# On some relationships between the centers and the derived ideal in Leibniz
3-algebras

P. Ye. Minaiev, O. O. Pypka
Oles Honchar Dnipro National University, Dnipro, Ukraine
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

One of the classic results of group theory is the so-called Schur theorem. It
states that if the central factor-group $G/\zeta(G)$ of a group $G$ is finite,
then its derived subgroup $[G,G]$ is also finite. This result has numerous
generalizations and modifications in group theory. At the same time, similar
investigations were conducted in other algebraic structures. In 2016, L.A.
Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for
Leibniz algebras: if central factor-algebra $L/\zeta(L)$ of Leibniz algebra
$L$ has finite dimension, then its derived ideal $[L,L]$ is also finite-
dimensional. Moreover, they also proved a slightly modified analogue of Schur
theorem: if the codimensions of the left $\zeta^{l}(L)$ and right
$\zeta^{r}(L)$ centers of Leibniz algebra $L$ are finite, then its derived
ideal $[L,L]$ is also finite-dimensional. One of the generalizations of
Leibniz algebras is the so-called Leibniz $n$-algebras. Therefore, the
question of proving analogs of the above results for this type of algebras
naturally arises. In this article, we prove the analogues of the two mentioned
theorems for Leibniz 3-algebras.

Key Words: Leibniz algebra, Leibniz 3-algebra, center, derived ideal, Schur
theorem.

2020 MSC: 17A32, 17A40.

##  1 Introduction.

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and
$[-,-]$. Then $L$ is called a left Leibniz algebra if it satisfies the left
Leibniz identity:

$[a,[b,c]]=[[a,b],c]+[b,[a,c]]$

for all $a,b,c\in L$. Leibniz algebras first appeared in the paper [2], but
the term “Leibniz algebra” appears in the book [20] and article [21]. In [22],
the authors conducted an in-depth study on Leibniz algebras properties. The
theory of Leibniz algebras has developed very intensely in many different
directions.

Note that Lie algebras present a partial case of Leibniz algebras. Conversely,
if $L$ is a Leibniz algebra in which $[a,a]=0$ for every element $a\in L$,
then it is a Lie algebra. Thus, Lie algebras can be characterized as
anticommutative Leibniz algebras.

One of the key tendencies in the development of Leibniz algebra theory is the
search for analogues of the basic results of Lie algebra theory. At the same
time, there are very significant differences between these two types of
algebras (see, for example, [4]).

We recall some necessary definitions. Let $L$ be a Leibniz algebra over a
field $F$. If $A,B$ are subspaces of $L$, then $[A,B]$ will denote a subspace
generated by all elements $[a,b]$ where $a\in A$, $b\in B$.

A subspace $A$ of $L$ is called a subalgebra of $L$, if $[a,b]\in A$ for every
$a,b\in A$. It follows that $[A,A]\leq A$. A subalgebra $A$ of $L$ is called a
left (respectively right) ideal of $L$, if $[b,a]\in A$ (respectively
$[a,b]\in A$) for every $a\in A$, $b\in L$. In other words, if $A$ is a left
(respectively right) ideal of $L$, then $[L,A]\leq A$ (respectively $[A,L]\leq
A$). A subalgebra $A$ of $L$ is called an ideal of $L$ (more precisely, two-
sided ideal) if it is both a left ideal and a right ideal. If $A$ is an ideal
of $L$, we can consider a factor-algebra $L/A$. It is not hard to see that
this factor-algebra also is a Leibniz algebra.

The left $\zeta^{l}(L)$ and right $\zeta^{r}(L)$ centers of a Leibniz algebra
$L$ are defined by the rules:

$\displaystyle\zeta^{l}(L)$ $\displaystyle=\\{a\in L|\ [a,b]=0\ \mbox{for all
}b\in L\\},$ $\displaystyle\zeta^{r}(L)$ $\displaystyle=\\{a\in L|\ [b,a]=0\
\mbox{for all }b\in L\\}.$

It is not hard to prove that the left center of $L$ is an ideal, but it is not
true for the right center. The right center is a subalgebra of $L$, and, in
general, the left and right centers are different. They even may have
different dimensions (see, for example, [10]).

The center $\zeta(L)$ of $L$ is the intersection of the left and right
centers, that is

$\zeta(L)=\\{a\in L|\ [a,b]=0=[b,a]\ \mbox{for all }b\in L\\}.$

Clearly, the center $\zeta(L)$ is an ideal of $L$. In particular, we can
consider the factor-algebra $L/\zeta(L)$.

There is a very close connection between the center $\zeta(L)$ of the Leibniz
algebra $L$ and its derived ideal $[L,L]$. In [12], the authors prove that if
the central factor-algebra $L/\zeta(L)$ has finite dimension $d$, then the
derived ideal $[L,L]$ is also finite-dimensional and $dim_{F}([L,L])\leq
d^{2}$. Moreover, they proved some modification of this result: if
$codim_{F}(\zeta^{l}(L))=d$ and $codim_{F}(\zeta^{r}(L))=r$ are finite, then
$dim_{F}([L,L])\leq d(d+r)$.

The first of the mentioned results is a direct analogue of the so-called [18]
Schur theorem. More precisely, Schur theorem states that if the central
factor-group $G/\zeta(G)$ of a group $G$ is finite, then the derived subgroup
$[G,G]$ is also finite. In this formulation, for the first time it appears in
the paper [23]. This theorem was obtained also in [1]. This theorem has
numerous generalizations and modifications in group theory (see, for, example,
[6, 7, 11, 13, 14, 24]). Furthermore, similar investigations were conducted in
other algebraic structures of a different nature. Thus, analogues of Schur
theorem were obtained for modules (see, for example, [19]), linear groups [5],
topological groups [28], $n$-groups [9], associative algebras [25], Lie
algebras [15, 26], Lie $n$-algebras [27], Lie rings [16], Poisson algebras
[17].

