# On the value of the fifth maximal projection constant

Beata Derȩgowska111B.D. is partially supported by National Science Center
(NCN) grant no. 2021/05/X/ST1/01212. For the purpose of Open Access, the
author has applied a CC-BY public copyright licence to any Author Accepted
Manuscript (AAM) version arising from this submission. Matthew Fickus222The
views expressed in this article are those of the authors and do not reflect
the official policy or position of the United States Air Force, Department of
Defense, or the U.S. Government. Simon Foucart333S.F. is partially supported
by grants from the NSF (DMS-2053172) and from the ONR (N00014-20-1-2787).
Barbara Lewandowska  Institute of Mathematics
Pedagogical University of Krakow, Podchorazych 2, Krakow, 30-084, Poland
Department of Mathematics and Statistics
Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA
Department of Mathematics, Texas A&M and Institute of Data Science
Texas A&M University, College Station, TX 77843, USA  Faculty of Mathematics
and Computer Science
Jagiellonian University, Lojasiewicza 6, Krakow, 30-048, Poland

###### Abstract

Let $\lambda(m)$ denote the maximal absolute projection constant over real
$m$-dimensional subspaces. This quantity is extremely hard to determine
exactly, as testified by the fact that the only known value of $\lambda(m)$
for $m>1$ is $\lambda(2)=4/3$. There is also numerical evidence indicating
that $\lambda(3)=(1+\sqrt{5})/2$. In this paper, relying on a new construction
of certain mutually unbiased equiangular tight frames, we show that
$\lambda(5)\geq 5(11+6\sqrt{5})/59\approx 2.06919$. This value coincides with
the numerical estimation of $\lambda(5)$ obtained by B. L. Chalmers, thus
reinforcing the belief that this is the exact value of $\lambda(5)$.

###### keywords:

maximal absolute projection constant , maximal relative projection constant ,
equiangular tight frames , real mutually unbiased equiangular tight frames

###### MSC:

41A65 , 41A44 , 46B20 , 15A42 , 42C15

††journal: Journal of Functional Analysis

##  1 Introduction

Let $X$ be a real Banach space and $Y\subset X$ be a finite-dimensional
subspace. Let $\mathcal{P}(X,Y)$ denote the set of all linear and continuous
projections from $X$ onto $Y$, recalling that an operator $P\colon
X\rightarrow Y$ is called a projection onto $Y$ if $P|_{Y}={\rm Id}_{Y}.$ We
define the relative projection constant of $Y$ by

$\lambda(Y,X):=\inf\\{\|P\|:\;P\in\mathcal{P}(X,Y)\\}$

and the absolute projection constant of $Y$ by

$\lambda(Y):=\sup\\{\lambda(Y,X):Y\subset X\\}.$ (1)

The literature also deals with the maximal absolute projection constant, which
is defined by

$\lambda(m):=\sup\\{\lambda(Y):\;\dim(Y)=m\\}.$

By the Kadec–Snobar theorem (see [17]), we have $\lambda(m)\leq\sqrt{m}$.
Moreover, it has been shown in [18] that this estimate is asymptotically the
best possible. However, the determination of the constant $\lambda(m)$ seems
to be difficult: apart from $\lambda(1)=1$, the only known value of
$\lambda(m)$ is $\lambda(2)=4/3$ — this is Grünbaum conjecture, formulated in
[14] and proved in [6]. Numerical computations presented in [13] indicate that
$\lambda(3)$ should equal $(1+\sqrt{5})/2$ — this was stated, with an
erroneous proof, in [19]. Other numerical experiments conducted by B. L.
Chalmers (and unfortunately unpublished) suggest that $\lambda(5)\approx
2.06919$. In this article, we show that

$\lambda(5)\geq 5(11+6\sqrt{5})/59\approx 2.06919.$

Viewed in isolation, this could seem anecdotal. However, several sources of
evidence hint that this is the actual value of $\lambda(5)$. This comes as a
surprise, because it was growingly believed that obtaining exact formulas for
$\lambda(m)$ was an unreasonable quest. Now there is hope that this quest
could be realized after all.

To establish the announced lower bound, we make a detour via maximal relative
projection constants. Recent results concerning maximal relative and absolute
projection constants can be found in [1, 2, 4, 13, 21]. Here, we only give the
definition of the maximal relative projection constant for $n\geq m$ as

$\lambda(m,n):=\sup\\{\lambda(Y,l_{\infty}^{(n)}):\;\dim(Y)=m\textrm{ and
}Y\subset l_{\infty}^{(n)}\\}.$

This is motivated by the fact that, in the expression (1) of $\lambda(m)$, it
suffices to take the supremum over finite-dimensional $l_{\infty}$ superspaces
(see e.g. [22, III.B.5]), so that the nondecreasing sequence
$(\lambda(m,n))_{n\geq m}$ converges to $\lambda(m)$. In reality, there even
is an $N\in\mathbb{N}$ such that $\lambda(m,n)=\lambda(m)$ for all $n\geq N$
(see [1, Theorem 1.4]). Our estimation of $\lambda(m,n)$ will rely on the
following result proved in [5].

###### Theorem 1.1

For integers $n\geq m$, one has

$\lambda(m,n)=\max\bigg{\\{}\sum_{i,j=1}^{n}t_{i}t_{j}|U^{\top}U|_{ij}:t\in\mathbb{R}^{n},\;\|t\|_{2}=1,U\in\mathbb{R}^{m\times
n},\;UU^{\top}={\rm I}_{m}\bigg{\\}}.$

Although this theorem provides an essential tool for estimating the maximal
relative projection constants, computing their exact values remains a
challenging problem, carried out in just a few cases (see e.g. [1, 5, 13]).
One particular situation where an explicit formula is available involves
equiangular tight frames. Let us recall that a system of unit (i.e.,
$l_{2}$-normalized) vectors $(v_{1},\dots,v_{n})$ in $\mathbb{R}^{m}$ is
called equiangular if there is a constant $c\geq 0$ such that

$|\langle v_{i},v_{j}\rangle|=c\qquad\textrm{ for all
}i,j\in\\{1,\dots,n\\},\;i\neq j.$

It is called a tight frame if

$VV^{\top}=\frac{n}{m}{\rm I}_{m},$

where $V$ is the matrix with columns $v_{1},\dots,v_{n}$. The system
$(v_{1},\dots,v_{n})$ of unit vectors is called an equiangular tight frame if
it is both equiangular and a tight frame. For an equiangular tight frame of
$n$ unit vectors in $\mathbb{R}^{m}$, it is well known (see e.g. [12, Theorem
5.7]) that

$|\langle v_{i},v_{j}\rangle|=\sqrt{\frac{n-m}{m(n-1)}}\qquad\textrm{ for all
}i,j\in\\{1,\dots,n\\},\;i\neq j.$

The above-mentioned explicit formula is presented as part of the result below.
Built from Theorems 1 and 2 of [20], it appeared in a slightly different form
as Theorem 5 in [13]. A new self-contained proof is included later as an
appendix.

###### Theorem 1.2

For integers $n\geq m$, the maximal relative projection constant
$\lambda(m,n)$ is upper bounded by

$\delta_{m,n}:=\frac{m}{n}\left(1+\sqrt{\frac{(n-1)(n-m)}{m}}\right).$

Moreover, the equality $\lambda(m,n)=\delta_{m,n}$ occurs if and only if there
is an equiangular tight frame for $\mathbb{R}^{m}$ consisting of $n$ unit
vectors.

###### Remark 1.1

We note in passing that $\delta_{m,n}<\sqrt{m}$ for $n\geq m>1$ (thus
providing another justification for Kadec–Snobar estimate). This is seen by
applying Cauchy–Schwarz inequality for the noncolinear vectors
$[1,\sqrt{n-1}]$ and $[1,\sqrt{(n-m)/m}]$ in

$\displaystyle\delta_{m,n}$
$\displaystyle=\frac{m}{n}\bigg{(}1+\sqrt{n-1}\sqrt{\frac{n-m}{m}}\bigg{)}<\frac{m}{n}\sqrt{1+n-1}\sqrt{1+\frac{n-m}{m}}=\sqrt{m}.$

In the rest of this paper, we present new explicit lower bounds for
$\lambda(m,n)$ under the condition that certain mutually unbiased equiangular
tight frames for $\mathbb{R}^{m}$ exist (see Theorem 2.3). We then provide a
construction of an infinite family of such mutually unbiased equiangular tight
frames (see Theorem 3.4). Finally, combining these two ingredients, we
highlight the resulting estimation of $\lambda(5,16)$ to arrive at the
promised lower bound for $\lambda(5)$, conjectured to be its true value.

##  2 The Lower Bound

Before stating the main result, we start with an observation about mutually
unbiased equiangular tight frames, formally defined below.

###### Definition 2.1

Two equiangular tight frames $(v_{1},\dots,v_{k})$ and $(w_{1},\dots,w_{l})$
for $\mathbb{R}^{m}$ are mutually unbiased if there exists $c\in\mathbb{R}$
such that

$|\langle v_{i},w_{j}\rangle|=c\qquad\mbox{for all }i\in\\{1,\dots,k\\}\mbox{
and }j\in\\{1,\dots,l\\}.$

This definition generalizes a concept introduced in [9] so as to permit the
case $k\neq l$. We point out that the scalar $c$ is uniquely determined, as
also noted in [3].

