2021

[1]Vasiliki Kouni

[1]Department of Informatics and Telecommunications, National and Kapodistrian
University of Athens, Athens, Greece 2]Chair for Mathematics of Information
Processing, RWTH Aachen University, Aachen, Germany

# Star DGT: a Robust Gabor Transform for Speech Denoising

<EMAIL_ADDRESS>Holger Rauhut<EMAIL_ADDRESS>Theoharis
Theoharis<EMAIL_ADDRESS>*  [

###### Abstract

In this paper, we address the speech denoising problem, where Gaussian, pink
and blue additive noises are to be removed from a given speech signal. Our
approach is based on a redundant, analysis-sparse representation of the
original speech signal. We pick an eigenvector of the Zauner unitary matrix
and –under certain assumptions on the ambient dimension– we use it as window
vector to generate a spark deficient Gabor frame. The analysis operator
associated with such a frame, is a (highly) redundant Gabor transform, which
we use as a sparsifying transform in denoising procedure. We conduct
computational experiments on real-world speech data, using as baseline three
Gabor transforms generated by state-of-the-art window vectors in time-
frequency analysis and compare their performance to the proposed Gabor
transform. The results show that our proposed redundant Gabor transform
outperforms all others, consistently for all signals and all examined types of
noise.

###### keywords:

Denoising, speech signal, Gabor transform, window vector, spark deficient
Gabor frame

##  1 Introduction

Noise is one of the main factors that affect the accuracy of the results in
audio processing. This explains why audio denoising is one of the most
extensively studied inverse problems in signal processing. The task consists
in recovering an audio signal $x\in\mathbb{R}^{L}$ from corrupted linear
observations

$y=x+e\in\mathbb{R}^{L}.$ (1)

Noise removal from audio signals is an important first step in applications
such as sound classification soundclass , sound event localization sel ,
speech recognition whisper , dereverberation (reverb, ), speech enhancement
enhance  and source separation soursep .

###  1.1 Related Work

In order to address the denoising problem, numerous approaches have emerged,
including statistical models bayes ; probability , empirical mode
decomposition (emd, ), spectral subtraction specsub ; colorsubtract ,
thresholding methods mallat ; durfler , neural networks wavenet ; relu ,
sparse and redundant representations plumbley ; lowrank , or a combination of
the aforementioned approaches (threshemd, ; aegan, ). Sparse and redundant
representations have shown very promising results fletcher ; thresh ,
especially when turning to analysis sparsity (also known as co-sparsity)
audascity ; genzel ; figu , which provides flexibility in modelling sparse
signals, since it leverages the redundancy of the involved analysis operators.
Albeit the authors of genzel  are mainly focused on analysis Compressed
Sensing rauhut , they state that using the analysis-prior formulation –with a
redundant analysis operator– in denoising (from now on we shall call this
framework analysis denoising) is fundamentally different from classical
denoising via soft thresholding. Similarly, figu  explores the superior
reconstruction produced by analysis denoising of 1D signals over its synthesis
counterpart elad .

###  1.2 Motivation

Our work is inspired by the articles mallat ; genzel ; figu ; durfler ; greedy
; pfander . These publications propose either analysis operators associated
with redundant frames (i.e. matrices whose atoms/rows form a frame of the
ambient space) with atoms in general linear position, or a finite difference
operator (associated to the popular method of total variation tv ), in which
many linear dependencies appear for large dimensions. Moreover, in mallat ;
durfler ; greedy , Gabor transforms are combined with thresholding methods for
audio denoising. In a similar spirit, we also deploy frames, but we
differentiate our approach by using spark deficient Gabor frames under the
_analysis denoising formulation_. The elements of spark deficient frames are
not in general linear position, while analysis denoising differentiates itself
–as already stated– from classical thresholding denoising. Our intuition
behind employing spark deficient frames is based on remarks of cosparse . In
the latter, the authors state that according to the union-of-subspaces model
union , it is desired to have analysis operators that exhibit high linear
dependencies among their rows; this is a condition satisfied by spark
deficient frames. Particularly, we choose spark deficient Gabor frames instead
of other classes of frames that may be spark deficient (e.g. equiangular tight
frames (equiangular, )) due to the fact that time-frequency representations
resemble the way that the human auditory system works (auditory, ; intensity,
); thus, they are better suited to the application of speech denoising. To
that end, we take advantage of an analysis operator star , namely star digital
Gabor transform (star-DGT), associated with a spark deficient Gabor frame
(SDGF). The latter can be generated111such a frame can also be generated by
the eigenvectors of certain unitaries belonging to the Clifford group by time-
frequency shifts of any eigenvector of the Zauner unitary matrix zauner ,
under certain assumptions on the signal’s dimension. To the best of our
knowledge, the efficiency of star-DGT when applied to denoising has not yet
been demonstrated. Therefore, it is intriguing to compare the robustness of
our proposed Gabor analysis operator to three other Gabor transforms, emerging
from state-of-the-art window vectors, by applying all four of them to analysis
denoising. Finally, we illustrate the practical importance of our method for
real-world speech signals.

###  1.3 Key Contributions

Our novelty is twofold: (a) we generate a SDGF based on a window vector,
associate this SDGF to a highly redundant Gabor analysis operator and use the
latter as a sparsifying transform in analysis denoising (b) we compare
numerically our proposed method with three other Gabor analysis operators,
based on common windows of time-frequency analysis, on real-world speech data,
arguing also about the selection of the lattice parameters. Our experiments
show that our method outperforms all others, consistently for all speech
signals and all types of noise – Gaussian, pink and blue.

###  1.4 Paper organization

The rest of the paper is outlined as follows. In Section 2, we give notation
and briefly present the setup of analysis denoising. Section 3 introduces
Gabor frames and extends to spark deficient ones, building the desirable SDGF
and its associated analysis operator. In Section 4, we describe the
experimental settings, while in Section 5, we present two sets of experiments,
with corresponding results and evaluation. Lastly, in Section 6 we make some
concluding remarks and give potential future directions.

##  2 Gabor denoising setup

###  2.1 Notation

  * •

For a set of indices $N=\\{0,1,\dots,N-1\\}$, we write $[N]$.

  * •

(Bra-kets) The set of (column) vectors $\lvert 0\rangle,\lvert
1\rangle,\dots,\lvert L-1\rangle$ is the standard basis of $\mathbb{C}^{L}$.

  * •

We write $\mathbb{Z}_{L}$ for the ring of residues $\mathrm{mod}L$, that is
$\mathbb{Z}_{L}=\\{0\mathrm{mod}L,1\mathrm{mod}L,\dots,(L-1)\mathrm{mod}L\\}$.

  * •

We write $a\equiv b(\mathrm{mod}L)$ for the congruence modulo, where
$a,\,b\in\mathbb{Z}$.

  * •

The support of a signal $x\in\mathbb{R}^{L}$ is denoted by
$\mathrm{supp}(x)=\\{i\in[L]:x_{i}\neq 0\\}$. For its cardinality, we write
$\lvert\mathrm{supp}(x)\rvert$ and if $\lvert\mathrm{supp}(x)\rvert\leq s<<L$,
we call $x$ $s$-sparse.

###  2.2 Analysis denoising formulation

As we mentioned in Section 1, the main idea of speech denoising is to
reconstruct a speech signal $x\in\mathbb{R}^{L}$ from

$y=x+e\in\mathbb{R}^{L},$ (2)

where $e\in\mathbb{R}^{L}$, $\|e\|_{2}\leq\eta$, corresponds to noise. To do
so, we first assume there exists a redundant sparsifying transform
$\Phi\in\mathbb{C}^{P\times L}$ ($P>L$) called the analysis operator, such
that $\Phi x$ is (approximately) sparse. This is an analysis sparsity model
for $x$.

On the other hand, the type of noise $e$ that is added on $x$ depends on the
speech recording method. Noise may originate from the microphones or the
environment, in the form e.g. of white noise, pink noise or babble noise. In
this paper, we examine three types of additive noises: zero-mean Gaussian with
standard deviation $\sigma$, pink and blue.

