Automorphisms of a Chevalley group of type $\mathbf{G}_{2}$

over a commutative ring $R$ with $1/3$

generated by the all invertible elements and $2R$

E. I. Bunina, M. A. Vladykina

Abstract.

In this paper we prove that every automorphism of a Chevalley group with the
root system $\mathbf{G}_{2}$ over a commutative ring $R$ with $1/3$, generated
by all its invertible elements and the ideal $2R$ is a composition of ring and
inner automorphisms.

## Introduction

Study of automorphisms of classical groups was started by the work of Schreier
and van der Warden [34] in 1928. They described all automorphisms of the group
${\,\mathrm{PSL}\,}_{n}$ ($n\geqslant 3$) over an arbitrary field.

Diedonne [22] (1951) and Rickart [33] (1950) introduced the involution method,
and with the help of this method described automorphisms of the group
${\,\mathrm{GL}\,}_{n}$ ($n\geqslant 3$) over a skewfield.

The first step in construction of the automorphism theory over rings, namely,
for the group ${\,\mathrm{GL}\,}_{n}$ ($n\geqslant 3$) over the ring of
integer numbers, was made by Hua Logen and Reiner [25] (1951), later some
papers over commutative integral domains appeared. The methods of the papers
mentioned above were based mostly on studying involutions in the corresponding
linear groups.

O’Meara [29] in 1976 invented very different (geometrical) method, which did
not use involutions. By its aid, O”Meara described automorphisms of the group
$\,\mathrm{GL}\,_{n}$ ($n\geqslant 3$) over domains.

In 1982 Petechuk [30] described automorphisms of the groups
${\,\mathrm{GL}\,}_{n},{\,\mathrm{SL}\,}_{n}(n\geqslant 4)$ over arbitrary
commutative rings. If $n=3$, automorphisms of given linear groups are not
always standard. They are standard either if in a ring $2$ is invertible, or
if a ring is a domain, or it is a semisimple ring.

Isomorphisms of the groups ${GL}_{n}(R)$ and ${GL}_{m}(S)$ over arbitrary
associative rings with $1/2$ for $n,m\geqslant 3$ were described in 1981 by
I.Z. Golubchik and A.V. Mikhalev [23] and independently by E.I. Zelmanov [40].
In 1997 I.Z Golubchik [24] described isomorphisms between these groups for
$n,m\geqslant 4$, but over arbitrary associative rings with $1$.

In 50-th years of the previous century Chevalley [20], Steinberg [35] and
others introduced the concept of Chevalley groups over commutative rings,
which includes classical linear groups (special linear ${\,\mathrm{SL}\,}$,
special orthogonal ${\,\mathrm{SO}\,}$, symplectic ${\,\mathrm{Sp}\,}$, spinor
${\,\mathrm{Spin}\,}$, and also projective groups connected with them) over
commutative rings.

Clear that isomorphisms and automorphisms of Chevalley groups were also
studied intensively. The description of isomorphisms of Chevalley groups over
fields was obtained by Steinberg for the finite case [36] and by Humphreys for
the infinite one [26]. Many papers were devoted to description of
automorphisms of Chevalley groups over different commutative rings, we can
mention here the papers of Borel–Tits [4], Carter–Chen Yu [18], Chen Yu [19],
E. Abe [1], A. Klyachko [28].

In the paper [6] Bunina proved that automorphisms of adjoint elementary
Chevalley groups with root systems
$\mathbf{A}_{l},\mathbf{D}_{l},\mathbf{E}_{l},$ $l\geqslant 2$, over local
rings with invertible $2$ can be represented as the composition of a ring
automorphism and an automorphism-conjugation by some matrix from the
normalizer of this group in ${\,\mathrm{GL}\,}(V)$. In the paper [8] according
to the results of [6] it was shown that every automorphism of an elementary
Chevalley group of the described type is standard, i. e., is represented by
the composition of ring, inner, central and graph automorphisms. In the same
paper the theorem describing the normalizer of Chevalley groups in their
adjoint representation, which also holds for local rings without $1/2$, was
obtained.

In the papers [7], [9], [12], by the same methods it was shown that all
automorphisms of Chevalley groups with the root systems
$\mathbf{F}_{4},\mathbf{G}_{2},\mathbf{B}_{l}$, $l\geqslant 2$, over local
rings with $1/2$ (in the case $\mathbf{G}_{2}$ also with $1/3$) are standard.
In the paper [13] it was proved that all automorphisms of Chevalley groups of
types $\mathbf{A}_{l},\mathbf{D}_{l},\mathbf{E}_{l},l\geqslant 3$, over local
rings without $1/2$, are standard.

In the paper [14] with the help of all previous results and localization
method the automorphisms of adjoint Chevalley groups over arbitrary
commutative rings were described, where a root system has rank $>1$, and for
$\mathbf{A}_{2},\mathbf{F}_{4},\mathbf{B}_{l},\mathbf{C}_{l}$ the ring
contains $1/2$, for $\mathbf{G}_{2}$ the ring contains $1/2$ and $1/3$. In the
recent paper [15] this result was extended to all Chevalley groups (not only
adjoint) with the same restrictions for rings.

In the papers [10], [11] E.I. Bunina and P.A. Weryovkin proved that every
automorphism of a Chevalley group of the type $\mathbf{G}_{2}$ over a local
ring with $1/3$ and without $1/2$ is standard.

Over fields of characteristic $2$ and the root systems $\mathbf{F}_{4}$ there
exists a non-standard automorphism of the corresponding Chevalley group (see
[35]). Over fields of characteristic $2$ and the root systems $\mathbf{B}_{l}$
and $\mathbf{C}_{l}$ here exists a non-standard isomorphism between the
corresponding Chevalley groups (also see [35]). In the case of local rings
without $1/2$ and the root system $\mathbf{A}_{2}$ there is a non-standard
automorphism of corresponding Chevalley group too (see [31] and [32]). For the
root system $\mathbf{G}_{2}$ non-invertible $3$ is a significant obstacle for
standardity of automorphisms, since even for fields of characteristic $3$ a
non-standard automorphism exists (see [35]), but as we see in the paper [11],
non-invertibility of $2$ does not interfere with standardity of automorphisms
for fields and for local rings.

So it is a natural goal to continue the result of standardity of automorphisms
of the type $\mathbf{G}_{2}$ for arbitrary commutative rings. At the moment,
this has been done for a commutative ring with $1/3$ generated by its
invertible elements and the ideal $2R$.

##  1 Definitions and main theorem.

We fix the root system of type $\mathbf{G}_{2}$ with the system of simple
roots
$\Delta(\mathbf{G}_{2})=\\{\alpha_{1}=e_{1}-e_{2},\alpha_{2}=-2e_{1}+e_{2}+e_{3}\\}$.

The system of positive roots :
$\mathbf{G}_{2}^{+}=\\{\alpha_{1},\alpha_{2},\alpha_{1}+\alpha_{2}=e_{1},2\alpha_{1}+\alpha_{2},3\alpha_{1}+\alpha_{2},3\alpha_{1}+2\alpha_{2}\\}$.

More details about root systems and their properties can be found in [5] and
[27].

Suppose now that we have some semisimple complex Lie algebra ${{\mathcal{L}}}$
of type $\mathbf{G}_{2}$ with Cartan subalgebra ${{\mathcal{H}}}$ (detailed
information about semisimple Lie algebras can be found in the book [27]). Then
we can choose a basis $\\{h_{1},...,h_{l}\\}\in{{\mathcal{H}}}$ and for every
$\alpha\in\mathbf{G}_{2}$ elements $x_{\alpha}\in{{\mathcal{L}}}_{\alpha}$ so
that $\\{h_{i};x_{\alpha}\\}$ form a basis in ${{\mathcal{L}}}$ and for every
two elements of this basis their commutator is an integral linear combination
of the elements of the same basis.

