# Neutrino Mass Matrices with one texture equality and one vanishing neutrino
mass

Jinzhong Han1, Ruihong Wang2, Weijian Wang3 , Xing-Ning Wei3
1 School of Physics and Telecommunications Engineering, Zhoukou Normal
University, Henan, 466001, China
2College of Information Science $\&$Technology, Hebei Agricultural University,
Baoding,China
3 Department of Physics, North China Electric Power University, Baoding
071003, P. R. China
<EMAIL_ADDRESS>
###### Abstract

In the light of latest data of neutrino oscillation experiments, we carry out
a systematic investigation on the texture structures of Majorana neutrino mass
matrix $M_{\nu}$, which contain one vanishing neutrino mass and an equality
between two matrix elements. Among 15 logically possible patterns, it is found
that for norm order ($m_{3}>m_{2}>m_{1}=0$) of neutrino masses only five of
them are compatible with recent experimental data at the $3\sigma$ level,
while for inverted order ($m_{2}>m_{1}>m_{3}=0$) ten patterns is
phenomenologically allowed. In the numerical analysis, we perform a scan over
the parameter space of all viable patterns to get a large sample of scattering
points. We present the implications of each allowed pattern for three mixing
angles $(\theta_{12},\theta_{2},\theta_{3})$, leptonic CP violation and
neutrinoless double-beta decay, predicting strong correlations between
oscillation parameters. The theoretical realization of a concrete example is
discussed in the framework of Froggatt-Nielsen mechanism.

##  1 Introduction

In spite of the evidences for massive neutrinos and the admixtures of the
flavor states [1], the origin of the lepton flavor structure remains an open
question. A popular approach to understand the leptonic mixing structure is
reducing the number of free parameters by adding Abelian or non-Abelian flavor
symmetries, leading to specific texture structures of neutrino mass matrix
($M_{\nu}$). Some common models include texture-zeros[2, 3, 4, 5, 6, 7],
hybrid textures[8, 9], zero trace[10], vanishing minors[11, 12, 13], two
traceless submatrices[14], equal elements or cofactors[15], hybrid
$M_{\nu}^{-1}$ textures[16], partial $\mu-\tau$ symmetry[17].

One can also reduce the number of free parameters of $M_{\nu}$ by assuming one
of neutrinos to be massless. It is well known that in type-I seesaw mechanism,
a vanishing neutrino mass is easily realized by assuming two families of heavy
right-handed neutrinos. Similar scenario also appears in radiative seesaw
model[18, 19]. In the framework of Affleck-Dine mechanism[20, 21, 22], an
extremely small neutrino mass($\approx 10^{-10}$eV) is required to
successfully produce the leptogenesis[23, 24]. Hence it is imperative to ask
if the lepton mass matrix with a specific texture structure and a vanishing
mass eigenvalue can survive under the current neutrino oscillation data.
Several attempts have been made in this direction. In Refs.[25, 26, 27],
particular attentions has been paid to the neutrino mass matrix structures
with one texture-zero and one massless eigenstate.

In the flavor basis where the charged lepton mass matrix is diagonal, we
investigate a specific class of neutrino mass matrices which contains an
vanishing neutrino mass and an texture equality between two independent
entries. The texture equality was proposed in many literatures. Bear in mind
the fact that an equality between ”12” and ”13” element of $M_{\nu}$ has
appeared in the the tri-bimaximal form of $M_{\nu}$

$M_{\nu}=\left(\begin{array}[]{ccc}a&b&b\\\ b&a-c&b+c\\\
b&b+c&a-c\end{array}\right)$ (1)

which can be diagonalized by tri-bimaximal form of PMNS matrix

$U_{\text{PMNS}}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\sqrt{\frac{1}{3}}&0\\\
-\sqrt{\frac{1}{6}}&\sqrt{\frac{1}{3}}&\sqrt{\frac{1}{2}}\\\
\sqrt{\frac{1}{6}}&\sqrt{\frac{1}{3}}&-\sqrt{\frac{1}{2}}\end{array}\right)$
(2)

and leads to the three neutrino masses eigenvalues
$m_{1}=a-b,m_{2}=a+2b,m_{3}=a-b-2c$. If $|a|\simeq|b|\ll|c|$, we obtain a
normal hierarchy for neutrino masses with $m_{1}\simeq 0$. Clearly, Eq.(2) is
not compatible with the observation of no-zero $\theta_{13}$ angle. Hence the
texture structure with an equality between ”12” and ”13” elements of $M_{\nu}$
can be considered as a perturbed generalization of Eq.(1), where the number of
free parameters is added but the texture equality still holds.

The main motivation of this work is three-fold:

  * •

From the phenomenological viewpoint, either a vanishing neutrino mass or an
equality between two nozero matrix elements imposes one constraint condition
on $M_{\nu}$ and reduces the number of free degrees by two. Thus texture
equalities are as predictive as the texture zeros.

  * •

On the experimental side, the absolute scale of lightest neutrino mass is
still unknown. Within the $\Lambda$CDM framework , an upper bound on the sum
of neutrino mass $\sum m_{i}<0.23$ eV at $95\%$ confidence level has been
reported by Plank Collaboration[28] . Recently, combined with the Planck,
TT,TE,EE+lowP+BAO+JLA+H0+Lensing data, a much tighter bound $\sum
m_{i}<0.105$, which is almost access to the lower limit of $\sum m_{i}$ for
inverted order spectrum of neutrino masses, is obtained in holographic dark
energy scenario[29]. Since we have stood the on the verge to distinguish the
mass spectrum (NO or IO) of three neutrinos through cosmological observations,
the phenomenology of some specific texture structures with a vanishing
neutrino mass deserves a detailed survey.

  * •

It is generally believed that the observed neutrino mixing pattern indicates
some underlying discrete flavor symmetries. The flavor symmetry realization
for all possible patterns is beyond the scope of this work. In the framework
of the Froggatt-Nielsen (FN) mechanism[30], we consider an explicit model
based on $S_{\mu-\tau}\times Z_{8}$ symmetry suggested by Ref.[17] to realize
one of the viable texture equality. Then we adopt the methodology in Ref.[31]
by assuming the broken FN symmetry a discrete $Z_{n}$. We will see that a
ultalight neutrino mass naturally arises as a good approximation to vanishing
neutrino masses. We expect that a phenomenological analysis may help us reveal
the underlying flavor structure of lepton mixing.

The rest of paper is organized as follows: In Sec.II, We describe some useful
notations and the framework used to obtain the constraint equations. In Sec.
III, the details of numerical analysis are presented. In Sec. IV, we discuss
the theoretical realization for a concrete example. The summary is given in
Sec. V.

Table 1: Fifteen possible texture structures with equality between two nozero elements where ”$\bigtriangleup$” denotes the nozero and equal elements, while ”$\times$” stands for the abitrary and nozero ones. $P1$ | $P2$ |  $P3$ |  $P4$ |  $P5$
---|---|---|---|---
$\left(\begin{array}[]{ccc}\bigtriangleup&\bigtriangleup&\times\\\ \bigtriangleup&\times&\times\\\ \times&\times&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\bigtriangleup&\times&\bigtriangleup\\\ \times&\times&\times\\\ \bigtriangleup&\times&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\bigtriangleup&\times\\\ \bigtriangleup&\bigtriangleup&\times\\\ \times&\times&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\times&\times\\\ \times&\bigtriangleup&\bigtriangleup\\\ \times&\bigtriangleup&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\times&\bigtriangleup\\\ \times&\times&\times\\\ \bigtriangleup&\times&\bigtriangleup\end{array}\right)$
$P6$ | $P7$ |  $P8$ |  $P9$ |  $P10$
$\left(\begin{array}[]{ccc}\times&\times&\times\\\ \times&\times&\bigtriangleup\\\ \times&\bigtriangleup&\bigtriangleup\end{array}\right)$ | $\left(\begin{array}[]{ccc}\bigtriangleup&\times&\times\\\ \times&\times&\bigtriangleup\\\ \times&\bigtriangleup&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\times&\bigtriangleup\\\ \times&\bigtriangleup&\times\\\ \bigtriangleup&\times&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\bigtriangleup&\times\\\ \bigtriangleup&\times&\times\\\ \times&\times&\bigtriangleup\end{array}\right)$ | $\left(\begin{array}[]{ccc}\bigtriangleup&\times&\times\\\ \times&\bigtriangleup&\times\\\ \times&\times&\times\end{array}\right)$
$P11$ | $P12$ |  $P13$ |  $P14$ |  $P15$
$\left(\begin{array}[]{ccc}\bigtriangleup&\times&\times\\\ \times&\times&\times\\\ \times&\times&\bigtriangleup\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\times&\times\\\ \times&\bigtriangleup&\times\\\ \times&\times&\bigtriangleup\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\bigtriangleup&\bigtriangleup\\\ \bigtriangleup&\times&\times\\\ \bigtriangleup&\times&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\bigtriangleup&\times\\\ \bigtriangleup&\times&\bigtriangleup\\\ \times&\bigtriangleup&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\times&\bigtriangleup\\\ \times&\times&\bigtriangleup\\\ \bigtriangleup&\bigtriangleup&\times\end{array}\right)$

##  2 Formalism and important relations

Assuming neutrinos the Majorana particles as proposed in various seesaw
models[32], the neutrino mass matrix $M_{\nu}$ is a symmetric and generally
complex matrix with six independent entries. We arrive at $C_{6}^{2}=15$
logically possible patterns to place an equality between two matrix elements,
which are shown in Table.I.

In following analysis, we conider the Pontecorvo-Maki-Nakagawa-Sakata matrix
$U_{\text{PMNS}}$[33] parameterized as

$U_{\text{PMNS}}=UP_{\nu}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}\\\
-c_{12}s_{23}s_{13}-s_{12}c_{23}e^{-i\delta}&-s_{12}s_{23}s_{13}+c_{12}c_{23}e^{-i\delta}&c_{13}s_{23}\\\
-c_{12}c_{23}s_{13}+s_{12}s_{23}e^{-i\delta}&-c_{23}s_{12}s_{13}-c_{12}s_{23}e^{i\delta}&c_{13}c_{23}\end{array}\right)\left(\begin{array}[]{ccc}e^{2i\rho}&0&0\\\
0&e^{2i\rho}&0\\\ 0&0&1\end{array}\right)$ (3)

where the abbreviations $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$
are used. The ($\rho$,$\sigma$) in $P_{\nu}$ denotes two Majorana CP-violating
phases and $\delta$ stands for Dirac CP-violating phase.