Using analysis sparsity in denoising, we wish to recover $x$ from $y$. A
common approach is the analysis basis pursuit denoising problem

$\min_{x\in\mathbb{R}^{L}}\|\Phi x\|_{1}\quad\text{subject
to}\quad\|x-y\|_{2}\leq\eta,$ (3)

or a regularized version222in terms of optimization, it is preferred to solve
(4) instead of (3) tfocs  of it:

$\min_{x\in\mathbb{R}^{L}}\|\Phi
x\|_{1}+\frac{\mu}{2}\|x-x_{0}\|_{2}^{2}\quad\text{subject
to}\quad\|x-y\|_{2}\leq\eta,$ (4)

where $x_{0}$ denotes an initial guess on $x$, $\mu>0$ is a smoothing
parameter and $\eta>0$ an estimate on the noise level.

We will devote the next Section to the construction of a suitable analysis
operator $\Phi$.

##  3 Gabor frames

###  3.1 Gabor systems

A discrete Gabor system $(g,a,b)$ dgs  is defined as a collection of time-
frequency shifts of the so-called window vector $g\in\mathbb{C}^{L}$,
expressed as

$\displaystyle g_{n,m}(l)$ $\displaystyle=e^{2\pi imbl/L}g(l-na),\quad
l\in[L],$ (5)

where $a,\,b$ denote time and frequency parameters (also known as lattice
parameters) respectively, $n\in[N]$ chosen such that $N=L/a\in\mathbb{N}$ and
$m\in[M]$ chosen such that $M=L/b\in\mathbb{N}$ denote time and frequency
shift indices respectively. If (5) spans $\mathbb{C}^{L}$, it is called Gabor
frame and an equivalent definition of a frame mal  is given below.

######  Definition 1.

Let $L\in\mathbb{N}$ and $(\phi_{p})_{p\in P}$ a finite subset of
$\mathbb{C}^{L}$. If the inequalities

$c_{1}\|x\|_{2}^{2}\leq\sum_{p\in P}\lvert\langle
x,\phi_{p}\rangle\rvert^{2}\leq c_{2}\|x\|_{2}^{2}$ (6)

hold true for all $x\in\mathbb{C}^{L}$, for some $0<c_{1}\leq c_{2}$ (frame
bounds), then $(\phi_{p})_{p\in P}$ is called a frame for $\mathbb{C}^{L}$.

######  Remark 1.

The number of elements in $(g,a,b)$ according to (5) is $P=MN=L^{2}/ab$ and if
$(g,a,b)$ is a frame, we have $ab<L$ (the so-called _oversampling_ case). A
crucial ingredient in order to have good time-frequency resolution of a signal
with respect to a Gabor frame, is the appropriate choice of the time-frequency
parameters $a$ and $b$. Apparently, this challenge can only be treated by
numerically experimenting with different choices of $a,\,b$ with respect to
$L$. In the following subsection, we associate two operators to a Gabor frame.

###  3.2 The analysis and synthesis operators associated with a Gabor frame

######  Definition 2.

Let $\Phi_{g}:\mathbb{C}^{L}\mapsto\mathbb{C}^{M\times N}$ denote the Gabor
analysis operator –also known as DGT333so we will interchangeably use both
terms from now on– whose action on a signal $x\in\mathbb{C}^{L}$ is defined as

$c_{m,n}=\sum_{l=0}^{L-1}x_{l}\overline{g(l-na)}e^{-2\pi imbl/L},$ (7)

for $m\in[M],\,n\in[N]$.

######  Definition 3.

The adjoint of the analysis operator defined in (7), is the Gabor synthesis
operator $\Phi_{g}^{T}:\mathbb{C}^{M\times N}\mapsto\mathbb{C}^{L}$, whose
action on the coefficients $c_{m,n}$ gives

$\Phi_{g}^{T}c_{m,n}=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}c_{m,n}g(l-na)e^{2\pi
imbl/L},$ (8)

for $l\in[L]$.

Since we will deal with analysis denoising in this paper, we will only focus
on $\Phi_{g}$ from now on.

###  3.3 Spark deficient Gabor frames

Let us first introduce some basic notions needed in this subsection.

######  Definition 4.

The symplectic group $\mathrm{SL}(2,\mathbb{Z}_{L})$ consists of all matrices

$G=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}$ (9)

such that $\alpha,\,\beta,\,\gamma,\,\delta\in\mathbb{Z}_{L}$ and

$\alpha\delta-\beta\gamma\equiv 1(\mathrm{mod}L).$ (10)

To each such matrix corresponds a unitary matrix given by the explicit formula
dang

$U_{G}=\frac{e^{i\theta}}{\sqrt{L}}\sum_{u,v=0}^{L-1}\tau^{\beta^{-1}(\alpha
v^{2}-2uv+\delta u^{2})}\lvert u\rangle\langle v\rvert,$ (11)

where $\theta$ is an arbitrary phase, $\beta^{-1}$ is the
inverse444$\beta\beta^{-1}\equiv 1\mathrm{mod}L$ of $\beta\mathrm{mod}L$ and

$\tau=-e^{\frac{i\pi}{L}}.$ (12)

######  Definition 5.

The spark of a set $F$ –denoted by $\mathrm{sp}(F)$– of $P$ vectors in
$\mathbb{C}^{L}$ is the size of the smallest linearly dependent subset of $F$.
A frame $F$ is full spark if and only if every set of $L$ elements of $F$ is a
basis, or equivalently $\mathrm{sp}(F)=L+1$, otherwise it is spark deficient.

Based on the previous definition, a Gabor frame with$P=L^{2}/ab$ elements of
the form (5) is full spark, if and only if every set of $L$ of its elements is
a basis. Now, as proven in mal , almost all window vectors generate full spark
Gabor frames, so the SDGFs are generated by exceptional window vectors.
Indeed, the following theorem was proven in dang  and informally stated in
sparkmal , for the Zauner matrix $\mathcal{Z}\in\mathrm{SL}(2,\mathbb{Z}_{L})$
given by

$\mathcal{Z}=\begin{pmatrix}0&-1\\\
1&-1\end{pmatrix}\equiv\begin{pmatrix}0&L-1\\\ 1&L-1\end{pmatrix}.$ (13)

######  Theorem 1 ((dang, )).

Let $L\in\mathbb{Z}$ such that $2\nmid L$, $3\,|\,L$ and $L$ is square-free.
Then, any eigenvector of the Zauner unitary matrix $U_{\mathcal{Z}}$ (produced
by combining (11) and (13)), generates a spark deficient Gabor frame for
$\mathbb{C}^{L}$.

Therefore, in order to produce a SDGF and apply its associated analysis
operator in (4), we must first choose an ambient dimension $L$ that fits the
assumptions of Theorem 1. Then, we calculate $U_{\mathcal{Z}}$ using (11) and
(13) and in the end, perform its spectral decomposition in order to acquire
its eigenvectors. Since all the eigenvectors of $U_{\mathcal{Z}}$ generate
SDGFs, we may choose an arbitrary one, which we call star window from now on
and denote it as $g_{*}$. We call the analysis operator associated with such a
SDGF star-DGT and denote it $\Phi_{g_{*}}$, in order to indicate the
dependance on $g_{*}$. We coin the term ”star”, due to the slight resemblance
of this DGT to a star when plotted in MATLAB, as it is demonstrated in the
example Fig. 1.

######  Remark 2.

A simple way to choose $L$, is by considering its prime factorization: take
$k$ prime numbers $p_{1}^{\alpha_{1}},\dots,p_{k}^{\alpha_{k}}$, with
$\alpha_{1},\dots,\alpha_{k}$ not all a multiple of 2 and $p_{1}=3,p_{i}\neq
2,i=2,\dots,k$, such that $L=3^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdot\dots\cdot
p_{k}^{\alpha_{k}}$. Since $a,b\,\lvert\,L$, we may also choose $a,b$ to be
one, or a multiplication of more than one, prime numbers from the prime
factorization of $L$. We have seen empirically that this method for fixing
$(L,a,b)$ produces satisfying results, as it is illustrated in the figures of
the next pages.