###### Lemma 2.1

The constant $c$ appearing in the definition of mutually unbiased equiangular
tight frames for $\mathbb{R}^{n}$ necessarily satisfies

$c=\frac{1}{\sqrt{m}}.$

Proof. Let $(v_{1},\dots,v_{k})$ and $(w_{1},\dots,w_{l})$ be mutually
unbiased equiangular tight frames for $\mathbb{R}^{m}$ and let
$V\in\mathbb{R}^{m\times k}$ be the matrix with columns $v_{1},\dots,v_{k}$.
For any $j\in\\{1,\dots,l\\}$, because the two frames are mutually unbiased,
we have

$\|V^{\top}w_{j}\|_{2}^{2}=\sum_{i=1}^{k}|\langle
v_{i},w_{j}\rangle|^{2}=\sum_{i=1}^{k}c^{2}=kc^{2}.$

Since $(v_{1},\dots,v_{k})$ is a tight frame for $\mathbb{R}^{m}$, we also
have $VV^{\top}=(k/m){\rm I}_{m}$, and so

$\|V^{\top}w_{j}\|_{2}^{2}=\langle V^{\top}w_{j},V^{\top}w_{j}\rangle=\langle
w_{j},VV^{\top}w_{j}\rangle=\Big{\langle}w_{j},\dfrac{k}{m}w_{j}\Big{\rangle}=\dfrac{k}{m}\|w_{j}\|_{2}^{2}=\dfrac{k}{m}.$

It follows that $kc^{2}=k/m$, and hence $c=1/{\sqrt{m}}$, as claimed.

We now present the main theorem of this section, whose statement involves the
quantity $\delta_{m,n}$ introduced in Theorem 1.2.

###### Theorem 2.3

If mutually unbiased equiangular tight frames $(v_{1},\dots,v_{k})$ and
$(w_{1},\dots,w_{l})$ for $\mathbb{R}^{m}$ exist, then the maximal relative
projection constant $\lambda(m,k+l)$ is bounded below as

$\lambda(m,k+l)\geq\frac{m-\delta_{m,k}\delta_{m,l}}{2\sqrt{m}-\delta_{m,k}-\delta_{m,l}}.$

Proof.  Let $V\in\mathbb{R}^{m\times k}$ be the matrix with columns
$v_{1},\dots,v_{k}$ and $W\in\mathbb{R}^{m\times l}$ the matrix with columns
$w_{1},\dots,w_{l}$. For any $\theta\in[0,\pi/2]$, let us consider the vector
$t_{\theta}\in\mathbb{R}^{k+l}$ and the matrix
$U_{\theta}\in\mathbb{R}^{m\times(k+l)}$ defined, in block notation, by

$t_{\theta}:=\begin{bmatrix}\cos\theta\dfrac{1}{\sqrt{k}}\mathbb{1}_{k}\\\
\hline\cr\sin\theta\dfrac{1}{\sqrt{l}}\mathbb{1}_{l}\end{bmatrix}\qquad\mbox{and}\qquad
U_{\theta}:=\begin{bmatrix}\;\cos\theta\sqrt{\dfrac{m}{k}}V&\vline&\sin\theta\sqrt{\dfrac{m}{l}}W\;\end{bmatrix},$
(2)

where $\mathbb{1}_{n}$ denotes the $n$-dimensional vector with all entries
equal to $1$. We observe that $\|t_{\theta}\|_{2}=1$, that

$U_{\theta}{U_{\theta}}^{\top}=\cos^{2}\theta\frac{m}{k}VV^{\top}+\sin^{2}\theta\frac{m}{k}WW^{\top}=\cos^{2}\theta\,{\rm
I}_{m}+\sin^{2}\theta\,{\rm I}_{m}={\rm I}_{m},$

and that

${U_{\theta}}^{\top}U_{\theta}=\begin{bmatrix}\cos^{2}\theta\dfrac{m}{k}V^{\top}V&\vline&\cos\theta\sin\theta\dfrac{m}{\sqrt{kl}}V^{\top}W\\\
\hline\cr\cos\theta\sin\theta\dfrac{m}{\sqrt{kl}}W^{\top}V&\vline&\sin^{2}\theta\dfrac{m}{l}W^{\top}W\end{bmatrix}.$
(3)

Therefore, according to the expression of $\lambda(m,n)$ from Theorem 1.1, we
can make use of the tight frame and unbiasedness properties of $U$ and $V$ to
obtain, with the shorthand notation $\phi_{m,n}:=\sqrt{(n-m)/(m(n-1))}$,

$\displaystyle\lambda(m,k+l)$
$\displaystyle\geq\sum_{i,j=1}^{k+l}(t_{\theta})_{i}(t_{\theta})_{j}|{U_{\theta}}^{\top}U_{\theta}|_{i,j}$
$\displaystyle=\cos^{2}\theta\frac{1}{k}\times\cos^{2}\theta\frac{m}{k}\times
k+\cos^{2}\theta\frac{1}{k}\times\cos^{2}\theta\frac{m}{k}\phi_{m,k}\times
k(k-1)$
$\displaystyle+\sin^{2}\theta\frac{1}{l}\times\sin^{2}\theta\frac{m}{l}\times
l+\sin^{2}\theta\frac{1}{l}\times\sin^{2}\theta\frac{m}{l}\phi_{m,l}\times
l(l-1)$
$\displaystyle+2\times\cos\theta\sin\theta\frac{1}{\sqrt{kl}}\times\cos\theta\sin\theta\frac{m}{\sqrt{kl}}\frac{1}{\sqrt{m}}\times
kl$
$\displaystyle=\cos^{4}\theta\bigg{(}\frac{m}{k}+\frac{m}{k}(k-1)\phi_{m,k}\bigg{)}+\sin^{4}\theta\bigg{(}\frac{m}{l}+\frac{m}{l}(l-1)\phi_{m,l}\bigg{)}$
$\displaystyle+2\cos^{2}\theta\sin^{2}\theta\sqrt{m}$
$\displaystyle=\bigg{(}\frac{1+\cos(2\theta)}{2}\bigg{)}^{2}\delta_{m,k}+\bigg{(}\frac{1-\cos(2\theta)}{2}\bigg{)}^{2}\delta_{m,l}+\big{(}\sin(2\theta)\big{)}^{2}\frac{\sqrt{m}}{2}.$

Since this is valid for any $\theta\in[0,\pi/2]$, after setting
$x:=\cos(2\theta)$, we arrive at

$\displaystyle\lambda(m,k+l)$
$\displaystyle\geq\max_{x\in[-1,1]}\left(\frac{\delta_{m,k}(1+2x+x^{2})}{4}+\frac{\delta_{m,l}(1-2x+x^{2})}{4}+\frac{\sqrt{m}}{2}(1-x^{2})\right)$
$\displaystyle=\frac{1}{4}\max_{x\in[-1,1]}\left(ax^{2}+2bx+c\right),$

where $a:=\delta_{m,k}+\delta_{m,l}-2\sqrt{m}$,
$b:=\delta_{m,k}-\delta_{m,l}$, and $c:=\delta_{m,k}+\delta_{m,l}+2\sqrt{m}$.
Taking momentarily for granted that $a<0$ and that $x_{*}:=-b/a\in[-1,1]$, we
deduce that

$\displaystyle\lambda(m,k+l)$
$\displaystyle\geq\frac{1}{4}\left(ax_{*}^{2}+2bx_{*}+c\right)=\frac{1}{4}\left(-\frac{b^{2}}{a}+c\right)=\frac{1}{4}\frac{b^{2}-ac}{-a}$
$\displaystyle=\frac{1}{4}\frac{(\delta_{m,k}-\delta_{m,l})^{2}+(2\sqrt{m}-\delta_{m,k}-\delta_{m,l})(2\sqrt{m}+\delta_{m,k}+\delta_{m,l})}{2\sqrt{m}-\delta_{m,k}-\delta_{m,l}}$
$\displaystyle=\frac{1}{4}\frac{4m-4\delta_{m,k}\delta_{m,l}}{2\sqrt{m}-\delta_{m,k}-\delta_{m,l}},$

which is the announced lower bound. It now remains to notice that $a<0$ and
that $-b/a\in[-1,1]$, but both follow from the general observation that
$\delta_{m,n}<\sqrt{m}$ for $n\geq m>1$, see Remark 1.1.

Before uncovering a family of mutually unbiased equiangular tight frames in
the next section, we emphasize here two noteworthy properties relating the
vector $t_{\theta}$ and the matrix $U_{\theta}$ that appeared in the above
proof.

###### Proposition 2.1

Let $\gamma_{m,k,l}$ be the lower bound for $\lambda(m,k+l)$ from Theorem 2.3
and let $\theta\in[0,\pi/2]$ be the angle used in its proof, i.e.,

$\gamma_{m,k,l}=\frac{m-\delta_{m,k}\delta_{m,l}}{2\sqrt{m}-\delta_{m,k}-\delta_{m,l}}\qquad\mbox{and}\qquad\cos(2\theta)=\frac{\delta_{m,k}-\delta_{m,l}}{2\sqrt{m}-\delta_{m,k}-\delta_{m,l}}.$

Then, with
$t_{\theta}\in\mathbb{R}^{k+l},U_{\theta}\in\mathbb{R}^{m\times(k+l)}$ defined
as in (2) and with $T_{\theta}:={\rm diag}[t_{\theta}]$, one has

$\displaystyle|U_{\theta}^{\top}U_{\theta}|\,t_{\theta}$
$\displaystyle=\gamma_{m,k,l}\,t_{\theta},$ (4) $\displaystyle T_{\theta}{\rm
sgn}(U_{\theta}^{\top}U_{\theta})T_{\theta}\,U_{\theta}^{\top}$
$\displaystyle=\frac{\gamma_{m,k,l}}{m}\,U_{\theta}^{\top}.$ (5)

Proof. When establishing both (4) and (5), it will be useful to keep in mind
that $\delta_{m,n}$ is tied to $\phi_{m,n}=\sqrt{(n-m)/(m(n-1))}$ via