In the flavor basis where $M_{l}$ is diagonal, the Majorana neutrino mass
$M_{\nu}$ is related to the diagonal mass matrix
$M^{D}=$diag$(m_{1},m_{2},m_{3})$ though the unitary transformation

$M_{\nu}=U_{\text{PMNS}}M^{D}U_{\text{PMNS}}^{T}$ (4)

and can be re-expressed as

$M_{\nu}=U\left(\begin{array}[]{ccc}\lambda_{1}&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}\end{array}\right)U^{T}$ (5)

where
$\lambda_{1}=m_{1}e^{2i\rho},\lambda_{2}=m_{2}e^{2i\sigma},\lambda_{3}=m_{3}$.
In this parametrization, the mass matrix elements are given by

$\begin{split}(M_{\nu})_{11}=&m_{1}c_{12}^{2}c_{13}^{2}e^{2i\rho}+m_{2}s_{12}^{2}c_{13}^{2}e^{2i\sigma}+m_{3}s_{13}^{2}\\\
(M_{\nu})_{12}=&m_{1}(-c_{12}^{2}s_{23}c_{13}s_{13}e^{2i\rho}-c_{12}s_{12}c_{23}c_{13}e^{i(2\rho-\delta)})+m_{2}(-s_{12}^{2}s_{23}c_{13}s_{13}e^{2i\sigma}+c_{12}s_{12}c_{23}c_{13}e^{i(2\sigma-\delta)})\\\
&+m_{3}s_{23}c_{13}s_{13}\\\
(M_{\nu})_{13}=&m_{1}(-c_{12}^{2}c_{23}c_{13}s_{13}e^{2i\rho}+c_{12}s_{12}s_{23}c_{13}e^{i(2\rho-\delta)})+m_{2}(-s_{12}^{2}c_{23}c_{13}s_{13}e^{2i\sigma}+c_{12}s_{12}s_{23}c_{13}e^{i(2\sigma-\delta)})\\\
&+m_{3}c_{23}c_{13}s_{13}\\\
(M_{\nu})_{22}=&m_{1}(c_{12}s_{23}s_{13}e^{2i\rho}+c_{23}s_{12}e^{i(\rho-\delta)})^{2}+m_{2}(s_{12}s_{23}s_{13}e^{2i\sigma}-c_{23}c_{12}e^{i(\sigma-\delta)})^{2}+m_{3}c_{13}^{2}s_{23}^{2}\\\
(M_{\nu})_{23}=&m_{1}(c_{12}^{2}c_{23}s_{23}s_{13}^{2}e^{2i\rho}+c_{12}s_{12}s_{13}(c_{23}^{2}-s_{23}^{2})e^{i(2\rho-\delta)}-c_{23}s_{23}s_{12}^{2}e^{2i(\rho-\delta)})\\\
&+m_{2}(s_{12}^{2}c_{23}s_{23}s_{13}^{2}e^{2i\sigma}+c_{12}s_{12}s_{13}(s_{23}^{2}-c_{23}^{2})e^{i(2\sigma-\delta)}-c_{23}s_{23}c_{12}^{2}e^{2i(\sigma-\delta)})+m_{3}c_{23}s_{23}c_{13}^{2}\\\
(M_{\nu})_{33}=&m_{1}(c_{12}c_{23}s_{13}e^{2i\rho}-s_{23}s_{12}e^{i(\rho-\delta)})^{2}+m_{2}(s_{12}c_{23}s_{13}e^{2i\sigma}+s_{23}c_{12}e^{i(\sigma-\delta)})^{2}+m_{3}c_{13}^{2}c_{23}^{2}\end{split}$
(6)

Table 2: The latest global-fit results of neutrino oscillation parameters given by Ref.[34] , with $\delta m^{2}=m_{2}^{2}-m_{1}^{2}$, $\Delta m^{2}=|m_{3}^{2}-\frac{1}{2}(m_{2}^{2}+m_{1}^{2}|)$.  Parameter | Order |  $1\sigma$ range |  $2\sigma$ range |  $3\sigma$ range
---|---|---|---|---
$\delta m^{2}/10^{-5}~{}\mathrm{eV}^{2}$ | NO, IO | 7.21 - 7.54 | 7.07 - 7.73 | 6.93 - 7.96
$\sin^{2}\theta_{12}/10^{-1}$ | NO, IO | 2.81 - 3.14 | 2.65 - 3.34 | 2.50 - 3.54
$|\Delta m^{2}|/10^{-3}~{}\mathrm{eV}^{2}$ | NO | 2.495 - 2.567 | 2.454 - 2.606 | 2.411 - 2.646
| IO | 2.473 - 2.539 | 2.430 - 2.582 | 2.390 - 2.624
$\sin^{2}\theta_{13}/10^{-2}$ | NO | 2.08 - 2.22 | 1.99 - 2.31 | 1.90 - 2.40
| IO | 2.07 - 2.24 | 1.98 - 2.33 | 1.90 - 2.42
$\sin^{2}\theta_{23}/10^{-1}$ | NO | 4.10 - 4.46 | 3.95 - 4.70 | 3.81 - 6.15
| IO | 4.17 - 4.48 $\oplus$ 5.67 - 6.05 | 3.99 - 4.83 $\oplus$ 5.33 - 6.21 | 3.84 - 6.36
$\delta/\pi$ | NO | 1.18 - 1.61 | 1.00 - 1.90 | 0 - 0.17 $\oplus$ 0.76 - 2
| IO | 1.12 - 1.62 | 0.92 - 1.88 | 0 - 0.15 $\oplus$ 0.69 - 2

In terms of the texture equality condition [e.g
$(M_{\nu})_{ab}=M_{\nu})_{cd}$], we obtain the constraint condition equation

$\sum_{n=1}^{3}(U_{ai}U_{bi}-U_{ci}U_{di})\lambda_{i}=0$ (7)

which, if the lightest neutrino is massless, leads to

$\begin{split}&\xi\equiv\frac{m_{2}}{m_{3}}=\left|\frac{U_{a3}U_{b3}-U_{c3}U_{d3}}{U_{a2}U_{b2}-U_{c2}U_{d2}}\right|\\\
&\sigma=-\frac{1}{2}arg\left(\frac{U_{a3}U_{b3}-U_{c3}U_{d3}}{U_{a2}U_{b2}-U_{c2}U_{d2}}\right)\end{split}$
(8)

for normal order (NO) spectrum. The three neutrino masses are given by

$m_{1}=0,\quad m_{2}=\sqrt{\delta m^{2}},\quad m_{3}=\frac{m_{2}}{\xi}$ (9)

It is clear that, the Majorana CP-violating phase $\rho$ becomes unphysical,
since $m_{1}$ is zero, and can be dropped out in numerical calculation.

For invert order (IO) spectrum, we obtain

$\begin{split}&\zeta\equiv\frac{m_{2}}{m_{1}}=\left|\frac{U_{a1}U_{b1}-U_{c1}U_{d1}}{U_{a2}U_{b2}-U_{c2}U_{d2}}\right|\\\
&\rho-\sigma=-\frac{1}{2}arg\left(\frac{U_{a1}U_{b1}-U_{c1}U_{d1}}{U_{a2}U_{b2}-U_{c2}U_{d2}}\right)\end{split}$
(10)

and three neutrino masses given by

$m_{3}=0,\quad m_{1}=\Delta m^{2}-\frac{\delta m^{2}}{2},\quad m_{2}=\zeta
m_{2}$ (11)

Note that only the difference $(\rho-\sigma)$ is physical in this case. Hence
the neutrino mass ratios and Majorana CP-violating phases are fully determined
in terms of the mixing angles $(\theta_{12},\theta_{23},\theta_{13})$ and the
Dirac-CP violating phase $\delta$. The three mixing angles
$(\theta_{12},\theta_{23},\theta_{13})$ as well as two independent neutrino
mass-squared differences $\delta m^{2}=m_{2}^{2}-m_{1}^{2}$, $\Delta
m^{2}=|m_{3}^{2}-\frac{1}{2}(m_{2}^{2}+m_{1}^{2})|$ are precisely measured by
many neutrino oscillation experiments. We summarize the latest global-fit
results[34] of neutrino oscillation parameters in Table.II. One can further
define the ratio of neutrino mass-squared difference as

$R_{\nu}=\frac{\delta m^{2}}{\Delta m^{2}}$ (12)

which, using Eq.(8) and (10), can be expressed as

$R_{\nu}=\left(\left|\frac{U_{a2}U_{b2}-U_{c2}U_{d2}}{U_{a3}U_{b3}-U_{c3}U_{d3}}\right|^{2}-1\right)^{-1}$
(13)

in case of normal order and

$R_{\nu}=1-\left|\frac{U_{a2}U_{b2}-U_{c2}U_{d2}}{U_{a1}U_{b1}-U_{c1}U_{d1}}\right|^{2}$
(14)

in case of inverted order.

The Dirac CP-violation in neutrino oscillation experiments can be described by
the Jarlskog rephasing invariant quantity

$J_{CP}=s_{12}s_{23}s_{13}c_{12}c_{23}c_{13}^{2}sin\delta$ (15)

On the other hand, the Majorana CP violation can be established if any signal
of neutrinoless double beta ($0\nu\beta\beta$) decay is observed. The rate of
$0\nu\beta\beta$ decay is determined by the effective Majorana neutrino mass
$m_{ee}$

$m_{ee}=\left|m_{1}c_{12}^{2}c_{13}^{2}e^{2i\rho}+m_{2}s_{12}^{2}c_{13}^{2}e^{2i\sigma}+m_{3}s_{13}^{2}\right|$
(16)

The next generation $0\nu\beta\beta$ experiments, with the aimed sensitivity
of $m_{ee}$ being up to 0.01 eV[35], will open the window to weigh both
neutrino masses and lepton number violation. Besides the $0\nu\beta\beta$
experiments, more severe constraints on the absolute scale of neutrino masses
is set from cosmology observation.

##  3 Numerical analysis

###  3.1 A benchmark point for P1 pattern

First of all, we point out that the requirement of one vanishing neutrino mass
is equivalent to the zero determinant of neutrino mass matrix Det$(M_{\nu})=0$
because

$\begin{split}\text{Det}(M_{\nu})&=\text{Det}(U_{\text{PMNS}}M^{D}U_{\text{PMNS}}^{T})=\text{Det}(U_{\text{PMNS}}^{T}U_{\text{PMNS}}M^{D})\\\
&=\text{Det}(U_{\text{PMNS}}^{T})\text{Det}(U_{\text{PMNS}})\text{Det}(M^{D})=0\end{split}$
(17)

In Ref.[36], the neutrino mass matrices with one texture equality and
Det$(M_{\nu})=0$ has been discussed. The authors claimed that such texture
structures are not allowed for $\theta_{13}\neq 0$. However, our analysis
demonstrates a different result. Here we take the P1 pattern as an
illustration.

For NO spectrum of neutrino masses, we set the benchmark points lying in
$3\sigma$ range of experimental data as

$\begin{split}(\theta_{12},\theta_{23},\theta_{13})=(32.2386^{\circ},40.9554^{\circ},8.4901^{\circ})\\\
(\delta,\sigma)=(83.7040^{\circ},-25.6372^{\circ})\\\ (\delta m^{2},\Delta
m^{2})=(7.4156\times 10^{-5}\text{eV}^{2},2.500\times
10^{-3}\text{eV}^{2})\end{split}$ (18)

then the corresponding neutrino mass matrix is

$M_{\nu}=\left(\begin{array}[]{ccc}0.0026-0.0019i&0.0026-0.0019i&0.0071+0.0020i\\\
0.0026-0.0019i&0.0187+0.0026i&0.0267-0.0019i\\\
0.0071+0.0020i&0.0267-0.0019i&0.0255+0.0012i\end{array}\right)$ (19)

with three neutrino masses given by

$m_{1}=0,\quad m_{2}=0.0086,\quad m_{3}=0.0500$ (20)

For IO spectrum of neutrino masses, if benchmark points are taken as following

$\begin{split}(\theta_{12},\theta_{23},\theta_{13})=(32.8056^{\circ},40.3032^{\circ},8.3978^{\circ})\\\
(\delta,\rho,\sigma)=(129.1235^{\circ},99.0824^{\circ},162.6441^{\circ})\\\
(\delta m^{2},\Delta m^{2})=(7.1394\times 10^{-5}\text{eV}^{2},2.500\times
10^{-3}\text{eV}^{2})\end{split}$ (21)

we can obtain the neutrino mass matrix

$M_{\nu}=\left(\begin{array}[]{ccc}-0.0209-0.0191i&-0.0209-0.0191i&0.0218+0.0199i\\\
-0.0209-0.0191i&0.0166+0.0156i&-0.0100-0.0096i\\\
0.0218+0.0199i&-0.0100-0.0096i&0.0043+0.0043i\end{array}\right)$ (22)

with three corresponding neutrino masses

$m_{1}=0.0501,\quad m_{2}=0.0508,\quad m_{3}=0$ (23)

It is clearly that, by direct calculation, the P1 pattern of neutrino mass
matrix with $M_{\nu 11}=M_{\nu 12}$ and a vanishing neutrino mass ($m_{1}$ or
$m_{3}=0$) is phenomenologically allowed if appropriate oscillation parameters
are chosen.