In [3], the authors introduced the concept of Leibniz $n$-algebras. Let $L$ be
an $n$-algebra over a field $F$ with the binary operations $+$ and an
$n$-linear bracket $[-,\ldots,-]$. Then $L$ is called a left Leibniz
$n$-algebra if it satisfies the following left Leibniz $n$-identity:

$[b_{1},\ldots,b_{n-1},[a_{1},\ldots,a_{n}]]=\sum_{i=1}^{n}[a_{1},\ldots,a_{i-1},[b_{1},\ldots,b_{n-1},a_{i}],a_{i+1},\ldots,a_{n}]$

for any $a_{1},\ldots,a_{n},b_{1},\ldots,b_{n-1}\in L$. The theory of Leibniz
$n$-algebras is much less developed than the theory of Leibniz algebras. Our
goal is to establish connections between these types of algebras. In
particular, in this paper we will prove analogs of the two above-mentioned
results from the theory of Leibniz algebras for Leibniz 3-algebras.

##  2 Preliminary results.

Let $L$ be a 3-algebra over a field $F$ with the binary operation $+$ and
ternary operation $[-,-,-]$. Then $L$ is called a Leibniz $3$-algebra (more
precisely, a left Leibniz $3$-algebra) if for all elements $a,b,c,x,y\in L$ it
satisfies the left Leibniz $3$-identity:

$[x,y,[a,b,c]]=[[x,y,a],b,c]+[a,[x,y,b],c]+[a,b,[x,y,c]].$

If $A,B,C$ are subspaces of $L$, then $[A,B,C]$ will denote a subspace
generated by all elements $[a,b,c]$ where $a\in A$, $b\in B$, $c\in C$. As
usual, a subspace $A$ of $L$ is called a subalgebra of $L$ if $[a,b,c]\in A$
for all elements $a,b,c\in A$. In other words, $[A,A,A]\leq A$. A subalgebra
$A$ is called a left (respectively middle, right) ideal of $L$ if $[x,y,a]\in
A$ (respectively $[x,a,y]\in A$, $[a,x,y]\in A$) for every $x,y\in L$, $a\in
A$. In other words, if $A$ is a left (respectively middle, right) ideal of
$L$, then $[L,L,A]\leq A$ (respectively $[L,A,L]\leq A$, $[A,L,L]\leq A$). A
subalgebra $A$ of $L$ is called an ideal of $L$ (or three-sided ideal) if it
is a left, middle and a right ideal of $L$. If $A$ is an ideal of $L$, we can
say about the factor-algebra $L/A$. It is not hard to see that $L/A$ is also a
Leibniz 3-algebra.

Let $L$ be a Leibniz 3-algebra over a field $F$, $M$ be a non-empty subset of
$L$, and $H$ be a subalgebra of $L$. Put

$\displaystyle Ann_{H}^{l}(M)$ $\displaystyle=\\{a\in H|\ [a,M,M]=\langle
0\rangle\\},$ $\displaystyle Ann_{H}^{m}(M)$ $\displaystyle=\\{a\in H|\
[M,a,M]=\langle 0\rangle\\},$ $\displaystyle Ann_{H}^{r}(M)$
$\displaystyle=\\{a\in H|\ [M,M,a]=\langle 0\rangle\\}.$

The subset $Ann_{H}^{l}(M)$ (respectively $Ann_{H}^{m}(M)$, $Ann_{H}^{r}(M)$)
is called the left (respectively middle, right) annihilator of $M$ in $H$. The
intersection

$Ann_{H}(M)=Ann_{H}^{l}(M)\cap Ann_{H}^{m}(M)\cap Ann_{H}^{r}(M)$

is called annihilator of $M$ in $H$. We note the following basic properties of
annihilators.

######  Lemma 2.1.

Let $L$ be a Leibniz $3$-algebra over a field $F$, $M$ be an ideal of $L$, and
$H$ be a subalgebra of $L$. Then
$Ann_{H}^{l}(M),Ann_{H}^{m}(M),Ann_{H}^{r}(M)$ are subalgebras of $L$.

Proof. Let $A=Ann_{H}^{l}(M)$, $a,b,c\in A$, $x,y\in M$. We have:

$[[a,b,c],x,y]=[a,b,[c,x,y]]-[c,[a,b,x],y]-[c,x,[a,b,y]].$

Since $c\in A$, $[c,x,y]=0$. The fact that $M$ is an ideal of $L$ implies that
$[a,b,x]\in M$ and $[a,b,y]\in M$, so that $[c,[a,b,x],y]=[c,x,[a,b,y]]=0$.
Thus, $[[a,b,c],x,y]=0$.

Now, let $B=Ann_{H}^{m}(M)$, $a,b,c\in B$, $x,y\in M$. We have:

$[x,[a,b,c],y]=[a,b,[x,c,y]]-[[a,b,x],c,y]-[x,c,[a,b,y]].$

Since $c\in B$, $[x,c,y]=0$. Furthermore, $M$ is an ideal of $L$, so that
$[a,b,x]\in M$ and $[a,b,y]\in M$. Therefore, $[[a,b,x],c,y]=[x,c,[a,b,y]]=0$.
Thus,

$[x,[a,b,c],y]=0.$

Finally, let $C=Ann_{H}^{r}(M)$, $a,b,c\in C$, $x,y\in M$. We have:

$[x,y,[a,b,c]]=[[x,y,a],b,c]+[a,[x,y,b],c]+[a,b,[x,y,c]].$

Since $a,b,c\in C$, $[x,y,a]=[x,y,b]=[x,y,c]=0$. Thus, $[x,y,[a,b,c]]=0$.