$\delta_{m,n}=\frac{m}{n}\bigg{(}1+(n-1)\phi_{m,n}\bigg{)}=\frac{m}{n}\bigg{(}1+\frac{n-m}{m}\frac{1}{\phi_{m,n}}\bigg{)}.$

Starting with the justification of (4), we notice that, since the matrix
$V^{\top}V$ has diagonal entries equal to $1$ and off-diagonal entries equal
to $\phi_{m,k}$ in absolute value, we have

$|V^{\top}V|=(1-\phi_{m,k}){\rm I}_{k}+\phi_{m,k}\mathbb{1}_{k,k},$

where $\mathbb{1}_{n,n^{\prime}}$ denotes the $n\times n^{\prime}$ matrix with
all entries equal to $1$. It follows that

$|V^{\top}V|\mathbb{1}_{k}=(1-\phi_{m,k})\mathbb{1}_{k}+k\phi_{m,k}\mathbb{1}_{k}=(1+(k-1)\phi_{m,k})\mathbb{1}_{k}=\frac{k}{m}\delta_{m,k}\mathbb{1}_{k}.$

Likewise, we can obtain

$|W^{\top}W|\mathbb{1}_{l}=\frac{l}{m}\delta_{m,l}\mathbb{1}_{l}.$

Moreover, since the matrices $V^{\top}W$ and $W^{\top}V$ have entries all
equal to $1/\sqrt{m}$ in absolute value, we have
$|V^{\top}W|=(1/\sqrt{m})\mathbb{1}_{k,l}$ and
$|W^{\top}V|=(1/\sqrt{m})\mathbb{1}_{l,k}$, so that

$|V^{\top}W|\mathbb{1}_{l}=\frac{l}{\sqrt{m}}\mathbb{1}_{k}\qquad\mbox{and}\qquad|W^{\top}V|\mathbb{1}_{k}=\frac{k}{\sqrt{m}}\mathbb{1}_{l}.$

Therefore, according to the block-forms of $t_{\theta}$ and
$U_{\theta}^{\top}U_{\theta}$ (see (2) and (3)), we observe that

$\displaystyle|U_{\theta}^{\top}U_{\theta}|\,t_{\theta}$
$\displaystyle=\begin{bmatrix}\cos^{2}\theta\dfrac{m}{k}\cos\theta\dfrac{1}{\sqrt{k}}\dfrac{k}{m}\delta_{m,k}\mathbb{1}_{k}+\cos\theta\sin\theta\dfrac{m}{\sqrt{kl}}\sin\theta\dfrac{1}{\sqrt{l}}\dfrac{l}{\sqrt{m}}\mathbb{1}_{k}\\\
\hline\cr\cos\theta\sin\theta\dfrac{m}{\sqrt{kl}}\cos\theta\dfrac{1}{\sqrt{k}}\dfrac{k}{\sqrt{m}}\mathbb{1}_{l}+\sin^{2}\theta\dfrac{m}{l}\sin\theta\dfrac{1}{\sqrt{l}}\dfrac{l}{m}\delta_{m,l}\mathbb{1}_{l}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\cos\theta\dfrac{1}{\sqrt{k}}\left(\cos^{2}\theta\delta_{m,k}+\sin^{2}\theta\sqrt{m}\right)\mathbb{1}_{k}\\\
\hline\cr\sin\theta\dfrac{1}{\sqrt{l}}\left(\cos^{2}\theta\sqrt{m}+\sin^{2}\theta\delta_{m,l}\right)\mathbb{1}_{l}\end{bmatrix}.$
(6)

Next, in view of

$\displaystyle\cos^{2}\theta$
$\displaystyle=\frac{1+\cos(2\theta)}{2}=\frac{\sqrt{m}-\delta_{m,l}}{2\sqrt{m}-\delta_{m,k}-\delta_{m,l}},$
$\displaystyle\sin^{2}\theta$
$\displaystyle=\frac{1-\cos(2\theta)}{2}=\frac{\sqrt{m}-\delta_{m,k}}{2\sqrt{m}-\delta_{m,k}-\delta_{m,l}},$

we easily derive that

$\cos^{2}\theta\delta_{m,k}+\sin^{2}\theta\sqrt{m}=\cos^{2}\theta\sqrt{m}+\sin^{2}\theta\delta_{m,l}=\gamma_{m,k,l}.$
(7)

When substituting the latter into (6), the identity (4) immediately follows.

Turning now to the justification of (5), recalling that the matrix $V^{\top}V$
has diagonal entries equal to $1$ and off-diagonal entries equal to
$\phi_{m,k}$ in absolute value, the diagonal entries of the matrix ${\rm
sgn}(V^{\top}V)$ are equal to $1$ and its off-diagonal entries are equal to
those of $V^{\top}V$ divided by $\phi_{m,k}$. In short, we see that ${\rm
sgn}(V^{\top}V)=(1-1/\phi_{m,k}){\rm I}_{k}+(1/\phi_{m,k})V^{\top}V$ holds,
and a similar identity holds for ${\rm sgn}(W^{\top}W)$. Moreover, we also
have ${\rm sgn}(V^{\top}W)=\sqrt{m}\,V^{\top}W$ and ${\rm
sgn}(W^{\top}V)=\sqrt{m}\,W^{\top}V$, as a consequence of all the entries of
$W^{\top}V$ and $W^{\top}V$ being equal to $1/\sqrt{m}$ in absolute value. All
in all, according to the block-form (3) of $U_{\theta}^{\top}U_{\theta}$, we
obtain

${\small{\rm
sgn}(U_{\theta}^{\top}U_{\theta})=\begin{bmatrix}\left(1-\dfrac{1}{\phi_{m,k}}\right){\rm
I}_{k}+\dfrac{1}{\phi_{m,k}}V^{\top}V&\vline&\sqrt{m}\,V^{\top}W\\\
\hline\cr\sqrt{m}\,W^{\top}V&\vline&\left(1-\dfrac{1}{\phi_{m,l}}\right){\rm
I}_{l}+\dfrac{1}{\phi_{m,l}}W^{\top}W\end{bmatrix}.}$

In turn, using the block-form of $T_{\theta}={\rm diag}[t_{\theta}]$, we
derive that $T_{\theta}{\rm sgn}(U_{\theta}^{\top}U_{\theta})T_{\theta}$ takes
the form

${\footnotesize\begin{bmatrix}\cos^{2}\theta\dfrac{1}{k}\left(\left(1-\dfrac{1}{\phi_{m,k}}\right){\rm
I}_{k}+\dfrac{1}{\phi_{m,k}}V^{\top}V\right)&\vline&\cos\theta\sin\theta\dfrac{1}{\sqrt{kl}}\sqrt{m}\,V^{\top}W\\\
\hline\cr\cos\theta\sin\theta\dfrac{1}{\sqrt{kl}}\sqrt{m}\,W^{\top}V&\vline&\sin^{2}\theta\dfrac{1}{l}\left(\left(1-\dfrac{1}{\phi_{m,l}}\right){\rm
I}_{l}+\dfrac{1}{\phi_{m,l}}W^{\top}W\right)\end{bmatrix}.}$

Multiplying on the right by the transpose of
$U_{\theta}={\small\begin{bmatrix}\;\cos\theta\sqrt{\dfrac{m}{k}}V&\vline&\sin\theta\sqrt{\dfrac{m}{l}}W\;\end{bmatrix}}$
and making use of the facts that $VV^{\top}=(k/m){\rm I}_{m}$ and
$WW^{\top}=(l/m){\rm I}_{m}$, the matrix $T_{\theta}{\rm
sgn}(U_{\theta}^{\top}U_{\theta})T_{\theta}\,U_{\theta}^{\top}$ becomes

$\displaystyle{\footnotesize\begin{bmatrix}\cos^{2}\theta\dfrac{1}{k}\cos\theta\sqrt{\dfrac{m}{k}}\left(\left(1-\dfrac{1}{\phi_{m,k}}\right)+\dfrac{k}{m}\dfrac{1}{\phi_{m,k}}\right)V^{\top}+\cos\theta\sin\theta\dfrac{1}{\sqrt{kl}}\sqrt{m}\sin\theta\sqrt{\dfrac{m}{l}}\dfrac{l}{m}V^{\top}\\\
\hline\cr\cos\theta\sin\theta\dfrac{1}{\sqrt{kl}}\sqrt{m}\cos\theta\sqrt{\dfrac{m}{k}}\dfrac{k}{m}W^{\top}+\sin^{2}\theta\dfrac{1}{l}\sin\theta\sqrt{\dfrac{m}{l}}\left(\left(1-\dfrac{1}{\phi_{m,l}}\right)+\dfrac{l}{m}\dfrac{1}{\phi_{m,l}}\right)W^{\top}\end{bmatrix}}$
$\displaystyle=\begin{bmatrix}\cos\theta\sqrt{\dfrac{m}{k}}\left(\dfrac{\cos^{2}\theta}{k}\left(1+\dfrac{k-m}{m}\dfrac{1}{\phi_{m,k}}\right)+\sin^{2}\theta\dfrac{1}{\sqrt{m}}\right)V^{\top}\\\
\hline\cr\sin\theta\sqrt{\dfrac{m}{l}}\left(\cos^{2}\theta\dfrac{1}{\sqrt{m}}+\dfrac{\sin^{2}\theta}{l}\left(1+\dfrac{l-m}{m}\dfrac{1}{\phi_{m,l}}\right)\right)W^{\top}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\cos\theta\sqrt{\dfrac{m}{k}}\left(\dfrac{\cos^{2}\theta}{m}\delta_{m,k}+\dfrac{\sin^{2}\theta}{\sqrt{m}}\right)V^{\top}\\\
\hline\cr\sin\theta\sqrt{\dfrac{m}{l}}\left(\dfrac{\cos^{2}\theta}{\sqrt{m}}+\dfrac{\sin^{2}\theta}{m}\delta_{m,l}\right)W^{\top}\end{bmatrix}.$

Similarly to (4), the identity (5) now simply follows by exploiting (7) again.