Table 3: The various predictions for $P1-P15$ patterns. The symbol $\times$ denotes that the corresponding pattern is not phenomenologically allowed under current experimental data. Textures | Specturm | $\theta_{23}$ | $\delta$ | Majorana phases |  $m_{ee}$(eV)
---|---|---|---|---|---
$P1$ | NO |  $38.22^{\circ}$ \- $52.31^{\circ}$ |  $78.8^{\circ}$ \- $300.2^{\circ}$ | ($-0.507^{\circ}$) - $0.502^{\circ}$ | 0.0028 - 0.0040
| IO |  $38.36^{\circ}$ \- $52.84^{\circ}$ |  $106.9^{\circ}$ \- $245.1^{\circ}$ | ($-89.6^{\circ}$) - ($-59.1^{\circ}$) $\oplus$ $58.0^{\circ}$ \- $89.4^{\circ}$ | 0.0226 - 0.0291
$P2$ | NO |  $37.85^{\circ}$ \- $52.84^{\circ}$ |  $0^{\circ}$ \- $113.7^{\circ}$ $\oplus$ $252.6^{\circ}$ \- $360^{\circ}$ | ($-0.513^{\circ}$) - ($-0.015^{\circ}$) $\oplus$ $0.002^{\circ}$ \- $0.503^{\circ}$ | 0.0028 - 0.0041
| IO |  $38.41^{\circ}$ \- $52.77^{\circ}$ |  $0^{\circ}$ \- $68.3^{\circ}$ $\oplus$ $300.1^{\circ}$ \- $360^{\circ}$ | ($-91.3^{\circ}$) - ($-53.2^{\circ}$) $\oplus$ $57.6^{\circ}$ \- $93.3^{\circ}$ | 0.0233 - 0.0297
$P3$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO |  $38.5^{\circ}$ \- $53.7^{\circ}$ |  $71.8^{\circ}$ \- $93.3^{\circ}$ $\oplus$ $271.4^{\circ}$ \- $291.0^{\circ}$ | ($-33.6^{\circ}$) - ($-18.7^{\circ}$) $\oplus$ $18.0^{\circ}$ \- $35.8^{\circ}$ | 0.0417 - 0.0475
$P5$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO |  $38.2^{\circ}$ \- $53.5^{\circ}$ |  $97.6^{\circ}$ \- $104.7^{\circ}$ $\oplus$ $254.2^{\circ}$ \- $268.8^{\circ}$ | ($-37.2^{\circ}$) - ($-19.1^{\circ}$) $\oplus$ $18.3^{\circ}$ \- $36.4^{\circ}$ | 0.0416 - 0.0472
$P4$ | NO |  $49.7^{\circ}$ \- $52.2^{\circ}$ |  $0^{\circ}$ \- $360^{\circ}$ | ($-1.534^{\circ}$) - $1.567^{\circ}$ | 0.0012 - 0.0040
| IO | $\times$ | $\times$ | $\times$ | $\times$
$P6$ | NO |  $38.4^{\circ}$ \- $40.2^{\circ}$ |  $0^{\circ}$ \- $360^{\circ}$ | ($-1.566^{\circ}$) - $1.565^{\circ}$ | 0.0011 - 0.0040
| IO | $\times$ | $\times$ | $\times$ | $\times$
$P7$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO | $\times$ | $\times$ | $\times$ | $\times$
$P8$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO |  $37.3^{\circ}$ \- $52.9^{\circ}$ |  $99.6^{\circ}$ \- $124.7^{\circ}$ $\oplus$ $227.6^{\circ}$ \- $265.3^{\circ}$ | ($-58.2^{\circ}$) - ($-23.0^{\circ}$) $\oplus$ $20.2^{\circ}$ \- $57.6^{\circ}$ | 0.0335 - 0.0467
$P9$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO |  $37.1^{\circ}$ \- $53.4^{\circ}$ |  $68.1^{\circ}$ \- $94.2^{\circ}$ $\oplus$ $277.0^{\circ}$ \- $308.1^{\circ}$ | ($-60.0^{\circ}$) - ($-21.8^{\circ}$) $\oplus$ $23.1^{\circ}$ \- $59.7^{\circ}$ | 0.0321 - 0.0459
$P10$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO | $\times$ | $\times$ | $\times$ | $\times$
$P11$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO | $\times$ | $\times$ | $\times$ | $\times$
$P12$ | NO |  $44.3^{\circ}$ \- $44.5^{\circ}$ $\oplus$ $45.6^{\circ}$ \- $52.8^{\circ}$ |  $0^{\circ}$ \- $360^{\circ}$ | ($-1.555^{\circ}$) - $1.546^{\circ}$ | 0.0011 - 0.0041
| IO |  $38.1^{\circ}$ \- $52.7^{\circ}$ |  $0^{\circ}$ \- $90.1^{\circ}$ $\oplus$ $99.6^{\circ}$ \- $269.9^{\circ}$ $\oplus$ $274.8^{\circ}$ \- $360^{\circ}$ | ($-52.5^{\circ}$) - $53.0^{\circ}$ | 0.0340 - 0.0500
$P13$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO |  $37.3^{\circ}$ \- $43.5^{\circ}$ $\oplus$ $47.8^{\circ}$ \- $53.4^{\circ}$ |  $0^{\circ}$ \- $83.6^{\circ}$ $\oplus$ $103.7^{\circ}$ \- $254.9^{\circ}$ $\oplus$ $289.3^{\circ}$ \- $360^{\circ}$ | ($-1.26^{\circ}$) - $1.26^{\circ}$ | 0.0476 - 0.0501
$P14$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO |  $38.4^{\circ}$ \- $53.7^{\circ}$ |  $103.5^{\circ}$ \- $128.2^{\circ}$ $\oplus$ $226.3^{\circ}$ \- $260.1^{\circ}$ | ($-58.3^{\circ}$) - ($-39.0^{\circ}$) $\oplus$ $37.9^{\circ}$ \- $59.3^{\circ}$ | 0.0318 - 0.0404
$P15$ | NO | $\times$ | $\times$ | $\times$ | $\times$
| IO |  $38.3^{\circ}$ \- $53.6^{\circ}$ |  $58.2^{\circ}$ \- $77.8^{\circ}$ $\oplus$ $288.0^{\circ}$ \- $309.5^{\circ}$ | ($-58.1^{\circ}$) - ($-37.9^{\circ}$) $\oplus$ $36.2^{\circ}$ \- $57.9^{\circ}$ | 0.0327 - 0.0409

###  3.2 Numerical results and discussion

We have performed a numerical analysis of all fifteen texture structures shown
in Table. I. For each pattern of $M_{\nu}$, the a set of random number inputs
are generated for the three mixing angles
$(\theta_{12},\theta_{23},\theta_{13})$ and the neutrino mass square
differences ($\delta m^{2},\Delta m^{2}$) in their $3\sigma$ range (Table.
II). Instead, we generate a random input of Dirac CP-violating phase $\delta$
in the range of $[0,2\pi)$. From Eqs.(13) and (14), $R_{\nu}$ is determined by
both ($\theta_{12},\theta_{23},\theta_{13},\delta$). This requires the input
scattering point of $(\theta_{12},\theta_{23},\theta_{13},\delta)$ is
empirically acceptable only when $R_{\nu}$ falls inside the $3\sigma$ range
$[\delta m^{2}_{min}/\Delta m^{2}_{max},\delta m^{2}_{max}/\Delta
m^{2}_{min}]$. From Eq.(9) and (11), we further get the three absolute scale
of neutrino masses $m_{1,2,3}$. Two Majorana CP-violating $\alpha$ and $\beta$
can be constrained by Eq. (13), (14) and the allowed range of $m_{ee}$ are
subsequently obtained. In the numerical analysis, we require the upper bound
on the sum of neutrino masses $\sum m_{i}<0.23$ eV being satisfied.

Before proceeding, one notes that the there exists a so-called $\mu-\tau$
permutation transformation that can relate one texture pattern to another
though

$\widetilde{M}_{\nu}=P_{23}M_{\nu}P_{23}$ (24)

where

$P_{23}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\ 0&1&0\end{array}\right)$
(25)

and the oscillation parameters between $M_{\nu}$ and $\widetilde{M}_{\nu}$
given by

$\widetilde{\theta}_{12}=\theta_{12},\quad\widetilde{\theta}_{13}=\theta_{13},\quad\widetilde{\theta}_{23}=\frac{\pi}{2}-\theta_{23},\quad\widetilde{\delta}=\pi-\delta$
(26)

One can also prove that two neutrino mass matrices related by $\mu-\tau$
permutation transformation share the same neutrino mass eigenvalues. It is
straightforward to verify that such a permutation symmetry exists between

$\begin{split}&P1\leftrightarrow P2\quad\quad P3\leftrightarrow P5\quad\quad
P4\leftrightarrow P6\quad\quad P7\leftrightarrow P7\quad\quad
P8\leftrightarrow P9\quad\quad P10\leftrightarrow P11\\\ &P12\leftrightarrow
P12\quad\quad P13\leftrightarrow P13\quad\quad P14\leftrightarrow
P15\end{split}$ (27)

Notice that the pattern P7, P12 and P13 transform into themselves under the
$\mu-\tau$ permutation. Thus among fifteen texture patterns we studied, only
nine of them is independent

We present the allowed range of oscillation parameters for each viable
patterns in Table.III and Fig.1-15. In the figures, the light blue bands
represent the 1$\sigma$ uncertainty in determination of
$\theta_{12},\theta_{23}$ and $\theta_{13}$ while they plus the beige bands
correspond to the 2$\sigma$ uncertainty. Some interesting observations are
summarized as follow

  * •

Among the fifteen possible patterns, the P7,P10 and P11 patterns are excluded
by the current experimental data at 3$\sigma$ confidence level. The P4
pattern, though allowed at $3\sigma$ level, is excluded at $2\sigma$ level.

  * •

The P1, P2 and P12 patterns are phenomenologically allowed for both NO and IO
spectrum of neutrino masses. The P4 and P6 patterns are phenomenologically
allowed only for NO spectrum of neutrino masses while P3, P5,P8,P9, P13, P14
and P15 patterns are phenomenologically allowed only for IO spectrum of
neutrino masses.