Since $Ann_{H}(M)=A\cap B\cap C$, $Ann_{H}(M)$ is also a subalgebra of $L$.
$\Box$

Let $L$ be a Leibniz 3-algebra over a field $F$. Put

$\zeta(L)=\\{a\in L|\ [a,x,y]=[x,a,y]=[x,y,a]=0\ \mbox{for all }x,y\in L\\}.$

The subset $\zeta(L)$ is called the center of $L$. We note that the center of
$L$ is the annihilator of $L$ in $L$. The left, middle and right annihilators
lead us to the following subsets. Put

$\displaystyle\zeta^{l}(L)$ $\displaystyle=\\{a\in L|\ [a,x,y]=0\ \mbox{for
all }x,y\in L\\},$ $\displaystyle\zeta^{m}(L)$ $\displaystyle=\\{a\in L|\
[x,a,y]=0\ \mbox{for all }x,y\in L\\},$ $\displaystyle\zeta^{r}(L)$
$\displaystyle=\\{a\in L|\ [x,y,a]=0\ \mbox{for all }x,y\in L\\}.$

The subset $\zeta^{l}(L)$ (respectively $\zeta^{m}(L)$, $\zeta^{r}(L)$) is
called the left (respectively middle, right) center of $L$. Obviously,

$\zeta(L)=\zeta^{l}(L)\cap\zeta^{m}(L)\cap\zeta^{r}(L).$

In particular, we can define the lm-center of $L$ by the rule

$\zeta^{lm}(L)=\zeta^{l}(L)\cap\zeta^{m}(L).$

We note that, in general, $\zeta^{l}(L),\zeta^{m}(L),\zeta^{r}(L)$ are not
ideals of $L$. However, we have the following result.

######  Lemma 2.2.

Let $L$ be a Leibniz $3$-algebra over a field $F$. Then the following
assertions hold.

  1. (i)

$\zeta^{l}(L),\zeta^{m}(L),\zeta^{r}(L),\zeta(L)$ are subalgebras of $L$.

  2. (ii)

$\zeta^{lm}(L),\zeta(L)$ are ideals of $L$.

Proof. (i) Since

$Ann_{L}^{l}(L)=\zeta^{l}(L),Ann_{L}^{m}(L)=\zeta^{m}(L),Ann_{L}^{r}(L)=\zeta^{r}(L),Ann_{L}(L)=\zeta(L),$

Lemma 2.1 shows that $\zeta^{l}(L),\zeta^{m}(L),\zeta^{r}(L),\zeta(L)$ are
subalgebras of $L$.

(ii) Let $D=\zeta^{lm}(L)$, $v\in D$, $a,b,x,y\in L$. We have:

$\displaystyle[[x,y,v],a,b]$
$\displaystyle=[x,y,[v,a,b]]-[v,[x,y,a],b]-[v,a,[x,y,b]],$
$\displaystyle[a,[x,y,v],b]$
$\displaystyle=[x,y,[a,v,b]]-[[x,y,a],v,b]-[a,v,[x,y,b]].$

Since $v\in D$,

$\displaystyle[[x,y,v],a,b]$ $\displaystyle=[x,y,0]-0-0=0,$
$\displaystyle[a,[x,y,v],b]$ $\displaystyle=[x,y,0]-0-0=0.$

This means that $D$ is a left ideal of $L$. Furthermore,

$\displaystyle[[x,v,y],a,b]$
$\displaystyle=[x,v,[y,a,b]]-[y,[x,v,a],b]-[y,a,[x,v,b]]$
$\displaystyle=0-[y,0,b]-[y,a,0]=0,$ $\displaystyle[a,[x,v,y],b]$
$\displaystyle=[x,v,[a,y,b]]-[[x,v,a],y,b]-[a,y,[x,v,b]]$
$\displaystyle=0-[0,y,b]-[a,y,0]=0,$

which shows that $D$ is a middle ideal of $L$. Finally, we have:

$\displaystyle[[v,x,y],a,b]$
$\displaystyle=[v,x,[y,a,b]]-[y,[v,x,a],b]-[y,a,[v,x,b]]$
$\displaystyle=0-[y,0,b]-[y,a,0]=0,$ $\displaystyle[a,[v,x,y],b]$
$\displaystyle=[v,x,[a,y,b]]-[[v,x,a],y,b]-[a,y,[v,x,b]]$
$\displaystyle=0-[0,y,b]-[a,y,0]=0.$

In other words, $D$ is a right ideal of $L$. Thus, $D$ is an ideal of $L$.

In the same way we can show that the center $\zeta(L)$ of $L$ is an ideal of
$L$. $\Box$

Let $L$ be a Leibniz 3-algebra over a field $F$. A linear transformation $f$
of $L$ is called a derivation of $L$ if

$f([a,b,c])=[f(a),b,c]+[a,f(b),c]+[a,b,f(c)]$

for all $a,b,c\in L$.