##  3 Construction of Mutually Unbiased Equiangular Tight Frames

To apply the result of Theorem 2.3 in practical situations, we evidently need
to uncover specific integers $k$, $l$, and $m$ allowing mutually unbiased
equiangular tight frames to exist. As a simple example, one can take $k=l=m$
and consider $(v_{1},\ldots,v_{k})$ to be the canonical basis for
$\mathbb{R}^{m}$ and $(w_{1},\ldots,w_{l})$ to be the columns of an $m\times
m$ Hadamard matrix — recall that $m\times m$ Hadamard matrices are conjectured
to exist when and only when $m$ is a multiple of $4$ (the ‘only when’ part
being acquired, of course). This would yield the lower bound
$\lambda(m)\geq(1+\sqrt{m})/2$, $m\in 4\mathbb{N}$, which is inferior to the
lower bounds reported in [13] for $m=4$ and $m=8$. As a slightly more
elaborate example, one can take $k=m$ and $(v_{1},\ldots,v_{k})$ to be the
canonical basis of $\mathbb{R}^{m}$, together with $l>m$ and
$(w_{1},\ldots,w_{l})$ to be a real equiangular tight frame for
$\mathbb{R}^{m}$ that is flat, in the sense that every entry of each vector
$w_{j}$ is either $1/\sqrt{m}$ or $-1/\sqrt{m}$. Real flat equiangular tight
frames are equivalent to binary codes achieving equality in the Grey–Rankin
bound and infinite families are known (see [16, 8]). This would yield the
lower bound $\lambda(m,m+l)\geq(m-\gamma_{m,l})/(2\sqrt{m}-1-\gamma_{m,l})$.
With $m=6$ and $l=16$, this provides the lower bound $\lambda(6)\gtrsim
2.2741$, which is superior to the lower bounds reported in [13] but inferior
to the numerical evaluation $\lambda(6)\approx 2.2857$ performed by B. L.
Chalmers and corroborated by our own computations. In order to apply Theorem
2.3 more effectively, we need further examples of mutually unbiased
equiangular tight frames. To this end, we now relate such frames to a type of
generalized Hadamard matrices.

###### Proposition 3.1

Given integers $k,l\geq m>1$, there are mutually unbiased equiangular tight
frames $(v_{1},\dots,v_{k})$ and $(w_{1},\dots,w_{l})$ for $\mathbb{R}^{m}$ if
and only if there is a $k\times l$ matrix $X$ with the following five
properties:

  1. (i)

$X_{ij}\in\\{-1,+1\\}\,$ for all $i\in\\{1,\dots,k\\}$ and
$j\in\\{1,\dots,l\\}$;

  2. (ii)

$XX^{\top}X=aX\,$ for some $a\in\mathbb{R}$;

  3. (iii)

$X$ has equiangular rows, i.e., $|XX^{\top}|_{i,i^{\prime}}$ is constant over
all $i\neq i^{\prime}$;

  4. (iv)

$X$ has equiangular columns, i.e., $|X^{\top}X|_{j,j^{\prime}}$ is constant
over all $j\neq j^{\prime}$;

  5. (v)

$X$ has rank $m$.

When this occurs, the following three quantities are necessarily integers:

$\frac{kl}{m},\qquad k\,\sqrt{\frac{l-m}{m(l-1)}},\qquad
l\,\sqrt{\frac{k-m}{m(k-1)}}.$ (8)

Proof. Firstly, let us assume that there are mutually unbiased equiangular
tight frames $(v_{1},\dots,v_{k})$ and $(w_{1},\dots,w_{l})$ for
$\mathbb{R}^{m}$. With $V\in\mathbb{R}^{m\times k}$ and
$W\in\mathbb{R}^{m\times l}$ denoting the matrices with columns
$v_{1},\dots,v_{k}$ and $w_{1},\dots,w_{l}$, respectively, we set

$X=\sqrt{m}\,V^{\top}W\in\mathbb{R}^{k\times l}.$

By Lemma 2.1, we have $|V^{\top}W|_{i,j}=|\langle
v_{i},w_{j}\rangle|=1/\sqrt{m}$ for all $i\in\\{1,\dots,k\\}$ and
$j\in\\{1,\dots,l\\}$, so Property (i) is immediate. In view of
$VV^{\top}=(k/m)\,{\rm I}_{m}$ and of $WW^{\top}=(l/m)\,{\rm I}_{m}$, it is
also straightforward to see that

$XX^{\top}=l\,V^{\top}V\qquad\mbox{and}\qquad X^{\top}X=k\,W^{\top}W.$ (9)

From here, using the fact that $VV^{\top}=(k/m)\,{\rm I}_{m}$ one more time,
we obtain that
$XX^{\top}X=(l\,V^{\top}V)(\sqrt{m}\,V^{\top}W)=(kl/m)\sqrt{m}\,V^{\top}W$,
i.e., $XX^{\top}X=aX$ with $a=kl/m$, so Property (ii) is satisfied. Properties
(iii) and (iv), too, are consequences of (9), since e.g. the off-diagonal
entries of $XX^{\top}$ are constant in absolute value because those of
$V^{\top}V$ are. Finally, Property (v) is also implied by (9) via ${\rm
rank}(X)={\rm rank}(XX^{\top})={\rm rank}(V^{\top}V)={\rm
rank}(VV^{\top})={\rm rank}({\rm I}_{m})~{}=~{}m$.

Conversely, let us assume that Properties (i)–(ii) are fulfilled by some
matrix $X\in\mathbb{R}^{k\times l}$. Consider the singular value decomposition
of this matrix written as $X=P\Sigma Q^{\top}$, where the diagonal matrix
$\Sigma\in\mathbb{R}^{m\times m}$ has positive entries (by (v)) and where the
matrices $P\in\mathbb{R}^{k\times m}$ and $Q\in\mathbb{R}^{l\times m}$ have
orthonormal columns, i.e., $P^{\top}P={\rm I}_{m}$ and $Q^{\top}Q={\rm
I}_{m}$. Property (ii) easily yields $\Sigma^{3}=a\,\Sigma$ and hence
$\Sigma=\sqrt{a}\,{\rm I}_{m}$. Then, looking at the squared Frobenius norm of
$X=\sqrt{a}\,PQ^{\top}$, we derive from (i) that $kl=am$, i.e., that $a=kl/m$.
We now set

$V=\sqrt{\frac{k}{m}}P^{\top}\in\mathbb{R}^{m\times k}\qquad\mbox{and}\qquad
W=\sqrt{\frac{l}{m}}Q^{\top}\in\mathbb{R}^{m\times l}$

and we claim that the columns $v_{1},\ldots,v_{k}$ of $V$ and
$w_{1},\ldots,w_{l}$ of $W$ are mutually unbiased equiangular tight frames for
$\mathbb{R}^{m}$. Indeed, using $V^{\top}V=(k/m)PP^{\top}$ and
$XX^{\top}=aPP^{\top}$, we see that $V^{\top}V=(1/l)XX^{\top}$, so that the
equiangularity of the system $(v_{1},\ldots,v_{k})$ is clear from (iii). Note
that each $v_{i}$ is a unit vector, since
$\|v_{i}\|_{2}^{2}=(V^{\top}V)_{i,i}=(1/l)(XX^{\top})_{i,i}=(1/l)\sum_{j=1}^{l}X_{i,j}^{2}=1$
by (i). The fact that these vectors form a tight frame is seen from
$VV^{\top}=(k/m)\,P^{\top}P=(k/m)\,{\rm I}_{m}$. Similar arguments (using
(iv)) would reveal that the system $(w_{1},\ldots,w_{l})$ is also an
equiangular tight frame. At last, to see that these systems are mutually
unbiased, it suffices to notice that
$V^{\top}W=(\sqrt{kl}/m)\,PQ^{\top}=(1/\sqrt{m})\,X$ and to invoke (i) once
again.

It finally remains to establish that the three quantities in (8) are integers.
For the first one, we have seen (in the proofs of both implications) that
$a=kl/m$ and (i)-(ii) show that $a$ is an integer: any entry of
$XX^{\top}X=aX$ is on the one hand an integer and on the other hand equal to
$\pm a$. For the third one, say, looking e.g. at (9), any off-diagonal entry
of $XX^{\top}=l\,V^{\top}V$ is on the one hand an integer and on the other
hand equal to $l$ times the common absolute inner product in a $k$-vector
equiangular tight frame for $\mathbb{R}^{m}$, i.e., to
$l\sqrt{(k-m)/(m(k-1))}$.

Although conditions (i)–(v) are restrictive, there are matrices $X$ satisfying
them with $m<k<l$. For instance, the $6\times 10$ matrix

$X=\small\left[\begin{array}[]{rrrrrrrrrr}1&1&1&1&1&1&1&1&1&1\\\
1&1&-1&1&-1&-1&1&-1&-1&-1\\\ 1&-1&1&-1&1&-1&-1&1&-1&-1\\\
-1&1&1&-1&-1&1&-1&-1&1&-1\\\ -1&-1&-1&1&1&1&-1&-1&-1&1\\\
-1&-1&-1&-1&-1&-1&1&1&1&1\end{array}\right]$

is one such matrix444As pointed out to us by Josiah Park, this same $6\times
10$ matrix appeared in a recent investigation of spherical half-designs (see
[15]).: it has $\pm 1$ entries, the identity $XX^{\top}X=aX$ is easily
verified (at least computationally), and it was already observed in [11] that
both its rows and its columns form equiangular tight frames for their
$5$-dimensional spans. Therefore, since $X$ fulfills the conditions of
Proposition 3.1 with $m=5$, $k=6$, and $l=10$, we are guaranteed the existence
of mutually unbiased equiangular tight frames $(v_{1},\dots,v_{6})$ and
$(w_{1},\dots,w_{10})$ for $\mathbb{R}^{5}$. Remarkably, this example is but
the first member of the infinite family presented below.