  * •

The P4 pattern predicts $\theta_{23}>45^{\circ}$ while for P6 we get
$\theta_{23}<45^{\circ}$. In particular, the allowed region of $\theta_{23}$
from P12 pattern is tightly located at around $45^{\circ}$. Taking the NO
spectrum as an example, we derive the analytical approximate formula up to the
second order of $\sin\theta_{13}$

$\xi=\frac{m_{2}}{m_{3}}\approx\frac{1}{\cos\theta_{12}}\big{(}1+\tan\theta_{12}\tan
2\theta_{23}\sin\theta_{13}-\frac{1}{4}\tan^{2}\theta_{12}\tan^{2}\theta_{23}\sin^{2}\theta_{13}\big{)}$
(28)

where, without loss of generality, we have set $\delta=0$. Clearly, the
smallness of mass ratio $\xi\approx\sqrt{\delta m^{2}/\Delta m^{2}}$ implies a
large cancellation in Eq.(28) which, as shown in Fig. 5, only happens when
$\theta_{23}$ approaches $\pi/4$ enough. Thus the two highly constrained
spaces for $\theta_{23}$ besides $45^{\circ}$ can be regarded as the two
solutions from Eq.(28). One can apply the similar analysis to IO spectrum of
neutrino masses, where in leading order of $\sin\theta_{13}$ we have

$\zeta=\frac{m_{2}}{m_{1}}\approx\tan^{2}\theta_{12}\Big{(}1+\frac{2\tan
2\theta_{23}\cos\delta\sin\theta_{13}}{\sin\theta_{12}\cos\theta_{12}}\Big{)}$
(29)

One can see that the value of $\theta_{23}$ should be located at around
$\pi/4$ in order to obtain the mass ratio $\zeta\approx 1$,

  * •

Although all the viable patterns admit a large Dirac CP violation, the P8, P9,
P10 P11, P14 and P15 patterns require a Dirac CP-violating phase $\delta$
highly located at around $90^{\circ}(270^{\circ})$, leading to a large Dirac
CP-violating effect $J_{CP}>0.02$. If the Dirac CP-violation is finally
observed to be very small, these patterned can be excluded. We mention that
the latest analysis gives a best-fit value of $1.38\pi$ for NO spectrum and
$1.31\pi$ for IO spectrum[34], which strengthens the trend in favor of
$\delta\sim 3\pi/2$.

  * •

The viable textures with NO spectrum of neutrino masses predict tiny $m_{ee}$
(at order of $0.001$eV), rendering it very challenging to be detected in
future $0\nu\nu\beta$ decay experiments. This is due to the fact that, in the
scenario of $m_{1}=0,m_{2}=\sqrt{\delta m^{2}}\ll m_{3}$, one easily obtains
$m_{ee}\approx\sqrt{\Delta m^{2}}s_{13}^{2}$ from Eq.(16), with the value of
$m_{ee}$ largely suppressed by small $s_{13}^{2}$ . For IO spectrum of
neutrino masses, on the other hand, each viable texture can predict $m_{ee}$
in the order of $0.01$eV, which is promising to be detected in the forthcoming
experiments.

Figure 1: The correlation plots for NO P1 pattern.  Figure 2: The correlation
plots for NO P2 pattern.  Figure 3: The correlation plots for NO P4 pattern.
Figure 4: The correlation plots for NO P6 pattern. Figure 5: The correlation
plots for NO P12 pattern. Figure 6: The correlation plots for IO P1 pattern.
Figure 7: The correlation plots for IO P2 pattern.  Figure 8: The correlation
plots for IO P3 pattern. Figure 9: The correlation plots for IO P5 pattern.
Figure 10: The correlation plots for IO P8 pattern.  Figure 11: The
correlation plots for IO P9 pattern. Figure 12: The correlation plots for IO
P12 pattern.  Figure 13: The correlation plots for IO P13 pattern  Figure 14:
The correlation plots for IO P14 pattern.  Figure 15: The correlation plots
for IO P15 pattern.

##  4 Theoretical realization for a concrete model

In this section, we present a detailed illustration on how flavor symmetry
gives rise to the desired texture structure. The neutrino mass matrices with
texture equalities have been realized by using a non-Abelian flavor symmetry
e.g $S_{3}$[9] or $A_{4}$[15]. On the other hand, one notices that as good
approximation of vanishing neutrino mass, a ultralight neutrino masse implies
a large mass hierarchy between three generations of neutrinos. To realize
hierarchies between neutrino masses, a popular approach is the Froggatt-
Nielsen mechanism[30] that was originally developed for quark sector while is
also suitable to understand the lepton mass and mixing pattern. In the
following, we just take $P12$ pattern as a concrete example, although a
complete realization for other patterns is also interesting and necessary.

We extend the particle content of Standard Model(SM) with three right-handed
neutrinos $\nu_{Ri}(i=1,2,3)$, three SM-like doublet scalars $H_{i}(i=1,2,3)$
responsible for charged lepton masses generation, four additional doublet
scalars $\phi_{i}(i=1,2,3,4)$ responsible for Dirac neutrino mass matrix
$M_{D}$, two singlet scalars $\sigma_{i}(i=1,2)$ giving rise to the right-
handed neutrino mass matrix $M_{R}$. Then the neutrino mass matrix is obtained
by the canonical type-I seesaw formula $M_{\nu}=M_{D}M_{R}^{-1}M_{D}$. The
general Lagrangian that producing to lepton masses is then given by

$\mathcal{L}=\Big{(}\frac{\left\langle\Phi\right\rangle}{\Lambda}\Big{)}^{Q_{L_{i}}+Q_{l_{Rj}}}y_{ij}^{(k)}\bar{L}_{i}H_{k}l_{Rj}+\Big{(}\frac{\left\langle\Phi\right\rangle}{\Lambda}\Big{)}^{Q_{L_{i}}+Q_{\nu_{Rj}}}h_{ij}^{(k)}\bar{L}_{i}\widetilde{\phi}_{k}^{\prime}\nu_{Rj}+\Big{(}\frac{\left\langle\Phi\right\rangle}{\Lambda}\Big{)}^{Q_{\nu_{Ri}}+Q_{\nu_{Rj}}}g_{ij}^{(k)}\sigma_{k}\nu_{Ri}\nu_{Rj}+h.c$
(30)

where $\widetilde{\phi}_{k}^{\prime}\equiv i\sigma_{2}\phi_{k}^{\prime}$ and
we denote $\epsilon\equiv\left\langle\Phi\right\rangle/\Lambda$. The
$Q_{\alpha}$($\alpha=L,l_{R},\nu_{R}$) are interpreted as the FN charges for
SM fermion ingredients under which different generations may be charged
differently. The flavon $\Phi$ obtains the vaccum (VEV)
$\left\langle\Phi\right\rangle$ that breaks the FN symmetry. We assign the FN
charges for lepton sector as

$\begin{split}\bar{L}_{1,2,3}:(a+1,a,a)\\\ l_{R1,2,3}:(0,1,2)\\\
\nu_{R1,2,3}:(d,c,b)\end{split}$ (31)

The realistic mass hierarchy between three charged leptons are successfully
reproduced by adopting the assignment of FN charges for $L_{i}$ and $l_{RI}$
with $\epsilon\simeq 0.05-0.1$[37]. The explicit forms of Dirac neutrino mass
matrix $M_{D}$ and right-handed neutrino mass matrix $M_{R}$ are

$\begin{split}M_{D}=\sum_{k=1}^{3}M_{D}^{(k)}&=\sum_{k=1}^{3}v_{k}\left(\begin{array}[]{ccc}\epsilon^{a+1}&0&0\\\
0&\epsilon^{a}&0\\\
0&0&\epsilon^{a}\end{array}\right)\left(\begin{array}[]{ccc}h_{11}^{(k)}&h_{12}^{(k)}&h_{13}^{(k)}\\\
h_{21}^{(k)}&h_{22}^{(k)}&h_{23}^{(k)}\\\
h_{31}^{(k)}&h_{32}^{(k)}&h_{33}^{(k)}\end{array}\right)\left(\begin{array}[]{ccc}\epsilon^{d}&0&0\\\
0&\epsilon^{c}&0\\\ 0&0&\epsilon^{b}\end{array}\right)\\\
&=\sum_{k=1}^{3}v_{k}\left(\begin{array}[]{ccc}h_{11}^{(k)}\epsilon^{d+a+1}&h_{12}^{(k)}\epsilon^{c+a+1}&h_{13}^{(k)}\epsilon^{b+a+1}\\\
h_{21}^{(k)}\epsilon^{d+a}&h_{22}^{(k)}\epsilon^{c+a}&h_{23}^{(k)}\epsilon^{b+a}\\\
h_{31}^{(k)}\epsilon^{d+a}&h_{32}^{(k)}\epsilon^{c+a}&h_{33}^{(k)}\epsilon^{b+a}\end{array}\right)\end{split}$
(32)

and

$\begin{split}M_{R}=\sum_{k=1}^{2}M_{R}^{(k)}=&\sum_{k=1}^{2}v_{k}^{\prime}\left(\begin{array}[]{ccc}\epsilon^{d}&0&0\\\
0&\epsilon^{c}&0\\\
0&0&\epsilon^{b}\end{array}\right)\left(\begin{array}[]{ccc}g_{11}^{(k)}&g_{12}^{(k)}&g_{13}^{(k)}\\\
g_{21}^{(k)}&g_{22}^{(k)}&g_{23}^{(k)}\\\
g_{31}^{(k)}&g_{32}^{(k)}&g_{33}^{(k)}\end{array}\right)\left(\begin{array}[]{ccc}\epsilon^{d}&0&0\\\
0&\epsilon^{c}&0\\\ 0&0&\epsilon^{b}\end{array}\right)\\\
&=\sum_{k=1}^{2}v_{k}^{\prime}\left(\begin{array}[]{ccc}g_{11}^{(k)}\epsilon^{2d}&g_{12}^{(k)}\epsilon^{d+c}&g_{13}^{(k)}\epsilon^{d+b}\\\
g_{21}^{(k)}\epsilon^{d+c}&g_{22}^{(k)}\epsilon^{2c}&g_{23}^{(k)}\epsilon^{c+b}\\\
g_{31}^{(k)}\epsilon^{d+b}&g_{32}^{(k)}\epsilon^{c+b}&g_{33}^{(k)}\epsilon^{2b}\end{array}\right)\end{split}$
(33)

Following the spirit of Ref.[17], we consider a cooperation of $\mu-\tau$
permutation symmetry and $Z_{8}$ horizontal symmetry i.e $S_{\mu-\tau}\times
Z_{8}$ under which the relevant particle fields transform as

$\begin{split}&(L_{e},L_{\mu},L_{\tau})\xrightarrow{S}(L_{e},L_{\mu},L_{\tau}),\quad(e_{R},\mu_{R},\tau_{R})\xrightarrow{S}(e_{R},\tau_{R},\mu_{R}),\quad(H_{1},H_{2},H_{3})\xrightarrow{S}(H_{1},H_{3},H_{2})\\\
&(\nu_{R1},\nu_{R2},\nu_{R3})\xrightarrow{S}(\nu_{R1},\nu_{R3},\nu_{R2}),\quad(\phi_{1},\phi_{2},\phi_{3},\phi_{4})\xrightarrow{S}(\phi_{1},\phi_{2},\phi_{3},-\phi_{4}),\quad(\sigma_{1},\sigma_{2})\xrightarrow{S}(\sigma_{1},\sigma_{2})\\\
&(L_{e},L_{\mu},L_{\tau})\xrightarrow{Z_{8}}(L_{e},-L_{\mu},-L_{\tau}),\quad(e_{R},\mu_{R},\tau_{R})\xrightarrow{Z_{8}}(e_{R},\tau_{R},-\mu_{R}),\quad(H_{1},H_{2},H_{3})\xrightarrow{Z_{8}}(H_{1},-H_{2},-H_{3})\\\
&(\nu_{R1},\nu_{R2},\nu_{R3})\xrightarrow{Z_{8}}(\omega\nu_{R1},\omega^{3}\nu_{R3},\omega^{3}\nu_{R2}),\quad(\phi_{1},\phi_{2},\phi_{3},\phi_{4})\xrightarrow{Z_{8}}(\omega\phi_{1},\omega^{3}\phi_{2},\omega^{7}\phi_{3},\omega^{3}\phi_{4})\\\
&(\sigma_{1},\sigma_{2})\xrightarrow{Z_{8}}(\omega^{6}\sigma_{1},\omega^{2}\sigma_{2})\end{split}$
(34)