Figure 1: Plotted star-DGT coefficients for the audio signal Glockenspiel
ltfat  (262144 samples, $(L,a,b)=(200583,19,23)$)

##  4 Experimental Setup

###  4.1 Signals’ description and preprocessing

We run experiments on 12 real-world, real-valued speech signals, all sampled
at $16$kHz, taken from LibriSpeech corpus libri . Their labels along with
short description can be found in Table 1. The true ambient dimension of each
real-world signal does not usually match the conditions of Theorem 1. Hence,
we load each signal and use Remark 2 to cut it off to a specific ambient
dimension (from now we shall refer to it as _artificial dimension_) $L$, being
as closer as it gets to its true dimension, in order to both denoise a
meaningful part of the signal and meet the conditions of Theorem 1.

Table 1: Signals’ details

$\\#$ | Labels | True ambient dimension | Artificial dimension $L$ | Types of noise added
---|---|---|---|---
1 | 251-136532-0014 | 36240 | $33915$ | Gaussian and pink
2 | 8842-304647-0007 | 27680 | $27531$ | Gaussian and blue
3 | 2035-147960-0013 | 42800 | $41769$ | Gaussian and pink
4 | 1462-170145-0020 | 34400 | $33915$ | Gaussian and blue
5 | 6241-61943-0002 | 43760 | $43605$ | Gaussian and blue
6 | 5338-284437-0025 | 31040 | $29835$ | Gaussian and blue
7 | 3752-4944-0042 | 51360 | $51051$ | Gaussian and blue
8 | 5694-64038-0013 | 52880 | $51051$ | Gaussian and pink
9 | 5895-34615-0001 | 52880 | $51051$ | Gaussian and pink
10 | 2428-83699-0035 | 43600 | $41769$ | Gaussian and pink
11 | 2803-154320-0006 | 34880 | $33915$ | Gaussian and blue
12 | 3752-4944-0008 | 31040 | $29835$ | Gaussian and pink

Table 2: 251-136532-0014 with $L=33915$ and $\sigma=0.001$

| MSEs
---|---
Windows $(a,b)$ | $(15,15)$ | $(5,17)$ | $(7,19)$ | $(17,19)$ | $(21,21)$
Gaussian | $8.4929\cdot 10^{-4}$ | $8.4846\cdot 10^{-4}$ | $8.4233\cdot 10^{-4}$ | $8.4488\cdot 10^{-4}$ | $8.4188\cdot 10^{-4}$
Hann | $8.4917\cdot 10^{-4}$ | $8.4858\cdot 10^{-4}$ | $8.4222\cdot 10^{-4}$ | $8.4502\cdot 10^{-4}$ | $8.4196\cdot 10^{-4}$
Hamming | $8.4936\cdot 10^{-4}$ | $8.4830\cdot 10^{-4}$ | $8.4234\cdot 10^{-4}$ | $8.4490\cdot 10^{-4}$ | $8.4199\cdot 10^{-4}$
Star | $\bf 7.4783\cdot 10^{-4}$ | $\bf 8.1694\cdot 10^{-4}$ | $\bf 7.5914\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}6.7464\cdot 10^{-4}$ | $\bf 7.0497\cdot 10^{-4}$

(a) Gaussian noise

| MSEs
---|---
Windows $(a,b)$ | $(15,15)$ | $(5,17)$ | $(7,19)$ | $(17,19)$ | $(21,21)$
Gaussian | $5.3213\cdot 10^{-4}$ | $5.3660\cdot 10^{-4}$ | $5.3533\cdot 10^{-4}$ | $5.3625\cdot 10^{-4}$ | $5.4130\cdot 10^{-4}$
Hann | $5.3217\cdot 10^{-4}$ | $5.3655\cdot 10^{-4}$ | $5.3521\cdot 10^{-4}$ | $5.3620\cdot 10^{-4}$ | $5.4122\cdot 10^{-4}$
Hamming | $5.3213\cdot 10^{-4}$ | $5.3655\cdot 10^{-4}$ | $5.3528\cdot 10^{-4}$ | $5.3632\cdot 10^{-4}$ | $5.4111\cdot 10^{-4}$
Star | $\bf 4.5283\cdot 10^{-4}$ | $\bf 4.7100\cdot 10^{-4}$ | $\bf 4.6333\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}3.5476\cdot 10^{-4}$ | $\bf 4.2386\cdot 10^{-4}$

(b) Pink noise

Table 3: 2035-147960-0013 with $L=41769$ and $\sigma=0.001$

| MSEs
---|---
Windows $(a,b)$ | $(9,9)$ | $(7,13)$ | $(9,17)$ | $(13,17)$ | $(17,17)$
Gaussian | $9.1225\cdot 10^{-4}$ | $9.1113\cdot 10^{-4}$ | $9.1134\cdot 10^{-4}$ | $9.0846\cdot 10^{-4}$ | $9.1484\cdot 10^{-4}$
Hann | $9.1230\cdot 10^{-4}$ | $9.1111\cdot 10^{-4}$ | $9.1126\cdot 10^{-4}$ | $9.0833\cdot 10^{-4}$ | $9.1492\cdot 10^{-4}$
Hamming | $9.1221\cdot 10^{-4}$ | $9.1108\cdot 10^{-4}$ | $9.1130\cdot 10^{-4}$ | $9.0838\cdot 10^{-4}$ | $9.1492\cdot 10^{-4}$
Star | $\bf 8.5635\cdot 10^{-4}$ | $\bf 8.0955\cdot 10^{-4}$ | $\bf 8.2828\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}7.7499\cdot 10^{-4}$ | $\bf 8.1464\cdot 10^{-4}$

(a) Gaussian noise

| MSEs
---|---
Windows $(a,b)$ | $(9,9)$ | $(7,13)$ | $(9,17)$ | $(13,17)$ | $(17,17)$
Gaussian | $5.8699\cdot 10^{-4}$ | $5.8925\cdot 10^{-4}$ | $5.8942\cdot 10^{-4}$ | $5.9071\cdot 10^{-4}$ | $5.8948\cdot 10^{-4}$
Hann | $5.8710\cdot 10^{-4}$ | $5.8921\cdot 10^{-4}$ | $5.8945\cdot 10^{-4}$ | $5.9080\cdot 10^{-4}$ | $5.8949\cdot 10^{-4}$
Hamming | $5.8703\cdot 10^{-4}$ | $5.8928\cdot 10^{-4}$ | $5.8961\cdot 10^{-4}$ | $5.9060\cdot 10^{-4}$ | $5.8943\cdot 10^{-4}$
Star | $\bf 5.3307\cdot 10^{-4}$ | $\bf 5.3072\cdot 10^{-4}$ | $\bf 4.9653\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}4.8565\cdot 10^{-4}$ | $\bf 4.8874\cdot 10^{-4}$

(b) Pink noise

Table 4: 5694-64038-0013 with $\sigma=0.001$ and Gaussian noise

| MSEs
---|---
Windows $(a,b)$ | $(11,11)$ | $(13,13)$ | $(13,21)$ | $(11,17)$ | $(13,17)$
Gaussian | $8.7630\cdot 10^{-4}$ | $8.7634\cdot 10^{-4}$ | $8.7914\cdot 10^{-4}$ | $8.7299\cdot 10^{-4}$ | $8.7429\cdot 10^{-4}$
Hann | $8.7643\cdot 10^{-4}$ | $8.7641\cdot 10^{-4}$ | $8.7918\cdot 10^{-4}$ | $8.7300\cdot 10^{-4}$ | $8.7419\cdot 10^{-4}$
Hamming | $8.7629\cdot 10^{-4}$ | $8.7639\cdot 10^{-4}$ | $8.7909\cdot 10^{-4}$ | $8.7301\cdot 10^{-4}$ | $8.7431\cdot 10^{-4}$
Star | $\bf 8.2926\cdot 10^{-4}$ | $\bf 8.0809\cdot 10^{-4}$ | $\bf 7.6079\cdot 10^{-4}$ | $\bf 7.5266\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}7.5009\cdot 10^{-4}$