Consider the mapping $l_{a,b}:L\rightarrow L$ defined by the rule
$l_{a,b}(x)=[a,b,x]$, $a,b\in L$. We note some basic properties of $l_{a,b}$.
Let $x,y,z\in L$, $\lambda\in F$. Then

$\displaystyle l_{a,b}(x+y)$
$\displaystyle=[a,b,x+y]=[a,b,x]+[a,b,y]=l_{a,b}(x)+l_{a,b}(y),$
$\displaystyle l_{a,b}(\lambda x)$ $\displaystyle=[a,b,\lambda
x]=\lambda[a,b,x]=\lambda l_{a,b}(x),$ $\displaystyle l_{a,b}([x,y,z])$
$\displaystyle=[a,b,[x,y,z]]$
$\displaystyle=[[a,b,x],y,z]+[x,[a,b,y],z]+[x,y,[a,b,z]]$
$\displaystyle=[l_{a,b}(x),y,z]+[x,l_{a,b}(y),z]+[x,y,l_{a,b}(z)].$

These equalities show that $l_{a,b}$ is a derivation of $L$.

##  3 Main result.

We now present the main result of this paper.

######  Theorem 3.1.

Let $L$ be a Leibniz $3$-algebra over a field $F$. If
$codim_{F}(\zeta^{lm}(L))=d$ and $codim_{F}(\zeta^{r}(L))=r$ are finite, then
$dim_{F}([L,L,L])\leq d^{2}(d+r)$.

Proof. We have: $L=\zeta^{lm}(L)\oplus E$ for some subspace $E$. Choose in $E$
a basis $\\{e_{1},\ldots,e_{d}\\}$. If $x,y,z$ are arbitrary elements of $L$,
then

$\displaystyle x$
$\displaystyle=\alpha_{1}e_{1}+\ldots+\alpha_{d}e_{d}+s_{1},$ $\displaystyle
y$ $\displaystyle=\beta_{1}e_{1}+\ldots+\beta_{d}e_{d}+s_{2},$ $\displaystyle
z$ $\displaystyle=\gamma_{1}e_{1}+\ldots+\gamma_{d}e_{d}+s_{3}$

for $\alpha_{i},\beta_{i},\gamma_{i}\in F$, $i\in\\{1,\ldots,d\\}$, and
$s_{j}\in\zeta^{lm}(L)$, $j\in\\{1,2,3\\}$. Then we have

$\displaystyle[x,y,z]$ $\displaystyle=\left[\sum_{1\leq i\leq
d}\alpha_{i}e_{i}+s_{1},\sum_{1\leq j\leq d}\beta_{j}e_{j}+s_{2},\sum_{1\leq
k\leq d}\gamma_{k}e_{k}+s_{3}\right]$ $\displaystyle=\sum_{1\leq i,j,k\leq
d}\alpha_{i}\beta_{j}\gamma_{k}[e_{i},e_{j},e_{k}]+\sum_{1\leq i,j\leq
d}\alpha_{i}\beta_{j}[e_{i},e_{j},s_{3}].$

It follows that the subspace $S$ generated by the elements
$[e_{i},e_{j},e_{k}]$, $1\leq i,j,k\leq d$, and the subspaces
$[e_{i},e_{j},\zeta^{lm}(L)]$, $1\leq i,j\leq d$, include $[L,L,L]$.

Put $Z=\zeta^{lm}(L)$ and let $a,b$ be arbitrary elements of $L$. We define a
mapping $l_{a,b}(x):Z\rightarrow Z$ by the rule $l_{a,b}(z)=[a,b,z]$, $z\in
Z$. As we noted above, this mapping is linear, $Im(l_{a,b})=[a,b,Z]$ and
$Ker(l_{a,b})=Ann_{Z}^{r}((a,b))$ where $(a,b)$ is an ordered pair of elements
$a$ and $b$. Hence

$[a,b,Z]=Im(l_{a,b})\cong_{F}Z/Ker(l_{a,b})=Z/Ann_{Z}^{r}((a,b)).$

Since $\zeta^{r}(L)\leq Ann_{L}^{r}((a,b))$, we have
$codim_{F}(Ann_{L}^{r}((a,b)))\leq r$ and so

$dim_{F}([a,b,Z])=dim_{F}(Z/Ann_{Z}^{r}((a,b)))\leq r.$

In particular, $dim_{F}([e_{i},e_{j},Z])\leq r$ for every $1\leq i,j\leq d$.
It follows that the subspace $S$ has dimension at most
$d^{3}+d^{2}r=d^{2}(d+r)$, as required. $\Box$

######  Corollary 3.2.

Let $L$ be a Leibniz $3$-algebra over a field $F$. If $codim_{F}(\zeta(L))=d$
is finite, then $dim_{F}([L,L,L])\leq d^{3}$.

Indeed, we have $\zeta(L)=\zeta^{lm}(L)\cap\zeta^{r}(L)$. Since
$codim_{F}(\zeta^{lm}(L))$ and $codim_{F}(\zeta^{r}(L))$ are finite,
$codim_{F}(\zeta(L))$ is finite. Then it suffices to take the proof of Theorem
3.1 into account to obtain Corollary 3.2.

Recall [8] that a 3-algebra $L$ over a field $F$ ($char(F)\neq 2$) with the
binary operations $+$ and a 3-linear bracket $[-,-,-]$ is called a Lie
$3$-algebra if it satisfies the following conditions:

(i) Lie 3-bracket is antisymmetric, that is
$[a_{1},a_{2},a_{3}]=sign(\sigma)[a_{\sigma(1)},a_{\sigma(2)},a_{\sigma(3)}]$;

(ii) Lie 3-bracket satisfies the generalized Jacobi identity (or Jacobi
3-identity), that is

$[x,y,[a,b,c]]=[[x,y,a],b,c]+[a,[x,y,b],c]+[a,b,[x,y,c]]$

for any $a,b,c,x,y\in L$ and any permutation $\sigma\in S_{3}$. Note that
since $char(F)\neq 2$ the first condition equivalent to
$[a_{1},a_{2},a_{3}]=0$ whenever $a_{i}=a_{j}$ for some $i\neq j$, $1\leq
i,j\leq 3$. Obviously, every Lie 3-algebra is a Leibniz 3-algebra. Therefore,
we can apply the previous results to this partial case. We have the following

######  Corollary 3.3.

Let $L$ be a Lie $3$-algebra over a field $F$. If $codim_{F}(\zeta(L))=d$ is
finite, then $dim_{F}([L,L,L])\leq\frac{d(d-1)(d-2)}{6}$.