As proposed in Ref.[17], one can further assume a hierarchy in $H_{i}$’s
vacuums ($\left\langle H_{3}\right\rangle\gg\left\langle
H_{1}\right\rangle,\left\langle H_{2}\right\rangle$). Then the
$S_{\mu-\tau}\times Z_{8}$-invariant Lagrangian relevant to the charged lepton
sector leads to the texture structure of $M_{l}M^{\dagger}_{l}$ given by

$M_{l}M^{\dagger}_{l}=\left(\begin{array}[]{ccc}A^{2}&0&0\\\ 0&D^{2}&DC\\\
0&DC&C^{2}\end{array}\right)$ (35)

with

$\frac{B}{C}\sim\frac{m_{e}}{m_{\mu}}=2.8\times
10^{-4}\quad\frac{D}{C}\sim\frac{m_{\mu}}{m_{\tau}}=5.9\times 10^{-2}$ (36)

As required, we construct the model in the flavor basis corrected by an
extremely small rotation angle less than $10^{-2}$

Now move to the neutrino sector. The $S_{\mu-\tau}\times Z_{8}$ symmetry
imposes $b=c$ and the form of the neutrino mass matrices as follows:

$\begin{split}&M_{D}^{(1)}=\left(\begin{array}[]{ccc}A_{1}\epsilon^{d+a+1}&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right)\quad
M_{D}^{(2)}=\left(\begin{array}[]{ccc}0&B_{2}\epsilon^{c+a+1}&-B_{2}\epsilon^{c+a+1}\\\
0&0&0\\\ 0&0&0\end{array}\right)\\\
&M_{D}^{(3)}=\left(\begin{array}[]{ccc}0&0&0\\\
0&C_{3}\epsilon^{c+a}&D_{3}\epsilon^{c+a}\\\
0&D_{3}\epsilon^{c+a}&C_{3}\epsilon^{c+a}\end{array}\right)\quad
M_{D}^{(4)}=\left(\begin{array}[]{ccc}0&B_{4}\epsilon^{c+a+1}&B_{4}\epsilon^{c+a+1}\\\
0&0&0\\\ 0&0&0\end{array}\right)\end{split}$ (37)

where

$A_{1}\equiv v_{1}h_{11}^{(1)},\quad B_{2}\equiv
v_{2}h_{12}^{(2)}=-v_{2}h_{13}^{(2)}\quad C_{3}\equiv
v_{3}h_{22}^{(3)}=v_{3}h_{33}^{(3)}\quad D_{4}\equiv
v_{4}h_{23}^{(3)}=v_{4}h_{32}^{(3)}$ (38)

and

$M_{R}^{(1)}=\left(\begin{array}[]{ccc}A_{R}\epsilon^{2d}&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)\quad M_{R}^{(2)}=\left(\begin{array}[]{ccc}0&0&0\\\
0&C_{R}\epsilon^{2c}&D_{R}\epsilon^{2c}\\\
0&D_{R}\epsilon^{2c}&C_{R}\epsilon^{2c}\end{array}\right)$ (39)

where

$A_{R}=v_{1}^{\prime}g_{11}^{(1)},\quad
C_{R}=v_{2}^{\prime}g_{22}^{(2)}=v_{2}^{\prime}g_{33}^{(2)},\quad
D_{R}=v_{2}^{\prime}g_{23}^{(2)}=v_{2}^{\prime}g_{32}^{(2)}$ (40)

Hence the Dirac neutrino mass matrix $M_{D}$ and the right-handed neutrino
mass matrix $M_{R}$ are given by

$M_{D}=\left(\begin{array}[]{ccc}A_{D}\epsilon^{d+a+1}&B_{D}^{\prime}\epsilon^{c+a+1}&-B_{D}\epsilon^{c+a+1}\\\
0&C_{D}\epsilon^{c+a}&D_{D}\epsilon^{c+a}\\\
0&D_{D}\epsilon^{c+a}&C_{D}\epsilon^{c+a}\end{array}\right)\quad
M_{R}=\left(\begin{array}[]{ccc}A_{R}\epsilon^{2d}&0&0\\\
0&C_{R}\epsilon^{2c}&D_{R}\epsilon^{2c}\\\
0&D_{R}\epsilon^{2c}&C_{R}\epsilon^{2c}\end{array}\right)$ (41)

where we have defined

$\begin{split}A_{D}\equiv A_{1},\quad&B_{D}^{\prime}\equiv B_{2}+B_{4},\quad
B_{D}\equiv-B_{2}+B_{4},\quad C_{D}\equiv C_{3},\quad D_{D}\equiv
D_{3}\end{split}$ (42)

With the help of Eq.(41) and seesaw formula
$M_{\nu}=M_{D}M_{R}^{-1}M_{D}^{T}$, the effective neutrino mass matrix
$M_{\nu}$ is obtained by a direct calculation, leading to the exact form of
$P_{12}$ pattern i.e

$M_{\nu}=\left(\begin{array}[]{ccc}M_{\nu 11}&M_{\nu 12}&M_{\nu 13}\\\ M_{\nu
12}&M_{\nu 22}&M_{\nu 23}\\\ M_{\nu 13}&M_{\nu 23}&M_{\nu
22}\end{array}\right)$ (43)

In order to get a ultralight neutrino mass, we adopt the methodology given in
Ref.[31] . The basic point is to suppose the broken FN symmetry is a discrete
$Z_{n}$ symmetry instead of continuous $U(1)_{\text{FN}}$. One further assumes
$n=2d$ and $c<d$. Then the right-handed neutrino mass matrix $M_{R}$ in
Eq.(41) is

$M_{R}=\left(\begin{array}[]{ccc}A_{R}&0&0\\\
0&C_{R}\epsilon^{2c}&D_{R}\epsilon^{2c}\\\
0&D_{R}\epsilon^{2c}&C_{R}\epsilon^{2c}\end{array}\right)$ (44)

We see that In $M_{R}$ the element $A_{R}$ is not suppressed by the power of
$\epsilon$ compared with $C_{R}$ and $D_{R}$, which indicates an extremely
large right-handed neutrino masses for $N_{1}$ and hence yields an ultralight
neutrino mass via type-I seesaw mechanism. The texture hierarchy in the power
of $\epsilon$ is derived as

$M_{\nu}\sim\epsilon^{2a}\left(\begin{array}[]{ccc}\epsilon^{2}&\epsilon&\epsilon\\\
\epsilon&1&1\\\ \epsilon&1&1\end{array}\right)$ (45)

Note that the the FN charges $d$ and $c$ are completely canceled out when
applying seesaw formula and $\epsilon^{2a}$ appears as a total factor in the
form of $M_{\nu}$. As firstly indicated in Ref.[37], the structure of neutrino
mass matrices give in Eq.(45) can naturally leads to
$\theta_{23}\simeq\frac{\pi}{4}$, which also appears as a remarkable feature
of $P_{12}$ pattern (Fig.5).

##  5 Conclusion

In this paper, we have performed an systematic investigation on the neutrino
mass matrix $M_{\nu}$ with and one vanishing neutrino mass
($m_{\text{lightest}}=0$) and one texture equality($M_{\nu ab}=M_{\nu cd}$).
Although the zero neutrino mass are strictly set in numerical analysis, our
results is also well suited for the scenarios where an ultralight neutrino
mass is assumed in the consideration of both experimental and theoretical
sides. Using the latest neutrino oscillation and cosmological data, a
phenomenological analysis are systematically proposed for all possible
patterns. It is found that twelve out of fifteen possible patterns are
compatible with the experimental data at $3\sigma$ confidential level. In Fig.
1-15 and Table.3, we show the numerical results of viable patterns for normal
order and inverted order of neutrino masses. For each viable pattern, allowed
regions of neutrino oscillation parameters
$(\theta_{12},\theta_{23},\theta_{13},\delta)$, effective Majorana mass
$m_{ee}$, and Jarlskog quantity $J_{CP}$ are presented. We have summarized the
main results in the numerical part. These interesting predictions are
promising to be explored in the upcoming long-baseline neutrino oscillation
experiment, neutrinoless double beta decay experiments and further
cosmological study to the sum of neutrino masses.

Finally, we discussed the flavor symmetry realization of texture structures,
where a concrete example has been illustrated in Ref.[17] based on
$S_{\mu-\tau}\times Z_{8}$ symmetry. Inspired by this, we construct the model
in the framework of Froggatt-Nielsen (FN) mechanism and subsequently realize
the $P12$ pattern that naturally includes an ultralight neutrino mass. The
theoretical realization of other patterns deserves further study. Anyway, we
except that a cooperation between theoretical study from the flavor symmetry
viewpoint and a phenomenology study will help us reveal the underlying
structure of massive neutrinos.

##  6 Acknowledgments

The work of Weijian Wang is supported by National Natural Science Foundation
of China under Grant Numbers 11505062, Special Fund of Theoretical Physics
under Grant Numbers 11447117 and Fundamental Research Funds for the Central
Universities.