###### Theorem 3.4

For any integer $s\geq 2$, there are mutually unbiased equiangular tight
frames $(v_{1},\dots,v_{k})$ and $(w_{1},\dots,w_{l})$ for $\mathbb{R}^{m}$,
where

$k=2^{s-1}(2^{s}-1),\qquad l=2^{s-1}(2^{s}+1),\qquad m=\frac{2^{2s}-1}{3}.$

Proof. For any such $s$, $k$, $l$ and $m$, the requisite matrix $X$ of
Proposition 3.1 is produced in the recent paper [7], albeit nonobviously so.
In brief, let $Q$ and $B$ be the canonical hyperbolic-quadratic and symplectic
forms on the binary vector space $\mathbb{F}_{2}^{2s}$, respectively:

$\displaystyle Q(x)=Q(x_{1},\dotsc,x_{2s})$
$\displaystyle:=\sum_{r=1}^{s}x_{2r-1}x_{2r},$ $\displaystyle
B(x,y)=B((x_{1},\dotsc,x_{2s}),(y_{1},\dotsc,y_{2s}))$
$\displaystyle:=\sum_{r=1}^{s}(x_{2r-1}y_{2r}+x_{2r}y_{2r-1}).$

Let $\Gamma$ be the corresponding character table of $\mathbb{F}_{2}^{2s}$,
defined by $\Gamma(x,y)=(-1)^{B(x,y)}$ for all $x,y\in\mathbb{F}_{2}^{2s}$.
Any submatrix of $\Gamma$ obviously satisfies (i) from Proposition 3.1. Let
$X$ be the specific submatrix of $\Gamma$ whose rows and columns are indexed
by $\\{x\in\mathbb{F}_{2}^{2s}:Q(x)=1\\}$ and
$\\{x\in\mathbb{F}_{2}^{2s}:Q(x)=0\\}$, respectively. By Lemma 4.2 of [7],
these two subsets of $\mathbb{F}_{2}^{2s}$ are difference sets for
$\mathbb{F}_{2}^{2s}$ of cardinality $k$ and $l$, respectively. As detailed in
[7], this means that the rows and columns of $X$ are equiangular, namely that
(iii) and (iv) hold. Theorem 4.4 of [7] moreover gives that these two
difference sets are paired, meaning that the columns of $X$ form a tight frame
for their span, so that (ii) holds. Theorem 3.3 of [7] then implies that the
rank of $X$ is indeed $m$, so that (v) holds.

We close this section by highlighting that real mutually unbiased equiangular
tight frames are rare objects. Precisely, we have obtained rather stringent
necessary conditions for their existence (not included here because too
detached from our main focus). For instance, these conditions imply that
mutually unbiased $k$-vector and $l$-vector equiangular tight frames for
$\mathbb{R}^{m}$ can only exist for at most thirteen triples of integers
$(m,k,l)$ with $l>k>m+1$ when $m\leq 1000$, and that they cannot exist when
$l=k>m$, in contrast with the complex setting.

##  4 Epilogue: the fifth maximal projection constant

By combining the main results derived in the two previous sections, namely
Theorems 2.3 and 3.4, and after some tedious algebraic manipulation, we can
state that the maximal relative projection constant at any $m$ of the form
$m=(2^{2s}-1)/3$ for some integer $s\geq 2$ is bounded below as

$\lambda(m,4^{s})\geq\frac{2^{2s}-1}{2^{3s}-3\,2^{s-1}+1}\left(\frac{2^{2s-1}+2^{s}-1}{3}+2^{s-1}\sqrt{m}\right).$
(10)

If this was to be an equality, then the vector
$t_{\theta}\in\mathbb{R}_{+}^{n}$, $n=4^{s}$, and the matrix
$U_{\theta}\in\mathbb{R}^{m\times n}$, $m=(2^{2s}-1)/3$, appearing in the
proof of Theorem 2.3 should be maximizers of the expression for $\lambda(m,n)$
from Theorem 1.1. For genuine maximizers $\bar{t}\in\mathbb{R}_{+}^{n}$ and
$\bar{U}\in\mathbb{R}^{m\times n}$, we emphasize the following two necessary
conditions:

  1. (a)

$\bar{t}$ is a maximizer of
$\sum_{i,j}t_{i}t_{j}|\bar{U}^{\top}\bar{U}|_{i,j}$ subject to $\|t\|_{2}=1$,
so is characterized by the fact that $\bar{t}$ is an eigenvector (in fact, the
leading eigenvector) of $|U^{\top}U|$ — this is indeed satisfied by
$t_{\theta}$ and $U_{\theta}$, according to (4);

  2. (b)

$\bar{U}$ is a maximizer of $\sum_{i,j}\bar{t}_{i}\bar{t}_{j}{\rm
sgn}(\bar{U}^{\top}\bar{U})_{i,j}(U^{\top}U)_{i,j}=\textrm{tr}(\bar{T}{\rm
sgn}(\bar{U}^{\top}\bar{U})\,\bar{T}U^{\top}U)$, $\bar{T}:={\rm
diag}[\bar{t}]$, subject to $UU^{\top}={\rm I}_{m}$, so is characterized by
the fact that the rows of $\bar{U}$ are eigenvectors corresponding to the $m$
largest eigenvalues of $\bar{T}{\rm sgn}(\bar{U}^{\top}\bar{U})\bar{T}$ — this
is indeed satisfied by $t_{\theta}$ and $U_{\theta}$, according to (5).

###### Remark 4.1

The necessary conditions (a)-(b) combine to show that the genuine maximizers
$\bar{t}$ and $\bar{U}$ obey the noteworthy relation

$\big{(}\bar{U}^{\top}\bar{D}\bar{U}\big{)}_{i,i}=\lambda(m,n)\,\bar{t}_{i}^{2}\qquad\mbox{for
all }i\in\\{1,\ldots,n\\},$

where $\bar{D}={\rm diag}[\bar{\mu}_{1},\ldots,\bar{\mu}_{m}]$ is the diagonal
matrix with the $m$ leading eignevalues
$\bar{\mu}_{1}\geq\cdots\geq\bar{\mu}_{m}$ of $\bar{T}{\rm
sgn}(\bar{U}^{\top}\bar{U})\bar{T}$ on its diagonal. Indeed, by (a), we have

$\displaystyle\lambda(m,n)\,\bar{t}_{i}^{2}$
$\displaystyle=\bar{t}_{i}\sum_{j=1}^{n}|\bar{U}^{\top}\bar{U}|_{i,j}\bar{t}_{j}=\sum_{j=1}^{n}(\bar{U}^{\top}\bar{U})_{i,j}(\bar{T}{\rm
sgn}(\bar{U}^{\top}\bar{U})\bar{T})_{i,j}$
$\displaystyle=\big{(}(\bar{U}^{\top}\bar{U})(\bar{T}{\rm
sgn}(\bar{U}^{\top}\bar{U})\bar{T})\big{)}_{i,i}.$ (11)

Now, by (b), we have $\bar{T}{\rm
sgn}(\bar{U}^{\top}\bar{U}\bar{T})\bar{U}^{\top}=\bar{U}^{\top}\bar{D}$, or
$\bar{U}\bar{T}{\rm sgn}(\bar{U}^{\top}\bar{U})\bar{T}=\bar{D}\bar{U}$ by
taking the transpose. Making use of the latter in (11) gives the expected
relation.

The observation that $t_{\theta}$ and $U_{\theta}$ do satisfy conditions
(a)-(b) supports the belief that (10) could be an equality. To the question of
whether the right-hand side of (10) also coincides with the value of the
maximal absolute projection constant $\lambda(m)$, $m=(2^{2s}-1)/3$, the
answer is in general no. Indeed, for $s=3$, hence for $m=21$, $k=28$, and
$l=36$, we have $\gamma_{21,28,36}\approx 3.9397$, while a real equiangular
tight frame for $\mathbb{R}^{21}$ made of $126$ vectors is known to exist (see
e.g. [10]), so Theorem 1.2 yields $\lambda(21)\geq\lambda(21,126)\gtrsim
4.3333$. However, for $s=2$, hence for $m=5$, $k=6$, and $l=10$, there are
convincing reasons to believe that $\gamma_{5,6,10}\approx 2.06919$ coincide
with the value of $\lambda(5)$. These reasons are the extensive numerical
investigations carried out B. L. Chalmers, as well as our own computations
(some of which can be found in a matlab reproducible available on the authors’
webpages). All these clues prompt us to conclude with the following assertion.

######  Theorem 4.5 (and Conjecture)

The fifth absolute projection constant satisfies

$\lambda(5)\geq\lambda(5,16)\geq\frac{5}{59}(11+6\sqrt{5})\approx 2.06919,$

and it is expected that the latter is indeed the true value of $\lambda(5)$.

## Appendix

As bonus material, we present here a new proof of Theorem 1.2 as a immediate
consequence of the technical result below coupled with Theorem 1.1.

###### Proposition 4.1

For integers $n\geq m>1$, one has

$\displaystyle\max\bigg{\\{}\sum_{i,j=1}^{n}$ $\displaystyle
t_{i}t_{j}|U^{\top}U|_{ij}:t\in\mathbb{R}^{n},\;\|t\|_{2}=1,U\in\mathbb{R}^{m\times
n},\;UU^{\top}={\rm I}_{m}\bigg{\\}}$
$\displaystyle\leq\frac{m}{n}\left(1+\sqrt{\frac{(n-1)(n-m)}{m}}\right),$ (12)

with equality if and only if there exists a matrix $U\in\mathbb{R}^{m\times
n}$ with $UU^{\top}={\rm I}_{m}$, $(U^{\top}U)_{i,i}=m/n$ for all
$i\in\\{1,\ldots,n\\}$, and $|U^{\top}U|_{i,j}=\sqrt{(n-m)m/(n-1)}/n$ for all
$i\not=j\in\\{1,\ldots,n\\}$.