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# Open-vocabulary Keyword-spotting with Adaptive Instance Normalization

###### Abstract

Open vocabulary keyword spotting is a crucial and challenging task in
automatic speech recognition (ASR) that focuses on detecting user-defined
keywords within a spoken utterance. Keyword spotting methods commonly map the
audio utterance and keyword into a joint embedding space to obtain some
affinity score. In this work, we propose AdaKWS, a novel method for keyword
spotting in which a text encoder is trained to output keyword-conditioned
normalization parameters. These parameters are used to process the auditory
input. We provide an extensive evaluation using challenging and diverse multi-
lingual benchmarks and show significant improvements over recent keyword
spotting and ASR baselines. Furthermore, we study the effectiveness of our
approach on low-resource languages that were unseen during the training. The
results demonstrate a substantial performance improvement compared to baseline
methods.

Index Terms—  user-defined keyword spotting, open vocabulary, adaptive
instance normalization

##  1 Introduction

Keyword spotting (KWS) is a critical component of many speech recognition
systems [1]. It involves identifying specific words or phrases within a
continuous audio stream. KWS is essential for various applications, from
automated transcription to voice-activated assistants [2]. Despite recent
advancements, KWS still faces unresolved challenges in adaptability and
customization. Keyword spotting methods are frequently trained to detect a
predefined set of target keywords [3, 4, 5]. This requires a large amount of
labeled data per keyword and widely restricts the usability of the models.
Importantly, adapting to custom target keywords generally requires retraining
or finetuning. Recently, few-shot learning and query-by-example approaches
have arisen as a more flexible alternative for KWS [6, 7, 8, 9, 10, 11, 12].
However, these works are still limited in performance on novel out-of-
vocabulary keywords and require obtaining several auditory examples, which can
be restricting and challenging, specifically in low-resource languages or
domains.

A fully adaptable keyword-spotting approach that can accurately detect user-
defined keywords in multiple languages, with no additional examples or
optimization, remains a challenging goal in KWS. Recently, several methods
have proposed open-vocabulary KWS systems that can generalize to keywords not
seen during training [13, 14, 15]. The methods rely on a text encoder for
aligning audio and text information in a joint latent space. While these
methods eliminate the need for auditory examples for novel keywords, they face
several limitations. First, a misalignment in audio and text representations
may arise from using two disjoint encoders [15]. Second, the reliance on a
phoneme model can limit the applicability in low-resource languages. Lastly,
these methods are evaluated on English benchmarks, and it is unclear how these
approaches generalize to diverse languages and dialects.

(a)

(b)

Fig. 1: The architecture for AdaKWS: (a) Overview: A text encoder outputs a
set of keyword-conditioned normalization parameters. The audio is processed
using the classifier which is conditioned on the keyword through the keyword-
adaptive modules. (b) A detailed view of the keyword adaptive module.

To address these issues, we take a different path to the open-vocabulary
problem and propose a novel approach for KWS, termed AdaKWS. Instead of
embedding textual and auditory information into a joint latent space, we
employ a character-based LSTM encoder which maps an input keyword into a set
of normalization parameters. These parameters are used to process the audio
signal in keyword-adaptive modules (see, Figure 1). The adaptive module is a
standard transformer encoder module in which we swap the Layer Normalization
layers with an Adaptive Instance Normalization (AdaIN) layers [16], where the
adaptive parameters are keyword-conditioned. AdaIN layers have proven highly
effective for multiple tasks and domains, including image harmonization,
image-to-image translation, and style editing.

To reduce the false detection of acoustically similar keywords and effectively
train our model, we introduce a new technique for mining hard negative
examples. Importantly, unlike previous recent works [13, 14, 15] which train
the KWS model on segmented audio samples containing the keyword alone, we
train our model on entire sentences (up to 30 seconds). This eliminates the
need for word-level alignment or expensive preprocessing and dramatically
increases the amount of available training data.

To summarize, we make the following contributions: (1) Propose a novel KWS
approach, named AdaKWS, based on adaptive instance normalization layers, (2)
Introduce a new technique for mining hard negative keyword examples, (3)
Demonstrate the benefits and effectiveness of our approach using a diverse set
of multilingual benchmarks, and generalization to unseen languages.

##  2 Keyword-spotting using Adaptive Instance normalization

The AdaKWS model is constructed of two main building blocks. A text encoder,
and an audio classifier. The audio classifier is in turn construct of two main
modules, an audio encoder, and a keyword-adaptive module (see Figure 1).

