# Generalizing Geometric Nonwindowed Scattering Transforms on Compact
Riemannian Manifolds

Albert Chua<EMAIL_ADDRESS>Yang Yang<EMAIL_ADDRESS>Department of
Mathematics, Michigan State University, East Lansing, MI, 48824 USA
Department of Computational Mathematics, Science & Engineering, Michigan State
University, East Lansing, MI, 48824 USA

###### Abstract

Let $\mathcal{M}$ be a compact, smooth, $n$-dimensional Riemannian manifold
without boundary. In this paper, we generalize nonwindowed geometric
scattering transforms, which we formulate as $\mathbf{L}^{q}(\mathcal{M})$
norms of a cascade of geometric wavelet transforms and modulus operators. We
then provide weighted measures for these operators, prove that these operators
are well-defined under specific conditions on the manifold, invariant to the
action of isometries, and stable to diffeomorphisms for $\lambda$-bandlimited
functions.

###### keywords:

Wavelet Scattering, Geometric Deep Learning, Spectral Geometry

##  1 Introduction

For the purposes of this paper $\mathcal{M}$ will be a compact, smooth,
$n$-dimensional Riemannian manifold without boundary contained in
$\mathbb{R}^{d}$, where $d\geq n$ with geodesic distance between two points
$x_{1},x_{2}\in\mathcal{M}$ given by $r(x_{1},x_{2})$ and Laplace-Beltrami
operator denoted as $\Delta$. The notation $\mathbf{L}^{q}(\mathcal{M})$
denotes the set of all functions $f:\mathcal{M}\to\mathbb{R}$ such that
$\int_{\mathcal{M}}|f(x)|^{q}\,d\mu(x)<\infty,$ where $d\mu(x)$ is integration
with respect to the Riemannian volume, whose measure is given by $\mu$. We use
the notation $\text{Isom}(\mathcal{M}_{1},\mathcal{M}_{2})$ be the set of
isometries between manifolds $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$. Lastly,
the set of diffeomorphisms on $\mathcal{M}$ will be denoted by
$\text{Diff}(\mathcal{M})$, and the maximum placement of
$\gamma\in\text{Diff}(\mathcal{M})$ will be given by
$\|\gamma\|_{\infty}:=\sup_{x\in\mathcal{M}}r(x,\gamma(x))$.

In recent years, deep convolutional neural networks have shown strong
performance on various vision-related tasks [1, 2, 3]. However, because of how
complex deep convolutional architectures are, it is not entirely clear what
mechanisms enable deep convolutional networks to get strong performance on
these tasks.

In an effort to better understand the properties of deep convolutional
architectures, Mallat proposed the scattering transform [4, 5], which uses a
cascade of unlearned filters and nonlinearities to mimic the behavior of a
deep convolutional neural network. Using a specific class of wavelet filters,
Mallat found that the scattering transform had many desirable properties for
machine learning tasks such as translation invariance and stability to small
deformations. Additionally, the scattering transform and its generalizations
have been found various applications as a general feature extractor, such as
in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Authors have also
explored extensions of the scattering transform to semi-discrete frames [19,
20, 21] and more general gabor frames [22, 23].

However, certain forms of data, such as point cloud data have a geometry than
is non-Euclidean, which motivate manifold learning models [24, 25, 26] and
geometric deep learning [27]. As an extension of the scattering transform,
authors have considered graph scattering transforms in [28, 29, 30, 31, 32]
via constructing graph wavelets [25, 33]; additionally [34, 35] extend
wavelets and the scattering transform to simplicial complexes; the overarching
idea is that these extensions of the scattering transform have similar
desirable stability properties, and present success on non-Euclidean datasets.

Extensions of the scattering transform have also been presented on compact
Riemannian manifolds via defining the wavelet transform on a compact
Riemannian manifold using eigenfunctions of the Laplace-Beltrami operator in
[36]. In particular, for the windowed geometric scattering transform, the
authors were able to prove that the representation was locally invariant to
isometries for all $\mathbf{L}^{2}(\mathcal{M})$ functions and stable to
deformations in $\text{Diff}(\mathcal{M})$ for $\mathbf{L}^{2}(\mathcal{M})$
under mild restrictions. The authors derive a representation, the nonwindowed
geometric scattering transform, which is invariant to isometries and stable to
deformations in $\text{Diff}(\mathcal{M})$.

Regarding nonwindowed geometric scattering transforms, their Euclidean
counterparts, nonwindowed scattering transforms, have been effective for
applications in quantum chemistry, audio synthesis, and physics have appeared
in [6, 7, 8, 9, 10]; theoretical results involving stability to deformations
have also been provided in [37]. The main idea behind nonwindowed scattering
transforms is that they provide a small number of descriptive features for
high dimensional data. As a natural extension, [38] used scattering moments on
manifolds for classifications tasks involving point cloud data. However, there
were limited theoretical results in [39] for the representation provided in
[38], which motivates this paper.

We provide a well-defined weighted measure for scattering moments when
$f\in\mathbf{L}^{q}(\mathcal{M})$ for $q\in(1,2)$, which is an improvement
upon requiring $f\in\mathbf{L}^{2}(\mathcal{M})$ . However, our weighted
measure is only defined for an aribtrary, finite number of layers, and
requires restrictions on the regularity of the manifold. Additionally, we
prove a diffeomorphism stability result in the spirit of [5, 36].

###  1.1 Invariance and Stability

In machine learning tasks, it is often necessary for a representation to have
some degree of invariance with respect to the action of a Lie group. For tasks
involving manifolds, one may like to have local isometry invariance. More
formally, let $V_{\xi}f(x)=f(\xi^{-1}x)$ for $\xi\in\text{Isom}(\mathcal{M})$
and consider a representation $\Phi:\mathcal{B}_{1}\to\mathcal{B}_{2}$, where
$\mathcal{B}_{1},\mathcal{B}_{2}$ are Banach Spaces. It would be desirable to
have a representation such that

$\|\Phi f-\Phi V_{\xi}f\|_{\mathcal{B}_{2}}\leq
2^{-dJ}\|\xi\|_{\infty}\|f\|_{\mathcal{B}_{1}},$

where $J$ controls the degree of invariance. A simple choice is to use an
averaging filter, but this potentially leads to the loss of information that
can be crucial for the task.

For other tasks, such as manifold classification, a fully rigid representation
may be required, and full isometry invariance is desirable. That is to say, we
have

$\|\Phi f\|_{\mathcal{B}_{2}}=\|\Phi V_{\xi}f\|_{\mathcal{B}_{2}}.$

In addition to invariance, it is necessary for a representation to also have
stability properties. Instead of considering an isometry, consider
$\xi\in\text{Diff}(\mathcal{M})$ and think of $V_{\xi}f$ as a small
deformation of $f$. We want

$\|\Phi f-\Phi V_{\xi}f\|_{\mathcal{B}_{2}}\leq
2^{-dJ}K(\xi)\|f\|_{\mathcal{B}_{1}},$

where $K(\xi)\to 0$ as $\|\xi\|_{\infty}\to 0$; this is to ensure that small
deformations of an input do not lead to large changes in the representation.

##  2 A Review of the Geometric Scattering Transform on Manifolds

###  2.1 Spectral Filters and the Geometric Wavelet Transform

The convolution of $f,g\in\mathbf{L}^{2}(\mathbb{R}^{n})$ is usually defined
in space as

$(f\ast g)(x)=\int_{\mathbb{R}^{n}}f(y)g(x-y)\,dy.$

However, for a general manifold, even under the conditions we have prescribed,
a notation of translation does not necessarily exist. Instead, one can
consider a spectral definition of convolution via the spectral decomposition
of $-\Delta$. Denote $\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. Because our
manifold is compact, it is well known that $-\Delta$ has a discrete spectrum,
and we can order the eigenvalues in increasing order and denote them as
$\\{\lambda_{n}\\}_{n\in\mathbb{N}_{0}}$. We will denote the corresponding
eigenfunctions as $\\{e_{n}(x)\\}_{n\in\mathbb{N}_{0}}$, which form an
orthonormal basis for $\mathbf{L}^{2}(\mathcal{M})$.

Suppose $f\in\mathbf{L}^{2}(\mathcal{M})$. Since the set of functions
$\\{e_{n}(x)\\}_{n\in\mathbb{N}_{0}}$ forms a basis in
$\mathbf{L}^{2}(\mathcal{M})$, we decompose

$f(x)=\sum_{n\in\mathbb{N}_{0}}\langle f,e_{n}\rangle
e_{n}(x)=\sum_{n\in\mathbb{N}_{0}}\left(\int_{\mathcal{M}}f(y)\overline{e_{n}(y)}\,d\mu(y)\right)e_{n}(x),$
(1)

which is similar to a Fourier series. Since $e_{n}(y)$ is a replacement for a
Fourier mode, it is natural to let

$\hat{f}(n)=\int_{\mathcal{M}}f(y)\overline{e_{n}(y)}\,d\mu(y)$ (2)

and define convolution on $\mathcal{M}$ between functions
$f,h\in\mathbf{L}^{2}(\mathcal{M})$ as

$f\ast h(x)=\sum_{n\in\mathbb{N}_{0}}\hat{f}(n)\hat{h}(n)e_{n}(x).$ (3)

Defining the operator $T_{h}f(x):=f\ast h(x)$, it is easy to verify that the
kernel for $T_{h}$ is given by

$\tilde{K}_{h}(x,y):=\sum_{n\in\mathbb{N}_{0}}\hat{h}(n)e_{n}(x)\overline{e_{n}(y)}.$
(4)

Similar to how convolution commutes with translations on $\mathbb{R}^{n}$, it
is important for convolution on $\mathcal{M}$ to be equivariant to a group
action on $\mathcal{M}$. The authors of [36] construct an operator by
convolving with functions that commute with isometries since the the geometry
of $\mathcal{M}$ should be preserved by a representation.