Proof. For $t\in\mathbb{R}^{n}$ satisfying $\|t\|_{2}=1$ and
$U\in\mathbb{R}^{m\times n}$ satisfying $UU^{\top}={\rm I}_{m}$, we use the
nonnegativity of $(U^{\top}U)_{i,i}$ (as the inner product of the $i$th column
of $U$ with itself) and Cauchy–Schwarz inequality to write

$\displaystyle\Sigma$
$\displaystyle:=\sum_{i,j=1}^{n}t_{i}t_{j}|U^{\top}U|_{i,j}=\sum_{i=1}^{n}t_{i}^{2}|U^{\top}U|_{i,i}+\sum_{\begin{subarray}{c}i,j=1\\\
i\not=j\end{subarray}}^{n}t_{i}t_{j}|U^{\top}U|_{i,j}$
$\displaystyle\leq\sum_{i=1}^{n}t_{i}^{2}(U^{\top}U)_{i,i}+\sqrt{\sum_{\begin{subarray}{c}i,j=1\\\
i\not=j\end{subarray}}^{n}t_{i}^{2}t_{j}^{2}}\sqrt{\sum_{\begin{subarray}{c}i,j=1\\\
i\not=j\end{subarray}}^{n}(U^{\top}U)_{i,j}^{2}}$
$\displaystyle=\sum_{i=1}^{n}t_{i}^{2}(U^{\top}U)_{i,i}+\sqrt{\sum_{i,j=1}^{n}t_{i}^{2}t_{j}^{2}-\sum_{i=1}^{n}t_{i}^{4}}\sqrt{\sum_{i,j=1}^{n}(U^{\top}U)_{i,j}^{2}-\sum_{i=1}^{n}(U^{\top}U)_{i,i}^{2}}$
$\displaystyle=\sum_{i=1}^{n}\alpha_{i}\beta_{i}+\sqrt{A-\sum_{i=1}^{n}\alpha_{i}^{2}}\sqrt{B-\sum_{i=1}^{n}\beta_{i}^{2}},$

where we have set $\alpha_{i}=t_{i}^{2}$, $\beta_{i}=(U^{\top}U)_{i,i}$,
$A=\big{(}\sum_{i}t_{i}^{2}\big{)}\big{(}\sum_{j}t_{j}^{2}\big{)}=\|t\|_{2}^{4}=1$,
and
$B=\sum_{i,j}(U^{\top}U)_{i,j}^{2}=\|U^{\top}U\|_{F}^{2}=\textrm{tr}(U^{\top}UU^{\top}U)=\textrm{tr}(UU^{\top}UU^{\top})=m$.
Setting also $a=\|t\|_{2}^{2}=1$,
$b=\textrm{tr}(U^{\top}U)=\textrm{tr}(UU^{\top})=m$, as well as

$x_{i}:=\frac{\alpha_{i}-a/n}{\sqrt{A-a^{2}/n}}\qquad\mbox{and}\qquad
y_{i}:=\frac{\beta_{i}-b/n}{\sqrt{B-b^{2}/n}},$

we notice that $\sum_{i=1}^{n}x_{i}=0$ and $\sum_{i=1}^{n}y_{i}=0$. We exploit
these identities a few times to derive

$\displaystyle\Sigma$
$\displaystyle\leq\sum_{i=1}^{n}\bigg{(}\frac{a}{n}+\sqrt{A-\frac{a^{2}}{n}}x_{i}\bigg{)}\bigg{(}\frac{b}{n}+\sqrt{B-\frac{b^{2}}{n}}y_{i}\bigg{)}$
$\displaystyle+\sqrt{A-\sum_{i=1}^{n}\bigg{(}\frac{a}{n}+\sqrt{A-\frac{a^{2}}{n}}x_{i}\bigg{)}^{2}}\sqrt{B-\sum_{i=1}^{n}\bigg{(}\frac{b}{n}+\sqrt{B-\frac{b^{2}}{n}}y_{i}\bigg{)}^{2}}$
$\displaystyle=\frac{ab}{n}+\sqrt{A-\frac{a^{2}}{n}}\sqrt{B-\frac{b^{2}}{n}}\sum_{i=1}^{n}x_{i}y_{i}$
$\displaystyle+\sqrt{A-\frac{a^{2}}{n}-\Big{(}A-\frac{a^{2}}{n}\Big{)}\sum_{i=1}^{n}x_{i}^{2}}+\sqrt{B-\frac{b^{2}}{n}-\Big{(}B-\frac{b^{2}}{n}\Big{)}\sum_{i=1}^{n}y_{i}^{2}}$
$\displaystyle=\frac{ab}{n}+\sqrt{A-\frac{a^{2}}{n}}\sqrt{B-\frac{b^{2}}{n}}\left[\sum_{i=1}^{n}x_{i}y_{i}+\sqrt{1-\sum_{i=1}^{n}x_{i}^{2}}\sqrt{1-\sum_{i=1}^{n}y_{i}^{2}}\right].$

The latter term in square brackets is nothing but the inner product of the
unit vectors $\tilde{x}:=\begin{bmatrix}x,\sqrt{1-\|x\|_{2}^{2}}\end{bmatrix}$
and $\tilde{y}:=\begin{bmatrix}y,\sqrt{1-\|y\|_{2}^{2}}\end{bmatrix}$, so it
is bounded by one. Thus, keeping the values of $a=1$, $b=m$, $A=1$, and $B=m$
in mind, we arrive at

$\sum_{i,j=1}^{n}t_{i}t_{j}|U^{\top}U|_{i,j}\leq\frac{m}{n}+\sqrt{1-\frac{1}{n}}\sqrt{m-\frac{m^{2}}{n}}.$

Taking the supremum over $t$ and $U$ leads to the desired inequality (12)
after some algebraic manipulation. This inequality turns into an equality if
the matrix $U\in\mathbb{R}^{m\times n}$ with $UU^{\top}={\rm I}_{m}$ satisfies
$(U^{\top}U)_{i,i}=m/n$ for all $i\in\\{1,\ldots,n\\}$ and
$|U^{\top}U|_{i,j}=\sqrt{(n-m)m/(n-1)}/n$ for all
$i\not=j\in\\{1,\ldots,n\\}$, simply by choosing $t\in\mathbb{R}^{n}$ with
entries $t_{i}=1/\sqrt{n}$ for all $i\in\\{1,\ldots,n\\}$.

Conversely, let us assume that (12) is an equality. Our goal is now to prove
that $(U^{\top}U)_{i,i}=m/n$ for all $i\in\\{1,\ldots,n\\}$ and
$|U^{\top}U|_{i,j}=\sqrt{(n-m)m/(n-1)}/n$ for all
$i\not=j\in\\{1,\ldots,n\\}$, where $U\in\mathbb{R}^{m\times n}$ satisfying
$UU^{\top}={\rm I}_{m}$ achieves the maximum, together with
$t\in\mathbb{R}^{n}$ satisfying $\|t\|_{2}=1$. We start by taking into account
that equality must hold throughout the first part of the argument. Equality in
Cauchy–Schwarz inequality implies the existence of $c\in\mathbb{R}$ such that

$t_{i}t_{j}=c\,|U^{\top}U|_{i,j}\qquad\mbox{for all
}i\not=j\in\\{1,\ldots,n\\}$

and equality in $\langle\tilde{x},\tilde{y}\rangle\leq 1$ yields $x=y$, i.e.,

$(U^{\top}U)_{i,i}-\frac{m}{n}=\frac{\sqrt{m-m^{2}/n}}{\sqrt{1-1/n}}\bigg{(}t_{i}^{2}-\frac{1}{n}\bigg{)}\qquad\mbox{for
all }i\in\\{1,\ldots,n\\}.$ (13)

Since the matrix $T{\rm sgn}(U^{\top}U)T$ has diagonal entries $(T{\rm
sgn}(U^{\top}U)T)_{i,i}=t_{i}^{2}$ and off-diagonal entries

$(T{\rm sgn}(U^{\top}U)T)_{i,j}=t_{i}t_{j}{\rm
sgn}(U^{\top}U)_{i,j}=c\,|U^{\top}U|_{i,j}{\rm
sgn}(U^{\top}U)_{i,j}=c\,(U^{\top}U)_{i,j},$

the necessary condition (b), written for all $i\in\\{1,\ldots,n\\}$ and
$h\in\\{1,\ldots,m\\}$ as

$\sum_{j=1}^{n}(T{\rm
sgn}(U^{\top}U)T)_{i,j}U^{\top}_{j,h}=\mu_{h}U^{\top}_{i,h},$

where $\mu_{1}\geq\cdots\geq\mu_{m}$ are the $m$ leading eigenvalues of
$T({\rm sgn}(U^{\top}U)T$, becomes

$t_{i}^{2}U^{\top}_{i,h}+\sum_{\begin{subarray}{c}j=1\\\
j\not=i\end{subarray}}^{n}c\,(U^{\top}U)_{i,j}U^{\top}_{j,h}=\mu_{h}U^{\top}_{i,h}.$

In other words, for all $i\in\\{1,\ldots,n\\}$ and $h\in\\{1,\ldots,m\\}$, we
have

