness of the restricted scattering operator controls the commutator terms throughout the induction.The restriction SJ[PJ,m] remains nonexpansive because SJ[PJ] is nonexpansive.

E Proof of Lemma 2.14

The proof decomposes wavelet commutators into regular and singular kernel parts and bounds both using decay, cancellation, Taylor expansions, and operator estimates. These bounds yield the lemma’s multiscale commutator inequality.

Positivity permits truncating the scale sum at −J, and inserting the commutator bounds proves the lemma’s upper bound.The same inequality remains valid for the finite scale range used by the scattering operator.
The wavelet commutator is represented by a kernel comparing convolution with its diffeomorphically transformed version.The kernel includes both the displaced argument and the Jacobian determinant of the deformation.
The regular kernel component is bounded using decay, cancellation, and Schur-type estimates.Vanishing integrals and decay of h and its derivatives control the associated self-adjoint operators.
The proof separates each kernel into a singular diagonal component and a remainder component before estimating their norms.The singular part is isolated as Kj,1, while Kj,2 is treated through Taylor remainders.
The resulting estimate gives ∥Kj∥ ≤ C∥∇τ∥∞ and supplies the bounds needed for the lemma.The proof combines the singular and remainder estimates, with Cotlar’s lemma used for cross-scale operator products.
The remainder component decays like 2^-j for j ≥ 0 and is controlled for j ≤ 0 by Taylor remainder estimates.Changes of variables and Schur’s lemma convert the pointwise kernel bounds into operator bounds.

F Proof of Lemma 3.6

The proof analyzes extensions of scattering paths across scales and shows that the resulting normalized coefficients can be rewritten through path integrals. This yields monotonicity and convergence of the scattering metric.

Extensions of a path p into the next scale are characterized recursively, and iterating this construction gives k-step extensions.The recursion tracks how paths in PJ generate paths in PJ+k.
Normalized coefficients are constant across compatible path extensions after accounting for the corresponding path-set measures.The coefficient relation permits the sum over extensions to be rewritten as a path integral.

G Proof of Lemma 4.8

The proof establishes the lemma for stationary processes by showing that the transformed second moment is spatially constant, then extends the result to sequences of operators.

Stationary-process argument: Restricting a stationary process to a finite hypercube reduces the argument to a finite-energy process while preserving the required second-moment calculation.The proof verifies that E(|KτX(x)|2) does not depend on x.
Lemma conclusion: The resulting bounds establish the lemma result (185) through the L2(R^d) operator norm definition and inequality (186).The proof computes the relevant quantity, integrates along x, and applies (186).
Stationary-process argument: Stationarity of X and the kernel hypothesis, together with independence of X and τ, show that E(|KτX(x)|2) does not depend on x.The change of variables v = x − u and v′ = x − u′ produces the spatial invariance.
Sequence extension: Lemma 4.8 extends to sequences of operators by replacing the L2(R^d) norm with the finite-energy sequence norm and using corresponding sup operator norms.The extension assumes each average bilinear kernel is stationary.

H Proof of Theorem 4.7

The proof of Theorem 4.7 applies the extended commutator lemma to wavelet operators. It verifies kernel stationarity and decay conditions before deriving the stated bounds and using modulus nonexpansiveness.

Final estimate: The theorem’s remaining inequalities follow by combining the commutator estimate with the established bounds and the nonexpansiveness of the modulus operator.The same argument is applied to the process mean-square norm by replacing function norms with expected squared magnitudes.
Commutator extension: The wavelet commutator Kτ is a sequence containing the averaging and wavelet commutators, so the extended Lemma 4.8 applies when their kernels satisfy conditions (193) and (194).This extension yields the bound used in the theorem proof.
Kernel stationarity: Joint stationarity of τ and ∇τ implies stationarity of the wavelet commutator kernels for both averaging and wavelet operators.The kernel is expressed using h = φ for [A_J, Lτ] and h(x) = ψ(r^-1x) for wavelet commutators.
Kernel bounds: If |h(x)| = O((1 + |x|)^(-d-2)) and ∥∇τ∥∞ ≤ 1/2 almost surely, the second kernel hypothesis follows from a scale change and bounded integral estimates.The proof obtains I_j ≤ C′ and then sums the scale-weighted terms to prove (203) and (194).