To accomplish this goal, we use a similar definition for spectral filters. A
filter $h\in\mathbf{L}^{2}(\mathcal{M})$ is a spectral filter if
$\lambda_{k}=\lambda_{\ell}$ implies $\hat{h}(k)=\hat{h}(\ell)$. One can prove
that there exists $H:[0,\infty)\to\mathbb{R}$ such that

$H(\lambda_{n})=\hat{h}(n),\qquad\forall n\in\mathbb{N}_{0}.$

Let $G:[0,\infty)\to\mathbb{R}$ be nonnegative and decreasing with $G(0)>0$. A
low-pass spectral filter $\phi$ is given in frequency as
$\hat{\phi}(n):=G(\lambda_{n})$ and its dilation at scale $2^{j}$ for
$j\in\mathbb{Z}$ is $\hat{\phi}_{j}(n):=G(2^{j}\lambda_{n})$.

Using the set of low pass filters, $\\{\hat{\phi}_{j}\\}_{j\in\mathbb{Z}}$, we
define wavelets by

$\hat{\psi}_{j}(n):=[|\hat{\phi}_{j-1}(n)|^{2}-|\hat{\phi}_{j}(n)|^{2}]^{1/2}=[|G(2^{j-1}n)|^{2}-|G(2^{j}n)|^{2}]^{1/2},$
(5)

which are diffusion wavelets [40].

Fix $J\in\mathbb{Z}$. Define the operators

$\displaystyle A_{J}f$ $\displaystyle:=f\ast\phi_{J},$
$\displaystyle\Psi_{j}f$ $\displaystyle:=f\ast\psi_{j},\qquad j\leq J.$

The windowed geometric wavelet transform is given by

$W_{J}f:=\\{A_{J}f,\Psi_{j}f:\quad j\leq J\\}$ (6)

and the nonwindowed geometric wavelet transform is given by

$Wf:=\\{\Psi_{j}f:\quad j\in\mathbb{Z}\\}.$ (7)

We have the following theorem, which provides a condition for when our wavelet
frame is a nonexpansive frame.

######  Theorem 1.

Let $G:[0,\infty)\to\mathbb{R}$ be nonnegative, decreasing, and continuous
with $0<G(0)=C$, $\lim_{x\to\infty}G(x)=0$, and
$\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ is a set of wavelets generated by using the
low pass filter $\hat{\phi}(k)=G(\lambda_{k})$ in Equation 5. Then we have

$\sum_{j\in\mathbb{Z}}\|f\ast\psi_{j}\|_{2}^{2}=C^{2}\|f\|_{2}^{2}.$ (8)

###### Proof.

For fixed $J>1$, we telescope to get

$\displaystyle\sum_{j=-J}^{J}|\hat{\psi}_{j}(k)|^{2}$
$\displaystyle=\sum_{j=-J}^{J}\left[|G(2^{j-1}\lambda_{k})|^{2}-|G(2^{j}\lambda_{k})|^{2}\right]$
$\displaystyle=|G(2^{-J-1}\lambda_{k})|^{2}-|G(2^{J}\lambda_{k})|^{2}.$

Since $\lim_{J\to\infty}|G(2^{-J-1}\lambda_{k})|^{2}$ and
$\lim_{J\to\infty}|G(2^{J}\lambda_{k})|^{2}$ both exist, it follows that

$\sum_{j\in\mathbb{Z}}|\hat{\psi}_{j}(k)|^{2}=\lim_{J\to\infty}|G(2^{-J-1}\lambda_{k})|^{2}-\lim_{J\to\infty}|G(2^{J}\lambda_{k})|^{2}=C^{2}.$

We can write

$\|f\ast\psi_{j}\|_{2}^{2}=\sum_{k\in\mathbb{N}_{0}}|\hat{\psi}_{j}(k)|^{2}|\hat{f}(k)|^{2}.$

Thus, it follows that

$\displaystyle\sum_{j\in\mathbb{Z}}\|f\ast\psi_{j}\|_{2}^{2}$
$\displaystyle=\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{N}_{0}}|\hat{f}(k)|^{2}|\hat{\psi}_{j}(k)|^{2}$
$\displaystyle=\sum_{k\in\mathbb{N}_{0}}|\hat{f}(k)|^{2}\left(\sum_{j\in\mathbb{Z}}|\hat{\psi}_{j}(k)|^{2}\right)$
$\displaystyle=C^{2}\|f\|_{2}^{2}.$

∎

###  2.2 The Geometric Scattering Transform

In an analogous manner to the Euclidean definition of the scattering
transform, it is valuable to find a representation that meaningfully encodes
high frequency information of a signal $f$. Define the propagator as

$U[j]f:=|\Psi_{j}f|\qquad\forall j\in\mathbb{Z},$ (9)

which is convolution of a wavelet and applying a nonlinearity; we can also
define the windowed propagator as

$U_{J}[j]f:=|\Psi_{j}f|\qquad\forall j\leq J.$ (10)

Similar to scattering transforms on Euclidean space, one can apply a cascade
of convolutions and modulus operators repeatedly. In particular, for
$m\in\mathbb{N}$, let $j_{1},\ldots,j_{m}\in\mathbb{Z}$. The $m$-layer
propagator is defined as

$U[j_{1},\ldots,j_{m}]f:=U[j_{m}]\cdots
U[j_{1}]f=|||f\ast\psi_{j_{1}}|\ast\psi_{j_{2}}|\cdots\ast\psi_{j_{m}}|$ (11)

and the $m$-layer windowed propagator is defined as

$U[j_{1},\ldots,j_{m}]f:=U[j_{m}]\cdots
U[j_{1}]f=|||f\ast\psi_{j_{1}}|\ast\psi_{j_{2}}|\cdots\ast\psi_{j_{m}}|,\qquad
j_{1},\ldots,j_{m}\leq J$ (12)

with $U[\emptyset]f=f$ and $U_{J}[\emptyset]f=f.$ To aggregate low frequency
information and get local isometry invariance, one can apply a low pass filter
in a manner similar to pooling to each windowed propagator to get windowed
scattering coefficients:

$S_{J}[j_{1},\ldots,j_{m}]=A_{J}U_{J}[j_{1},\ldots,j_{m}]f=U_{J}[j_{1},\ldots,j_{m}]f\ast\phi_{J},$

where we defined $S_{J}[\emptyset]f=f\ast\phi_{J}$. The windowed geometric
scattering transform is given by

$S_{J}f=\\{S_{j}[j_{1},\ldots,j_{m}]f:\,m\geq 0,\quad j_{i}\leq J\quad\forall
1\leq i\leq m\\}.$ (13)

The authors of [36] were able to prove that this windowed scattering operator
was nonexpansive, invariant to isometries up to the scale of the low pass
filter, and stable to diffeomporhisms under mild assumptions.

In addition, the authors consider a nonwindowed scattering transform, which
removes the low pass filtering. For applications such as manifold
classification, one desires full isometry invariance instead of isometry
invariance up to scale $2^{J}$. We see that

$\lim_{J\to\infty}S[j_{1},\ldots,j_{m}]f=\text{vol}(\mathcal{M})^{-1/2}\|U[j_{1},\ldots,j_{m}]f\|_{1}.$
(14)

As a proxy, one can consider

$\overline{S}f[j_{1},\ldots,j_{m}]=\|U[j_{1},\ldots,j_{m}]f\|_{1},$ (15)

This motivates [36] defining the nonwindowed geometric scattering transform as

$\overline{S}f=\\{\overline{S}[j_{1},\ldots,j_{m}]f:\,m\geq 0,\quad
j_{i}\in\mathbb{Z},\quad\forall 1\leq i\leq m\\}.$ (16)

Instead of considering $\mathbf{L}^{1}(\mathcal{M})$ norms of $m$-layer
propagators, we can instead consider $\mathbf{L}^{q}(\mathcal{M})$ norms of
$m$-layer propagators for $q\in(1,2]$, which we define as $m$-layer
$q$-nonwindowed geometric scattering coefficients (which are also referred to
as scattering moments in other works):

$\overline{S}_{q}^{m}[j_{1},\ldots,j_{m}]f=\|U[j_{1},\ldots,j_{m}]f\|_{\mathbf{L}^{q}(\mathcal{M})}\qquad\forall(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m},$
(17)

which has seen application in quantum chemistry [6, 7, 8] and for point cloud
data [38]. As shorthand notation, we will use the following notation for one
layer coefficients:

$\overline{S}_{q}[j]f=\|U[j]f\|_{\mathbf{L}^{q}(\mathcal{M})}\qquad\forall
j\in\mathbb{Z}.$ (18)

To measure stability, invariance, and equivariance, we define the following
norm for $q$-nonwindowed geometric scattering coefficients:

$\|\overline{S}_{q}^{m}f\|_{\mathbf{\ell}^{2}(\mathbb{Z}^{m})}^{q}:=\left(\sum_{j_{m}\in\mathbb{Z}}\ldots\sum_{j_{1}\in\mathbb{Z}}|\overline{S}_{q}^{m}[j_{1},\ldots,j_{m}]f|^{2}\right)^{q/2},$
(19)

which follows the definition in [37]. Since many of the results in [37] rely
on bounds of Littlewood Paley $g$-functions, we will provide extensions of
these results to compact manifolds, with some extra restrictions added.