Let us introduce elementary Chevalley groups (see, for example,[35]). Let
${{\mathcal{L}}}$ be a semisimple Lie algebra (over ${{\mathcal{L}}}$) with a
root system $\mathbf{G}_{2},\,\pi:{{\mathcal{L}}}\rightarrow gl_{n}(V)$ be its
finitely dimensional faithful representation (of dimension $n$). If
${{\mathcal{H}}}$ is a Cartan subalgebra of ${{\mathcal{L}}}$, then a
functional $\lambda\in{{\mathcal{H}}}^{*}$ is called a _weight_ of a given
representation, if there exists a nonzero vector $v\in V$ (called a _weight
vector_) such that for any $h\in{{\mathcal{H}}}\,\pi(h)v=\lambda(h)v$ in the
space $V$ there exists a basis of weight vectors such that all operators
$t^{k}\pi(x_{\alpha})^{k}/k!\quad k\in\mathbb{N}$ are written as integral
(nilpotent) matrices. This basis is called a _Chevalley basis_. An integral
matrix also can be considered as a matrix over an arbitrary commutative ring
with $1$. Let $R$ be such a ring. Consider matrices $n\times n$ over $R$,
matrices $t^{k}\pi(x_{\alpha})^{k}/k!$ for
$\alpha\in\mathbf{G}_{2},k\in\mathbb{N}$ are included in $M_{n}(R)$.

Now consider automorphisms of the free module $\,\mathrm{Aut}\,(R^{n})$ of the
form

$x_{\alpha}(t):=\exp(tx_{\alpha})=1+t\pi(x_{\alpha})+\dots+t^{k}\pi(x_{\alpha})^{k}/k!+\dots$

Since all matrices $x_{\alpha}$ are nilpotent, this series is finite.
Automorphisms $x_{\alpha}(t)$ are called _elementary root elements_. The
subgroup $\,\mathrm{Aut}\,(R^{n})$, generated by
$x_{\alpha}(t),\alpha\in\mathbf{G}_{2}$, is called an _elementary adjoint
Chevalley group_.

The action of $x_{\alpha}(t)$ on the Chevalley basis is described in [39].

All weights of a given representation generate (by addition) a lattice (a free
Abelian group, where every $\mathbb{Z}$-basis is also a $\mathbb{C}$-basis in
${{\mathcal{H}}}^{*}$), which is called the _weight lattice_ $\Lambda_{\pi}$.

Elementary Chevalley groups are defined not even by a representation of the
Chevalley groups, but just by its _weight lattice_. More precisely, up to an
abstract isomorphism an elementary Chevalley group is completely defined by a
root system $\Phi$, a commutative ring $R$ with $1$ and a weight lattice
$\Lambda_{\pi}$.

Among all lattices we mark the lattice corresponding to the adjoint
representation, it is generated by all roots (the _root lattice_
$\Lambda_{ad}$). The corresponding (elementary) Chevalley group is called
_adjoint_.

Note that for the root system $\mathbf{G}_{2}$ there exists only one weight
lattice, which is simultaneously simply connected and adjoint, therefore for
every ring $R$ there exists only one Chevalley group of the type
$\mathbf{G}_{2}$: $G(R)=G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$.

In elementary Chevalley group it is possible to consider the following
important elements:

— $w_{\alpha}(t)=x_{\alpha}(t)x_{-\alpha}(-t^{-1})x_{\alpha}(t),\;t\in
R^{\star}$;

— $h_{\alpha}(t)=w_{\alpha}(t)w_{\alpha}(1)^{-1}$.

Every elementary Chevalley group satisfies the following relations:

(R1) $\forall\alpha\in\mathbf{G}_{2},\forall t,u\in
R,\;x_{\alpha}(t)x_{\alpha}(u)=x_{\alpha}(t+u);$

(R2) If $\\{\alpha,\beta\\}$ are simple roots of the system $\mathbf{G}_{2}$,
$\alpha$ is long, $\beta$ is short, $t,u\in R$, then

$\displaystyle\left[x_{\alpha}(t),x_{\beta}(u)\right]$
$\displaystyle=x_{\alpha+\beta}(tu)x_{\alpha+3\beta}(-tu^{3})x_{\alpha+2\beta}(-tu^{2})x_{2\alpha+3\beta}(t^{2}u^{3}),$
$\displaystyle\left[x_{\alpha+\beta}(t),x_{\beta}(u)\right]$
$\displaystyle=x_{\alpha+2\beta}(2tu)x_{\alpha+3\beta}(-3tu^{2})x_{2\alpha+3\beta}(3t^{2}u^{3});$
$\displaystyle\left[x_{\alpha}(t),x_{\alpha+3\beta}(u)\right]$
$\displaystyle=x_{2\alpha+3\beta}(tu),$
$\displaystyle\left[x_{\alpha+2\beta}(t),x_{\beta}(u)\right]$
$\displaystyle=x_{\alpha+3\beta}(-3tu),$
$\displaystyle\left[x_{\alpha+\beta}(t),x_{\alpha+2\beta}(u)\right]$
$\displaystyle=x_{2\alpha+3\beta}(3tu);$

(R3) $\forall\alpha\in\mathbf{G}_{2}\,\,\,w_{\alpha}:=w_{\alpha}(1)$;

(R4) $\forall\alpha,\beta\in\mathbf{G}_{2},\forall t\in
R^{\star}\;w_{\alpha}h_{\beta}(t)w_{\alpha}^{-1}=h_{w_{\alpha}(\beta)}(t)$;

(R5) $\forall\alpha,\beta\in\mathbf{G}_{2}\,\,\,\forall t\in
R\;w_{\alpha}x_{\beta}(t)w_{\alpha}^{-1}=x_{w_{\alpha}(\beta)}(ct);$, where
$c=c(\alpha,\beta)=\pm 1$;

(R6) $\forall\alpha,\beta\in\mathbf{G}_{2}\,\,\,\forall t\in
R^{\star}\;\forall u\in
R\;h_{\alpha}(t)x_{\beta}(u)h_{\alpha}(t)^{-1}=x_{\beta}(t^{\langle\beta,\alpha\rangle}u)$.

Introduce now Chevalley groups. For more details see [35], [20], [39], [21],
[3], [17], [38].

Consider semisimple linear algebraic groups over algebraically closed fields.
These are precisely elementary Chevalley groups $E_{\pi}(\Phi,K)$.

All these groups are defined in $\,\mathrm{SL}\,_{n}(K)$ as common set of
zeros of polynomials of matrix entries $a_{ij}$ with integer coefficients It
is clear now that multiplication and taking inverse element are defined by
polynomials with integer coefficients. Therefore, these polynomials can be
considered as polynomials over an arbitrary commutative ring with a unit. Let
some elementary Chevalley group $E$ over $\mathbb{C}$ be defined in
$\,\mathrm{SL}\,_{n}(\mathbb{C})$ by polynomials
$p_{1}(a_{ij}),\dots,p_{m}(a_{ij})$. For a commutative ring $R$ with a unit
let us consider the group

$G(R)=\\{(a_{ij})\in\,\mathrm{SL}\,_{n}(R)\mid\widetilde{p}_{1}(a_{ij})=0,\dots,\widetilde{p}_{m}(a_{ij})=0\\},$

where $\widetilde{p}_{1}(\dots),\dots\widetilde{p}_{m}(\dots)$ are polynomials
having the same coefficients as $p_{1}(\dots),\dots,p_{m}(\dots)$, but
considered over $R$.