## References

  * [1] Q.R. Ahmad et al.(SNO Collaboration), Phys. Rev. Lett 89, 011301(2002); K. Eguchi et al. (KamLAND Collaboration), Phys. Rev. Lett 90, 021802(2003); M.H. Ahn et al. (K2K Collaboration), Phys. Rev. Lett 90, 041801(2003). F.P. An et al. (DAYA-BAY Xollaboration), Phys. Rev. Lett. 108, 171803(2012); J.K. Ahn et al. (RENO Collaboration), Phys. Rev. D108, 191802(2012).
  * [2] P.H. Frampton, S. L. Glashow, and D. Marfatia, Phys. Lett. B536, 79(2002); H. Fritzsch, Z.-z. Xing, and S. Zhou, J. High Energy Phys. 09 (2011)083.
  * [3] W. Grimus, A.S. Joshipura, L. Lavoura, M. Tanimoto, Eur. Phys. J. C36, 227(2004).
  * [4] Z,Z. Xing, Phys. Lett. B530, 159(2002); A. Merle, and W. Rodejohann, Phys. Rev. D73, 073012(2006); S. Dev, S. Kumar, S. Verma, and S. Gupta, Phys. Rev. D76, 013002(2007); S. Dev, S. Kumar, S. Verma, and S. Gupta, Nucl. Phys. B784, 103(2007);S. Dev, S. Kumar, Mod. Phys, Lett. A22, 1401(2007); S. Kumar, Phys. Rev. D84, 077301(2011); P.O. Ludl, S. Morisi, and E. Peinado, Nucl. Phys. B857, 411(2012); W. Grimus. and P.O. Ludl, arXiv:1208.4515; D. Meloni, and G. Blankenburg, Nucl. Phys. B867, 749(2013);
  * [5] G. C. Branco, D. Emmannuel-Costa, R. Gonz$\acute{a}$lez Felipe, and H. Ser$\hat{o}$dio, Phys. Lett. B670, 340(2009);
  * [6] M. Randhawa, G. Ahuja, and M. Gupta, Phys. Lett. B643, 175(2006); G. Ahuja, S. Kumar, M. Randhawa, M. Gupta, and S. Dev, Phys. Rev. D76, 013006(2007); G. Ahuja, M. Gupta, M. Randhawa, and R. Verma, Phys. Rev. D79, 093006(2009); J. A. Acosta, Alfredo Aranda, M. A. Buen-Abad, A. D. Rojas, Phys. Lett. B18, 1413(2013) S. Sharma, P. Fakay, G. Ahuja, M. Gupta, arXiv: 1402.0628; S. Sharma, P. Fakay, G. Ahuja, M. Gupta, J. Phys. G 40, 055003 (2013); S. Dev, L. Singh and D. Raj, Eur. Phys. J. C 75, no. 8, 394 (2015).
  * [7] M. Gupta, G. Ahuja; Int. Jour. Mod. Phys. A23, 1270033(2012);
  * [8] S. Kaneko, H. Sawanaka, and M. Tanimoto, J. High Energy Phys. 08 (2005)073; S. Dev, S. Verma, and S. Gupta, Phys. Lett. B687, 53(2010); S. Goswami, S. Khan, and A. Watanable, Phys. Lett. B687, 53(2010), W. Grimus, and P. O. Ludl, arXiv: 1208.4515.
  * [9] J.-Y. Liu and S. Zhou, Phys. Rev. D87, 093010(2013).
  * [10] X.-G. He and A. Zee, Phys. Rev. D68, 037302(2003).
  * [11] L. Lavoura, Phys. Lett. B609, 317(2005); E.I. Lashin and N. Chamoun, Phys. Rev. D78, 073002(2008); E.I. Lashin and N. Chamoun, Phys. Rev. D80, 093004(2009);
  * [12] S. Dev, S. Gupta, and R.R. Gautam, Mod. Phys, Lett. A26, 501(2011); S. Dev, S. Gupta, R.R. Gautam, and L. Singh, Phys. Lett. B706, 168(2011); T. Araki, J. Heeck, and J. Kubo, J. High Energy Phys. 07 (2012)083; S. Verma, Nucl. Phys. B854, 340(2012); S. Dev, R.R. Gautam, and L. Singh, Phys. Rev. D89, 013006(2014)S. Dev, Radha Raman Gautam, Lal Singh, Phys. Rev. D89, 013006(2014); W. Wang, Phys. Lett. B 733, 320 (2014); W. Wang, Phys. Rev. D 90, 033014 (2014); W. Wang, S. Y. Guo and Z. G. Wang,Mod. Phys. Lett. A 31, 1650080 (2016).
  * [13] S. Dev, S. Verma, S. Gupta, and R.R. Gautam, Phys. Rev. D81, 053010(2010); J. Liao,D. Marfatia, K. Whisnant, J. High Energy Phys. 09, 013(2014).
  * [14] H.A. Alhendi, E.I. Lashin, and A.A. Mudlej, Phys. Rev. D77, 013009(2008).
  * [15] S. Dev, R.R. Gautam, and L. Singh, Phys. Rev. D87, 073011(2013).
  * [16] W. Wang, Eur. Phys. J. C73, 2551(2013); S. Dev, R.R. Gautam, and L. Singh, Phys. Rev. D88, 033008(2013).
  * [17] E. I. Lashin, N. Chamoun, C. Hamzaoui and S. Nasri, Phys. Rev. D 89, no. 9, 093004 (2014)
  * [18] W. Wang and Z. L. Han, Phys. Rev. D 92, 095001 (2015)
  * [19] W. Wang, R. Wang, Z. L. Han and J. Z. Han, arXiv:1705.00414 [hep-ph].
  * [20] I. Affleck and M. Dine, Nucl. Phys. B 249, 361 (1985).
  * [21] H. Murayama and T. Yanagida, Phys. Lett. B 322, 349 (1994)
  * [22] M. Dine, L. Randall and S. D. Thomas, Nucl. Phys. B 458, 291 (1996)
  * [23] T. Asaka, M. Fujii, K. Hamaguchi and T. Yanagida, Phys. Rev. D 62, 123514 (2000)
  * [24] M. Fujii, K. Hamaguchi and T. Yanagida, Phys. Rev. D 64, 123526 (2001)
  * [25] S. Antusch, S. F. King, C. Luhn and M. Spinrath, Nucl. Phys. B 856, 328 (2012)
  * [26] W. Rodejohann, M. Tanimoto and A. Watanabe, Phys. Lett. B 710, 636 (2012)
  * [27] R. R. Gautam, M. Singh and M. Gupta, Phys. Rev. D 92, no. 1, 013006 (2015)
  * [28] P.A.R. Ade et al. (Planck Collaboration), arXiv: 1303.5076.
  * [29] S. Wang, Y. F. Wang, D. M. Xia and X. Zhang, Phys. Rev. D 94, no. 8, 083519 (2016)
  * [30] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B174(1979) 227
  * [31] M. Fujii, K. Hamaguchi and T. Yanagida, Phys. Rev. D 63, 123513 (2001)
  * [32] H. Fritzsch, M. Gell-Mann, and P. Minkowski, Phys. Lett. B59, 256(1975); P. Minkowski, Phys. Lett. B67, 421(1977); T. Yanagida, in Proceedings of Workshop on Unified Theory and the Baryon Number of the Universe, edited by O. Sawada and A. Sugamoto(KEK, Tsukuba, 1979), p. 95; M. Gell-Mann, P. Ramond, and Slansky, in Supergravity, edited by P. van. Nieuwenhuizen and D.Z. Freeman (North-Holland, Amsterdam,1979), p. 315; R.N. Mohapatra and G. Senjanovic, Phys, Rev. Lett. 44, 912(1980); J. Schechter and J. W. F. Valle, Phys. Rev. D22, 2227(1980); J. Schechter and J. W. F. Valle, Phys. Rev. D25, 774(1982).
  * [33] B. Pontecorvo, Zh. Eksp. Teor. Fiz. 33, 549(1957); Z. Maki, M. Nakagawa, and N. Sakata, Prog. Theor. Phys. 28, 870(1962). .
  * [34] F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Melchiorri and A. Palazzo, arXiv:1703.04471 [hep-ph].
  * [35] S.M. Bilenky and C. Giunti, Mod. Phys, Lett. A16, 1230015(2012).
  * [36] B. C. Chauhan, J. Pulido and M. Picariello, Phys. Rev. D 73, 053003 (2006)
  * [37] J. Sato and T. Yanagida, Phys. Lett. B 493, 356 (2000)


1School of Computing, University of Eastern Finland, Joensuu, Finland
2Université de Lorraine, CNRS, Inria, LORIA, F-54000, Nancy, France

# Learnable Nonlinear Compression for Robust Speaker Verification

###### Abstract

In this study, we focus on nonlinear compression methods in spectral features
for speaker verification based on deep neural network. We consider different
kinds of channel-dependent (CD) nonlinear compression methods optimized in a
data-driven manner. Our methods are based on power nonlinearities and dynamic
range compression (DRC). We also propose multi-regime (MR) design on the
nonlinearities, at improving robustness. Results on VoxCeleb1 and VoxMovies
data demonstrate improvements brought by proposed compression methods over
both the commonly-used logarithm and their static counterparts, especially for
ones based on power function. While CD generalization improves performance on
VoxCeleb1, MR provides more robustness on VoxMovies, with a maximum relative
equal error rate reduction of 21.6%.

Index Terms—  Speaker Verification, Nonlinear Compression, Multi-Regime
Compression.

##  1 Introduction

_Automatic speaker verification_ (ASV) [1, 2] is the task of verifying a
person’s identity using his or her voice. Modern ASV systems consists of three
main components: acoustic feature extractor, speaker embedding extractor, and
back-end classifier. In recent years, substantial improvement has been
achieved by using _deep neural networks_ (DNNs) to implement, especially the
last two components. Concerning speaker embedding extractor, statistical
models such as _i-vectors_ [3] have been replaced by deep models such as _x-
vector_ with _time-delayed neural network (TDNN)_ [4]. As for the back-end,
recent studies have replaced _probabilistic linear discriminant analysis_
(PLDA) [5] with neural approaches [6].

Concerning features, however, many ASV systems still use _mel-frequency
cepstral coefficients_ (MFCCs) [7], which are not specialized for ASV and
neglects information such as phase and temporal characteristics [8, 9, 10,
11]. Meanwhile, spectrograms are also widely used [12, 13, 14, 15]. There are
multiple types of spectrograms such as raw one where no filter is applied [12,
16] and more widely-used spectral energies output with mel filters [17, 18].
Even if the spectral representations are usually higher-dimensional (hence,
more expressive) than MFCCs, problems of lacking specialization and missing
information remain.

There are also attempts to replace hand-crafted features with neural networks
[19, 20, 21, 22]. However, such design may be hard to interpret. Moreover,
many state-of-the-art DNN extractors are based on convolutional kernels, whose
modeling capability on variabilities across different frequency (_channel_ or
_subband_) components have been questioned [19, 23]. These potential
shortcomings motivate the idea of optimizing signal processing modules of
feature extractor, including spectrogram-based features. Such topic has been
addressed recently for audio representation learning [24] and ASV [25], but
expanding and optimizing nonlinear compression module has received less
attention.

This study, motivated by the above, addresses _channel-dependent_ (CD)
nonlinear compression of spectrogram energies. This is realized, as presented
in Section 2, by expanding the nonlinearity from a channel-independent to
channel-dependent operation. Similar ideas on mel spectrogram have been
effective in keyword spotting [26], audio classification [27], and far-field
speaker verification [28]. However, to the best of our knowledge, it has not
been applied to various nonlinear compression methods in the task we consider.

Our main contributions are summarized in two folds: 1) We leverage the power
of such channel-dependent setting by revisiting two established nonlinear
compression methods that have been efficient in previous works and generalized
them to be channel-dependent, namely power function and dynamic range
compression; 2) In order to capture different level of variabilities and
compromise instabilities during the joint optimization, we propose a _multi-
regime_ (MR) design based on CD.

##  2 Nonlinear Compression in Acoustic Feature Extraction

When using spectrogram to extract features, as illustrated in Fig. 1, we
typically apply logarithmic compression to spectral energies. However,
logarithm has a singularity at zero. This problem is often addressed by adding
a small positive offset: $\log(x+\mathrm{offset})$. Even if it avoids the
singularity, the ad-hoc design still lacks specificity to a given task and has
unpredictable impacts for different kinds of input [26]. We consider two
alternative parameterized methods and further make them to be channel-
dependent, which are described below.

###  2.1 Power Function

The concept of applying power nonlinearity to compress the signal amplitude is
inspired by human-auditory processing [29, 30]. By using $X$ and $Y$ to denote
the input and output magnitude spectra respectively, power nonlinearity is
expressed as:

$\displaystyle Y[t,f]=X[t,f]^{1/\alpha},$ (1)

where $\alpha$ is known as _temperature coefficient_ for the compression. $t$
and $f$ are the time and channel indices, respectively, for spectrogram
energies. Experimentally, two particular values of $\alpha$ have been popular
in speech front-ends. The first one is $\alpha=3$, known as _cube-root_ [31,
32, 33]. The other one is $\alpha=15$, known as _power-law_ [34]. Setting
higher values of $\alpha$ can provide better recognition performance in the
presence of white noise, while lower values may be required for maintaining
accuracy for cleaner speech [11].

###  2.2 Dynamic Range Compression

Power nonlinearity neither addresses foreground-background noise nor other
variations. These problems can be addressed by applying dynamic range
compression (DRC). It was proposed and applied originally to far-field keyword
spotting as part of _per-channel energy normalization_ (PCEN) [26, 35]. PCEN
has also been applied recently in audio event detection [36]. Using the same
notation as above, the DRC operation is defined by:

$\displaystyle Y[t,f]=(X[t,f]+\delta)^{r}-\delta^{r},$ (2)

where $\delta>0$ is a positive bias and $r$ is the exponential offset. This
method bears resemblance to _spectral subtraction_ [37] in speech denoising.
In the context of PCEN, it is applied to the spectral energies processed with
_automatic gain control_ (AGC) [38].. In this study, under the framework of
DNN-based ASV, we investigate the efficiency of DRC by directly applying it to
spectral energies.