(a) Gaussian noise

| MSEs
---|---
Windows $(a,b)$ | $(11,11)$ | $(13,13)$ | $(13,21)$ | $(11,17)$ | $(13,17)$
Gaussian | $5.5950\cdot 10^{-4}$ | $5.5945\cdot 10^{-4}$ | $5.6133\cdot 10^{-4}$ | $5.5808\cdot 10^{-4}$ | $5.5888\cdot 10^{-4}$
Hann | $5.5949\cdot 10^{-4}$ | $5.5943\cdot 10^{-4}$ | $5.6131\cdot 10^{-4}$ | $5.5812\cdot 10^{-4}$ | $5.5890\cdot 10^{-4}$
Hamming | $5.5949\cdot 10^{-4}$ | $5.5958\cdot 10^{-4}$ | $5.6128\cdot 10^{-4}$ | $5.5800\cdot 10^{-4}$ | $5.5897\cdot 10^{-4}$
Star | $\bf 5.1971\cdot 10^{-4}$ | $\bf 4.9725\cdot 10^{-4}$ | $\bf 4.4889\cdot 10^{-4}$ | $\bf 4.7244\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}4.3773\cdot 10^{-4}$

(b) Pink noise

Table 5: 5338-284437-0025 with $L=29835$ and $\sigma=0.001$

| MSEs
---|---
Windows $(a,b)$ | $(9,9)$ | $(15,15)$ | $(5,13)$ | $(5,17)$ | $(13,17)$
Gaussian | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$
Hann | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$
Hamming | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$ | $12\cdot 10^{-4}$
Star | $\bf 11\cdot 10^{-4}$ | $\bf 9.8065\cdot 10^{-4}$ | $\bf 11\cdot 10^{-4}$ | $\bf 11\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}9.7897\cdot 10^{-4}$

(a) Gaussian noise

| MSEs
---|---
Windows $(a,b)$ | $(9,9)$ | $(15,15)$ | $(5,13)$ | $(5,17)$ | $(13,17)$
Gaussian | $7.8626\cdot 10^{-4}$ | $7.8715\cdot 10^{-4}$ | $7.8627\cdot 10^{-4}$ | $7.8690\cdot 10^{-4}$ | $7.8674\cdot 10^{-4}$
Hann | $7.8629\cdot 10^{-4}$ | $7.8689\cdot 10^{-4}$ | $7.8650\cdot 10^{-4}$ | $7.8689\cdot 10^{-4}$ | $7.8696\cdot 10^{-4}$
Hamming | $7.8627\cdot 10^{-4}$ | $7.8745\cdot 10^{-4}$ | $7.8623\cdot 10^{-4}$ | $7.8676\cdot 10^{-4}$ | $7.8677\cdot 10^{-4}$
Star | $\bf 6.8879\cdot 10^{-4}$ | $\bf 6.6695\cdot 10^{-4}$ | $\bf 7.0199\cdot 10^{-4}$ | $\bf 6.6884\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}6.3867\cdot 10^{-4}$

(b) Blue noise

Table 6: 6241-61943-0002 with $L=43605$ and $\sigma=0.001$

| MSEs
---|---
Windows $(a,b)$ | $(9,9)$ | $(5,17)$ | $(9,17)$ | $(17,19)$ | $(19,19)$
Gaussian | $5.0929\cdot 10^{-4}$ | $5.1122\cdot 10^{-4}$ | $5.1164\cdot 10^{-4}$ | $5.0961\cdot 10^{-4}$ | $5.1202\cdot 10^{-4}$
Hann | $5.0929\cdot 10^{-4}$ | $5.1119\cdot 10^{-4}$ | $5.1166\cdot 10^{-4}$ | $5.0961\cdot 10^{-4}$ | $5.1202\cdot 10^{-4}$
Hamming | $5.0928\cdot 10^{-4}$ | $5.1121\cdot 10^{-4}$ | $5.1160\cdot 10^{-4}$ | $5.0957\cdot 10^{-4}$ | $5.1201\cdot 10^{-4}$
Star | $\bf 4.9231\cdot 10^{-4}$ | $\bf 4.6263\cdot 10^{-4}$ | $\bf 4.7426\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}4.5764\cdot 10^{-4}$ | $\bf 4.6173\cdot 10^{-4}$

(a) Gaussian noise

| MSEs
---|---
Windows $(a,b)$ | $(9,9)$ | $(5,17)$ | $(9,17)$ | $(17,19)$ | $(19,19)$
Gaussian | $2.6371\cdot 10^{-4}$ | $2.6358\cdot 10^{-4}$ | $2.6336\cdot 10^{-4}$ | $2.6356\cdot 10^{-4}$ | $2.6356\cdot 10^{-4}$
Hann | $2.6373\cdot 10^{-4}$ | $2.6360\cdot 10^{-4}$ | $2.6338\cdot 10^{-4}$ | $2.6359\cdot 10^{-4}$ | $2.6359\cdot 10^{-4}$
Hamming | $2.6371\cdot 10^{-4}$ | $2.6359\cdot 10^{-4}$ | $2.6340\cdot 10^{-4}$ | $2.6359\cdot 10^{-4}$ | $2.6359\cdot 10^{-4}$
Star | $\bf 2.3890\cdot 10^{-4}$ | $\bf 2.3582\cdot 10^{-4}$ | $\bf 2.2962\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}1.9016\cdot 10^{-4}$ | $\bf 2.2279\cdot 10^{-4}$

(b) Blue noise

Table 7: 8842-304647-0007 with $L=27531$ and $\sigma=0.001$

| MSEs
---|---
Windows $(a,b)$ | $(9,9)$ | $(9,23)$ | $(7,19)$ | $(19,19)$ | $(19,23)$
Gaussian | $7.2315\cdot 10^{-4}$ | $7.2621\cdot 10^{-4}$ | $7.1806\cdot 10^{-4}$ | $7.2532\cdot 10^{-4}$ | $7.1924\cdot 10^{-4}$
Hann | $7.2314\cdot 10^{-4}$ | $7.2605\cdot 10^{-4}$ | $7.1806\cdot 10^{-4}$ | $7.2535\cdot 10^{-4}$ | $7.1915\cdot 10^{-4}$
Hamming | $7.2321\cdot 10^{-4}$ | $7.2609\cdot 10^{-4}$ | $7.1803\cdot 10^{-4}$ | $7.2519\cdot 10^{-4}$ | $7.1919\cdot 10^{-4}$
Star | $\bf 6.7449\cdot 10^{-4}$ | $\bf 6.1704\cdot 10^{-4}$ | $\bf 6.0620\cdot 10^{-4}$ | $\bf 6.2294\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}5.4646\cdot 10^{-4}$

(a) Gaussian noise

| MSEs
---|---
Windows $(a,b)$ | $(9,9)$ | $(9,23)$ | $(7,19)$ | $(19,19)$ | $(19,23)$
Gaussian | $4.1860\cdot 10^{-4}$ | $4.1906\cdot 10^{-4}$ | $4.1881\cdot 10^{-4}$ | $4.1925\cdot 10^{-4}$ | $4.1887\cdot 10^{-4}$
Hann | $4.1863\cdot 10^{-4}$ | $4.1903\cdot 10^{-4}$ | $4.1879\cdot 10^{-4}$ | $4.1935\cdot 10^{-4}$ | $4.1887\cdot 10^{-4}$
Hamming | $4.1864\cdot 10^{-4}$ | $4.1899\cdot 10^{-4}$ | $4.1885\cdot 10^{-4}$ | $4.1932\cdot 10^{-4}$ | $4.1884\cdot 10^{-4}$
Star | $\bf 3.6969\cdot 10^{-4}$ | $\bf 2.8825\cdot 10^{-4}$ | $\bf 3.3767\cdot 10^{-4}$ | $\bf 3.2135\cdot 10^{-4}$ | $\bf\color[rgb]{0.59,0,0.66}2.8677\cdot 10^{-4}$

(b) Blue noise

###  4.2 Experimental settings

  1. 1.