$t_{i}^{2}U^{\top}_{i,h}+c\,(U^{\top}UU^{\top})_{i,h}-c\,(U^{\top}U)_{i,i}U^{\top}_{i,h}=\mu_{h}U^{\top}_{i,h},$

or equivalently, in view of $UU^{\top}={\rm I}_{m}$,

$\left(t_{i}^{2}+c-c\,(U^{\top}U)_{i,i}\right)U^{\top}_{i,h}=\mu_{h}U^{\top}_{i,h}.$
(14)

This actually shows that $\mu_{h}$ is independent of $h\in\\{1,\ldots,m\\}$
and — thanks to the alternate expression $\lambda(m,n)=\mu_{1}+\ldots+\mu_{m}$
(see e.g. [13, Theorem 1]) — one must have $\mu_{h}=\lambda(m,n)/m$. Now (14)
reduces (say, by multiplying by $U^{\top}_{i,h}$, summing over $h$, and
simplifying) to $t_{i}^{2}+c-c\,(U^{\top}U)_{i,i}=\lambda(m,n)/m$. Summing
over $i\in\\{1,\ldots,n\\}$) yields

$1+c\,(n-m)=\frac{n}{m}\lambda(m,n)=1+\sqrt{\frac{(n-1)(n-m)}{m}},$

which shows that

$c=\sqrt{\frac{n-1}{m(n-m)}}.$

Invoking Remark 4.1, we notice that $(U^{\top}U)_{i,i}=mt_{i}^{2}$ for all
$i\in\\{1,\ldots,n\\}$, and therefore (13) becomes
$m(t_{i}^{2}-1/n)=\sqrt{m(n-m)/(n-1)}(t_{i}^{2}-1/n)$. Given that
$m~{}\not=~{}\sqrt{m(n-m)/(n-1)}$ when $m>1$, we consequently obtain
$t_{i}^{2}=1/n$ for all $i~{}\in~{}\\{1,\ldots,n\\}$. In turn, we deduce from
$(U^{\top}U)_{i,i}=mt_{i}^{2}$ that $(U^{\top}U)_{i,i}=m/n$ for all
$i\in\\{1,\ldots,n\\}$ and from $c|U^{\top}U|_{i,j}=t_{i}t_{j}$ that
$|U^{\top}U|_{i,j}=\sqrt{m(n-m)/(n-1)}/n$ for all
$i\not=j\in\\{1,\ldots,n\\}$. The proof is now complete.

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# A Benchmarking on Cloud based Speech-To-Text Services for French Speech and
Background Noise Effect††thanks: 6th National Conference on Practical
Applications of Artificial Intelligence, 2021, Bordeaux, France

Binbin Xu1*, Chongyang Tao1+, Zidu Feng1+, Youssef Raqui2, Sylvie Ranwez1*
1 EuroMov Digital Health in Motion, Univ Montpellier, IMT Mines Al s
2 DiappyMed

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This study presents a large scale benchmarking on cloud based Speech-To-Text
systems: Google Cloud Speech-To-Text, Microsoft Azure Cognitive Services,
Amazon Transcribe, IBM Watson Speech to Text. For each systems, $40\,158$
clean and noisy speech files about $101$ hours are tested. Effect of
background noise on STT quality is also evaluated with 5 different Signal-to-
noise ratios from $40\text{\,}\mathrm{dB}$ to $0\text{\,}\mathrm{dB}$. Results
showed that Microsoft Azure provided lowest transcription error rate $9.09\%$
on clean speech, with high robustness to noisy environment. Google Cloud and
Amazon Transcribe gave similar performance, but the latter is very limited for
time-constraint usage. Though IBM Watson could work correctly in quiet
conditions, it is highly sensible to noisy speech which could strongly limit
its application in real life situations.

### R sum

Alors que les applications de reconnaissance vocale se sont impos es dans
notre quotidien, il existe peu d’ tudes grande chelle pour comparer les
performances des solutions de l’ tat de l’art. Ceci est d’autant plus vrai
dans une langue autre que la langue anglaise. Cet article propose une telle
analyse comparative bas e sur 17 heures d’enregistrement en Fran ais. Quatre
syst mes sont analys s : Google Cloud Speech-To-Text, Microsoft Azure
Cognitive Services, Amazon Transcribe, et IBM Watson Speech to Text. Chacun
ayant t mis l’ preuve de cinq niveaux de bruit de fond, c’est l’ quivalent de
400 heures de discours qui sont analys es. Microsoft Azure Cognitive Services
a montr les meilleurs r sultats en terme de taux d’erreur et une bonne r
sistance au bruit, tandis que la sensibilit au bruit d’IBM Watson Speech to
Text compromet son usage en situation r elle.

### Keywords

Speech-To-Text, Benchmarking, French language, Google Cloud, Microsoft Azure
Cognitive Services, Amazon Transcribe, IBM Watson

##  1 Introduction

Lots applications with automated speech recognition (ASR) or Speech-To-Text
(STT) over the past few years have been developed to improve our daily life
like personal voice assistant, or have been deeply integrated in many of
business chains. Thanks to the substantial development of deep neural network
(DNN), the performance of STT has been drastically improved. Like other deep
neural network applications, today it is not surprising that in some
situations, current STT can even outperform humans. The IBM/Appen human
transcription study [1] showed that word error rate of human parity is about
$5.1\%$. Microsoft Research is the first team reaching this milestone.
However, the outstanding performances in DNN is based on large amount of
labeled training data. This is also the case for DNN models on STT. For
languages other than English, there’s much less high quality audio data like
in English. In consequence, the performances of STT on other languages are in
general lower than for English, especially for languages featuring rich
morphology like French.

Though many public Deep Neural Network models are available for offline use,
retraining or regular updating require extensive computing power which
prevents individuals or small business from accessing these models or using
them in an efficient way. The choice will be the cloud-based API services.
Actually, the most powerful STT systems are all cloud-based. Integrating these
systems in an application or a product line requires at first a benchmarking
on their performance. There exist many benchmarking studies on the performance
of cloud-based STT services. However, they are often conducted with very small
or small sample size, for example, 20–60 sentences or hundreds of sentences.
The benchmarking on English from Picovoice is one of the few large scale tests
on STT, which contains 2620 audio files (5h24m) from LibriSpeech dataset [2].
Benchmarking of cloud-based STT on French is even less studied. Another major
negative factor on STT performance is the background noise. Very often, only
clean speech record is processed. However, for most real-life application, the
background noise can hardly be avoided. This should be taken into account in
STT benchmarking as well.

The objective of this study is to benchmark four most used Speech-To-Text API
(Application Programming Interface) with a large French dataset : 6693 files,
about 17hours speech record. Five levels of common background noise are added
in the clean speech and evaluated additionally. In total, more than 400 hours
speech are transcribed.

##  2 Speech-To-Text system and data

###  2.1 Cloud based STT services

Four cloud based Speech-To-Text services are evaluated in this work:

  * •

Amazon Transcribe, is part of Cloud Computing Services from the Amazon Web
Services (AWS) which holds currently the largest share in Cloud Computing
market. Their recent speech recognition model on English reached State-of-the-
Art word error rate at $6.2\%$ [3]. To convert speech files to text, the data
needs to be at first uploaded to Amazon Simple Storage Service (Amazon S3).
Then Transcribe call the objects from S3 for transcription. Though Transcribe
jobs can be treated on batch mode (up to 100 parallel jobs). This S3
requirement adds additional complexity for the transcription tasks. Actually
Amazon Transcribe is the only STT requiring storage. The other three services
can be feed directly with audio files.

  * •

Google Cloud Speech-to-Text, is integrated in the widely used platform Google
Cloud. In 2012, Google Research had achieved word error rate at $15\%$ for
English broadcast news transcription. This error rate dropped considerably to
$5.6\%$ with updated model trained on over $12\,500$ hours audio in 2018 [3].
Their STT model is one of the most powerful in the market, and the performance
is continuously improving.

  * •

IBM Watson Speech-to-Text. IBM Watson is a conventional top player in speech
recognition. In 2015, their speech recognition system beat other models with a
word error rate at $8\%$ [4]. Two years later, their system reached $5.5\%$
[1]. It’s now among the most popular STT services and provides similar
features as other cloud STT.

  * •

Microsoft Azure Cognitive Services. Microsoft’s speech recognition is now one
of the leading STT service. In 2017, their model reached a historical human
parity milestone on conversational telephony speech transcription, with 5.1%
word error rate in benchmarked Switchboard task [5]. As all the other STT
systems, Microsoft’s STT system is also integrated in the Cloud Computing
platform.

All the four STT services offer the possibility to customize (like domain-
specific) or retrain the Speech-to-Text models. However, since they are all
black-boxed APIs, the background models and architectures are unknown, it is
difficult to benchmark the customized models with different configurations in
a fair way. So, only the basic models (APIs) are called.

###  2.2 Speech corpus

The basic audio dataset in this work is from WCE-SLT-LIG [6, 7]. This corpus
contains 6693 speech utterances recorded by 42 native speakers.

Figure 1: Main topic of WCE-SLT-LIG corpus

They come from French news media with main topic on European economy. The
total audio duration is 16h52. The ground-truth transcriptions are also
available, which makes the benchmarking possible.

The number of word in this corpus is $22\pm 12.8$ (median $\pm$ standard
deviation), with audio duration $8.4\pm 4.6$ seconds as shown in Figure 2.

Figure 2: Distribution of WCE-SLT-LIG corpus

###  2.3 Environmental noise corpus

In real world cases, most speech takes place in noisy environments. This is
one of the main challenges in Speech-to-Text applications. To evaluate the
effects on the STT quality, we introduce another recently released
environmental noise dataset: Microsoft Scalable Noisy Speech Dataset (MS-SNSD)
[8]. The dataset provides a variety of common environmental noise, which can
be mixed on clean speech data. The signal-to-noise (SNR) in
$\text{\,}\mathrm{dB}$ can be configured as well.