This group is called _the Chevalley group_
$G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ of the type $\mathbf{G}_{2}$ over the
ring $R$, and it coincides with the elementary Chevalley group for every
algebraically closed field $K$.

Define two types of automorphisms of a Chevalley group
$G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$, we call them _standard_.

Ring automorphisms. Let $\rho:R\to R$ be an automorphism of the ring $R$. The
mapping $(a_{i,j})\mapsto(\rho(a_{i,j}))$ from
$G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ onto itself is an automorphism of the
group $G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$, denoted by the same letter
$\rho$. It is called a _ring automorphism_ of the group
$G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$. Note that for all
$\alpha\in\mathbf{G}_{2}$ and $t\in R$ an element $x_{\alpha}(t)$ is mapped to
$x_{\alpha}(\rho(t))$.

Inner automorphisms. Let $S$ be some ring containing $R$, $g$ be an element of
$G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},S)$, that normalizes the subgroup
$G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$. Then the mapping $x\mapsto gxg^{-1}$
is an automorphism of the group $G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$,
denoted by $i_{g}$. It is called an _inner automorphism_ , _induced by the
element_ $g\in G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},S)$. If $g\in
G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$, then we call $i_{g}$ a _strictly
inner_ automorphism.

These two automorphisms are called _standard_. Also central and graph
automorphisms are called standard, but in the case under consideration
nontrivial central and graph automorphisms do not exist, therefore we will
call an automorphism of a group $G(R)$ _standard_ , if it is a composition of
two introduced types of automorphisms.

Our goal is to prove the following theorem:

######  Theorem 1.

Let $G=G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ or
$G=E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ be a Chevalley group of the type
$\mathbf{G}_{2}$ or its elementary subgroup, $R$ be a commutative ring with
$1/3$, generated by its invertible elements and the ideal $2R$. Then any
automorphism of the group $G$ is standard, i. e., is a composition of a ring
and strictly inner automorphisms.

##  2 Notions and theorems, necessary for the proof.

###  2.1 Localization of rings and modules; injection of a ring into the
product of its localizations.

###### Definition 1.

Let $R$ be a commutative ring. A subset $Y\subset R$ is called
_multiplicatively closed_ in $R$, if $1\in Y$ and $Y$ is closed under
multiplication.

Introduce an equivalence relation $\sim$ on the set of pairs $R\times Y$ as
follows:

$\frac{a}{s}\sim\frac{b}{t}\Longleftrightarrow\exists u\in Y:\ (at-bs)u=0.$

By $\frac{a}{s}$ we denote the whole equivalence class of the pair $(a,s)$, by
$Y^{-1}R$ we denote the set of all equivalence classes. On the set $S^{-1}R$
we can introduce the ring structure by

$\frac{a}{s}+\frac{b}{t}=\frac{at+bs}{st},\quad\frac{a}{s}\cdot\frac{b}{t}=\frac{ab}{st}.$

###### Definition 2.

The ring $Y^{-1}R$ is called the _ring of fractions of $R$ with respect to
$Y$_.

Let $\mathfrak{p}$ be a prime ideal of $R$. Then the set
$Y=R\setminus{\mathfrak{p}}$ is multiplicatively closed (it is equivalent to
the definition of the prime ideal). We will denote the ring of fractions
$Y^{-1}R$ in this case by $R_{\mathfrak{p}}$. The elements $\frac{a}{s}$,
$a\in\mathfrak{p}$, form an ideal $\mathfrak{M}$ in $R_{\mathfrak{p}}$. If
$\frac{b}{t}\notin\mathfrak{M}$, then $b\in Y$, therefore $\frac{b}{t}$ is
invertible in $R_{\mathfrak{p}}$. Consequently the ideal $\mathfrak{M}$
consists of all non-invertible elements of the ring $R_{\mathfrak{p}}$, i. e.,
$\mathfrak{M}$ is the greatest ideal of this ring, so $R_{\mathfrak{p}}$ is a
local ring.

The process of passing from $R$ to $R_{\mathfrak{p}}$ is called _localization
at ${\mathfrak{p}}$._

The construction $S^{-1}A$ can be easily carried trough with an $A$-module
$M$. Let $m/s$ denote the equivalence class of the pair $(m,s)$, the set
$S^{-1}M$ of all such fractions is made as a module $S^{-1}M$ with obvious
operations of addition and scalar multiplication. As above we will write
$M_{\mathfrak{p}}$ instead of $S^{-1}M$ for $S=A\setminus{\mathfrak{p}}$,
where $\mathfrak{p}$ is a prime ideal of $A$.

######  Proposition 1.

Every commutative ring $R$ with $1$ can be naturally embedded in the cartesian
product of all its localizations by maximal ideals

$S=\prod\limits_{{\mathfrak{m}}\text{ is a maximal ideal of
}A}A_{\mathfrak{m}}$

by diagonal mapping, which corresponds every $a\in R$ to the element

$\prod\limits_{\mathfrak{m}}\left(\frac{a}{1}\right)_{\mathfrak{m}}$

of the ring $S$.

###  2.2 Isomorphisms of Chevalley groups over fields.

We will need the description of isomorphisms between Chevalley groups over
fields.

######  Theorem 2 (see Theorems 30 and 31 from [35]).

Let $G$, $G^{\prime}$ be Chevalley groups, constructed with root system
$\mathbf{G}_{2}$ and fields $k,k^{\prime}$, respectively. Suppose that
$k,k^{\prime}$ have characteristics not equal to three, $\varphi:G\to
G^{\prime}$ be a group isomorphism. Then the fields $k$ and $k^{\prime}$ are
isomorphic, and the isomorphism $\varphi$ is a composition of a ring
isomorphism between $G$ and $G^{\prime}$, and inner automorphism of
$G^{\prime}$.

###  2.3 The subgroup $E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ is
characteristic in the $G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$

###### Definition 3.

A subgroup $H$ of the group $G$ is called _characteristic_ , if it it is
mapped into itself under any automorphism of group $G$. In particular, any
characteristic subgroup is normal.

######  Theorem 3 ([37]).

If the rank of an indecomposable root system $\Phi$ is more than one, then the
elementary group $E_{\,\mathrm{ad}\,}(G_{2},R)$ is characteristic in the
Chevalley group $G_{\,\mathrm{ad}\,}(G_{2},R)$.

###  2.4 Normal structure of Chevalley groups over commutative rings.

If $R$ is a ring, $I$ is its ideal, then by
$\lambda_{I}:G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)\to
G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R/I)$
($E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)\to
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R/I)$) we denote a homomorphism, obtained
by corresponding every matrix $A\in G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ to
its image under the natural homomorphism $R\to R/I$.

We denote the inverse image of the center of $G_{\,\mathrm{ad}\,}(R/I)$ under
the homomorphism $\lambda_{I}$ by $Z_{I}$

######  Theorem 4.

([2]) If a subgroup ${{\mathcal{H}}}$ of
$E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ is normal in
$E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ then

$E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R,I)\leq{{\mathcal{H}}}\leq
Z_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R,I)\cap
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$

for some uniquely defined ideal $I$ of the ring $R$.

Note, that the center of Chevalley group of type $\mathbf{G}_{2}$ is trivial.