##  3 Generalization of the Littlewood Paley $g$-function

Let $\mathcal{B}$ be a Banach space, suppose that
$g:\mathcal{M}\to\mathcal{B}$, and $x\to\|g(x)\|_{\mathcal{B}}$ is measurable
with respect to the measure induced by the Riemannian volume. Define
$\mathbf{L}^{p}_{\mathcal{B}}(\mathcal{M})$ for $1\leq p<\infty$ to be

$\|g\|_{\mathbf{L}^{p}_{\mathcal{B}}(\mathcal{M})}^{p}=\int_{\mathcal{M}}\|g(x)\|_{\mathcal{B}}^{p}\,d\mu(x)\,.$

Also, for $1\leq p<\infty$, define

$\|g\|_{\mathbf{L}_{\mathcal{B}}^{p,\infty}(\mathcal{M})}=\sup_{\delta>0}\delta\cdot\mu(\\{x\in\mathcal{M}:\|g(x)\|_{\mathcal{B}}>\delta\\})^{1/p}\,,$

It is clear that

$\|g\|_{\mathbf{L}_{\mathcal{B}}^{p,\infty}(\mathcal{M})}\leq\|g\|_{\mathbf{L}^{p}_{\mathcal{B}}(\mathcal{M})}\,.$

In particular, let our Banach space be $\ell^{2}(\mathbb{Z})$, and consider
the operator $\vec{T}f:\mathcal{M}\to\ell^{2}(\mathbb{Z})$ given by

$\vec{T}(f)(x):=\\{T_{j}f(x)\\}_{j\in\mathbb{Z}}=\left\\{\int_{\mathcal{M}}K_{2^{-j/2}}(x,y)f(y)\,d\mu(y)\right\\}_{j\in\mathbb{Z}}.$

with kernel given by $\vec{K}=\\{K_{2^{-j/2}}\\}_{j\in\mathbb{Z}}$ associated
to the wavelets $\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ generated using a low pass
filter $G\in\mathcal{S}(\mathbb{R}^{+})$ satisfying the conditions satisfying
the conditions of Theorem 1 for Equation 5. This can be seen as an extension
of the Littlewood Paley $g$-function for a compact Riemannian manifold since

$\|\vec{T}(f)(x)\|_{\ell^{2}(\mathbb{Z})}=\left(\sum_{j\in\mathbb{Z}}|f\ast\psi_{j}(x)|^{2}\right)^{1/2}.$

From Theorem 1,

$\|\vec{T}f\|_{\mathbf{L}^{2}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}\leq\|f\|_{\mathbf{L}^{2}(\mathcal{M})}.$

Our goal is to extend this operator and prove that for all $q\in(1,2)$, there
exists $C_{q}$ such that

$\|\vec{T}f\|_{\mathbf{L}^{q}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}\leq
C_{q}\|f\|_{\mathbf{L}^{q}(\mathcal{M})}.$

Before providing any proofs, we will state preliminary lemmas that will be
vital to our approach. The first few lemmas concern the kernel of our
convolution operator.

######  Lemma 2 ([41]).

Suppose that $F\in\mathcal{S}(\mathbb{R}^{+})$, the space of Schwartz
functions restricted to $\mathbb{R}^{+}$ and $F(0)=0$. For the kernel

$K_{t}(x,y)=\sum_{n\in\mathbb{N}}F(t^{2}\lambda_{n})e_{n}(x)\overline{e_{n}(y)},$

the following pointwise bound holds for some $C_{n}>0$ and for all $t>0$ and
all $x,y\in\mathcal{M}$:

$|K_{t}(x,y)|\leq\frac{C_{n}t^{-n}}{\left(1+\tfrac{r(x,y)}{t}\right)^{n+1}},$

We remind the reader that $n$ is defined as the dimension of $\mathcal{M}$.

Let $\mathcal{U}$ be an open cover for $\mathcal{M}$. A number $\gamma>0$ is
called a Lebesgue number for $\mathcal{U}$ if and only if for all
$x\in\mathcal{M}$, there exists $\mathcal{U}_{x}\in\mathcal{U}$ such that
$B(x,\gamma)\subset\mathcal{U}_{x}$, where we define

$B(x,r_{0}):=\\{y\in X\,:\,r(x,y)<r_{0}\\}.$ (20)

######  Lemma 3 ([42]).

Cover $\mathcal{M}$ with a finite collection of open sets $P_{i}$ with $1\leq
i\leq I$ such that the following properties hold for each index $i$:

  1. 1.

there exists a chart $(V_{i},\varphi_{i})$ with $\overline{P}_{i}\subset
V_{i}$

  2. 2.

$\varphi_{i}(P_{i})$ is a ball in $\mathbb{R}^{n}$.

Choose $\delta>0$ so that $3\delta$ is a Lebesgue number for the covering
$\\{P_{i}\\}$. Then there exist $c_{1},c_{2}>0$ such that for any
$x\in\mathcal{M}$ and any $B(x,3\delta)\subset P_{i}$, the following
statements hold in the coordinate system on $P_{i}$ obtained from
$\varphi_{i}$:

  1. 1.

For all $y,z\in P_{i}$, we have $r(y,z)\leq c_{2}|y-z|$.

  2. 2.

For all $y,z\in B(x,\delta)$, we have $r(y,z)\geq c_{1}|y-z|$.

###### Proof.

This proof follows the idea for [42, Proposition 3.1] yet with explicitly
identified constants $c_{1},c_{2}$ to demonstrate their continuous dependence
on the metric $g$. Using the coordinates given by $\varphi_{i}$, we identify
$g(x)$ with an $n\times n$ smooth matrix for each $x\in\overline{P}_{i}$. We
write $|\cdot|_{g}$ for the norm induced by $g$, and $|\cdot|$ for the
Euclidean norm in the $\varphi_{i}$-coordinates.

For each $x\in\overline{P}_{i}$ and a tangent vector $v_{x}\in
T_{x}\mathcal{M}$, we have

$\underline{\Lambda}(x)|v_{x}|^{2}\leq|v_{x}|^{2}_{g(x)}=\sum^{n}_{\alpha,\beta=1}v_{x}^{\alpha}g_{\alpha\beta}(x)v_{x}^{\beta}\leq\overline{\Lambda}(x)|v_{x}|^{2}.$

Here $\underline{\Lambda}(x),\overline{\Lambda}(x)$ denote the minimum and
maximum eigenvalues of the matrix $g(x)$ respectively. We conclude

$\left(\min_{x\in\overline{P}_{i}}\underline{\Lambda}(x)\right)|v_{x}|^{2}\leq|v_{x}|^{2}_{g(x)}\leq\left(\max_{x\in\overline{P}_{i}}\overline{\Lambda}(x)\right)|v_{x}|^{2}$

uniformly for $(x,v)\in T\overline{P}_{i}$. As both $\underline{\Lambda}(x)$
and $\overline{\Lambda}(x)$ are positive continuous functions of $x$ on the
compact subset $\overline{P}_{i}$, their minimum and maximum are strictly
positive.

For the first statement, take $y,z\in P_{i}$ and define
$\gamma:[0,1]\rightarrow P_{i}$, $\gamma(t):=tz+(1-t)y$ where the coordinates
of $y,z$ are given by $\varphi_{i}$. Then $\gamma$ is a smooth curve
connecting $y,z$ and $|\gamma^{\prime}(t)|=|y-z|$ for all $t$. By the
definition of the Riemannian distance $r(y,z)$, we have

$r(y,z)\leq\text{ length of
}\gamma=\int^{1}_{0}|\gamma^{\prime}(t)|_{g(\gamma(t))}\,dt\leq\left(\max_{x\in\overline{P}_{i}}\overline{\Lambda}(x)\right)^{\frac{1}{2}}\int^{1}_{0}|\gamma^{\prime}(t)|\,dt=\left(\max_{x\in\overline{P}_{i}}\overline{\Lambda}(x)\right)^{\frac{1}{2}}|y-z|.$

One choice for $c_{2}$ is
$c_{2}:=\max_{i=1,\dots,I}\left(\max_{x\in\overline{P}_{i}}\overline{\Lambda}(x)\right)^{\frac{1}{2}}$.

For the second statement, take $y,z\in B(x,\delta)$ and let
$\gamma_{k}:[0,1]\rightarrow\mathcal{M}$ be a sequence of piecewise $C^{1}$
curves connecting $y,z$ such that their lengths $\ell(\gamma_{k})\rightarrow
r(y,z)$ as $k\rightarrow\infty$. For large $k$, we have

$r(\gamma_{k}(t),x)\leq r(\gamma_{k}(t),y)+r(y,x)\leq r(z,y)+\delta\leq
r(z,x)+r(x,y)+\delta\leq 3\delta$

for all $t\in[0,1]$, hence $\gamma_{k}\subset P_{i}$. Therefore, we have for
large $k$ that

$\ell(\gamma_{k})=\int^{1}_{0}|\gamma^{\prime}_{k}(t)|_{g}\,dt\geq\left(\min_{x\in\overline{P}_{i}}\underline{\Lambda}(x)\right)^{\frac{1}{2}}\int^{1}_{0}|\gamma^{\prime}_{k}(t)|\,dt\geq\left(\min_{x\in\overline{P}_{i}}\underline{\Lambda}(x)\right)^{\frac{1}{2}}\left|\int^{1}_{0}\gamma^{\prime}_{k}(t)\,dt\right|\geq\left(\min_{x\in\overline{P}_{i}}\underline{\Lambda}(x)\right)^{\frac{1}{2}}|y-z|.$

Let $k\rightarrow\infty$ proves the statement, and one choice for $c_{1}$ is
$c_{1}:=\min_{i=1,\dots,I}\left(\min_{x\in\overline{P}_{i}}\underline{\Lambda}(x)\right)^{\frac{1}{2}}$.

∎

For the rest of this paper, we fix collections $\\{P_{i}\\}$, $\\{V_{i}\\}$,
$\\{\varphi_{i}\\}$, and constants $\delta,c_{1},c_{2}$ from the previous
Lemma.