We first focus on the audio classifier, $f$. As the audio encoder, we use a
pre-trained Whisper transformer encoder [17]. In our experiments, we keep the
audio encoder frozen. The audio is first processed using the audio encoder,
and the resulting audio representation is then passed to two sequential
keyword-adaptive modules. Each module is a standard transformer encoder block
in which we swap the Layer Normalization layers with Adaptive Instance
Normalization layers (AdaIN). An AdaIn is a normalization layer of the form,

$\text{AdaIN}(z,v)=\sigma_{v}\left(\frac{z-\mu_{z}}{\sigma_{z}}\right)+\mu_{v}.$
(1)

In our case, $z$ is the audio representation and $v$ is the target keyword.
Finally, the keyword-conditioned audio representation is max-pooled and fed
into a linear classifier. To map the target keyword to the corresponding
normalization parameters, we employ a lightweight text encoder, $h$. The text
encoder is a 4-layers character-based LSTM with a $256$ hidden dimension.

More formally, let $\phi$ denote the parameters of the text encoder $h$, and
let $\theta$ denote the collection of parameters in $f$ which are shared among
keywords (i.e., all parameters that are not in the AdaIN layers). To detect a
keyword $v$ in an auditory utterance $x$, we first process $v$ using $h$ to
obtain the keyword conditioned normalization parameters
$h(v;\phi)=\theta_{v}$. Here, $\theta_{v}$ denote collection of all
normalization parameters $\mu(v),\sigma(v)$ from Eq. 1 Then, we determine the
existence of $v$ in $x$ according to $f(x;\theta,\theta_{v}):=P(v|x)$.

Note that under the above setup, the trainable parameters are $\phi,\theta$.
The text encoder parameters can be updated according to the chain rule,

$\nabla_{\phi}\mathcal{L}(x,v)=(\nabla_{\phi}\theta_{v})^{T}\nabla_{\theta_{v}}\mathcal{L}(x,v)$
(2)

where $\mathcal{L}$ denotes the classification loss (i.e., cross-entropy) for
$(x,v)$. We summarize our training procedure in Alg. 1.

Our method, AdaKWS, provides a natural way for sharing information across
keywords, through the shared parameters $(\phi,\theta)$, while maintaining the
flexibility of generating diverse keyword-specific classifiers.

Algorithm 1 AdaKWS

input: $\alpha$ — Text encoder learning rate, $\eta$ — classifier learning
rate, $\theta$ — Audio classifier parameters, $\phi$ — Text encoder parameters

for $r=1,...,n$ do

sample a batch of pos/neg examples $\\{(x_{i},v_{i})\\}_{i=1}^{B}.$

compute $\theta_{v_{i}}=h(v_{i};\phi)$

$\theta=\theta-\eta\frac{1}{B}\sum_{i}\nabla_{\theta}\mathcal{L}(x_{i},v_{i})$

$\phi=\phi-\alpha\frac{1}{B}\sum_{i}(\nabla_{\phi}\theta_{v_{i}})^{T}\nabla_{\theta_{v_{i}}}\mathcal{L}(x_{i},v_{i})$

return: $\phi,\theta$

Table 1: VoxPopuli: F1 results for the VoxPopuli test dataset. | CS | DE | EN | ES | ET | FI | FR | HR | HU | IT | LT | NL | PL | RO | SK | SL | Overall | # Params
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Whisper-Tiny | $48.5$ | $77.0$ | $83.4$ | $85.6$ | $39.5$ | $58.4$ | $78.3$ | $48.6$ | $47.0$ | $72.2$ | $38.4$ | $67.8$ | $74.7$ | $52.0$ | $43.8$ | $46.0$ | $69.3$ |  $39$M
Whisper-Small | $77.9$ | $89.3$ | $84.1$ | $86.3$ | $55.2$ | $81.5$ | $90.5$ | $72.5$ | $72.5$ | $85.4$ | $62.5$ | $85.8$ | $89.7$ | $77.3$ | $68.8$ | $64.3$ | $83.3$ |  $244$M
Whisper-Large-V2 | $91.0$ | $93.2$ | $80.5$ | $87.8$ | $78.5$ | $89.1$ | $93.2$ | $67.9$ | $87.4$ | $87.7$ | $76.1$ | $\mathbf{92.6}$ | $93.8$ | $87.7$ | $84.0$ | $80.0$ | $88.4$ |  $1550$M
AdaKWS-Tiny | $91.8$ | $94.4$ | $95.5$ | $95.1$ | $86.2$ | $92.1$ | $94.9$ | $89.3$ | $90.6$ | $90.8$ | $\mathbf{85.6}$ | $91.1$ | $93.0$ | $91.9$ | $91.7$ | $85.6$ | $92.8$ |  $15$M
AdaKWS-Base | $92.3$ | $95.2$ | $95.9$ | $95.1$ | $85.2$ | $92.2$ | $\mathbf{96.3}$ | $90.4$ | $91.3$ | $92.0$ | $82.1$ | $91.5$ | $94.3$ | $92.9$ | $93.9$ | $88.5$ | $93.7$ |  $31$M
AdaKWS-Small | $\mathbf{94.4}$ | $\mathbf{95.7}$ | $\mathbf{96.3}$ | $\mathbf{95.6}$ | $\mathbf{91.1}$ | $\mathbf{94.8}$ | $95.9$ | $\mathbf{93.0}$ | $\mathbf{92.7}$ | $\mathbf{92.8}$ | $78.5$ | $92.2$ | $\mathbf{95.5}$ | $\mathbf{94.8}$ | $\mathbf{94.5}$ | $\mathbf{89.9}$ | $\mathbf{94.6}$ |  $109$M

##  3 Negative Sampling

In this section, we present our negative sampling approach which we employ for
creating a diverse set of hard negative examples per batch.