###### Definition 4.

Let $N_{I}$ denotes the subgroup $\,\mathrm{ker}\,\lambda_{I}\cap
E(\mathbf{G}_{2},R)$.

######  Proposition 2.

(see [14]) Let $\varphi$ be an arbitrary automorphism of a group
$E(\mathbf{G}_{2},R)$, $I$ be a maximal ideal of the ring $R$. Then there
exists a maximal ideal $J$ of the ring $R$ such that $\varphi(N_{I})=N_{J}$.

##  3 Formulation of main steps of the proof.

Consider a ring $R$ and its maximal ideal $I$. We denote the localization $R$
with $I$ by $R_{I}$, and its radical by ${\,\mathrm{Rad}\,}R_{I}$. Note, that
there are two isomorphic fields: $R/I$ and $R_{I}/{\,\mathrm{Rad}\,}R_{I}$:

$\begin{CD}R@>{}>{}>R_{I}\\\
@V{\lambda_{I}}V{}V@V{}V{\lambda_{\,\mathrm{Rad}\,R_{I}}}V\\\
R/I@>{\mu_{I}}>{}>R_{I}/\,\mathrm{Rad}\,R_{I}\end{CD}$

Let now $\varphi$ be an arbitrary automorphism of
$E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$. Proposition 2 gives us a possibility
to consider the commutative diagram

$\begin{CD}E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)@>{\varphi}>{}>E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)\\\
@V{r_{I}}V{}V@V{}V{r_{J}}V\\\
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R_{I})E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R_{J})\\\
@V{\lambda_{\,\mathrm{Rad}\,R_{I}}}V{}V@V{}V{\lambda_{\,\mathrm{Rad}\,R_{J}}}V\\\
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R_{I}/\,\mathrm{Rad}\,R_{I})E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R_{J}/\,\mathrm{Rad}\,R_{J})\\\
@V{s_{I}}V{}V@V{}V{s_{J}}V\\\
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R/I)@>{\overline{\varphi}}>{}>E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R/J)\end{CD}$
(1)

The groups $E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R/I)$ and
$E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R/J)$ are just elementary Chevalley groups
over fields, their isomorphism were described in Theorem 2.

Recall, that the fields $R/I$ and $R/J$ are isomorphic (as above let us denote
the corresponding isomorphism by $\rho$), and

$\overline{\varphi}(A)=i_{g}\circ\rho(A)\ \forall A\in
E_{\,\mathrm{ad}\,}(G_{2},R/I),\quad g\in
G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R/J)$

The description of automorphisms of $E(\mathbf{G}_{2},R)$ can be done by the
following scheme. the ring $R$ is embedded into the ring $S=\prod R_{I}$: the
Cartesian product of all local rings $R_{I}$, obtained by localization of $R$
by different maximal ideals $I$.

Clear that $E(\mathbf{G}_{2},R)$ is embedded into

$G(\mathbf{G}_{2},S)=G(\mathbf{G}_{2},\prod R_{I}).$

First step.

We prove, that for every maximal ideal $J$ we have

$r_{J}\varphi(x_{\alpha}(1))=i_{g_{J}}r_{J}(x_{\alpha}(1)),$

where $g_{J}\in G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},\overline{R_{J}})$ (an
extesion of $R_{J}$).

Second step.

We prove, that an inner automorphism of the group
$G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},S)$, generated by $g=\prod g_{J}$, induces
an automorphism of the group $G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ and in
fact is strictly inner.

Further, we show, that if we take a composition of the initial automorphism
and the inner automorphism $i_{g^{-1}}$, the obtained automorphism is a ring
one.

##  4 Proof of the first step.

Consider an arbitrary element $x_{\alpha}(1)\in
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$, $\alpha\in\mathbf{G}_{2}$. Its image
under the mapping $r_{I}$ is $x_{\alpha}(1)=x_{\alpha}(1/1)\in
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R_{I})$. In the field $R/I$ its image has
the same form. The element $x_{\alpha}^{\prime}=\varphi(x_{\alpha}(1))\in
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ under the factorization by the ideal
$J$ gives $\overline{\varphi}(x_{\alpha}(1))=i_{\overline{g}}(x_{\alpha}(1))$,
where $\overline{g}\in G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R/J)$.

Now choose $g\in G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R_{J})$ such that under
the factorization of ring $R_{J}$ by its radical $g$ is mapped to
$\overline{g}$.

Then consider the following mapping
$\psi:E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)\to
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R_{J})$:

$\psi=i_{g^{-1}}\circ r_{J}\circ\varphi.$

Under the mapping $\psi$ all $x_{\alpha}(1)$, $\alpha\in\mathbf{G}_{2}$ are
mapped to $x_{\alpha}^{\prime}$ such that
$x_{\alpha}(1)-x_{\alpha}^{\prime}\in M_{N}(\,\mathrm{Rad}\,R_{J})$.

Therefore, we now have a set of elements
$\\{x_{\alpha}^{\prime}\mid\alpha\in\mathbf{G}_{2}\\}\subset
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R_{J})$, satisfying the same relations as
$\\{x_{\alpha}(1)\mid\alpha\in\mathbf{G}_{2}\\}$, and equivalent to the
corresponding $x_{\alpha}(1)$ modulo radical $R_{J}$.

It is precisely the situation of the paper [10], in which for a local ring $R$
and the root system $\mathbf{G}_{2}$ if $3\in R^{*}$ without any additional
conditions for the ring it is proved, that if in the group
$E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ some elements $x_{\alpha}^{\prime}$
are the images of the corresponding $x_{\alpha}(1)$,
$\alpha\in\mathbf{G}_{2}$, and also $x_{\alpha}(1)-x_{\alpha}^{\prime}\in
M_{N}(\,\mathrm{Rad}\,R)$, then there exists $g^{\prime}\in
G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ , $g^{\prime}-E\in
M_{N}(\,\mathrm{Rad}\,R)$ such that for any $\alpha\in\mathbf{G}_{2}$

$x_{\alpha}(1)=i_{g^{\prime}}(x_{\alpha}^{\prime}).$

Thus, the first step of the theorem completely follows from the statements of
the previous paragraph. $\square$

Now embedding the initial ring $R$ into the ring $S=\prod\limits_{J}R_{J}$, we
see that

$\varphi(x_{\alpha}(1))=g(x_{\alpha}(1))g^{-1},$

where $g=\prod\limits_{J}g_{J}$.

##  5 Proof of the second step of the theorem.

Conjugation by an element $g\in G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},S)$ can be
extended up to the automorphism of the whole matrix ring $M_{N}(S)$.

Let us prove the following lemma:

######  Lemma 1.

Any matrix $x_{\alpha}(t),\alpha\in\mathbf{G}_{2}$, $t\in R^{*}$ or $t\in 2R$,
under conjugation by $g$ is mapped into a matrix from
$\,\mathrm{GL}\,_{N}(R)$.

###### Proof.

For $x_{\alpha}(1)$, where $\alpha\in\mathbf{G}_{2}$, this statement is true
(see the first step).

Let $\alpha$ — be a long root, $\alpha\in\mathbf{G}_{2}$. For the long roots
let us consider

$x_{\alpha}(1)=E+X_{\alpha}+\frac{X_{\alpha}^{2}}{2},\quad$

where $\frac{X_{\alpha}^{2}}{2}=E_{(\alpha,-\alpha)}.$ Lets us take two long
roots $\gamma$, $\beta\in\mathbf{G}_{2}$ such that $\gamma+\beta=\alpha$. Note
that

$(x_{\gamma}(1)x_{\beta}(1)-x_{\gamma}(1)-x_{\beta}(1)+E)^{2}=\pm
E_{(\alpha,-\alpha)},$

it easily gives $X_{\alpha}$.