######  Lemma 4.

Suppose that $r(y,z)<\min\left\\{\tfrac{1}{2}r(x,y),\delta\right\\}$ so that
$y$ and $z$ lie on the same ball of the covering. Assume that there exist
$c_{1}$ and $c_{2}$ from Lemma 3 with $c_{1}c_{2}<2.$ Then there exists a
constant $C_{\delta}$ such that

$|K_{t}(x,y)-K_{t}(x,z)|\leq
C_{\delta}\frac{r(y,z)t^{-n-1}}{\left(1+\frac{r(x,y)}{t}\right)^{n+1}}.$

###### Proof.

Using the proof of Theorem 5.5 in [42], for each $x\in\mathcal{M}$, there
exists a point $w_{x}$ on the segment connecting $y$ to $z$ such that

$|K_{t}(x,y)-K_{t}(x,z)|\leq
C_{1}\frac{r(y,z)t^{-n-1}}{\left(1+\frac{r(x,w_{x})}{t}\right)^{n+1}}.$

Now notice that triangle inequality implies that

$r(x,y)\leq r(x,w_{x})+r(y,w_{x}).$

By Lemma 3, since $w_{x}$ lies on the line segment between $y$ and $z$, we see
that

$r(y,w_{x})\leq c_{2}|y-w_{x}|\leq c_{2}|y-z|\leq c_{1}c_{2}r(y,z).$

It follows that

$r(x,y)-c_{1}c_{2}r(y,z)\leq r(x,w_{x}).$

Since $c_{1}c_{2}<2$ and $r(x,y)\geq 2r(y,z)$, we see that

$r(x,y)-\frac{c_{1}c_{2}}{2}r(x,y)\leq r(x,w_{x})$

so that $1-\tfrac{c_{1}c_{2}}{2}>0$. This leads to $r(x,y)\leq
C_{r}r(x,w_{x})$ for some constant $C_{r}$ independent of $x$. Finally, we
have

$|K_{t}(x,y)-K_{t}(x,z)|\leq
C_{\delta}\frac{r(y,z)t^{-n-1}}{\left(1+\frac{r(x,y)}{t}\right)^{n+1}}.$

∎

Now we provide the necessary tools from classical harmonic analysis for
extension. The idea is similar to the proof in the Euclidean case; we wish to
prove a weak-type $(1,1)$ bound and extend by interpolation. Let $X$ be a set,
$\beta$ a quasimetric, and $\mu$ a measure on $X$ such that
$0<\mu(B(x,r))<\infty$ for all $x\in X$ and $r>0$. We say that a space
$(X,\beta,\mu)$ is of homogeneous type, though often with the quasi-metric and
measure omitted when implied, if for all $x\in X$ and $r>0$ there exists a
constant $C_{D}$ such that

$\mu(B(x,2r))\leq C_{D}\mu(B(x,r)),$ (21)

where $B(x,r)$ is a ball of radius $r$ centered at $x$ for $(X,\beta).$ The
property above is also known as the doubling property. It is well known that a
$C^{\infty}$ compact Riemannian manifold using the standard Riemannian metric
and volume is of homogeneous type.

The first result we will need is a Calderon-Zygmund decomposition:

######  Theorem 5 ([43]).

Suppose that $X$ is a space of homogeneous type. Suppose that
$f\in\mathbf{L}^{1}(X)$ and choose $\alpha>0$ such that
$\alpha^{-1}\|f\|_{1}<\mu(X)$. Then we can decompose $f:=g+b$ such that

$\displaystyle\|g\|_{\mathbf{L}^{2}(\mathcal{M})}^{2}$ $\displaystyle\leq
C_{1}\alpha\|f\|_{\mathbf{L}^{1}(\mathcal{M})},$ $\displaystyle b$
$\displaystyle=\sum_{i}b_{i},$

where $C_{1}>0$ is a constant, $\text{supp}(b_{i})\subset B(x_{i},r_{i})$ for
some countable collection of balls $\\{B(x_{i},r_{i})\\}$, and each $b_{i}$
satisfies

$\displaystyle\int_{X}b_{i}(x)d\mu(x)$ $\displaystyle=0,$
$\displaystyle\|b_{i}\|_{1}$ $\displaystyle\leq C\alpha\mu(B(x_{i},r_{i})),$
$\displaystyle\sum_{i}\mu(B(x_{i},r_{i}))$ $\displaystyle\leq
C\alpha^{-1}\|f\|_{\mathbf{L}^{1}(\mathcal{M})}.$

######  Lemma 6.

Suppose that we choose wavelets $\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ generated
using $G\in\mathcal{S}(\mathbb{R}^{+})$ in Equation 5 that satisfy the
conditions of Theorem 1 and $c_{1}c_{2}<2$ in Lemma 3. The following weak
$(1,1)$ bound holds for some $A>0$:

$\|\vec{T}f\|_{\mathbf{L}^{1,\infty}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}\leq
A\|f\|_{\mathbf{L}^{1}(\mathcal{M})}.$

###### Proof.

First, for any $\alpha$ such that
$\alpha^{-1}\|f\|_{\mathbf{L}^{1}(\mathcal{M})}>\mu(\mathcal{M})$, we see that

$\displaystyle\mu(\\{x\in\mathcal{M}:\|\vec{T}f(x)\|_{\mathcal{B}}>\alpha\\})\leq\mu(\mathcal{M})\leq\alpha^{-1}\|f\|_{\mathbf{L}^{1}(\mathcal{M})}.$

Now, we consider the case where
$\alpha^{-1}\|f\|_{\mathbf{L}^{1}(\mathcal{M})}\leq\mu(\mathcal{M})$. We use
our Caulderon-Zygmund decomposition and write $f=g+b$. It follows that

$\displaystyle\mu(\\{x\in\mathcal{M}:\|\vec{T}f(x)\|_{\mathcal{B}}>\alpha\\})$
$\displaystyle\leq\mu(\\{x\in\mathcal{M}:\|\vec{T}g(x)\|_{\mathcal{B}}>\alpha/2\\})+\mu(\\{x\in\mathcal{M}:\|\vec{T}b(x)\|_{\mathcal{B}}>\alpha/2\\})$
$\displaystyle:=I_{1}+I_{2}.$

For $I_{1}$, we apply Chebyshev inequality,
$\mathbf{L}^{2}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})$ boundedness of $\vec{T}$,
and our assumption on $g$ to find that

$\displaystyle\mu(\\{x\in\mathcal{M}:\|\vec{T}g(x)\|_{\mathcal{B}}>\alpha/2\\})$
$\displaystyle\leq\frac{4}{\alpha^{2}}\|\vec{T}g\|_{\mathbf{L}^{2}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}^{2}$
$\displaystyle\leq\frac{4}{\alpha^{2}}\|\vec{T}\|_{\mathbf{L}^{2}(\mathcal{M})\to\mathbf{L}^{2}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}^{2}\|g\|_{\mathbf{L}^{2}(\mathcal{M})}^{2}$
$\displaystyle\leq\frac{4C_{1}}{\alpha}\|\vec{T}\|_{\mathbf{L}^{2}(\mathcal{M})\to\mathbf{L}^{2}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}^{2}\|f\|_{\mathbf{L}^{1}(\mathcal{M})}.$

For $I_{2}$, let $B=\bigcup_{i}B(x_{i},2r_{i})$. Then it follows that

$\displaystyle I_{2}$ $\displaystyle\leq\mu(B)+\mu(\\{x\in
B^{c}:\|\vec{T}b(x)\|_{\mathcal{B}}>\alpha/2\\}$
$\displaystyle\leq\frac{C_{D}}{\alpha}\|f\|_{\mathbf{L}^{1}(\mathcal{M})}+\frac{2}{\alpha}\|\vec{T}b\|_{\mathbf{L}^{1}_{\ell^{2}(\mathbb{Z})}(B^{c})}$
$\displaystyle\leq\frac{C_{D}}{\alpha}\|f\|_{\mathbf{L}^{1}(\mathcal{M})}+\frac{2}{\alpha}\sum_{i}\|\vec{T}b_{i}\|_{\mathbf{L}^{1}_{\ell^{2}(\mathbb{Z})}(B(x_{i},2r_{i})^{c})},$

where the constant $C_{D}$ comes from the fact that our measure has the
doubling property.

To estimate
$\|\vec{T}b_{i}\|_{\mathbf{L}^{1}_{\ell^{2}(\mathbb{Z})}(B(x_{i},2r_{i})^{c})}$,
we notice that

$\displaystyle\|\vec{T}b_{i}\|_{\mathbf{L}^{1}_{\ell^{2}(\mathbb{Z})}(B(x_{i},2r_{i})^{c})}$
$\displaystyle=\int_{B(x_{i},2r_{i})^{c}}\|\vec{T}b_{i}(x)\|_{\ell^{2}(\mathbb{Z})}\,d\mu(x)$
$\displaystyle=\int_{B(x_{i},2r_{i})^{c}}\left\|\int_{B(x_{i},r_{i})}\vec{K}(x,y)b_{i}(y)\,dy\right\|_{\ell^{2}(\mathbb{Z})}\,d\mu(x)$
$\displaystyle=\int_{B(x_{i},2r_{i})^{c}}\left\|\int_{B(x_{i},r_{i})}(\vec{K}(x,y)-\vec{K}(x,x_{i}))b_{i}(y)\,d\mu(y)\right\|_{\ell^{2}(\mathbb{Z})}\,d\mu(x)$
$\displaystyle=\int_{B(x_{i},r_{i})}\left(\int_{B(x_{i},2r_{i})^{c}}\left(\sum_{j\in\mathbb{Z}}\left|{K}_{2^{-j/2}}(x,y)-{K}_{2^{-j/2}}(x,x_{i})\right|^{2}|b_{i}(y)|^{2}\right)^{1/2}\,d\mu(x)\right)\,d\mu(y)$
$\displaystyle\leq\int_{B(x_{i},r_{i})}|b_{i}(y)|\int_{B(x_{i},2r_{i})^{c}}\sum_{j\in\mathbb{Z}}\left|{K}_{2^{-j/2}}(x,y)-{K}_{2^{-j/2}}(x,x_{i})\right|\,d\mu(x)\,d\mu(y)$
$\displaystyle=\int_{B(x_{i},r_{i})}|b_{i}(y)|\left(\sum_{j\in\mathbb{Z}}\int_{B(x_{i},2r_{i})^{c}}\left|{K}_{2^{-j/2}}(x,y)-{K}_{2^{-j/2}}(x,x_{i})\right|\,d\mu(x)\right)\,d\mu(y),$

where $\vec{K}$ is the kernel defined on page 7. Now we consider the term
inside the parentheses. We will break this argument into cases. First,
consider if $2r_{i}\geq\delta$. We see that