We examine different pairs of time-frequency parameters $(a,b)$, deploying
Remark 2.

  2. 2.

We use the power iteration method power  which yields the largest in magnitude
eigenvalue and corresponding eigenvector of $U_{\mathcal{Z}}$, then set this
eigenvector as our desired window vector $g_{*}$.

  3. 3.

We construct –using the MATLAB package LTFAT ltfat – four different Gabor
frames with their associated analysis operators/DGTs, which go as follows:
$\Phi_{g_{1}}$, $\Phi_{g_{2}}$, $\Phi_{g_{3}}$ and $\Phi_{g_{*}}$,
corresponding to a Gaussian, a Hann, a Hamming and the star window vector
respectively. Since we process real-valued signals, we alter the four analysis
operators to compute only the DGT coefficients of positive frequencies instead
of the full DGT coefficients.

  4. 4.

We consider zero-mean Gaussian noise with standard deviation $\sigma$ either
varying, i.e. $\sigma\equiv\bm{\sigma}=\text{linspace}(0.001,0.01,100)$
(MATLAB function _linspace_ generates a row vector of 100 evenly spaced points
between 0.001 and 0.01), or fixed scalar $\sigma=0.001$. Similarly, we
consider pink and blue noises with amplitude either varying and controlled by
a scaling factor $\bm{\sigma}=\text{linspace}(0.001,0.01,100)$, or fixed and
controlled by the scalar $\sigma=0.001$. Note that for
$\bm{\sigma_{100}}=0.01$, the amplitude of the coloured555in the rest of the
paper, when we speak of coloured noises, we mean the examined cases of pink
and blue noise noises is almost equal to the amplitude of the signals.

  5. 5.

We take noisy measurements

$y=x+e,$ (14)

where $e$ denotes either Gaussian or coloured noise.

  6. 6.

We solve –using the Matlab package TFOCS tfocs – four different instances of
(4), one for each of the four DGTs. For TFOCS, we set $x_{0}=0,\,z_{0}=[\,]$;
for each of the instances $i=1,2,3,*$, we set the smoothing parameter
$\mu_{i}=10^{-1}\|\Phi_{i}x\|_{\infty}$, since we noticed an improved
performance of the solving algorithm when $\mu$ is a function of $\Phi_{i}$
(the scaling factor $10^{-1}$ and the function $\|\cdot\|_{\infty}$ are simply
chosen empirically) and employ the solver_BPDN_W solver.

  7. 7.

From the aforementioned procedure, we obtain four different estimators for
$x$, namely $\hat{x}_{1}$, $\hat{x}_{2}$, $\hat{x}_{3}$, $\hat{x}_{*}$ and
their corresponding MSEs, i.e.$\mathbb{E}(\|x-\hat{x}_{i}\|_{2}^{2})$,
$i=1,2,3,*$.

(a) 251-136532-0014, $(L,a,b)=(33915,51,19)$, Gaussian noise

(b) 251-136532-0014, $(L,a,b)=(33915,51,19)$, pink noise

(c) 8842-304647-0007, $(L,a,b)=(27531,19,23)$, Gaussian noise

(d) 8842-304647-0007, $(L,a,b)=(27531,19,23)$, blue noise

(e) 2035-147960-0013, $(L,a,b)=(41769,21,17)$, Gaussian noise

(f) 2035-147960-0013, $(L,a,b)=(41769,21,17)$, pink noise

(g) 1462-170145-0020, $(L,a,b)=(33915,51,19)$, Gaussian noise

(h) 1462-170145-0020, $(L,a,b)=(33915,51,19)$, blue noise

Figure 2: Rate of denoising success for 4 speech signals with different
parameters $(L,a,b)$, contaminated by Gaussian (left) and coloured (right)
noise. Note that 3 of the 4 methods roughly coincide.

(a) 6241-61943-0002, $(L,a,b)=(43605,51,19)$, Gaussian noise

(b) 6241-61943-0002, $(L,a,b)=(43605,51,19)$, pink noise

(c) 5338-284437-0025, $(L,a,b)=(29835,17,13)$, Gaussian noise

(d) 5338-284437-0025, $(L,a,b)=(29835,17,13)$, blue noise

(e) 3752-4944-0042, $(L,a,b)=(51051,33,17)$, Gaussian noise

(f) 3752-4944-0042, $(L,a,b)=(51051,33,17)$, blue noise

(g) 5694-64038-0013, $(L,a,b)=(51051,33,17)$, Gaussian noise

(h) 5694-64038-0013, $(L,a,b)=(51051,33,17)$, pink noise

Figure 3: Rate of denoising success for 4 speech signals with different
parameters $(L,a,b)$, contaminated by Gaussian (left) and coloured (right)
noise. Note that 3 of the 4 methods roughly coincide.

(a) 5895-34615-0001, $(L,a,b)=(51051,33,17)$, Gaussian noise

(b) 5895-34615-0001, $(L,a,b)=(51051,33,17)$, pink noise

(c) 2428-83699-0035, $(L,a,b)=(41769,21,17)$, Gaussian noise

(d) 2428-83699-0035, $(L,a,b)=(41769,21,17)$, pink noise

(e) 2803-154320-0006, $(L,a,b)=(33915,51,19)$, Gaussian noise

(f) 2803-154320-0006, $(L,a,b)=(33915,51,19)$, blue noise

(g) 3752-4944-0008, $(L,a,b)=(29835,17,13)$, Gaussian noise

(h) 3752-4944-0008, $(L,a,b)=(29835,17,13)$, pink noise

Figure 4: Rate of denoising success for 4 speech signals with different
parameters $(L,a,b)$, contaminated by Gaussian (left) and coloured (right)
noise. Note that 3 of the 4 methods roughly coincide.

##  5 Experiments and Results

We present two sets of experiments in the following subsections.

###  5.1 Fixed $(L,a,b)$ with varying $\sigma$

We fix for each of the 12 signals the lattice parameters $a,b$, with respect
to each artificial dimension $L$. We add to all signals zero-mean Gaussian
noise with varying standard deviation, using the vector $\bm{\sigma}$, and
perform analysis denoising for each entry of $\bm{\sigma}$. For the coloured
noise cases, we randomly split the set of 12 signals into two subsets, of 6
signals each. We add to the signals of the first and second subset blue and
pink noise respectively, with varying amplitude controlled by the vector
$\bm{\sigma}$, and perform analysis denoising for each entry of $\bm{\sigma}$.
The left column in Fig. 2-4 demonstrates for different signals, how the 4
resulting MSEs scale in the case of the Gaussian noise, as its standard
deviation increases. Clearly, our proposed DGT outperforms the rest of DGTs,
consistently for all signals and for different choices of artificial dimension
with time-frequency parameters. Similarly, the right column in Fig. 2-4
demonstrates for different signals, how the 4 resulting MSEs scale in the case
of blue and pink noise, as the scaling factor of each coloured noise’s
amplitude increases. We observe that star-DGT is more robust than the rest of
DGTs, even when the amplitude of each coloured noise is almost equal to the
amplitude of the speech signal to which it is added.