Then any matrix $x_{\alpha}(t)$ under the conjugation by $g$ is mapped into a
matrix from $M_{N}(R)$.

Now consider some invertible element from the ring $R$. For any invertible
element $t\in R^{*}$

$h_{\alpha}(t)=x_{\alpha}(t)x_{-\alpha}(-t^{-1})x_{\alpha}(t)x_{\alpha}(1)^{-1}x_{-\alpha}(-1)^{-1}x_{\alpha}(1)^{-1}.$

Therefore, conjugation $g$ maps the element $h_{\alpha}(t)$ to a matrix from
$M_{N}(R)$.

Now let us consider an arbitrary short root $\beta$ and a long root $\alpha$,
such that

$\langle\beta,\alpha\rangle=1.$

Then

$x_{\beta}(t^{\langle\beta,\alpha\rangle})=h_{\alpha}(t)x_{\beta}(1)h_{\alpha}(t)^{-1},$

i. e., $x_{\beta}(t)$ for any invertible element $t\in R^{*}$ is mapped under
conjugation by $g$ into a matrix with coefficients from $R$.

Now let again $\beta$ be some short root. Then

$x_{\beta}(\pm 1)=E\pm
X_{\beta}+\frac{X_{\beta}^{2}}{2}\pm\frac{X_{\beta}^{3}}{6},$

hence

$x_{\beta}(-1)+x_{\beta}(1)=2E+X_{\beta}^{2}\Longrightarrow
X_{\beta}^{2}=x_{\beta}(-1)+x_{\beta}(1)-2E.$ $None$

Similarly

$x_{\beta}(1)-x_{\beta}(-1)=2X_{\beta}+\frac{X_{\beta}^{3}}{3}\Longrightarrow\frac{X_{\beta}^{3}}{3}=x_{\beta}(1)-x_{\beta}(-1)-2X_{\beta}.$
$None$

Besides,

$x_{\beta}(2)=E+2X_{\beta}+2X_{\beta}^{2}+\frac{4X_{\beta}^{3}}{3}=\\\
=E+2X_{\beta}+2x_{\beta}(1)+2x_{\beta}(-1)-4E+4x_{\beta}(1)-4x_{\beta}(-1)-8X_{\beta}=\\\
=-3E-6X_{\beta}+6x_{\beta}(1)-2x_{\beta}(-1),$

therefore, using the invertibility of $3$, we obtain

$2X_{\beta}=\frac{-x_{\beta}(2)-2x_{\beta}(-1)}{3}-E+2x_{\beta}(1).$ $None$

From (2) and (3) it now follows

$\frac{X_{\beta}^{3}}{3}=-x_{\beta}(1)-\frac{x_{\beta}(-1)}{3}+\frac{x_{\beta}(2)}{3}+E.$
$None$

Then any $x_{\beta}(t)$, where $t\in 2R$, can be expressed through
$x_{\beta}(1)$ and $t$, therefore, it also mapped into a matrix with
coefficients from $R$ under the conjugation by $g$.

Therefore all $x_{\beta}(t)$, where $t$ is either invertible, or is divided by
$2$, are mapped into matrices with coefficients from $R$ under the conjugation
by $g$. ∎

If (as in our case) $R$ is generated by invertible elements and the ideal
$2R$, then

$gG_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)g^{-1}\subseteq M_{N}(R)\cap
G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},S)=G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R),$

Therefore

$gG_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)g^{-1}=G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)\;\text{and}\;gE_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)g^{-1}=E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R).$

Consequently, if we take composition of the initial automorphism
$\varphi\in\,\mathrm{Aut}\,(G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R))$ and
conjugation $i_{g^{-1}}$ by the element $g^{-1}$, we get some automorphism
$\rho\in\,\mathrm{Aut}\,(G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R))$, for which
$\rho(x_{\alpha}(1))=x_{\alpha}(1)$ for any $\alpha\in\mathbf{G}_{2}$.

######  Lemma 2.

Any automorphism
$\rho\in\,\mathrm{Aut}\,(G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R))$ (or
$\rho\in\,\mathrm{Aut}\,(E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R))$), for which
for all $\alpha\in\mathbf{G}_{2}$ $\rho(x_{\alpha}(1))=x_{\alpha}(1)$, is a
ring automorphism of the Chevalley group (respectively, its elementary
subgroup).

###### Proof.

For a given $\alpha\in\mathbf{G}_{2}$ by $\Gamma_{\alpha}$ let us denote the
set of all $x_{\beta}(1)$ such that $[x_{\alpha}(1),x_{\beta}(1)]=e$. In the
paper [16] it is proved, that the centralizer of $\Gamma_{\alpha}$ in the
Chevalley group (or in its elementary subgroup) coincides with $CX_{\alpha}$,
where $C$ is the center of the Chevalley group, and
$X_{\alpha}=\\{x_{\alpha}(t)\mid t\in R\\}$. In our root system
$\mathbf{G}_{2}$ center is trivial (since the group is adjoint), so we get
$X_{\alpha}$.

If under the action of $\rho$ the elements $x_{\beta}(1)$ are mapped into
itself, so the centralizer of any its set also is mapped into itself, so for
any $t\in R$ there exists $s\in R$ such that
$\rho(x_{\alpha}(t))=x_{\alpha}(s)$.

Let us show, that the obtained mapping $t\mapsto s$ does not depend of
choosing a root $\alpha\in\mathbf{G}_{2}$. Actually, it must coincide on roots
of the same length, since if $\alpha_{1}$ and $\alpha_{2}$ have the same
length, there exists an element from the Weil group $w$ (generated only by
$x_{\beta}(1)$) such that

$\forall t\in R\quad wx_{\alpha_{1}}(t)w^{-1}=x_{\alpha_{2}}(t).$

Let us denote the obtained mapping on long roots by $\rho_{1}$, and on short
roots by $\rho_{2}$.

Apply the automorphism $\rho$ to the expression
$[x_{\alpha}(t),x_{\beta}(1)]$, where $\alpha,\beta$ are short roots of the
system $\mathbf{G}_{2}$, $\alpha$ is long, $\beta$ is short:

$\rho([x_{\alpha}(t),x_{\beta}(1)])=[\rho(x_{\alpha}(t)),\rho(x_{\beta}(1))]\Longleftrightarrow\\\
\Longleftrightarrow
x_{\alpha+\beta}(\rho_{2}(t))x_{\alpha+3\beta}(\rho_{1}(-t))x_{\alpha+2\beta}(\rho_{2}(-t))x_{2\alpha+3\beta}(\rho_{1}(t^{2}))=\\\
=x_{\alpha+\beta}(\rho_{1}(t))x_{\alpha+3\beta}(-\rho_{1}(t))x_{\alpha+2\beta}(-\rho_{1}(t))x_{2\alpha+3\beta}(\rho_{1}(t^{2})),$

so $\rho_{1}$ and $\rho_{2}$ coincide.

Denote the obtained mapping also by $\rho:R\to R$, we just need to prove that
this mapping is an automorphism of $R$.

Actually, its bijectivity follows from the fact that the initial automorphism
$\rho$ is bijective.