$\displaystyle\sum_{j\in\mathbb{Z}}\int_{B(x_{i},2r_{i})^{c}}|K_{2^{-j/2}}(x,y)-K_{2^{-j/2}}(x,x_{i})|\,d\mu(x)$
$\displaystyle\leq
C\sum_{j\in\mathbb{Z}}\int_{B(x_{i},2r_{i})^{c}}\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}+\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,y)\right)^{n+1}}\,d\mu(x).$

Now, since $x_{i}$ is the center of $B(x_{i},r_{i})$, if $x\in
B(x_{i},2r_{i})^{c}$, then $r(x,x_{i})\geq 2r_{i}\geq\delta$. Similarly, since
$y\in B(x_{i},r_{i})$, it follows that $r(y,x_{i})<r_{i}$ and we have
$2r(y,x_{i})\leq r(x,x_{i})$. Apply triangle inequality to get

$r(x,x_{i})\leq r(x,y)+r(y,x_{i})\leq r(x,y)+r_{i}\leq
r(x,y)+\frac{1}{2}r(x,x_{i}),$

which means that $r(x,x_{i})\leq 2r(x,y).$

Going back to the integral, there exists $C_{1}$ such that

$\displaystyle\sum_{j\in\mathbb{Z}}\int_{B(x_{i},2r_{i})^{c}}\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,y)\right)^{n+1}}+\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}\,d\mu(x)$
$\displaystyle\leq C_{1}\sum_{j\in\mathbb{Z}}\int_{r(x,x_{i})\geq
2r(y,x_{i})}\frac{2^{nj}}{\left(1+2^{j}r(x,x_{i})\right)^{n+1}}\,d\mu(x)$
$\displaystyle=C_{1}\sum_{j\geq 0}\int_{r(x,x_{i})\geq
2r(y,x_{i})}\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}\,d\mu(x)+C_{1}\sum_{j<0}\int_{r(x,x_{i})\geq
2r(y,x_{i})}\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}\,d\mu(x)$
$\displaystyle:=J_{1}+J_{2}.$

For $J_{1}$, since $r(x,x_{i})>\delta$,

$\sum_{j\geq 0}\int_{r(x,x_{i})\geq
2r(y,x_{i})}\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}\,d\mu(x)\leq\sum_{j\geq
0}2^{-j/2}\delta^{n+1}<\infty.$

For $J_{2}$, it is routine to see that

$\sum_{j<0}\int_{r(x,x_{i})\geq
2r(y,x_{i})}\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}\,d\mu(x)\leq\sum_{j<0}2^{nj/2}\mu(\mathcal{M})<\infty.$

Now we consider the case where $2r_{i}<\delta$. In this case, we see that
$r(y,x_{i})<r_{i}<\delta$, and we still have $2r(y,x_{i})<r(x,x_{i})$. Thus,
the bound

$|K_{2^{-j/2}}(x,y)-K_{2^{-j/2}}(x,x_{i})|\leq
C\frac{2^{nj/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}$

still applies. We can also apply Lemma 4 to get

$|K_{2^{-j/2}}(x,y)-K_{2^{-j/2}}(x,x_{i})|\leq
C_{\delta}\frac{r(y,x_{i})2^{(n+1)j/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}.$

Taking the geometric mean, for any $s\in[0,1]$, we have

$|K_{2^{-j/2}}(x,y)-K_{2^{-j/2}}(x,x_{i})|\leq
C_{2}\frac{2^{nj/2}(2^{j/2}r(y,x_{i}))^{s}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}$

for some constant $C_{2}$. It now follows that for
$C_{3}=\max\\{C_{\delta},C_{2}\\}$, we have

$\displaystyle\sum_{j\in\mathbb{Z}}|K_{2^{-j/2}}(x,y)-K_{2^{-j/2}}(x,x_{i})|$
$\displaystyle\leq\sum_{2^{j/2}<\frac{2}{r(x,x_{i})}}|K_{2^{-j/2}}(x,y)-K_{2^{-j/2}}(x,x_{i})|+\sum_{2^{j/2}\geq\frac{2}{r(x,x_{i})}}|K_{2^{-j/2}}(x,y)-K_{2^{-j/2}}(x,x_{i})|$
$\displaystyle\leq
C_{\delta}\sum_{2^{j/2}<\frac{2}{r(x,x_{i})}}\frac{r(y,x_{i})2^{(n+1)j/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}+C_{2}\sum_{2^{j/2}\geq\frac{2}{r(x,x_{i})}}\frac{2^{nj/2}(2^{j/2}r(y,x_{i}))^{1/2}}{\left(1+2^{j/2}r(x,x_{i})\right)^{n+1}}$
$\displaystyle\leq
C_{3}\left(r(x_{i},y)\sum_{2^{j/2}<\frac{2}{r(x,x_{i})}}2^{(n+1)j/2}+r(x_{i},y)^{1/2}\sum_{2^{j/2}\geq\frac{2}{r(x,x_{i})}}2^{(n+1/2)j/2}(2^{j/2}r(x,x_{i}))^{-n-1}\right)$
$\displaystyle\leq
C_{3}(r(x_{i},y)r(x,x_{i})^{-n-1}+r(x_{i},y)^{1/2}r(x,x_{i})^{-n-1/2}).$

Integrating over $2r(x_{i},y)\leq r(x,x_{i})$ yields a constant independent of
$r_{i}$. It now follows that

$\|\vec{T}b_{i}\|_{\mathbf{L}^{1}_{\ell^{2}(\mathbb{Z})}(B(x_{i},r_{i})^{c})}\leq
C_{4}\|b_{i}\|_{\mathbf{L}^{1}(\mathcal{M})}$

for some constant $C_{4}$. Using the Caulderon-Zygmund decomposition,

$I_{2}\leq\left(\frac{C_{D}}{\alpha}+\frac{2C_{4}}{\alpha}\right)\|f\|_{\mathbf{L}^{1}(\mathcal{M})}.$

∎

Recall the following result, which is a vector-valued version of Marcinkiewicz
Interpolation:

######  Lemma 7 ([44]).

Let $\mathcal{A}_{1},\mathcal{A}_{2}$ be Banach spaces,
$T:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be quasilinear on
$\mathbf{L}^{p_{0}}_{\mathcal{A}_{1}}(X)$ and
$\mathbf{L}^{p_{1}}_{\mathcal{A}_{1}}(X)$ with $0<p_{0}<p_{1}$. If $T$
satisfies

$\|Tf\|_{\mathbf{L}^{p_{i},\infty}_{\mathcal{A}_{2}}(X)}\leq
M_{i}\|f\|_{\mathbf{L}_{\mathcal{A}_{1}}^{p_{i}}(X)}$

for $i=0,1$, then

$\|Tf\|_{\mathbf{L}^{p}_{\mathcal{A}_{2}(X)}}\leq
N_{p}\|f\|_{\mathbf{L}_{\mathcal{A}_{1}}^{p}(X)}\qquad\forall
p\in(p_{0},p_{1}),$

where $N_{p}$ is dependent on $p$.

The following corollary is a direct result of interpolation now:

######  Corollary 8.

Suppose that we choose wavelets $\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ generated by
using $G\in\mathcal{S}(\mathbb{R}^{+})$ in Equation 5, $G$ satisfies the
conditions of Theorem 1, and $c_{1}c_{2}<2$. We have

$\|\vec{T}f\|_{\mathbf{L}^{q}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}^{q}\leq
C_{q}\|f\|_{\mathbf{L}^{q}(\mathcal{M})}^{q}$

for some constant $C_{q}>0$, where $q\in(1,2)$.

By duality, the result of Corollary 8 actually holds for $q\in(1,\infty)$.
However, for generalizing the nonwindowed geometric scattering transform, we
only need results for $q\in(1,2)$ since
$\mathbf{L}^{2}(\mathcal{M})\subset\mathbf{L}^{q}(\mathcal{M})$. For $q>2$,
since our manifold is compact, we have
$\mathbf{L}^{q}(\mathcal{M})\subset\mathbf{L}^{2}(\mathcal{M})$, so previous
results in [36] are applicable, and further theoretical analysis is not as
significant.

Additionally, although the result of Corollary 8 seems restrictive because one
needs $c_{1}c_{2}<2$, the result applies for a variety of different manifolds.
If one finds a metric where the condition above holds, a class of metrics can
be found by perturbing the metric. This is because the choice of the constants
$c_{1},c_{2}$ in the proof of Lemma 3 depend continuously on the metric $g$.
Thus if $c_{1}c_{2}<2$ for $g$, the same strict inequality holds for all
metrics that are sufficiently close to $g$. We provide a simple example below
where the conditions of Lemma 3 hold. The result of the example below can also
be extended to $n$-torii without much difficulty.