Let $\\{(x_{i},\mathbf{v}_{i})\\}$ for $i=1,...,N$ denotes the training data.
Here $x$ denotes the speech utterance and $\mathbf{v}=(v_{1},...,v_{M})$
denotes the corresponding transcripts, with $v_{i}\in\mathcal{V}$ for all $i$.
Here $\mathcal{V}$ denotes a set of training keywords. During training, given
an example $(x,\mathbf{v})$, we sample a random positive keyword
$v^{+}\in\mathbf{v}$ to form a positive training example $(x,v^{+})$. In order
to generate a negative training example, we consider several alternatives.
Random negative. Here we simply sample a random keyword from
$v^{-}\in\mathcal{V}\setminus\mathbf{v}$. We empirically found that this
simple approach is not sufficient for training a KWS model that can accurately
separate acoustic similar words. Intuitively, the random sampled keywords are
acoustically far from the keywords in $\mathbf{v}$.

To address this issue, we introduce several approaches for constructing hard
negative examples.
Character substitution. Here, we alter a positive keyword $v^{+}$, by
substituting one or more of its characters. The new character can be chosen
randomly, or according to an a priori mapping of acoustically similar
characters (“s”$\to$“z”,“p”$\to$“b”, etc.).
Keyword concatenation. Here we form a negative keyword by concatenating a
random keyword $v\in\mathcal{V}\setminus\mathbf{v}$ to a positive keyword
$v^{+}$, i.e., $v^{-}=v\circ v^{+}$ or $v^{-}=v^{+}\circ v$, where $\circ$
denotes the concatenation operator.
Nearest keyword (NK). To obtain negative keyword $v^{-}$ that are acoustically
similar to a reference keyword $v^{+}$, we sample $v^{-}$ according to the
text embedding representation $e(v)$. Specifically, we use the last hidden
layer of $h$ to form the embedding representation. To ensure efficiency, we
sample negative examples within each training batch, by querying for the
keyword with the smallest cosine distance, i.e.,
$v^{-}=\arg\min_{v\in\mathcal{V}_{\mathcal{B}}\setminus
v^{+}}d(e(v^{+}),e(v))$, where $\mathcal{V}_{\mathcal{B}}$ denotes the
keywords in batch $\mathcal{B}$.

Fig. 2: Model predictions. We illustrate the predicted probability of a queried keyword (y-axis) across different time windows. Note that higher probability scores are aligned with the temporal occurrence of the word within the utterance (x-axis). Table 2: LibriPhrase: AUC and EER results for LibriPhrase hard (LH) and easy (LE) test splits. |  AUC (%) $\uparrow$ |  EER (%) $\downarrow$
---|---|---
| LH | LE | LH | LE
Triplet [12] | $54.88$ | $63.53$ | $44.36$ | $32.75$
Attention [11] | $62.65$ | $78.74$ | $41.95$ | $28.74$
DONUT [10] | $54.88$ | $63.53$ | $44.36$ | $32.75$
CMCD [13] | $73.58$ | $96.70$ | $32.90$ | $8.42$
EMKWS [14] | $84.21$ | $97.83$ | $23.36$ | $7.36$
CED [15] | $92.70$ | $\mathbf{99.84}$ | $14.40$ | $1.70$
AdaKWS-Tiny | $93.75$ | $99.80$ | $13.47$ | $1.61$
AdaKWS-Base | $94.39$ | $99.81$ | $12.60$ | $1.37$
AdaKWS-Small | $\mathbf{95.09}$ | $99.82$ | $\mathbf{11.48}$ | $\mathbf{1.21}$

##  4 Experiments

Table 3: Multilingual-LibriSpeech: Zero-shot F1 results for the Multilingual-LibriSpeech test dataset. | DE | EN | ES | FR | IT | NL | PL | Overall | # Params |  Inference Time (MS) $\downarrow$
---|---|---|---|---|---|---|---|---|---|---
Whisper-Tiny | $77.6$ | $85.8$ | $83.6$ | $73.4$ | $72.4$ | $69.2$ | $76.0$ | $77.9$ |  $39$M | $260$
Whisper-Small | $90.8$ | $91.3$ | $92.9$ | $88.2$ | $85.8$ | $86.1$ | $91.5$ | $89.6$ |  $244$M | $621$
Whisper-Large-V2 | $\mathbf{95.0}$ | $\mathbf{93.8}$ | $\mathbf{95.1}$ | $93.8$ | $91.7$ | $\mathbf{92.3}$ | $\mathbf{95.2}$ | $\mathbf{93.9}$ |  $1550$M | $1836$
AdaKWS-Tiny | $92.6$ | $91.4$ | $92.1$ | $91.9$ | $92.3$ | $87.9$ | $92.6$ | $91.3$ |  $15$M | $6$
AdaKWS-Base | $93.8$ | $92.6$ | $92.2$ | $92.8$ | $92.7$ | $90.0$ | $92.8$ | $92.4$ |  $31$M | $7$
AdaKWS-Small | $94.6$ | $\mathbf{93.8}$ | $94.4$ | $\mathbf{94.1}$ | $\mathbf{94.2}$ | $91.8$ | $94.4$ | $93.8$ |  $109$M | $11$