Its additivity follows from the formula

$x_{\alpha}(\rho(t_{1})+\rho(t_{2}))=x_{\alpha}(\rho(t_{1}))x_{\alpha}(\rho(t_{2}))=\rho(x_{\alpha}(t_{1}))\cdot\rho(x_{\alpha}(t_{2}))=\\\
=\rho(x_{\alpha}(t_{1})x_{\alpha}(t_{2}))=\rho(x_{\alpha}(t_{1}+t_{2})),$

and multiplicativity follows from formula $(R2)$

$x_{2\alpha+3\beta}(\rho(t_{1})\rho(t_{2}))=[x_{\alpha}(\rho(t_{1})),x_{\alpha+3\beta}(\rho(t_{2}))]=\rho([x_{\alpha}(t_{1}),x_{\alpha+3\beta}(t_{2})])=\\\
=\rho(x_{2\alpha+3\beta}(t_{1}t_{2}))=x_{2\alpha+3\beta}(\rho(t_{1}t_{2}))$

where $\alpha,\beta$ are simple roots of $\mathbf{G}_{2}$.

Thus, $\rho$ is a ring automorphism on all $x_{\alpha}(t)$,
$\alpha\in\mathbf{G}_{2}$, $t\in R$, therefore it is a ring automorphism of
the elementary Chevalley group $E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$. If an
initial automorphism was considered only on the elementary subgroup, lemma is
proved.

Then consider the general Chevalley group
$G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$. If an automorphism $\rho$ coincides
with ring automorphism on an elementary subgroup, so if we take the
composition of it and the inverse mapping of the ring automorphism, we obtain
the automorphism $\rho^{\prime}$ of the Chevalley group, which is identical on
its elementary subgroup. In this case for any $g\in
G_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ and all $x\in
E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$ we have

$gxg^{-1}=x^{\prime}\Longrightarrow\rho^{\prime}(g)x\rho^{\prime}(g)^{-1}=x^{\prime}\Longrightarrow\rho^{\prime}(g)x\rho^{\prime}(g)^{-1}=gxg^{-1}\Longrightarrow(g^{-1}\rho^{\prime}(g))x=x(g^{-1}\rho^{\prime}(g)).$

As this equality is true for $x\in E_{\,\mathrm{ad}\,}(\mathbf{G}_{2},R)$, and
the centralizer of the elementary subroup of an adjoint Chevalley group is
trivial, then $g^{-1}\rho^{\prime}(g)=e$ and $\rho^{\prime}$ is a trivial
automorphism.

Therefore, $\rho$ is a ring automorphism of the Chevalley group, and the
initial automorphism $\varphi$ is a composition of a ring and inner
automorphisms, what was required.

∎

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[]JihwanLee []AdityaKommineni []TiantianFeng []KleanthisAvramidis []XuanShi
[]Sudarsana ReddyKadiri []ShrikanthNarayanan

# Toward Fully-End-to-End Listened Speech Decoding from EEG Signals

###### Abstract

Speech decoding from EEG signals is a challenging task, where brain activity
is modeled to estimate salient characteristics of acoustic stimuli. We propose
FESDE, a novel framework for Fully-End-to-end Speech Decoding from EEG
signals. Our approach aims to directly reconstruct listened speech waveforms
given EEG signals, where no intermediate acoustic feature processing step is
required. The proposed method consists of an EEG module and a speech module
along with a connector. The EEG module learns to better represent EEG signals,
while the speech module generates speech waveforms from model representations.
The connector learns to bridge the distributions of the latent spaces of EEG
and speech. The proposed framework is both simple and efficient, by allowing
single-step inference, and outperforms prior works on objective metrics. A
fine-grained phoneme analysis is conducted to unveil model characteristics of
speech decoding. The source code is available here: github.com/lee-jhwn/fesde.

###### keywords:

speech decoding, speech synthesis, EEG, neural activity, brain signals

##  1 Introduction

Brain-computer interfaces (BCI), particularly those targeting decoding
listened or articulated speech from neural (brain activity) signals such as
electroencephalography (EEG) and electrocorticography (ECoG), hold promise for
improving life quality and rehabilitation outcomes for patients with
communication disorders. Speech decoding technologies from brain activity also
offer possibilities for seamless, immersive, and interactive entertainment,
where command and control are coordinated without compromising sensitive
information accessible to others. For instance, a research group recently
successfully prototypes a digital avatar of a paralyzed patient relying on
understanding and interpretation of ECoG signals [1] enabling the patient to
communicate more freely.

Several approaches have been proposed to decode speech or text from neural
activities such as EEG or ECoG signals [1, 2, 3, 4, 5, 6]. For example, many
efforts have proposed the use of convolution neural networks (CNN) to decode
speech from EEG [7, 8] and ECoG signals [9] especially to reconstruct speech
signals or representations, like mel-spectrogram, from brain activity signals.
Furthermore, several methods to decode imagined speech from neural activities
have also been proposed by [10, 11].

Decoding speech from EEG signals rather than ECoG signals is more practical
and economical in real world scenarios. Acquiring ECoG signals is challenging
and limited to certain types of patients, as a surgical intervention is
necessary to implant the sensing electrodes. On the other hand, obtaining EEG
signals only requires participants to wear a cap-like device. Hence, speech
decoding frameworks based on EEG signals can easily be applied to larger group
of users, while ECoG based approaches are primarily focused on patients with
specific clinical conditions such as epilepsy or certain movement disorders.

However, modeling and decoding information from EEG recordings has been a
significant signal processing challenge, due to the low signal-to-noise ratio
(SNR) that is inherent in EEG measurements and sparsity of measurements. EEG
devices attempt to measure electrical activity in the brain through non-
invasive electrodes, spatially arranged on the scalp. Hence, the sensed
signals are weakened, with low spatial resolution, and contaminated by various
biological, environmental noises, such as muscle activity, cardiac activity,
eye movements, power lines, or nearby electronic devices [12, 13]. Various
approaches have been proposed to deal with these sources of variability,
ranging from sophisticated pre-processing and decomposition [12] algorithms to
representation learning frameworks that are trained in a label-agnostic, self-
supervised regime [14, 15].

Decoding speech waveforms directly from EEG signals has numerous advantages
when compared to multi-step approaches that usually incorporate signal-to-text
estimation as an intermediate step. Speech contains rich information such as
prosody including rhythm, intonation, emotion, and speaker identity that are
lost in the typical written lexical form. Hence, a pipelined approach to
processing EEG signals that first goes through text decoding, followed by
text-to-speech (TTS), faces challenges in decoding high-fidelity speech from
the loss of important prosodic and other paralinguistic information.

Recent development in TTS technology may facilitate direct speech decoding
from EEG signals. Fully-end-to-end TTS systems, such as VITS [16], are TTS
systems that directly synthesize speech in waveform, rather than intermediate
acoustic features like mel-spectrograms or mel-frequency cepstral coefficients
(MFCC) [17]. Fully-end-to-end TTS systems have numerous advantages over other
TTS systems that require any additional intermediate acoustic feature mapping
step. As the intermediate step is skipped, fully-end-to-end TTS systems are
faster, simpler, and less vulnerable to error accumulation, compared to TTS
systems that require an extra step of converting acoustic features to final
waveforms. With such recent developments in the TTS domain, we believe the
intermediate acoustic feature extraction step can be omitted.