######  Example 1.

Take $R=1$ and use the charts:

$\begin{array}[]{ll}V_{1}:=\\{(x_{1},x_{2}):x_{1}^{2}+x_{2}^{2}=1,\;x_{1}>0\\},&\qquad\varphi_{1}:V_{1}\rightarrow\mathbb{R},\;(x_{1},x_{2})\mapsto
x_{2}\\\
V_{2}:=\\{(x_{1},x_{2}):x_{1}^{2}+x_{2}^{2}=1,\;x_{2}>0\\},&\qquad\varphi_{2}:V_{2}\rightarrow\mathbb{R},\;(x_{1},x_{2})\mapsto
x_{1}\\\
V_{3}:=\\{(x_{1},x_{2}):x_{1}^{2}+x_{2}^{2}=1,\;x_{1}<0\\},&\qquad\varphi_{3}:V_{3}\rightarrow\mathbb{R},\;(x_{1},x_{2})\mapsto
x_{2}\\\
V_{4}:=\\{(x_{1},x_{2}):x_{1}^{2}+x_{2}^{2}=1,\;x_{2}<0\\},&\qquad\varphi_{4}:V_{4}\rightarrow\mathbb{R},\;(x_{1},x_{2})\mapsto
x_{1}.\end{array}$

These are clearly diffeomorphisms. Choose

$P=\\{(-\tfrac{\pi}{3}+\omega,\tfrac{\pi}{3}-\omega),(\tfrac{\pi}{6}+\omega,\tfrac{5\pi}{6}-\omega),(\tfrac{2\pi}{3}+\omega,\tfrac{4\pi}{3}-\omega),(\tfrac{7\pi}{6}+\omega,\tfrac{11\pi}{6}-\omega)\\}:=\\{P_{1},P_{2},P_{3},P_{4}\\}.$

Here $\omega\in(0,\frac{\pi}{12})$ is a small angle. The covers $P_{i}$
clearly satisfy the first two conditions laid out in Lemma 3. Equip
$\mathbb{S}^{1}$ with the standard metric induced by the inclusion
$\mathbb{S}^{1}\hookrightarrow\mathbb{R}^{2}$. We can verify the desired
estimates in Lemma 3 as follows: It is clear that any arc of $\mathbb{S}^{1}$
with length less than $\tfrac{\pi}{6}$ is contained in one of
$P_{1},\dots,P_{4}$. If we choose $\delta\in(0,\tfrac{\pi}{36})$, then for any
$x\in\mathbb{S}^{1}$, $B(x,3\delta)\subset P_{i}$ for some $i$. Suppose

$\displaystyle y=(y_{1},y_{2})=(\cos\theta,\sin\theta),$ $\displaystyle
z=(z_{1},z_{2})=(\cos\tilde{\theta},\sin\tilde{\theta}),$
$\displaystyle\theta,\tilde{\theta}\in(-\tfrac{\pi}{3}+\omega,\tfrac{\pi}{3}-\omega),$
$\displaystyle
y_{2},z_{2}\in(\sin(-\tfrac{\pi}{3}+\omega),\sin(\tfrac{\pi}{3}-\omega)).$

The $\varphi_{1}$-coordinates of $y,z$ are $y_{2},z_{2}$, respectively. Hence,

$\displaystyle|y-z|$
$\displaystyle=|y_{2}-z_{2}|=|\sin\theta-\sin\tilde{\theta}|$ $\displaystyle
r(y,z)$ $\displaystyle=|\theta-\tilde{\theta}|.$

By the mean value theorem:

$|y-z|=|\sin\theta-\sin\tilde{\theta}|=|\cos\xi|_{\xi\in(-\tfrac{\pi}{3}+\omega,\tfrac{\pi}{3}-\omega)}|\theta-\tilde{\theta}|\leq|\theta-\tilde{\theta}|=r(y,z).$

and

$\displaystyle r(y,z)$ $\displaystyle=|\theta-\tilde{\theta}|$
$\displaystyle=|\arcsin y_{2}-\arcsin z_{2}|$
$\displaystyle=\left|\frac{1}{\sqrt{1-\eta^{2}}}\right|_{\eta\in(\sin(-\tfrac{\pi}{3}+\omega),\sin(\tfrac{\pi}{3}-\omega))}|y_{2}-z_{2}|$
$\displaystyle\leq\frac{1}{\sqrt{1-\sin^{2}\left(\frac{\pi}{3}-\omega\right)}}|y_{2}-z_{2}|$
$\displaystyle=\frac{1}{\sqrt{1-\sin^{2}\left(\frac{\pi}{3}-\omega\right)}}|y-z|.$

This suggests the choice $c_{1}=1$ and
$c_{2}=\frac{1}{\sqrt{1-\sin^{2}\left(\frac{\pi}{3}-\omega\right)}}$. We have

$c_{1}c_{2}=\frac{1}{\sqrt{1-\sin^{2}\left(\frac{\pi}{3}-\omega\right)}}<\frac{1}{\sqrt{1-\sin^{2}\left(\frac{\pi}{3}\right)}}=2.$

The analysis for $y,z\in P_{3}$ is similar, only with the difference that
$\theta,\tilde{\theta}\in(\tfrac{2\pi}{3}+\omega,\tfrac{4\pi}{3}-\omega)$ and
we have
$y_{2},z_{2}\in(\sin(\tfrac{4\pi}{3}-\omega),\sin(\tfrac{2\pi}{3}+\omega))=(\sin(-\tfrac{\pi}{3}+\omega),\sin(\tfrac{\pi}{3}-\omega))$.

Next, consider $y,z\in P_{2}$. Suppose

$\displaystyle y=(y_{1},y_{2})=(\cos\theta,\sin\theta),$ $\displaystyle
z=(z_{1},z_{2})=(\cos\tilde{\theta},\sin\tilde{\theta}),$
$\displaystyle\theta,\tilde{\theta}\in(\tfrac{\pi}{6}+\omega,\tfrac{5\pi}{6}-\omega),$
$\displaystyle
y_{1},z_{1}\in(\cos(\tfrac{5\pi}{6}-\omega),\cos(\tfrac{\pi}{6}+\omega)).$

The $\varphi_{2}$-coordinates of $y,z$ are $y_{1},z_{1}$, respectively. Hence,

$\displaystyle|y-z|$
$\displaystyle=|y_{1}-z_{1}|=|\cos\theta-\cos\tilde{\theta}|$ $\displaystyle
r(y,z)$ $\displaystyle=|\theta-\tilde{\theta}|.$

By the mean value theorem:

$|y-z|=|\cos\theta-\cos\tilde{\theta}|=|\sin\xi|_{\xi\in(\tfrac{\pi}{6}+\omega,\tfrac{5\pi}{6}-\omega)}|\theta-\tilde{\theta}|\leq|\theta-\tilde{\theta}|=r(y,z).$

and

$\displaystyle r(y,z)$ $\displaystyle=|\theta-\tilde{\theta}|$
$\displaystyle=|\arccos y_{1}-\arccos z_{1}|$
$\displaystyle=\left|\frac{1}{\sqrt{1-\eta^{2}}}\right|_{\eta\in(\cos(\tfrac{5\pi}{6}-\omega),\cos(\tfrac{\pi}{6}+\omega))}|y_{1}-z_{1}|$
$\displaystyle\leq\frac{1}{\sqrt{1-\cos^{2}(\tfrac{\pi}{6}+\omega)}}|y_{1}-z_{1}|$
$\displaystyle=\frac{1}{\sqrt{1-\cos^{2}(\tfrac{\pi}{6}+\omega)}}|y-z|.$

This suggests the choice $c_{1}=1$ and
$c_{2}=\frac{1}{\sqrt{1-\cos^{2}(\tfrac{\pi}{6}+\omega)}}$. We have

$c_{1}c_{2}=\frac{1}{\sqrt{1-\cos^{2}(\tfrac{\pi}{6}+\omega)}}<\frac{1}{\sqrt{1-\cos^{2}\left(\frac{\pi}{6}\right)}}=2.$

Note that the choice agrees with the case $y,z\in P_{1}$. The analysis for
$y,z\in P_{4}$ is similar as well, which proves the desired claim.

##  4 Generalizing Nonwindowed Geometric Scattering

Now that we have developed the machinery necessary for the rest of the paper,
we prove the $q$-nonwindowed scattering transforms are bounded operators with
respect to (19) and outline basic properties of the representation.

###  4.1 The $2$-nonwindowed Geometric Scattering Norm

We start by proving that 2-nonwindowed scattering transforms are bounded
operators with respect to (19).

######  Theorem 9.

Suppose that $G$ satisfies the conditions of Theorem 1 and
$\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ be a set of spectral filters generated by
using $G$ in Equation 5. Then we have

$\|\overline{S}_{2}^{m}f-\overline{S}_{2}^{m}g\|^{2}_{\ell^{2}(\mathbb{Z}^{m})}\leq
C^{2m}\|f-g\|_{\mathbf{L}^{2}(\mathcal{M})}^{2}$

for all $f,g\in\mathbf{L}^{2}(\mathcal{M})$.