Fig. 1: Feature extraction with log and proposed compression methods, using
power function as example.

##  3 Proposed Method

###  3.1 Channel-Dependent Design

While the control parameters in Eq. (1) and (2) can be set by hand, this may
lead to suboptimal performance in a recognition task. Related prior studies on
channel-dependent compression utilize information such as loudness and signal-
to-noise ratio (SNR) [39], which motivates data-driven settings. From the
equations, we can see that the parameters $\alpha$, $\delta$, and $r$ are
differentiable. Therefore, we propose to optimize them as part of the neural
network by generalizing to their channel-dependent (CD) counterparts:
$\boldsymbol{\alpha}=\alpha(f),\boldsymbol{\delta}=\delta(f),\boldsymbol{r}=r(f)$,
where $f$ is the channel index. This design follows the proposal from [26].
The generalized parameters are then jointly optimized with the neural network.
Furthermore, we employ _kernelized initialization_ where the parameters are
initialized from their static counterparts [25].

###  3.2 Multi-Regime Design

Learnable parameters are tuned and selected by the training data during the
learning process, thus might be suboptimal if domain mismatch between training
and testing data is large, due to joint training which may let the parameters
suffer from the overparameterized DNN models [40]. CD generalization of the
parameters with kernel initialization may scrutinize such problem by larger
search space and proper starting point, but it still may fail to have a wide-
enough coverage of different level of speech variabilities. This is especially
the case when the DNN has large number of layers, which may cause the problem
of vanishing gradient when being back-propagated to first early layers, then
nonlinearities [41].

Therefore, inspired by the design of _multiple_ feature maps in image
processing [42] and audio event detection [36], we use a _multi-regime_ (MR)
design by passing the spectrum to multiple submodules, with shared compression
algorithm, but different initialized parameters, as shown in Fig. 1. The
output of each module is averaged to form the input for further operations.
Using power function as an example:

$\displaystyle Y[t,f]=\frac{1}{N}\sum_{i=1}^{N}(X[t,f])^{1/\alpha_{i}(f)}$ (3)

We define the initial values by defining minimum and maximum to create $N$
evenly spaced values:
$\alpha_{i}=\alpha_{\mathrm{min}}+(\alpha_{\mathrm{max}}-\alpha_{\mathrm{min}})*i/(N-1),i=1,...,N$,
where $\alpha_{\mathrm{max}}$ and $\alpha_{\mathrm{min}}$ are maximum and
minimum reference values and $N$ denotes number of intermediate spectrograms
generated. The setup for this work is shown in Table 1. For all cases in this
work, $N=3$. Further tuning of number of intermediates and parameter search is
left as future work.

Method | CD | MR-CD
---|---|---
_cube root_ [31] | $\alpha=3$ | $\alpha_{\mathrm{max}}=3,\alpha_{\mathrm{min}}=1$
_power law_ [33] | $\alpha=15$ | $\alpha_{\mathrm{max}}=15,\alpha_{\mathrm{min}}=1$
_DRC_ | $\delta=2.0$ | $\delta_{\mathrm{max}}=2.0,\delta_{\mathrm{min}}=1.0$
$r=0.5$ | $r_{\mathrm{max}}=1.0,r_{\mathrm{min}}=0.0$
Table 1: Parameter settings for kernel initialization. DRC values in CD are from [26] while for MR-CD it was hand-crafted based on [35] and pilot experiments. |  | VoxCeleb1-E | VoxCeleb1-H | VoxMovies
---|---|---|---|---
Method | Design | EER(%) | minDCF | EER(%) | minDCF | E-1 | E-2 | E-3 | E-4 | E-5 | pooled
$\log$ | - | 2.23 | 0.2676 | 4.43 | 0.5371 | 10.55 | 16.82 | 15.17 | 24.73 | 19.55 | 20.64
$\log(x+\mathrm{offset})$ | - | 2.41 | 0.2920 | 4.93 | 0.6219 | 13.3 | 16.49 | 14.41 | 25.44 | 18.38 | 19.00
_cube-root_ | - | 1.93 | 0.2377 | 4.02 | 0.5090 | 10.68 | 14.74 | 13.00 | 25.21 | 16.61 | 18.21
CD | 1.91 | 0.2329 | 3.84 | 0.5071 | 10.84 | 14.79 | 12.93 | 25.71 | 16.67 | 18.33
MR-CD | 2.24 | 0.2876 | 4.52 | 0.5421 | 10.92 | 12.99 | 13.77 | 15.86 | 15.79 | 14.27
_power-law_ | - | 2.14 | 0.2542 | 4.38 | 0.5239 | 11.09 | 15.15 | 13.58 | 25.96 | 17.13 | 18.70
CD | 2.14 | 0.2505 | 4.29 | 0.5186 | 10.25 | 13.44 | 13.23 | 21.76 | 15.82 | 16.39
MR-CD | 2.78 | 0.3141 | 5.31 | 0.5683 | 12.07 | 14.27 | 15.35 | 18.05 | 16.92 | 15.80
_DRC_ | - | 2.25 | 0.2598 | 4.60 | 0.5629 | 11.09 | 14.07 | 14.14 | 22.20 | 15.16 | 16.89
CD | 2.67 | 0.2993 | 5.20 | 0.6408 | 11.95 | 14.07 | 14.45 | 19.78 | 16.24 | 16.38
MR-CD | 2.90 | 0.3526 | 5.81 | 0.6442 | 12.96 | 15.48 | 16.70 | 19.55 | 16.94 | 17.03
Table 2: Speaker verification results on VoxCeleb and VoxMovies. ‘CD’ means
channel-dependent and ‘MR’ means multi-regime setups. Rows with slight grey
shades are regarded as baselines. ‘-’ indicates the system with static
nonlinearity without learning involved. For VoxMovies all results are reported
in EER(%).

##  4 Experimental Protocol

Data. For all experiments, we train the DNN speaker embedding extractor using
the _dev_ set of VoxCeleb2 [16], which consists of 5994 speakers. We report
the performance of different methods on two evaluation sets: 1) The two test
sets from VoxCeleb1 [12] following [43], known as _VoxCeleb1-E_ and
_VoxCeleb1-H_. 2) The recent _VoxMovies_ [44], which overlaps with VoxCeleb1
in terms of speakers and contains various levels of mismatch between the
enrollment and test utterances. It consists of five trial sets, denoted E-1
(easiest) through E-5 (hardest). Besides condition-specific results, we also
report the _pooled_ performance over the all five sets.

Features. We use raw magnitude spectrogram obtained using _short-time Fourier
transform_ (STFT) as the time-frequency representation, to which different
compression methods are applied, as illustrated in Fig. 1. The number of
frequency bins $N_{\mathrm{STFT}}=512$ for all systems. The sampling rate is
16 kHz, and the STFT is computed using a 25 ms Hamming window every 10 ms.
Additionally, we include a system where the logarithm is factorized by an
offset as part of the baseline, as mentioned in Section 2:
$\log(x+\mathrm{offset})$. The offset is parameterized by an exponential
function $\mathrm{offset}=exp(\boldsymbol{\beta})$, where
$\boldsymbol{\beta}=\beta(f)$ is CD and initialized with normal distribution.

Speaker embeddings. We use x-vector with _extended TDNN_ to generate speaker
embeddings, following the design choice from [45] with two main modifications:
1) We replace the statistics pooling layer with attentive statistics pooling
[46]; 2) Instead of multi-class cross-entropy, we use _additive angular
softmax_ [47] as the training objective. We set the scaling factor $s=30$ and
the margin $m=0.2$. We extract the embedding vectors from the first fully-
connected layer after the pooling layer. The extracted vectors are centered
and projected via a 150-dimensional _linear discriminant classifier_ (LDA).

Evaluation. For both VoxCeleb1 and VoxMovies, we train _probabilistic LDA_
(PLDA) classifier using VoxCeleb1. We report ASV performance in terms of
_equal error rate_ (EER) and _minimum detection cost function_ (minDCF). For
minDCF, the target speaker prior is $p_{\mathrm{tar}}=0.01$ and detection
costs were $C_{\text{fa}}=C_{\text{miss}}=1.0$.

##  5 Results

###  5.1 Speaker Verification

The results are presented in Table 2. Let us first focus on VoxCeleb1. The
EERs for both of the two power functions are improved from both the logarithm
baseline and their static counterparts (marked as ’-’ in the ‘Design’ column
of the table) by the CD design. As part of the baseline, applying CD
exponential offset on the logarithm compression degrades the performance for
both test sets. The lowest EER on both test sets is obtained using _cube-root_
with CD, outperforming the baseline logarithm by 14.3% and 13.3%,
respectively. This indicates the usefulness of CD. Nevertheless, the same
design does not work well with DRC, which contradicts the findings reported in
[26] with mel spectrogram (in a different task, though). This indicates that
in ASV with spectrogram input, DRC may not combine well with CD compression.
For all compression methods, the MR-CD design degrades performance and fails
to show substantial improvement over the logarithm. One reason could be
suboptimal parameter initialization as in [26], where the DRC parameters are
set for far-field keyword spotting.

On the other hand, the trend is different for VoxMovies with more severe
mismatch. Both _pooled_ and condition-specific indicate that improvements from
CD generalization are modest, as opposed with the observations from VoxCeleb1.
For _cube-root_ , CD actually degrades the performance for pooled and
individual trial sets apart from E-3, where 14.7% relative EER reduction is
obtained over the logarithm. However, for _power-law_ CD improves upon its
static counterpart, with lowest EER on E-1 across all systems.

Generalization via both MR and CD brings substantial improvements on
nonlinearities based on power function. Lowest EER of pooled, E-2 and E-4 is
obtained using _cube-root_ with MR and CD. Its pooled performance outperforms
its static counterpart by relatively 21.6%. We notice the same for _power-law_
, whose pooled performance with MR and CD outperform its static version by
relatively 15.5%. This indicates usefulness of MR in enhancing the robustness.

Nevertheless, the behavior of DRC is different from the power function. While
its static setting reduced the EER from the logarithm by relative 18.2%,
applying CD results in only slight relative EER reduction (3.1%) on pooled
results and does not lead to better performance for the individual trial sets
apart from E-4. Generalizing it with MR degrades the performance, which agrees
with our observations for VoxCeleb. However, its static setting reaches the
best performance across all methods. This indicates the parameters for DRC
(bias and offset) being not suitable to cope with CD and MR, at least not
within the DRC framework itself. Further investigation is needed on its
parameterization.

###  5.2 Representation Analysis

We illustrate the learnt temperature representation of the two power nonlinear
functions from VoxCeleb2 _dev_ set (as described in Section 4) in Fig. 2. Note
that for power function, larger magnitude of temperature parameters will
result in higher compressing effect.

As shown in the figure, applying only CD on power functions casts more
compression on both low and high frequency regions (higher temperature values
imply more aggressive compression, according to Eq.(1)). Meanwhile, applying
MR results in relatively less compression on some of the middle frequency
components as well as low frequency regions, while relatively maintaining its
pattern on high frequency components. Interestingly, we see that both power
function methods result in similar numerical range, even if their initialized
values are very different (Table 1).

Fig. 2: Learnt temperature values of nonlinear compression based on power
function. The x-axis denotes frequency (channel) bin index and y-axis measures
the values.