In this paper, we propose FESDE, a novel framework for Fully-End-to-end Speech
Decoding from EEG signals. Our proposed framework directly generates waveform
of listened speech given EEG signals, without any additional intermediate
acoustic feature mapping step, such as to mel-spectrogram. To the best of our
knowledge, this is the first framework that directly generates speech waveform
given respective EEG signals. FESDE consists of three parts: the EEG module,
the speech module, and the connector. The EEG module learns to provide a
better representation of EEG signals. The speech module aims to generate the
speech waveform from the speech embeddings. The connector learns to map and
convert the distribution of EEG embedding to speech embedding. Figure 1
illustrates the overall structure of the proposed framework. During inference,
only the EEG encoder and the speech decoder are utilized, along with the
connector.

The proposed framework has various advantages over previous approaches. It
enables single-step inference and does not require any additional pipelines
such as a vocoder. Hence, it is faster and more straightforward compared to
previous multi-step approaches, while outperforming the previous approach in
terms of objective metrics.

The main contributions of the paper are as follows:

  * •

We propose a fully-end-to-end speech decoding framework for EEG signals, by
incorporating EEG module and speech module, reducing multiple steps into one.
To our best knowledge, this is the first attempt to directly reconstruct
listened speech waveform from EEG signals.

  * •

The proposed framework allows single-step inference, which is simpler and
faster, while also outperforming the previous approaches in objective
measures.

  * •

We conduct phoneme analysis and present the characteristics of phonemes that
are easy or challenging to decode by this approach.

##  2 Proposed Method

###  2.1 Model Architecture

Figure 1: The overall schematic of the proposed method. The EEG module is
trained to produce descriptive representations of EEG signals. The speech
module aims to generate speech waveform from the speech embeddings. The
connector converts the distribution of the EEG embedding into the speech
embedding. During inference, only the EEG encoder and the speech decoder are
utilized, along with the connector.

As shown in Figure 1, the proposed framework consists of three parts: the EEG
module, the speech module, and the connector. The EEG and speech signals are
handled by their respective modules. The connector bridges the two
intermediate embeddings from EEG and speech. During inference, only the EEG
encoder, the connector, and the speech decoder are used. For detailed
implementation, refer to the source code.

####  2.1.1 EEG Module

The EEG module is based on [15], a self-supervised learning framework for EEG
signal representation. The EEG module consists of an encoder, that learns to
encode input EEG signals into intermediate EEG embeddings, and a decoder, that
learns to reconstruct EEG signals from those embeddings. The EEG encoder
consists of convolution blocks and Structured State Space Sequence (S4) layers
[18]. Each convolution block is composed of a 1D convolution layer, a dropout
layer, layer normalization, and GELU activation. Each S4 layer consists of an
S4 kernel estimator, gated linear unit (GLU) activation, dropout, and layer
normalization. The S4 layers show superior performance in encoding long range
temporal dependencies which make them suitable for encoding EEG signal [15].
The EEG decoder is composed of deconvolution blocks, where each block contains
a 1D transpose convolution layer along with a dropout layer, layer
normalization, and GELU activation.

####  2.1.2 Speech Module

We base the speech module on VITS [16], one of the well performing fully-end-
to-end TTS frameworks. The speech module consists of two parts: the speech
encoder and the speech decoder. The speech encoder takes linear spectrograms
as input and outputs the intermediate speech embeddings. It is composed of
non-causal WaveNet [19] residual blocks, as in [16, 20, 21], and a projection
layer that outputs the mean and the variance of the distribution of the speech
embeddings. We adopt HiFi-GAN V1 [22] as the speech decoder, as in [16].

####  2.1.3 Connector

The connector consists of two parts: the prenet and the flow. As in [16], the
prenet consists of the transformer encoder [23] and a linear projection layer.
It takes the intermediate EEG embeddings as input and outputs mean and
variance of the distribution. The flow is identical to the normalizing flow
network in [16], with a stack of affine coupling layers [24]. The flow works
as an invertible function between the two distributions of the EEG and speech
embeddings. There is a gradient stop between the intermediate EEG embeddings
and the connector. We empirically observed that the gradient stop stabilizes
the training of the EEG module, as it is not affected by the performance of
the speech module in early stages of training.

###  2.2 Training Objectives

Employing the approach in [15], the cosine similarity loss is used for
training the EEG module, as in Eq. (1):

$\displaystyle
L_{\text{EEG}}(x,\hat{x})=1-\frac{1}{N_{\text{ch}}}\sum_{i=1}^{N_{\text{ch}}}\frac{x_{i}^{T}\cdot\hat{x_{i}}}{\|x_{i}\|\|\hat{x_{i}}\|}$
(1)

where $N_{\text{ch}}$ is the number of EEG channels, and $x$ and $\hat{x}$
represent the input and the reconstructed EEG signals, respectively.

We adopt training objectives for the speech module from [16, 22]. The
reconstruction loss $L_{\text{mel}}$ is defined as the L1 loss between the
mel-spectrograms of generated and actual speech waveform. The KL-divergence
loss $L_{\text{KL}}$ helps map the distributions of the intermediate
embeddings of speech and EEG to one another, as in Eq. (2).

$\displaystyle L_{\text{KL}}(p,q)=\log{q(z|y)}-\log{p(z|x)}$ (2)

where $y$ is the input speech and $z$ is the intermediate speech embedding.
Hence, $q(z|y)$ and $p(z|x)$ represent the distribution of the intermediate
speech embeddings given $y$ (speech) and $x$ (EEG signals), respectively. The
total loss for the speech module $L_{\text{speech}}$ is as follows:

$\displaystyle\vspace{-0.5cm}L_{\text{speech}}=L_{\text{mel}}+L_{\text{KL}}+L_{\text{GAN}}\vspace{-0.5cm}$
(3)

where $L_{\text{GAN}}$ consists of the adversarial loss and the feature
matching loss from HiFi-GAN V1 [22]. Note that gradient stop is applied
between the EEG module and the connector, the training of the EEG module is
not affected by the speech module.

Table 1: MCD (dB) and Mel-Corr (%) of each configuration, with 95% confidence intervals. Model | MCD (dB) $\downarrow$ | Mel-Corr (%) $\uparrow$
---|---|---
unseen audio | unseen subject | unseen both | unseen audio | unseen subject | unseen both
baseline [7] | $13.00\pm 0.12$ | $13.20\pm 0.12$ | $12.82\pm 0.39$ | $13.27\pm 0.66$ | $12.66\pm 0.74$ | $13.07\pm 2.06$
Ours | vanilla | $11.67\pm 0.12$ | $11.72\pm 0.11$ | $11.56\pm 0.36$ | $13.75\pm 0.75$ | $12.76\pm 0.68$ | $13.74\pm 2.15$
pt-audio | $11.58\pm 0.12$ | $11.61\pm 0.11$ | $11.68\pm 0.34$ | $14.34\pm 0.68$ | $13.84\pm 0.68$ | $13.31\pm 2.25$
pt-audio-fz | $11.71\pm 0.11$ | $11.71\pm 0.10$ | $11.57\pm 0.31$ | $12.80\pm 0.71$ | $12.97\pm 0.68$ | $11.11\pm 2.47$
pt-audio-eeg | $11.67\pm 0.12$ | $11.64\pm 0.11$ | $11.69\pm 0.34$ | $\bm{14.71\pm 0.68}$ | $14.42\pm 0.70$ | $\bm{14.46\pm 2.42}$
pt-audio-eeg-fz | $\bm{11.45\pm 0.10}$ | $\bm{11.46\pm 0.10}$ | $\bm{11.43\pm 0.32}$ | $14.56\pm 0.70$ | $\bm{14.50\pm 0.69}$ | $14.04\pm 2.06$

##  3 Experiments

###  3.1 Dataset

We conducted experiments on the N400 dataset [25], which contains EEG signals
that were recorded from 24 subjects, while each subject was listening to 440
sentences in English. The speech samples are around 2 - 3 seconds long and
contain 5 to 8 words. The 128 channel EEG was recorded at a sampling rate of
$512$ Hz, while the participants listened to the sentences. Four
subjects111subject # 5, 10, 15, and 18. were excluded owing to unreliable
data, as suggested by [25]. The test set consists of two subjects222subject #
23 and 24. and 40 sentences. As a result, the train set contains 7,200
pairs33318 subjects with 400 sentences each. and the remaining 1,600 pairs
were selected for the test set, where 720 and 800 of them are unseen audio and
subject, respectively, and 80 are unseen in both audio and subject.