###### Proof.

In the case of $m=1$, we see that

$\displaystyle\sum_{j\in\mathbb{Z}}|\overline{S}_{2}f[j]-\overline{S}_{2}g[j]|^{2}$
$\displaystyle=\sum_{j\in\mathbb{Z}}|\|f\ast\psi_{j}\|_{2}-|g\ast\psi_{j}\|_{2}|^{2}$
$\displaystyle\leq\sum_{j\in\mathbb{Z}}\|f\ast\psi_{j}-g\ast\psi_{j}\|_{2}^{2}$
$\displaystyle=\sum_{j\in\mathbb{Z}}\|(f-g)\ast\psi_{j}\|_{2}^{2}$
$\displaystyle\leq C^{2}\|f-g\|_{\mathbf{L}^{2}(\mathcal{M})}^{2}.$

For the $m+1$ case, we proceed recursively and see that

$\displaystyle\sum_{(j_{1},\ldots,j_{m+1})\in\mathbb{Z}^{m+1}}\left|\overline{S}_{2}^{m+1}f[j_{1},\ldots,j_{m+1}]-\overline{S}_{2}^{m+1}g[j_{1},\ldots,j_{m+1}]\right|^{2}$
$\displaystyle=\sum_{(j_{1},\ldots,j_{m+1})\in\mathbb{Z}^{m+1}}\left|\|U[j_{1},\ldots,j_{m}]f\ast\psi_{j+1}\|_{2}-\|U[j_{1},\ldots,j_{m}]g\ast\psi_{j+1}\|_{2}\right|^{2}$
$\displaystyle\leq\sum_{(j_{1},\ldots,j_{m+1})\in\mathbb{Z}^{m+1}}\|(U[j_{1},\ldots,j_{m}]f-U[j_{1},\ldots,j_{m}]g)\ast\psi_{j+1}\|_{2}^{2}$
$\displaystyle\leq
C^{2}\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\|U[j_{1},\ldots,j_{m}]f-U[j_{1},\ldots,j_{m}]g\|_{2}^{2}$
$\displaystyle=C^{2}\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\||U[j_{1},\ldots,j_{m-1}]f\ast\psi_{j_{m}}|-|U[j_{1},\ldots,j_{m-1}]g\ast\psi_{j_{m}}|\|_{2}^{2}$
$\displaystyle\leq
C^{2}\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\|U[j_{1},\ldots,j_{m-1}]f\ast\psi_{j_{m}}-U[j_{1},\ldots,j_{m-1}]g\ast\psi_{j_{m}}\|_{2}^{2}$
$\displaystyle\leq
C^{4}\sum_{(j_{1},\ldots,j_{m-1})\in\mathbb{Z}^{m-1}}\|U[j_{1},\ldots,j_{m-1}]f-U[j_{1},\ldots,j_{m-1}]g\|_{2}^{2}$
$\displaystyle\vdots$ $\displaystyle\leq
C^{2(m+1)}\|f-g\|_{\mathbf{L}^{2}(\mathcal{M})}^{2}.$

Thus, the claim is proven. ∎

######  Corollary 10.

Suppose that $G$ satisfies the conditions of Theorem 1 and let
$\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ be a set of wavelets generated by using $G$
in Equation 5. Then we have

$\|\overline{S}_{2}^{m}f\|^{2}_{\ell^{2}(\mathbb{Z}^{m})}\leq
C^{2m}\|f\|_{\mathbf{L}^{2}(\mathcal{M})}^{2}$

for all $f\in\mathbf{L}^{2}(\mathcal{M})$ and all $m\geq 1$.

For proper invariance, we provide a theorem that demonstrates that the
$2$-nonwindowed geometric scattering transform is invariant to isometries.

######  Theorem 11.

Let $\xi\in\text{Isom}(\mathcal{M},\mathcal{M}^{\prime})$, and let
$f\in\mathbf{L}^{2}(\mathcal{M})$. Define $f^{\prime}=V_{\xi}f$ and let
$\left(\overline{S}_{2}^{m}\right)^{\prime}$ be the corresponding
$2$-nonwindowed geometric scattering transform on $\mathcal{M}^{\prime}$
produced by a littlewood paley wavelet satisfying the conditions described in
Theorem 1. We have
$\left(\overline{S}_{2}^{m}\right)^{\prime}f^{\prime}=\overline{S}_{2}^{m}f$.

###### Proof.

We see that $\overline{S}_{2}[\emptyset]f=\|f\|_{2}=\|V_{\xi}f\|_{2}$ since
$V_{\xi}$ is an isometry. Now suppose that we consider
$p=(j_{1},\ldots,j_{m})$. Then since convolution using a spectral filter
commutes with isometries and modulus operators (see Theorem 2.1 in [36]),

$\displaystyle\overline{S}_{2}^{m}[j_{1},\ldots,j_{m}]f$
$\displaystyle=\|U[p]f\|_{\mathbf{L}^{2}(\mathcal{M})}$
$\displaystyle=\|V_{\xi}U[p]f\|_{\mathbf{L}^{2}(\mathcal{M})}$
$\displaystyle=\|U[p]V_{\xi}f\|_{\mathbf{L}^{2}(\mathcal{M})}$
$\displaystyle=\|U[p]f^{\prime}\|_{\mathbf{L}^{2}(\mathcal{M})}$
$\displaystyle=\left(\overline{S}_{2}^{m}\right)^{\prime}[j_{1},\ldots,j_{m}]f^{\prime}.$

Thus, we can see that each layer is isometry invariant. ∎

###  4.2 The $q$-nonwindowed Geometric Scattering Norm

Now we prove the $q$-nonwindowed Geometric Scattering Transforms, for
$q\in(1,2)$, are bounded operators with respect to (19) under mild
assumptions.

######  Theorem 12.

Suppose that we choose wavelets $\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ generated by
using $G\in\mathcal{S}(\mathbb{R}^{+})$ in Equation 5, $G$ satisfies the
conditions of Theorem 1, and $c_{1}c_{2}<2$. Then

$\|\overline{S}_{q}^{m}f-\overline{S}_{q}^{m}g\|^{q}_{\ell^{2}(\mathbb{Z}^{m})}\leq
C_{q}^{m}\|f-g\|_{\mathbf{L}^{q}(\mathcal{M})}^{q}$

for all $f,g\in\mathbf{L}^{q}(\mathcal{M})$, for all $m\geq 1$, and some
constant $C_{q}$ dependent on $q$.

###### Proof.

We start by providing a proof for the case of $m=1$:

$\displaystyle\|\overline{S}_{q}f-\overline{S}_{q}g\|^{q}_{\ell^{2}(\mathbb{Z}^{m})}$
$\displaystyle=\left(\sum_{j\in\mathbb{Z}}|\overline{S}_{q}[j]f-\overline{S}_{q}[j]g|^{2}\right)^{q/2}$
$\displaystyle=\left(\sum_{j\in\mathbb{Z}}|\|U[j]f\|_{q}-\|U[j]g\|_{q}|^{2}\right)^{q/2}$
$\displaystyle\leq\left(\sum_{j\in\mathbb{Z}}\|U[j]f-U[j]g\|_{q}^{2}\right)^{q/2}$
$\displaystyle=\left(\sum_{j\in\mathbb{Z}}\left(\int_{\mathcal{M}}|U[j]f(x)-U[j]g(x)|^{q}\,d\mu(x)\right)^{2/q}\right)^{q/2}$

Via Minkowski’s Integral Intequality,

$\displaystyle\left(\sum_{j\in\mathbb{Z}}\left(\int_{\mathcal{M}}|U[j]f(x)-U[j]g(x)|^{q}\,d\mu(x)\right)^{2/q}\right)^{q/2}$
$\displaystyle\leq\int_{\mathcal{M}}\left(\sum_{j\in\mathbb{Z}}|U[j]f(x)-U[j]g(x)|^{2}\right)^{q/2}\,d\mu(x)$
$\displaystyle\leq\int_{\mathcal{M}}\left(\sum_{j\in\mathbb{Z}}|(f\ast\psi_{j})(x)-(g\ast\psi_{j})(x)|^{2}\right)^{q/2}\,d\mu(x)$
$\displaystyle\leq\|\vec{T}(f-g)\|_{\mathbf{L}^{q}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}^{q}.$

Now apply Corollary 8 to get

$\|\vec{T}(f-g)\|_{\mathbf{L}^{q}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}^{q}\leq
C_{q}\|f-g\|_{\mathbf{L}^{q}(\mathcal{M})}^{q}.$

Now assume that for some $m\geq 1$, we have

$\|\overline{S}_{q}^{m}f-\overline{S}_{q}^{m}g\|^{q}_{\ell^{2}(\mathbb{Z}^{m})}\leq
C_{q}^{m}\|f-g\|_{\mathbf{L}^{q}(\mathcal{M})}^{q}.$

Similar to above, when we consider the case with $m+1$, we can mimic the steps
above to get

$\displaystyle\|\overline{S}_{q}^{m+1}f-\overline{S}_{q}^{m+1}g\|^{q}_{\ell^{2}(\mathbb{Z}^{m})}$
$\displaystyle=\left(\sum_{j_{m+1}\in\mathbb{Z}}\cdots\sum_{j_{1}\in\mathbb{Z}}|\overline{S}_{q}^{m+1}[j_{1},\ldots,j_{m+1}]f-\overline{S}_{q}^{m+1}[j_{1},\ldots,j_{m+1}]g|^{2}\right)^{q/2}$
$\displaystyle=\left(\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\cdots\sum_{j_{m+1}\in\mathbb{Z}}\left(\int_{\mathcal{M}}|U[j_{1},\ldots,j_{m+1}]f(x)-U[j_{1},\ldots,j_{m+1}]g(x)|^{q}\,d\mu(x)\right)^{2/q}\right)^{q/2}$
$\displaystyle=\left(\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\left(\sum_{j_{m+1}\in\mathbb{Z}}\left(\int_{\mathcal{M}}|U[j_{1},\ldots,j_{m+1}]f(x)-U[j_{1},\ldots,j_{m+1}]g(x)|^{q}\,d\mu(x)\right)^{2/q}\right)^{\frac{q}{2}\cdot\frac{2}{q}}\right)^{q/2}$
$\displaystyle\leq\left(\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\left(\int_{\mathcal{M}}\left(\sum_{j_{m+1}\in\mathbb{Z}}|U[j_{1},\ldots,j_{m+1}]f(x)-U[j_{1},\ldots,j_{m+1}]g(x)|^{2}\right)^{q/2}\,d\mu(x)\right)^{2/q}\right)^{q/2}$
$\displaystyle\leq\left(\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\left(\int_{\mathcal{M}}\left(\sum_{j_{m+1}\in\mathbb{Z}}|(U[j_{1},\ldots,j_{m}]f\ast\psi_{j_{m+1}})(x)-(U[j_{1},\ldots,j_{m}]g\ast\psi_{j_{m+1}})(x)|^{2}\right)^{q/2}\,d\mu(x)\right)^{2/q}\right)^{q/2}$
$\displaystyle=\left(\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\|\vec{T}(U[j_{1},\ldots,j_{m}]f-U[j_{1},\ldots,j_{m}]g)\|_{\mathbf{L}^{q}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}^{2}\right)^{q/2}$
$\displaystyle=C_{q}\left(\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\|U[j_{1},\ldots,j_{m}]f-U[j_{1},\ldots,j_{m}]g\|_{\mathbf{L}^{q}(\mathcal{M})}^{2}\right)^{q/2}.$