###  5.2 Fixed $L$ and $\sigma$, with varying $(a,b)$

We randomly pick 6 out of the 12 speech signals (we prefer to examine signals
with different artificial dimensions). We alter for each signal the time-
frequency parameters $a,b$ with respect to its artificial dimension. We
consider the fixed scalar $\sigma=0.001$ serving as both the standard
deviation of the Gaussian noise and the scaling factor controlling the
coloured noises’ amplitude. For different pairs of $(a,b)$, we add Gaussian
noise to all six signals, blue noise to three of the six signals and pink
noise to the rest of them. Finally, we denoise all signals for all types of
noise and present the resulting MSEs in Tables 2(b)-7(b). For all choices of
$(a,b)$, star-DGT (indicated in bold in each subtable) outperforms the
baseline DGTs, consistently for all signals, for both Gaussian and coloured
noises. Additionally, we see that among all examined pairs of lattice
parameters, star-DGT achieves the smallest MSE (indicated in purple in each
subtable) when $a,b$ are chosen as the two largest factors in the prime
factorization of $L$; the rest of DGTs do not seem to benefit much from this
selection. On the other hand, among all examined choices of $(a,b)$, star-DGT
performs slightly worse when $a=b$. For example, as indicated in Tables 4(b)
and 7(b), star-DGT reaches a slightly bigger MSE when $a=b=11$ and
$a=b=3^{2}$, respectively.

##  6 Conclusion and Future Directions

In the present paper, we took advantage of a window vector to generate a spark
deficient Gabor frame and introduced a redundant analysis Gabor operator/DGT,
namely the star-DGT, associated with this SDGF. We then applied the star-DGT
to analysis denoising, along with three other DGTs generated by state-of-the-
art window vectors in the field of Gabor Analysis. First, we fixed the ambient
dimension and the time-frequency parameters, and altered the standard
deviation of the Gaussian noise and the amplitude of the coloured noises.
Second, we examined how different pairs of lattice parameters, with fixed
standard deviation and amplitude of the Gaussian and coloured noises
respectively, affect the performance of analysis denoising. All experiments
confirm improved robustness: the increased amount of linear dependencies
provided by this SDGF, yields for all speech signals a lower MSE for the
proposed method. Future directions will include the combination of the present
framework with deep learning methods gabdl , as well as the examination of the
robustness of the weighted combination

$\begin{split}\text{minimize}\quad&\|\Phi
x\|_{1}+\|x\|_{\mathrm{TV}}+\frac{\mu}{2}\|x-x_{0}\|_{2}^{2}\\\ \text{subject
to}\quad&\|x-y\|_{2}\leq\eta,\end{split}$ (15)

where $\|x\|_{\mathrm{TV}}$ is the total-variation norm of $x$. Finally, it
would be interesting to develop a rigorous mathematical proof, explaining why
star-DGT benefits more when the lattice parameters are chosen as the two
largest primes in the prime factorization of a signal’s dimension.

## Acknowledgments

V. Kouni would like to thank G. Paraskevopoulos for his valuable advice and
insightful discussions around the framework presented in this paper.

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$\displaystyle\chi(\cdot\in\widehat{C}){\,\overset{\mathsf{epi}}{\longrightarrow}\,}\chi(\cdot\in
C)\text{ \quad in probability}$ (31)

The claim is equivalent to $d_{AW}(\widehat{C},C){\,\overset{{p}}{\to}\,}0$,
where $d_{AW}$ is defined in Eq 26. For any $\epsilon>0$, the event
$\\{d_{AW}(\widehat{C},C)>\epsilon\\}$ is equivalent to
$\\{\widehat{I}_{+}\neq I_{+}\\}$. First we bound
${\mathbb{P}}(\widehat{I}_{+}\neq I_{+})$.

$\displaystyle{\mathbb{P}}(\widehat{I}_{+}\neq I_{+})$
$\displaystyle={\mathbb{P}}(\exists i\in
I_{+},1-\beta^{\gamma}_{i}>\delta_{t},\text{ or }\exists
i\in{I^{c}},1-\beta^{\gamma}_{i}<\delta_{t})$ $\displaystyle\leq\sum_{i\in
I_{+}}{\mathbb{P}}(1-\beta^{\gamma}_{i}>\delta_{t})+\sum_{i\in{I^{c}}}{\mathbb{P}}(1-\beta^{\gamma}_{i}<\delta_{t})$

For $i\in I_{+}$, by $\beta^{\gamma}_{i}-1=o_{p}(\frac{1}{\sqrt{t}})$, we have
${\mathbb{P}}(1-\beta^{\gamma}_{i}>\delta_{t})={\mathbb{P}}(o_{p}(1)>\delta_{t}\sqrt{t})\to
0$ since $\delta_{t}\sqrt{t}\to c>0$. For $i\in{I^{c}}$, since
${\beta^{*}_{i}}-1<0$,
${\mathbb{P}}(1-\beta^{\gamma}_{i}<\delta_{t})={\mathbb{P}}({\beta^{*}_{i}}-\beta^{\gamma}_{i}<{\beta^{*}_{i}}-1+\delta_{t})={\mathbb{P}}(o_{p}(1)<{\beta^{*}_{i}}-1+\delta_{t})\to
0$ by $\delta_{t}\downarrow 0$. We conclude ${\mathbb{P}}(\widehat{I}_{+}\neq
I_{+})\to 0$ and so
$\chi(\cdot\in\widehat{C}){\,\overset{\mathsf{epi}}{\longrightarrow}\,}\chi(\cdot\in
C)$ in probability.

Step 2. Next, we show $\beta^{b}{\,\overset{{p}}{\to}\,}\beta^{*}$. Since
$\epsilon_{t}\to 0$ and $G^{b}=O_{p}(1)$, we have
$\epsilon_{t}G^{b}=o_{p}(1)$. Then for each $\beta\in{\mathbb{R}^{n}}$

$\displaystyle\epsilon_{t}G^{b}+\frac{1}{2}\|\beta^{\gamma}-\beta\|_{\widehat{\mathscr{H}}}^{2}+\chi(\beta\in\widehat{B})$
$\displaystyle=\frac{1}{2}\|\beta^{*}-\beta\|_{\widehat{\mathscr{H}}}^{2}+\chi(\beta\in\widehat{B})+\bigg{(}\epsilon_{t}G^{b}+(\beta^{*}-\beta)^{\scriptscriptstyle\mathsf{T}}\widehat{\mathscr{H}}(\beta^{\gamma}-\beta^{*})+\frac{1}{2}\|\beta^{\gamma}-\beta^{*}\|_{\widehat{\mathscr{H}}}^{2}\bigg{)}$
$\displaystyle=\frac{1}{2}\|\beta^{*}-\beta\|_{\widehat{\mathscr{H}}}^{2}+\chi(\beta\in\widehat{B})+o_{p}(1)$
$\displaystyle{\,\overset{{p}}{\to}\,}\frac{1}{2}\|\beta^{*}-\beta\|_{{\mathscr{H}}}^{2}+\chi(\beta\in
B^{\prime})$

where $B^{\prime}=\\{\beta\in[0,1]^{n}:{\beta_{i}}=1,i\in I_{+}\\}$. By
convexity, it implies uniform convergence on compact sets in probability.
Since $\beta^{*}$ uniquely minimize
$\|\beta^{*}-\beta\|_{{\mathscr{H}}}^{2}+\chi(\beta\in B)$, we conclude
$\beta^{b}{\,\overset{{p}}{\to}\,}\beta^{*}$ (Newey & McFadden, 1994, Theorem
2.7).

Step 3. Identify the limit distribution of $\beta^{b}$.

Note
$\epsilon_{t}^{{-1}}(\beta^{b}-\beta^{\gamma})=\epsilon_{t}^{{-1}}(\beta^{b}-\beta^{*})+\epsilon_{t}^{{-1}}(\beta^{*}-\beta^{\gamma})=\epsilon_{t}^{{-1}}(\beta^{b}-\beta^{*})+o_{p}(1)$,
it suffices to show
$\epsilon_{t}^{{-1}}(\beta^{b}-\beta^{*})\,{\stackrel{{\scriptstyle
p}}{{\rightsquigarrow}}}\,{\mathscr{J}}_{\mathsf{{FPPE}}}$.