##  6 Conclusion

In this work, we have investigated the alternative nonlinearities for
spectrogram compression and their dynamic, channel-dependent variants. We have
extended their representation via channel-wise manner and utilized a multi-
regime design based on it. Initialization on relevant parameters has been
based on the corresponding static known values. We have evaluated the
performance of proposed extended dynamic compression methods for different
degree of mismatch conditions. Results demonstrates the efficacy of the
proposed methods on power nonlinearities, with a maximum of 21.6% pooled EER
reduction on VoxMovies. Future work may focus on: 1) extending the framework
with other types of spectrogram; 2) exploring more advanced design and
appropriate initialization and tuning methods, especially for DRC.

##  7 Acknowledgements

This work was partially supported by Inria Nancy Grand Est.

##  8 References

## References

  * [1] J.H.L. Hansen and T. Hasan “Speaker Recognition by Machines and Humans: A tutorial review” In _IEEE Sig. Proc. Mag._ 32.6, 2015, pp. 74–99 DOI: 10.1109/MSP.2015.2462851
  * [2] Z. Bai and X. Zhang “Speaker recognition based on deep learning: An overview” In _Neural Networks_ 140, 2021, pp. 65–99 DOI: https://doi.org/10.1016/j.neunet.2021.03.004
  * [3] N. Dehak et al. “Front-End Factor Analysis for Speaker Verification” In _IEEE Transactions on Audio, Speech, and Language Processing_ 19.4, 2011, pp. 788–798 DOI: 10.1109/TASL.2010.2064307
  * [4] D. Snyder “X-Vectors: Robust DNN Embeddings for Speaker Recognition” In _Proc. ICASSP_ , 2018, pp. 5329–5333 DOI: 10.1109/ICASSP.2018.8461375
  * [5] S. Ioffe “Probabilistic Linear Discriminant Analysis” In _ECCV 2006_ , 2006, pp. 531–542
  * [6] S. Ramoji “NPLDA: A Deep Neural PLDA Model for Speaker Verification” In _Proc. Odyssey 2020_ , 2020, pp. 202–209 DOI: 10.21437/Odyssey.2020-29
  * [7] S. Davis and P. Mermelstein “Comparison of parametric representations for monosyllabic word recognition in continuously spoken sentences” In _IEEE Transactions on Acoustics, Speech, and Signal Processing_ 28.4, 1980, pp. 357–366 DOI: 10.1109/TASSP.1980.1163420
  * [8] Erfan Loweimi et al. “Robust Source-Filter Separation of Speech Signal in the Phase Domain” In _Proc. Interspeech_ , 2017, pp. 414–418 DOI: 10.21437/Interspeech.2017-210
  * [9] Leigh D. Alsteris and Kuldip K. Paliwal “Short-time phase spectrum in speech processing: A review and some experimental results” In _Digital Signal Processing_ 17.3, 2007, pp. 578–616 DOI: https://doi.org/10.1016/j.dsp.2006.06.007
  * [10] Zhizheng Wu “Synthetic speech detection using temporal modulation feature” In _Proc. ICASSP_ , 2013, pp. 7234–7238 DOI: 10.1109/ICASSP.2013.6639067
  * [11] C. Kim and R. Stern “Power-Normalized Cepstral Coefficients (PNCC) for Robust Speech Recognition” In _IEEE/ACM Trans. on Audio, Speech, and Language Processing_ 24.7, 2016, pp. 1315–1329 DOI: 10.1109/TASLP.2016.2545928
  * [12] A. Nagrani, J. Chung and A. Zisserman “VoxCeleb: A Large-Scale Speaker Identification Dataset” In _Proc. Interspeech_ , 2017, pp. 2616–2620 DOI: 10.21437/Interspeech.2017-950
  * [13] Suwon Shon “VoiceID Loss: Speech Enhancement for Speaker Verification” In _Proc. Interspeech_ , 2019, pp. 2888–2892 DOI: 10.21437/Interspeech.2019-1496
  * [14] Chunlei Zhang “Towards Robust Speaker Verification with Target Speaker Enhancement” In _Proc. ICASSP_ , 2021, pp. 6693–6697 DOI: 10.1109/ICASSP39728.2021.9414017
  * [15] Weidi Xie “Utterance-level Aggregation for Speaker Recognition in the Wild” In _Proc. ICASSP_ , 2019, pp. 5791–5795 DOI: 10.1109/ICASSP.2019.8683120
  * [16] J. Chung “VoxCeleb2: Deep Speaker Recognition” In _Proc. Interspeech_ , 2018, pp. 1086–1090 DOI: 10.21437/Interspeech.2018-1929
  * [17] Daniel Garcia-Romero, Greg Sell and Alan Mccree “MagNetO: X-vector Magnitude Estimation Network plus Offset for Improved Speaker Recognition” In _Proc. Odyssey 2020 The Speaker and Language Recognition Workshop_ , 2020, pp. 1–8 DOI: 10.21437/Odyssey.2020-1
  * [18] Brecht Desplanques “ECAPA-TDNN: Emphasized Channel Attention, Propagation and Aggregation in TDNN Based Speaker Verification” In _Proc. Interspeech_ , 2020, pp. 3830–3834 DOI: 10.21437/Interspeech.2020-2650
  * [19] M. Ravanelli and Y. Bengio “Speaker Recognition from Raw Waveform with SincNet” In _2018 IEEE Spoken Language Technology Workshop (SLT)_ , 2018, pp. 1021–1028 DOI: 10.1109/SLT.2018.8639585
  * [20] N. Zeghidour et al. “Learning Filterbanks from Raw Speech for Phone Recognition” In _Proc. ICASSP_ , 2018, pp. 5509–5513 DOI: 10.1109/ICASSP.2018.8462015
  * [21] Jee-weon Jung et al. “RawNet: Advanced end-to-end deep neural network using raw waveforms for text-independent speaker verification” In _Proc. Interspeech_ , 2019, pp. 1268–1272
  * [22] W. Lin and M.W. Mak “Wav2Spk: A Simple DNN Architecture for Learning Speaker Embeddings from Waveforms” In _Proc. Interspeech_ , 2020, pp. 3211–3215 DOI: 10.21437/Interspeech.2020-1287
  * [23] Erfan Loweimi, Peter Bell and Steve Renals “On Learning Interpretable CNNs with Parametric Modulated Kernel-Based Filters” In _Proc. Interspeech_ , 2019, pp. 3480–3484 DOI: 10.21437/Interspeech.2019-1257
  * [24] M. Won et al. “Data-Driven Harmonic Filters for Audio Representation Learning” In _Proc. ICASSP_ , 2020, pp. 536–540 DOI: 10.1109/ICASSP40776.2020.9053669
  * [25] X. Liu, M. Sahidullah and T. Kinnunen “Learnable MFCCs for Speaker Verification” In _2021 IEEE International Symposium on Circuits and Systems (ISCAS)_ , 2021, pp. 1–5 DOI: 10.1109/ISCAS51556.2021.9401593
  * [26] Y. Wang “Trainable frontend for robust and far-field keyword spotting” In _Proc. ICASSP_ , 2017, pp. 5670–5674 DOI: 10.1109/ICASSP.2017.7953242
  * [27] C. Ick and B. McFee “Sound Event Detection in Urban Audio with Single and Multi-Rate Pcen” In _Proc. ICASSP_ , 2021, pp. 880–884 DOI: 10.1109/ICASSP39728.2021.9414697
  * [28] X Liu, M Sahidullah and T Kinnunen “Parameterized Channel Normalization for Far-field Deep Speaker Verification” In _2021 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU) (to appear)_ , 2021
  * [29] Richard M. Stern et al. “Signal Processing for Robust Speech Recognition” In _Automatic Speech and Speaker Recognition: Advanced Topics_ Boston, MA: Springer US, 1996, pp. 357–384
  * [30] C. Kim and R.. Stern “Power-Normalized Cepstral Coefficients (PNCC) for robust speech recognition” In _Proc. ICASSP_ , 2012, pp. 4101–4104 DOI: 10.1109/ICASSP.2012.6288820
  * [31] H. Hermansky and N. Morgan “RASTA processing of speech” In _IEEE Transactions on Speech and Audio Processing_ 2.4, 1994, pp. 578–589 DOI: 10.1109/89.326616
  * [32] M.. Alam “Multitaper MFCC and PLP features for speaker verification using i-vectors” In _Speech Communication_ 55.2, 2013, pp. 237–251 DOI: https://doi.org/10.1016/j.specom.2012.08.007
  * [33] Seyed Omid Sadjadi and John H.L. Hansen “Mean Hilbert envelope coefficients (MHEC) for robust speaker and language identification” In _Speech Communication_ 72, 2015, pp. 138–148 DOI: https://doi.org/10.1016/j.specom.2015.04.005
  * [34] C. Kim “Signal processing for robust speech recognition motivated by auditory processing”, 2010
  * [35] V. Lostanlen “Per-Channel Energy Normalization: Why and How” In _IEEE Sig. Pro. Lett._ 26.1, 2019, pp. 39–43 DOI: 10.1109/LSP.2018.2878620
  * [36] C. Ick and B. McFee “Sound Event Detection in Urban Audio with Single and Multi-Rate PCEN” In _Proc. ICASSP_ , 2021, pp. 880–884 DOI: 10.1109/ICASSP39728.2021.9414697
  * [37] J. Porter and S. Boll “Optimal estimators for spectral restoration of noisy speech” In _Proc. ICASSP_ 9, 1984, pp. 53–56 DOI: 10.1109/ICASSP.1984.1172545
  * [38] J. Prez “Automatic Gain Control: Techniques and Architectures for RF Receivers” Springer-Verlag New York, 2011
  * [39] Babak Nasersharif and Ahmad Akbari “A framework for robust MFCC feature extraction using SNR-dependent compression of enhanced mel filter bank energies” In _Proc. Interspeech_ , 2006
  * [40] Chaoyue Liu, Libin Zhu and Misha Belkin “On the linearity of large non-linear models: when and why the tangent kernel is constant” In _Advances in Neural Information Processing Systems_ 33 Curran Associates, Inc., 2020, pp. 15954–15964 URL: https://proceedings.neurips.cc/paper/2020/file/b7ae8fecf15b8b6c3c69eceae636d203-Paper.pdf
  * [41] Erfan Loweimi, Peter Bell and Steve Renals “On the Robustness and Training Dynamics of Raw Waveform Models” In _Proc. Interspeech_ , 2020, pp. 1001–1005 DOI: 10.21437/Interspeech.2020-17
  * [42] Rafael C. Gonzalez and Richard E. Woods “Digital image processing” Upper Saddle River, N.J.: Prentice Hall, 2008 URL: http://www.amazon.com/Digital-Image-Processing-3rd-Edition/dp/013168728X
  * [43] J. Chung et al. “VoxSRC 2019: The first VoxCeleb Speaker Recognition Challenge” In _ISCA archive_ , 2019
  * [44] A. Brown et al. “Playing a Part: Speaker Verification at the movies” In _Proc. ICASSP_ , 2021, pp. 6174–6178 DOI: 10.1109/ICASSP39728.2021.9413815
  * [45] D. Snyder “Speaker Recognition for Multi-speaker Conversations Using X-vectors” In _Proc. ICASSP_ , 2019, pp. 5796–5800 DOI: 10.1109/ICASSP.2019.8683760
  * [46] K. Okabe “Attentive Statistics Pooling for Deep Speaker Embedding” In _Proc. Interspeech_ , 2018, pp. 2252–2256 DOI: 10.21437/Interspeech.2018-993
  * [47] F. Wang et al. “Additive Margin Softmax for Face Verification” In _IEEE Signal Processing Letters_ 25.7, 2018, pp. 926–930 DOI: 10.1109/LSP.2018.2822810