Now we see that we can apply the induction hypothesis to get

$\displaystyle\left(\sum_{(j_{1},\ldots,j_{m})\in\mathbb{Z}^{m}}\|U[j_{1},\ldots,j_{m}]f-U[j_{1},\ldots,j_{m}]g\|_{\mathbf{L}^{q}_{\ell^{2}(\mathbb{Z})}(\mathcal{M})}^{2}\right)^{q/2}$
$\displaystyle=\|\overline{S}_{q}^{m}[j_{1},\ldots,j_{m}]f-\overline{S}_{q}^{m}[j_{1},\ldots,j_{m}]g\|^{q}_{\ell^{2}(\mathbb{Z}^{m})}$
$\displaystyle\leq C_{q}^{m}\|f-g\|_{\mathbf{L}^{q}(\mathcal{M})}^{q}.$

∎

######  Corollary 13.

Suppose that we choose wavelets $\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ generated by
using $G\in\mathcal{S}(\mathbb{R}^{+})$ in Equation 5, $G$ satisfies the
conditions of Theorem 1, and $c_{1}c_{2}<2$. Then

$\|\overline{S}_{q}^{m}f\|^{q}_{\ell^{2}(\mathbb{Z}^{m})}\leq
C_{q}^{m}\|f\|_{\mathbf{L}^{q}(\mathcal{M})}^{q}$

for all $f\in\mathbf{L}^{q}(\mathcal{M})$, for all $m\geq 1$, and some
constant $C_{q}$ dependent on $q$.

For the next theorem, we omit the proof since it is identical to the case when
$q=2$, but we state it for completeness.

######  Theorem 14.

Let $\xi\in\text{Isom}(\mathcal{M},\mathcal{M}^{\prime})$, and let
$f\in\mathbf{L}^{q}(\mathcal{M})$. Define $f^{\prime}=V_{\xi}f$ and let
$\left(\overline{S}_{q}^{m}\right)^{\prime}$ be the corresponding
$q$-nonwindowed geometric scattering transform on $\mathcal{M}^{\prime}$
produced by wavelets $\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ using
$G\in\mathcal{S}(\mathbb{R}^{+})$ in Equation 5, $G$ satisfies the conditions
of Theorem 1, and $c_{1}c_{2}<2$ in Lemma 3. We have
$\left(\overline{S}_{q}^{m}\right)^{\prime}f^{\prime}=\overline{S}_{q}^{m}f$.

##  5 Diffeomorphism Stability

In this section, we provide diffeomorphism stability results for a
generalization of bandlimited functions, $\lambda$-bandlimited functions,
which are defined as functions which satisfy $\hat{f}(k)=\langle
f,\phi_{k}\rangle=0$ whenever $\lambda_{k}\geq\lambda$.

######  Lemma 15 ([36]).

Suppose $\xi\in\text{Diff}(\mathcal{M})$. If $f\in\mathbf{L}^{2}(\mathcal{M})$
is $\lambda$-bandlimited, then

$\|f-V_{\xi}f\|_{\mathbf{L}^{2}(\mathcal{M})}\leq
C(\mathcal{M})\lambda^{n}\|\xi\|_{\infty}\|f\|_{\mathbf{L}^{2}(\mathcal{M})}$

for some constant $C(\mathcal{M})$.

######  Theorem 16.

Suppose $\xi\in\text{Diff}(\mathcal{M})$. Let
$f\in\mathbf{L}^{2}(\mathcal{M})$, and assume that $\psi$ is a wavelet family
satisfying the conditions of Theorem 1. Then

$\|\overline{S}_{2}^{m}f-\overline{S}_{2}^{m}V_{\xi}f\|_{\ell^{2}(\mathbb{Z}^{m})}\leq
C(\mathcal{M})\lambda^{n}\|\xi\|_{\infty}\|f\|_{\mathbf{L}^{2}(\mathcal{M})}.$

###### Proof.

We apply Theorem 9, so Lemma 15 gives the desired result. ∎

###  5.1 Stability Results for the $q$-nonwindowed Geometric Scattering Norm

######  Lemma 17.

Suppose $\xi\in\text{Diff}(\mathcal{M})$. If $f\in\mathbf{L}^{q}(\mathcal{M})$
is $\lambda$-bandlimited, then

$\|f-V_{\xi}f\|_{\mathbf{L}^{q}(\mathcal{M})}\leq
C(\mathcal{M})\lambda^{n}\|\xi\|_{\infty}\|f\|_{\mathbf{L}^{q}(\mathcal{M})}$

for some constant $C_{q}(\mathcal{M})$.

###### Proof.

Since $f$ is $\lambda$-bandlimited, $f\in\mathbf{L}^{2}(\mathcal{M})$ as well,
and the proof is nearly identical to the proof of the case when $q=2$, but we
provide the steps for completeness. We define $\pi_{\lambda}$ be the operator
that projects a function $f\in\mathbf{L}^{2}(\mathcal{M})$ onto the eigenspace
$E_{\lambda}$ and define the projection operator

$P_{\lambda}:=\sum_{\lambda_{n}\leq\lambda}\pi_{\lambda_{n}}$

with kernel

$K^{(\lambda)}(x,y)=\sum_{\lambda_{n}\leq\lambda}e_{n}(x)\overline{e_{n}(y)}.$

We have $P_{\lambda}f=f$ $\mu$-almost-everywhere. Thus, via Holder’s
inequality,

$\displaystyle|f(x)-V_{\xi}f(x)|$
$\displaystyle=|P_{\lambda}f(x)-V_{\xi}P_{\lambda}f(x)|$
$\displaystyle=\left|\int_{\mathcal{M}}K^{(\lambda)}(x,y)f(y)\,dy-\int_{\mathcal{M}}K^{(\lambda)}(\xi^{-1}(x),y)f(y)\,dy\right|$
$\displaystyle\leq\left|\int_{\mathcal{M}}(K^{(\lambda)}(x,y)-K^{(\lambda)}(\xi^{-1}(x),y))f(y)\,dy\right|$
$\displaystyle\leq\|f\|_{\mathbf{L}^{q}(\mathcal{M})}\left(\int_{\mathcal{M}}|K^{(\lambda)}(x,y)-K^{(\lambda)}(\xi^{-1}(x),y)|^{p}\,dy\right)^{1/p}$
$\displaystyle\leq
C_{q,\text{Vol}}(\mathcal{M})\|f\|_{\mathbf{L}^{q}(\mathcal{M})}\|\xi\|_{\infty}\|\nabla
K^{(\lambda)}\|_{\infty}$

for some constant $C_{q,\text{Vol}}(\mathcal{M})$ dependent on $q$ and the
volume of the manifold. Now, by Lemma H.1 in [36], we have

$\|\nabla K^{(\lambda)}\|_{\infty}\leq C_{q}(\mathcal{M})\lambda^{n}.$

Thus, the proof is complete. ∎

######  Theorem 18.

Suppose $\xi\in\text{Diff}(\mathcal{M})$. Let
$f\in\mathbf{L}^{q}(\mathcal{M})$ be $\lambda$-bandlimited. Additionally,
suppose that we choose wavelets $\\{\psi_{j}\\}_{j\in\mathbb{Z}}$ generated by
using $G\in\mathcal{S}(\mathbb{R}^{+})$ in Equation 5, $G$ satisfies the
conditions of Theorem 1, and $c_{1}c_{2}<2$. Then

$\|\overline{S}_{q}^{m}f-\overline{S}_{q}^{m}V_{\xi}f\|_{\ell^{2}(\mathbb{Z}^{m})}\leq
C(\mathcal{M})\lambda^{n}\|\xi\|_{\infty}\|f\|_{\mathbf{L}^{q}(\mathcal{M})}$

for some constant $C(\mathcal{M})$.

###### Proof.

We apply Theorem 12 to get

$\|\overline{S}_{q}^{m}f-\overline{S}_{q}^{m}V_{\xi}f\|_{\ell^{2}(\mathbb{Z}^{m})}\leq
C_{q}\|f-V_{\xi}f\|_{\mathbf{L}^{q}(\mathcal{M})}.$

By Lemma 17, we have

$\|f-V_{\xi}f\|_{\mathbf{L}^{q}(\mathcal{M})}\leq
C(\mathcal{M})\lambda^{n}\|\xi\|_{\infty}\|f\|_{\mathbf{L}^{q}(\mathcal{M})},$

which gives the desired result. ∎

##  6 Conclusions and Future Work

We have provided a framework for understanding nonwindowed scattering
coefficients. In particular, we provide a weighted measure for distortion
between nonwindowed scattering coefficients, showed our weighted measure is
well-defined mapping for $\mathbf{L}^{q}(\mathcal{M})$ functions, and showed
that nonwindowed scattering coefficients are stable to diffeomorphisms for
$\lambda$-bandlimited functions.

For future work, it is of interest to see if it is possible to extend our
results to manifolds that are not restricted the conditions present in
Sections 4 and 5. Additionally, what are other manifolds that satisfy the
conditions present in sections 4 and 5? These questions will be left to future
work.

##  7 Acknowledgements

We would like to thank Michael Perlmutter for providing feedback, which
improved the quality of this initial draft.

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