First with a change of variable $h=\epsilon_{t}^{{-1}}(\beta-\beta^{*})$ and
so
$\epsilon_{t}^{{-1}}(\beta-\beta^{\gamma})=h+\epsilon_{t}^{{-1}}(\beta^{*}-\beta^{\gamma})$,
dividing the objective function by $\epsilon_{t}^{2}$, the bootstrap estimator
in Eq 17, with probability approaching one, can be written as

$\displaystyle\frac{\beta^{b}-\beta^{*}}{\epsilon_{t}}=\operatorname*{arg\,min}_{h\in{\mathbb{R}^{n}}}\bigg{\\{}\underbrace{(G^{b})^{\scriptscriptstyle\mathsf{T}}(h+\frac{\beta^{*}-\beta^{\gamma}}{\epsilon_{t}})+\frac{1}{2}\|h+\frac{\beta^{*}-\beta^{\gamma}}{\epsilon_{t}}\|_{\widehat{\mathscr{H}}}^{2}}_{:=X_{t}(h)}+\chi(h\in\widehat{C})\bigg{\\}}$

######  Claim 4.

$X_{t}(h)\rightsquigarrow(\,{\stackrel{{\scriptstyle
p}}{{\rightsquigarrow}}}\,)G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h$
in $\ell^{\infty}(K)$ for any compact $K\subset{\mathbb{R}^{n}}$.

By Knight (1999), we know convergence in distribution with respect to the
topology of uniform convergence on compact sets implies weak epi-convergence,
and so
$X_{t}{\,\stackrel{{\scriptstyle\mathsf{epi}}}{{\rightsquigarrow}}\,}G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h$.
Combining
$X_{t}{\,\stackrel{{\scriptstyle\mathsf{epi}}}{{\rightsquigarrow}}\,}G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h$,
Eq 31 and Lemma 1, we have

$\displaystyle
X_{t}(h)+\chi(h\in\widehat{C}){\,\stackrel{{\scriptstyle\mathsf{epi}}}{{\rightsquigarrow}}\,}G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}\|h\|_{\mathscr{H}}^{2}+\chi(h\in
C)$

The above implies conditional weak epi-convergence in probability of the
processes $h\mapsto X_{t}(h)+\chi(h\in\widehat{C})$. A bootstrap version of
the continuous mapping theorem can be stated for conditional weak epi-
convergence in probability (Hong & Li, 2020). Then
$\epsilon_{t}^{{-1}}(\beta^{b}-\beta^{*})\,{\stackrel{{\scriptstyle
p}}{{\rightsquigarrow}}}\,\operatorname*{arg\,min}_{h\in
C}G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h$,
which is exactly ${\mathscr{J}}_{\mathsf{{FPPE}}}$. ∎

###### Proof of 4.

First, we show for any compact set $K\in{\mathbb{R}^{n}}$,

$\displaystyle\sup_{h\in
K}|X_{t}(h)-((G^{b})^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}\|h\|_{{\mathscr{H}}}^{2})|=o_{p}(1)$
(32)

Note the LHS can be upper bounded by

$\displaystyle(G^{b})^{\scriptscriptstyle\mathsf{T}}(\beta^{\gamma}-\beta^{*})\epsilon_{t}^{{-1}}+\frac{1}{2}\|\beta^{\gamma}-\beta^{*}\|_{{\mathscr{H}}}^{2}\epsilon_{t}^{-2}+\sup_{K}h^{\scriptscriptstyle\mathsf{T}}\widehat{\mathscr{H}}(\beta^{\gamma}-\beta^{*})\epsilon_{t}^{{-1}}+\sup_{K}h^{\scriptscriptstyle\mathsf{T}}(\widehat{\mathscr{H}}-{\mathscr{H}})h$

We just need to show
$(G^{b})^{\scriptscriptstyle\mathsf{T}}(\beta^{\gamma}-\beta^{*})=o_{p}(\epsilon_{t})$,
$\|\beta^{\gamma}-\beta^{*}\|_{\widehat{\mathscr{H}}}=o_{p}(\epsilon_{t}^{2})$
and
$\sup_{K}h^{\scriptscriptstyle\mathsf{T}}\widehat{\mathscr{H}}(\beta^{\gamma}-\beta^{*})=o_{p}(\epsilon_{t})$
and
$\sup_{K}|h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h-h^{\scriptscriptstyle\mathsf{T}}\widehat{\mathscr{H}}h|=o_{p}(1)$
(Thm 5). This holds by $G^{b}=O_{p}(1)$,
$\beta^{\gamma}-\beta^{*}=O_{p}(1/\sqrt{t})$,
$\|{\mathscr{H}}-\widehat{\mathscr{H}}\|_{2}=o_{p}(1)$ and
$1/\sqrt{t}=o(\epsilon_{t})$.

Next we show for any compact $K\subset{\mathbb{R}^{n}}$,

$\displaystyle(G^{b})^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}\|h\|_{\widehat{\mathscr{H}}}^{2}\rightsquigarrow
G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}\|h\|_{\mathscr{H}}^{2}\text{\quad
in $\ell^{\infty}(K)$}$ (33)

where $G\sim{\mathscr{N}}(0,{\operatorname{Cov}}(\nabla
F({\cdot},\beta^{*})))$. It suffices to show $G^{b}{\,\overset{{d}}{\to}\,}G$
and $\widehat{\mathscr{H}}{\,\overset{{p}}{\to}\,}{\mathscr{H}}$. In Thm 5 is
has been shown that $\widehat{\mathscr{H}}-{\mathscr{H}}=o_{p}(1)$. Note
$G^{b}=\sqrt{t}({P^{b}_{t}}-P_{t})D_{F}({\cdot},\beta^{\gamma})=\sqrt{t}({P^{b}_{t}}-P_{t})(D_{F}({\cdot},\beta^{\gamma})-D_{F}({\cdot},\beta^{*}))+\sqrt{t}({P^{b}_{t}}-P_{t})D_{F}({\cdot},\beta^{*})=\sqrt{t}({P^{b}_{t}}-P_{t})D_{F}({\cdot},\beta^{*})+o_{p}(1)$
by Wellner & Zhan (1996, Lemma 4.1). We conclude
$G^{b}{\,\overset{{d}}{\to}\,}G$ and $X_{t}\rightsquigarrow
h^{\scriptscriptstyle\mathsf{T}}G+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h$.
To show conditional convergence $\,{\stackrel{{\scriptstyle
p}}{{\rightsquigarrow}}}\,$, one can use arguments analogous to Theorem 2.9.6
in Van der Vaart (2000). ∎

###  D.8 Proof of Thm 6

###### Proof of Thm 6.

Let $T^{b,\infty}$ be the conditional limit distribution of $T^{b}$, and
$T^{\infty}$ be the limit distribution of $T^{\gamma}(\beta^{*},\delta^{*})$.
The proof relies on the following result. For two real-valued random variables
$X$ and $Y$, we say $X$ is stochastically dominated by $Y$, denoted
$X\leq_{st}Y$ if ${\mathbb{P}}(X>x)\leq{\mathbb{P}}(Y>x)$ for all
$x\in{\mathbb{R}}$.

######  Theorem 10.

For all $\kappa\in(0,\infty)$, $T^{\infty}\leq_{st}T^{b,\infty}$. When
$\kappa=\infty$, $T^{\infty}=T^{b,\infty}$.

We recall a result regarding quantiles.

######  Lemma 2.

Let $F(x)={\mathbb{P}}(X\leq x)$ be the CDF of a real-valued random variable
$X$. And let $F^{-1}(p)=\inf(x\in{\mathbb{R}}:F(x)\geq p)$ for $p\in(0,1)$ be
the quantile. Then $F(F^{-1}(p))\geq p$ for $p\in(0,1)$.

Recall the following condition

the CDF of $T^{\infty}$ is continuous at the $(1-\alpha)$-th quantile of
$T^{\infty}$. (34)

Let $c_{1-\alpha}$ be the $(1-\alpha)$-th quantile of $T^{\infty}$. Then Lemma
2 implies that ${\mathbb{P}}(T^{\infty}\leq c_{1-\alpha-\epsilon})\geq
1-\alpha-\epsilon$ for all $\epsilon\geq 0$ small enough. The assumption in Eq
34 implies ${\mathbb{P}}(T^{\infty}\leq c_{1-\alpha})=1-\alpha$.