In this section, we compare AdaKWS with various KWS approaches on different
learning setups and datasets. The experiments show the superiority of AdaKWS
over previous KWS methods.
Datasets. We use several datasets. VoxPopuli [18]: A large-scale multilingual
speech corpus collected from the European parliament, with $\sim 1800$ hours
of transcribed utterances from $16$ languages. LibriPhrase [13]: A recent
benchmark for KWS based on the LibriSpeech [19] dataset. It consists of two
splits, LibriPhrase Hard (LH) and LibriPhrase Easy (LE). Furthermore, we
evaluate our model using two additional datasets, Multilingual LibriSpeech and
Fluers [20]. To construct our evaluation set, we first randomly sample
positive keywords. Then, we sample random, concat, and swap negatives as
described in Section 3, with equal probability. This results in a diverse and
challenging open-vocabulary KWS benchmark.
Experimental setup. The AdaKWS models are trained using the VoxPopuli dataset
for $25$ epochs with batch-size $144$ and $1e-4$ learning rate. We train three
variants with varying audio encoder sizes, Tiny, Base, and Small,
corresponding to the frozen Whisper encoder (audio encoder, Figure 1). We
report common metrics in the KWS field, namely F1 score, Area Under the Curve
(AUC), and Equal Error Rate (ERR).
Data preprocessing. We follow the data processing procedure from [17]. Each
audio instance is resampled to a frequency of 16KHz, then an 80-channel log-
magnitude Mel spectrogram representation is generated using 25-millisecond
windows and 10-millisecond strides.
Baselines. We compare AdaKWS with recent keyword spotting and ASR works. The
compared methods include: (1) Whisper [17] \- We transcribe the input audio
into text and subsequently conduct a search to ascertain the presence of the
desired keyword within the output text. (2) CED [15] \- flexible KWS method
using audio-compliant text encoder. (3) EMKWS [14] \- contrastive learning
approach using text-audio embedding matching. (4) CMCD [13] \- fusing the text
and audio representation to shared latent space using cross-modal attention.
Additional query by example methods (5) DONUT [10], (6) Attention [11], and
(7) Triplet [12].

Table 4: Zero-shot performance for low resource, novel languages: F1 results for four novel and low resources languages from the Fleurs test dataset. | Icelandic | Maltese | Swahili | Uzbek | Overall
---|---|---|---|---|---
Whisper-Tiny | $42.3$ | $37.6$ | $43.9$ | $34.2$ | $37.8$
Whisper-Small | $52.0$ | $34.9$ | $55.6$ | $35.4$ | $40.4$
Whisper-Large-V2 | $69.2$ | $38.6$ | $\mathbf{74.9}$ | $36.6$ | $48.1$
AdaKWS-Tiny | $69.2$ | $\mathbf{76.7}$ | $70.1$ | $\mathbf{67.5}$ | $\mathbf{71.9}$
AdaKWS-Base | $\mathbf{71.5}$ | $\mathbf{76.7}$ | $71.6$ | $65.7$ | $71.6$
AdaKWS-Small | $70.6$ | $76.0$ | $69.4$ | $66.1$ | $70.9$

###  4.1 Main Results

We start with evaluating AdaKWS on the VoxPopuli dataset. The results are
presented in Table 1. AdaKWS outperforms the Whisper baselines by a notable
margin, over $6\%$ w.r.t the best performing baseline. Moreover, AdaKWS
outperforms the baselines in $15$ of $16$ languages with $\sim 10\times$ fewer
parameters. We follow the protocol from [13, 15, 14] and evaluate AdaKWS on
the LibriPhrase dataset. We use AdaKWS pre-trained on VoxPopuli and fine-tune
it using $30K$ samples from the LibriPhrase training set for an additional
$10$ epochs. The results are presented in Table 2. AdaKWS significantly
outperforms the current SOTA method on the challenging LibriPhrase Hard
benchmark.

###  4.2 Generalization to Novel Languages and Datasets

We further investigate the generalization ability of AdaKWS in two ways:
Adapting to novel, low-resource languages, and generalizing to datasets that
are not utilized during training. For both experiments, we use AdaKWS trained
on the VoxPopuli dataset from Section 4.1 _without fine-tuning_. First, we
evaluate the pre-trained AdaKWS on a subset of $4$ languages randomly drawn
from Fleurs [20], which share the same character set observed during training.
Second, we evaluate how well AdaKWS generalizes to unseen utterances from the
multilingual LibriSpeech (MLS) [19] dataset. We also report the inference time
for each model estimated using a single V$100$ Nvidia GPU, averaged using
$1000$ random samples. The results are presented in Tables 4 and 3. For MLS,
our AdaKWS-Small model achieves on-par performance with $\sim 160$x faster
inference time, w.r.t the Whisper large-v2 baseline.

###  4.3 Ablation Study

We empirically investigate the effects of distinct negative sampling methods
on the performance of our model on the Voxpopuli dataset. The results
presented in Table 5 showcase the importance of the proposed negative sampling
approach for AdaKWS performance.

Table 5: Negative sampling: Results for the AdaKWS-tiny model, trained using different negative sampling approaches, on the VoxPopuli test dataset. |  F1 $\uparrow$ |  AUC $\uparrow$ |  EER $\downarrow$
---|---|---|---
Random | $79.37$ | $88.09$ | $19.37$
Random + NK | $81.24$ | $88.79$ | $18.80$
Random + NK + Cat. | $84.99$ | $91.20$ | $15.91$
Random + NK + Cat. + Swap | $\mathbf{92.87}$ | $\mathbf{97.78}$ | $\mathbf{7.16}$

##  5 Conclusion

In this paper, we present AdaKWS, a novel approach for keyword spotting. Our
model employs a keyword adaptive instance normalization layer which allows the
model to adapt to novel keywords during inference. Along with the introduced
negative keyword mining approach, AdaKWS achieves state-of-the-art results in
challenging open-vocabulary multilingual setups. Furthermore, we demonstrate
the generalization abilities of AdaKWS to adapt to unseen languages and
datasets. The impressive achievements of AdaKWS not only advance the current
state-of-the-art but also pave the way for future research in this domain.

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