The pre-processing pipeline for EEG is as follows: First, a notch filter at
$60$ Hz was employed in order to remove the powerline noise. Then, bandpass
filtering with frequency limits of low $0.5$ Hz and high $50$ Hz was applied
in order to preserve the bands relevant to EEG spectral information, followed
by eye blink removal using independent component analysis (ICA). The signals
were then resampled to $256$ Hz.

All of the speech samples were down-sampled to $22,050$ Hz, to match the
sampling rate of pre-trained speech models. For linear spectrograms, the
following short-time Fourier transform (STFT) parameters were used: $1024$ for
both FFT and window size, and $256$ for the hop size. For mel-spectrograms,
the same parameters were utilized with $80$ mel-bands.

###  3.2 Experimental Setup

Five different training configurations were considered as below:

  * •

vanila: all of the modules were trained from scratch, without any pre-trained
parameters.

  * •

pt-audio: only the speech module was initialized with pre-trained parameters
for training.

  * •

pt-audio-fz: the pre-trained parameters were used for the speech module, but
frozen, that is, the speech module was not trained.

  * •

pt-audio-eeg: both of the EEG module and the speech module were initialized
with pre-trained parameters.

  * •

pt-audio-eeg-fz: the EEG module and the speech module adopted the pre-trained
parameters, but they were not trained.

For the pre-trained speech module,
pretrained_ljs.pth444https://github.com/jaywalnut310/vits was adopted. In pt-
audio-eeg and pt-audio-eeg-fz, the EEG module, that had been pre-trained for
300 epochs, was later frozen when combined with the speech module. The
compared baseline model is VLAAI [7], which consists of a stack of convolution
layers with skip connections. The final layer of the baseline model was
modified to generate 80 band mel-spectrograms. One Nvidia A40 GPU was utilized
for each training configuration. All of the FESDE training configurations were
trained for 100k iterations with the AdamW optimizer [26]. The baseline model
was trained for 68 epochs.

(a) MCD (dB) of each consonant (blue) and vowel (red) group. (b) Mel-Corr (%)
of each consonant (blue) and vowel (red) group.

Figure 2: MCD (dB) and Mel-Corr (%) of each phoneme group. The lower MCD and
higher Mel-Corr indicate better performance. The consonants (blue) are
clustered by three criteria: manner, place, and tenseness of articulation. The
vowels (red) are clustered by its tongue position and tenseness.

##  4 Results and Discussion

###  4.1 Evaluation

The performance was measured in the following two objective metrics: mel-
cepstral distortion (MCD) [27] and mel-spectrogram correlation (Mel-Corr). MCD
measures the distance between two mel-cepstral coefficients as Eq (4):

$\displaystyle\vspace{-0.5cm}\text{MCD}=\alpha\sqrt{\sum_{i=1}^{N_{\text{MCC}}}(\text{MCC}_{i}-\widehat{\text{MCC}_{i}})^{2}}\vspace{-0.5cm}$
(4)

where $\alpha=\frac{10\sqrt{2}}{\ln 10}$ and $N_{\text{MCC}}$ is the number of
mel-cepstral coefficients. Mel-Corr is calculated as the Pearson correlation
coefficient between two mel-spectrograms. For convenience, the reported
numbers are multiplied with 100. Lower MCD and higher Mel-Corr values indicate
better performance.

Table 1 shows the MCD and Mel-Corr values for the baseline model and the
proposed five models. From the results in the table, the best performance was
achieved when both of the EEG and speech modules are pre-trained separately
(i.e., pt-audio-eeg-fz). Also, almost all of the proposed models perform
better than the baseline model.

###  4.2 Phoneme Analysis

In order to analyze the effectiveness of proposed models in decoding speech at
the level of phoneme, a fine-grained analysis was carried out by calculating
some objective metrics. This analysis will reveal which of the category of
phonemes are difficult or easy to decode. The Montreal forced aligner (MFA)
ver3 [28] was used to acquire the timestamps of each phoneme. The pre-trained
english_us_arpa configuration was used for both acoustic model and the
dictionary. As MFA does not perform accurately for reconstructed speech, the
timestamps acquired from the ground-truth speech are assumed to be identical
to the reconstructed speech.

The consonants are grouped according to their manner of articulation, place of
articulation, and tenseness of articulation. Similarly, the vowels are grouped
based on position of the tongue and tenseness. As illustrated in Figures 2,
the consonants that are nasal, dental, or lenis tend to be relatively easily
decoded. Also, an interesting tendency is observed wherein more closed vowels
are easier to decode.

##  5 Ablation Study

###  5.1 EEG Channels

As recent studies [29] suggest, we explore the utilization of only the
parietal and temporal regions of EEG that are known to contain rich auditory
information. Instead of an increase in performance, a drop in performance was
observed, as shown in Table 2, especially when the speech module was frozen
during training. This result may suggest that non-auditory parts of the brain
may be involved in speech decoding, however, further investigation is
necessary for a clearer explanation.

Table 2: Performance drop (%) when only the parietal and temporal regions of EEG signals are used. Model | Performance Drop (%)
---|---
| MCD | Mel-Corr
pt-audio-eeg | $-0.13$ | $-5.99$
pt-audio-eeg-fz | $-1.44$ | $-14.64$

###  5.2 Text-Spotting

Apart from evaluating the performance of speech reconstruction from EEG, we
investigate whether our proposed EEG encoder captures textual information. A
binarized word-spotting experiment was conducted to validate whether our
proposed EEG presentation can be used to identify certain words in a speech
sample. The 30 most frequently occurring nouns among all sentences are chosen
as the set of keywords, and the task is to detect whether any of these
keywords exist in a speech. The pre-trained EEG module was utilized. The
results across different test conditions are presented in Table 3. We
demonstrate the feasibility of text detection from EEG signals, and it remains
a challenging and open endeavor.

Table 3: Performance of predicting binarized word-spotting. The performance score is measured in the unweighted F1 score. unseen audio | unseen subject | unseen both
---|---|---
$60.12$ | $56.74$ | $57.78$

##  6 Conclusion

In this work, we introduce FESDE, a framework to decode listened speech
waveforms directly from EEG signals. The proposed approach is faster and
simpler by enabling single-step inference and it also outperforms the baseline
model that involves an intermediate conversion to text representation. We also
explore the characteristics of phonemes that are easily or challenging to be
decoded by the proposed framework. Our approach is currently limited to
listened speech, however, in the foreseeable future, we plan to extend our
research scope to speech production tasks, such as imagined or phonated speech
decoding.

##  7 Acknowledgements

This project was partially supported by a USC Annenberg Graduate Fellowship
and also by the Defense Advanced Research Projects Agency (DARPA) under
cooperative agreement No. N660012324006. The content of the information does
not necessarily reflect the position or the policy of the Government, and no
official endorsement should be inferred.

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