For any $\epsilon>0$, let $A$ be the event $\\{\gamma:{\mathbb{P}}(T^{b}\leq
c_{1-\alpha}\,|\,\gamma)-(1-\alpha)\leq\epsilon\\}$. Let $c^{b}_{1-\alpha}$ be
that the $(1-\alpha)$-th quantile of $T^{b}$ conditional on $\gamma$.

First we show ${\mathbb{P}}(A)\to 1$. By $T^{b,\infty}\geq_{st}T^{\infty}$ we
know ${\mathbb{P}}(T^{b,\infty}\leq
c_{1-\alpha})\leq{\mathbb{P}}(T^{\infty}\leq c_{1-\alpha})=1-\alpha$. Then

$\displaystyle{\mathbb{P}}(A^{c})$
$\displaystyle={\mathbb{P}}({\mathbb{P}}(T^{b}\leq
c_{1-\alpha}\,|\,\gamma)-(1-\alpha)>\epsilon)$
$\displaystyle\leq{\mathbb{P}}({\mathbb{P}}(T^{b}\leq
c_{1-\alpha}\,|\,\gamma)-{\mathbb{P}}(T^{b,\infty}\leq
c_{1-\alpha})>\epsilon)\to 0$

due to $T^{b}\,{\stackrel{{\scriptstyle
p}}{{\rightsquigarrow}}}\,T^{b,\infty}$. Under event $A$, we have
$c_{1-\alpha-\epsilon}\leq c^{b}_{1-\alpha}$. To see this note
${\mathbb{P}}(T^{\infty}\leq
c^{b}_{1-\alpha}\,|\,\gamma)\geq{\mathbb{P}}(T^{b}\leq
c^{b}_{1-\alpha})-\epsilon\geq 1-\alpha-\epsilon$. Then

$\displaystyle{\mathbb{P}}\big{(}(\beta^{*},\delta^{*})\in
C^{\gamma}(c^{b}_{1-\alpha})\big{)}$
$\displaystyle={\mathbb{P}}(T^{\gamma}(\beta^{*},\delta^{*})\leq
c^{b}_{1-\alpha})$
$\displaystyle\geq{\mathbb{P}}(T^{\gamma}(\beta^{*},\delta^{*})\leq
c^{b}_{1-\alpha},A)$
$\displaystyle\geq{\mathbb{P}}(T^{\gamma}(\beta^{*},\delta^{*})\leq
c_{1-\alpha-\epsilon},A)$
$\displaystyle\geq{\mathbb{P}}(T^{\gamma}(\beta^{*},\delta^{*})\leq
c_{1-\alpha-\epsilon})-{\mathbb{P}}(A^{c})\geq 1-\alpha-\epsilon+o(1)$

where the last line follows from
${\mathbb{P}}(T^{\gamma}(\beta^{*},\delta^{*})\leq
c_{1-\alpha-\epsilon})\to{\mathbb{P}}(T^{\infty}\leq
c_{1-\alpha-\epsilon})\geq 1-\alpha-\epsilon$ and ${\mathbb{P}}(A^{c})\to 0$.
Since $\epsilon>0$ is arbitrary, we conclude
$\liminf{\mathbb{P}}((\beta^{*},\delta^{*})\in
C^{\gamma}(c^{b}_{1-\alpha}))\geq 1-\alpha$. ∎

###### Proof of Thm 10.

We study the asymptotic distributions of $T^{\gamma}(\beta^{*},\delta^{*})$
and $T^{b}$. Recall $\nabla H(\beta^{*})=-\delta^{*}$.

Step 1. We will show

$\displaystyle
t(L_{t}(\beta^{*}+\tfrac{h}{\sqrt{t}},\delta^{*})-L_{t}(\beta^{*},\delta^{*}))\rightsquigarrow
h^{\scriptscriptstyle\mathsf{T}}G+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h$

in $\ell^{\infty}(K)$ for any compact $K\subset{\mathbb{R}^{n}}$.

$\displaystyle
t(L_{t}(\beta^{*}+\tfrac{h}{\sqrt{t}},\delta^{*})-L_{t}(\beta^{*},\delta^{*}))$
$\displaystyle=t(H_{t}(\beta^{*}+\tfrac{h}{\sqrt{t}})-H_{t}(\beta^{*})+(\delta^{*})^{\scriptscriptstyle\mathsf{T}}(\tfrac{h}{\sqrt{t}}))$
$\displaystyle=\sqrt{t}(P_{t}-P)(\nabla
F({\cdot},\beta^{*}))^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h+o_{p}(1)$
$\displaystyle\rightsquigarrow
h^{\scriptscriptstyle\mathsf{T}}G+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h\text{
in $\ell^{\infty}(K)$}$

where $G\sim{\mathscr{N}}(0,{\operatorname{Cov}}(\nabla
F({\cdot},\beta^{*})))$, $o_{p}(1)$ term is uniform over $h\in K$ by 3.
Applying a continuous mapping theorem

$\displaystyle T^{\gamma}(\beta^{*},\delta^{*})$
$\displaystyle=-\inf_{h\in{\mathbb{B}}_{\kappa}}t(L_{t}(\beta^{*}+\tfrac{h}{\sqrt{t}})-L_{t}(\beta^{*}))$
$\displaystyle{\,\overset{{d}}{\to}\,}-\inf_{h\in{\mathbb{B}}_{\kappa}}(G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h)=:T$

Step 2. In 4, we have shown

$\displaystyle X^{b}(\beta^{*}+\epsilon_{t}h)\,{\stackrel{{\scriptstyle
p}}{{\rightsquigarrow}}}\,G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h$

in $\ell^{\infty}(K)$ for any compact $K\in{\mathbb{R}^{n}}$. Next we study
the asymptotic distribution of $T^{b}$.

$\displaystyle T^{b}$
$\displaystyle=-\inf_{\beta\in{\mathbb{R}^{n}_{+}}}X^{b}(\beta)$
$\displaystyle=-\inf_{h\in({\mathbb{R}^{n}_{+}}-\beta^{*})/{\epsilon_{t}}}X^{b}(\beta^{*}+\epsilon_{t}h)$
$\displaystyle\,{\stackrel{{\scriptstyle
p}}{{\rightsquigarrow}}}\,-\inf_{h\in{\mathbb{R}^{n}}}(G^{\scriptscriptstyle\mathsf{T}}h+\frac{1}{2}h^{\scriptscriptstyle\mathsf{T}}{\mathscr{H}}h)=:T^{b,\infty}$

where the last line follows due to $\beta^{*}$ lying in the interior of
${\mathbb{R}^{n}_{+}}$, and a bootstrap version of continuous mapping theorem;
see Theorem 10.8 in Kosorok (2008). We can see for each draw of $G$, we have
the dominance relationship $T^{b,\infty}\geq T$. We conclude the
$(1-\alpha)$-quantile of $T^{b,\infty}$ is greater than or equal to that of
$T$.

The claim that when $\kappa=\infty$, $T^{b}\,{\stackrel{{\scriptstyle
p}}{{\rightsquigarrow}}}\,T^{\infty}$ is obvious.

######  Remark 1.

One could also use the statistic

$\displaystyle T^{\gamma}(\beta)=\inf_{0\leq\delta\leq
b,\delta^{\scriptscriptstyle\mathsf{T}}(1_{n}-\beta)=0}\bigg{(}L_{t}(\beta,\delta)-\inf_{h\in{\mathbb{B}}_{\kappa}}L_{t}(\beta+h/\sqrt{t},\delta)\bigg{)}$

and the region $\\{\beta\in(0,1]^{n}:T^{\gamma}(\beta)\leq\iota\\}$ to do
inference on just $\beta$. Noting $T^{\gamma}(\beta^{*})\leq
L_{t}(\beta^{*},\delta^{*})-\inf_{h\in{\mathbb{B}}_{\kappa}}L_{t}(\beta^{*}+h/\sqrt{t},\delta^{*})$,
we can estimate an upper bound of the quantile of its limit distribution by
bootstrap. ∎