${\displaystyle\mathrm{SNR_{dB}}=10\log_{10}\left({\frac{P_{\mathrm{signal}}}{P_{\mathrm{noise}}}}\right).}$

where $P_{\mathrm{signal}}$, $P_{\mathrm{noise}}$ are the power of signal and
background noise. We set 5 SNR cases here: $40\text{\,}\mathrm{dB}$,
$30\text{\,}\mathrm{dB}$, $20\text{\,}\mathrm{dB}$, $10\text{\,}\mathrm{dB}$
and $0\text{\,}\mathrm{dB}$ (1:1 signal vs. noise).

The raw MS-SNSD contains 181 noise files. However, many of them are recorded
with strong conversations in other languages (English, German etc.). Some of
noise type are also less common. So, these noises are excluded. We’d like to
evaluate the effect of noise type on the performance of STT, to make sure that
some types are not over-presented, 96 noise files in 18 types are kept.

AirConditioner | Kitchen | SqueakyChair
---|---|---
AirportAnnouncements | LivingRoom | Station
Babble | Munching | Traffic
Cafe | Restaurant | Typing
CafeTeria | ShuttingDoor | VacuumCleaner
CopyMachine | Square | WasherDryer
Table 1: Types of background noise used in this work. 96 noises in 18 types

###  2.4 Evaluation metrics

In Speech-to-Text, the most commonly used metric to evaluate the performance
is Word error rate (WER). Other metrics exist, like Match error rate (MER);
Word information lost (WIL) or Word information preserve (WIP) [9].

$\displaystyle WER$ $\displaystyle=\frac{S+D+I}{N1=H+S+D}$ (1) $\displaystyle
MER$ $\displaystyle=\frac{S+D+I}{N=H+S+D+I}$ (2) $\displaystyle WIP$
$\displaystyle=\frac{H}{N_{1}}\cdot\frac{H}{N_{2}}\cong\frac{I(X,Y)}{H(Y)},$
(3) $\displaystyle WIL$ $\displaystyle=1-WIP$ (4)

where $H$, $S$, $D$ and $I$ correspond to the total number of word hits,
substitutions, deletions and insertions. $N_{1}$ and $N_{2}$ are respectively
the number of words in ground-truth text and the output transcripts. The lower
are WER, MER and WIL, the better the performance is.

| Amazon | Google | IBM | Microsoft
---|---|---|---|---
WER | 11.76% | 14.29% | 14.81% | 9.09%
MER | 11.54% | 14.29% | 14.29% | 9.09%
WIL | 0.19 | 0.25 | 0.24 | 0.16

Table 2: Evaluation on clean audio. Upper, WER distributions; lower, median
values. (STTs accessed in February 2021)

##  3 Results

###  3.1 Clean speech

For clean speech, Microsoft Azure performed quite well, with a WER at $9.09\%$
which is close to the advertised rate. Amazon Transcribe took the second place
with WER $11.76\%$. Google Cloud and IBM Waston gave similar WER ($14.29\%$
and $14.81\%$). These WER are actually very good already. According to the
public DeepSpeech model [10] from Mozila, trained with a mixed French dataset
”CommonVoice + CssTen + LinguaLibre + Mailabs + Tatoeba + Voxforge”, the WER
on test dataset is $19.5\%$ (result retrieved on March 10th 2021) [11]. The
gain with cloud STT API is between $24\%-53\%$.

###  3.2 Noisy speech

After mixing five different levels of environmental noise, Microsoft Azure
gave a quite good global WER $11.11\%$ (Table 3). Amazon Transcribe and Google
Cloud showed the same WER at $20\%$. But IBM Waston failed at certain point.
Its global WER is $29.63\%$, with a word-information-lost rate at $43\%$
(0.43) which is unfortunately high.

| Amazon | Google | IBM | Microsoft
---|---|---|---|---
WER | 20.00% | 20.00% | 29.63% | 11.11%
MER | 19.64% | 20.00% | 28.57% | 11.11%
WIL | 0.31 | 0.33 | 0.43 | 0.19
Table 3: Evaluation on all noisy audio (5 SNR levels combined), median values

At individual SNR level, as shown in Figure 3, Microsoft Azure is the most
robust to noise. The variation across different noise levels is quite small.
In highly noisy environment, the WER from transcription by IBM Waston can be
more than $100\%$. While other STTs would be at worst less than $50\%$.

Figure 3: Evaluation on mixed noisy speech by five signal-noise-ratio levels;
upper, wer distributions; lower, median value for each level. (STTs accessed
in February 2021)

The exceptional STT performance of Microsoft Azure is due to that Microsoft
has been working intensively on Artificial Intelligence based noise
suppression. This environmental noise dataset MS-SNSD comes from Microsoft.
The noise suppression should be already in the pipeline of their Speech-to-
Text models. Actually, in December 2020 Microsoft introduced background noise
suppression functionality in Microsoft Teams meetings [12]. To achieve this,
they used 760 hours of clean speech data and 180 hours of noise data. These
data are now released for Interspeech 2021 Deep Noise Suppression Challenge
[13].

Figure 4: WER by different noise types; median WER values from tests on all
the five SNR noisy speech. (STTs accessed in February 2021)

The performance of STT depends also on the noise types. All the STT services
are sensible to noise type _Restaurant_. IBM Waston’s WER reached $46.51\%$;
Amazon Transcribe had also high WER for this type of noise. Google Cloud and
Microsoft Azure dealt it better without shape WER changes. Background noise in
environment _Restaurant_ could be a mixture of different noises (babble,
conversation, munching, traffic etc.) which make it be more difficult for
Speech-to-Text tasks. In general, Google Cloud and Microsoft Azure are more
robust to environmental noise (variation and standard deviation of the median
WER are $6.5\%$ and $2.6\%$ for Google Cloud; $1.4\%$, $1.2\%$ for Microsoft
Azure); Amazon Transcribe can be placed in the second rank with $24\%$ and
$4.9\%$. As for IBM Waston, as shown previously, it can fail in many cases
when the background noises are too strong. It suffered also strong performance
variation $53.7\%$ and $7.3\%$ of standard deviation of the median WER.

###  3.3 Main STT errors

The main source of errors contributed to WER is the substitution. For clean
speech, or less noisy speech, the percentage of substitution $S$ is generally
much higher than deletion $D$ and insertion $I$. When speech becomes highly
noisy (SNR lower than $10\text{\,}\mathrm{dB}$), deletion $D$ percentage
increased much more. STT service from Microsoft Azure is quite robust to noisy
environment, there’s practically no change for SNR from
$40\text{\,}\mathrm{dB}$ to $10\text{\,}\mathrm{dB}$. Only in the tested case
when mixing directly noise and speech $0\text{\,}\mathrm{dB}$, the deletion
$D$ and substitution $S$ increased slightly. However, the changes are much
more significant for other three STT services, especially for IBM Waston.

Figure 5: Main transcription errors distribution (mean values. The median
percentage values for lower SNR are zero, less meaningful for presentation)

There’s also inter-speakers difference of WER. Amount the 42 speakers, all the
four STTs had more difficulty to transcribe speech from speaker L23_P08.

Figure 6: Word Error Rate by speakers (median values) on clean speech for all
the four STT systems.

###  3.4 Transcription job time

In a production application, the STT service must be as responsive as
possible. Google Cloud is the fastest about the four tested APIs, with a
median value at 1.76 second par job. Microsoft Azure is also fast, 3.51 second
per transcription job. IBM Waston is slower and require 5.43 second to
complete the job. It’s not surprising that Amazon Transcribe is the slowest
STT service, with 27 second per job. Some transcription jobs can take up to
200 second. Even it’s possible to send up to 100 jobs in parallel, single job
waiting is not acceptable for any real world application. This time
requirement does not include the data transfer time to Amazon S3 storage: with
upload speed 100-700 kbps, for a large amount of data, this can take already
quite some time to complete. Though it’s possible to call Amazon Transcribe
for steaming usage, it’s not convenient for non-real-time scenario.

One of the potential reasons of the additional seconds from Google Cloud and
IBM Waston, could be that Microsoft Azure’s returns less complete
transcription information than the other three.

Figure 7: Transcriptio job time in second for all the four STT systems

Another observation is the server responsiveness: job completion time with
Microsoft Azure is almost linear to the audio duration. The variation is also
very tight. But for other APIs, though the relationship could be regarded as
linear, the variation is much larger. Speeches with same length would require
2 to 4 times more execution time to complete the task.

##  4 Discussion

In this work, we evaluated the four most used Speech-to-Text API on French
speech from four Computing Cloud: Amazon Transcribe, Google Cloud, IBM Waston
and Microsoft Azure. 5 levels of different environment noises are mixed with
6690 clean speeches (17 hours). 100 hours speech tests per STT API gave 400
hours speech transcription.

The results showed that Microsoft Azure’s STT service provided the lowest Word
Error Rate (median 9%). It’s also very robust to common environment noise,
even in strong noise environment, the median WERs are only around 16.67%. STT
from Amazon Transcribe and Google Cloud performed well, their WER are
respectively at 11.76% and 14%. Amazon Transcribe works better in relatively
quiet environment while Google Cloud is better for noisy speech. IBM Waston’s
STT service can provide reasonable results with a median WER at 14.29%. But
when the speech is recorded in noisy environment, the WER can go up to around
70% which is difficult to be used. In general, when the signal-to-noise ratio
is higher than $20\text{\,}\mathrm{dB}$, the WERs are still acceptable.
However, if SNR drops lower than $20\text{\,}\mathrm{dB}$, except Microsoft
Azure, all the three APIs will have difficulties to recognize correctly the
speech. Among the 18 environment noise types, Restaurant type is the most
difficult one to deal with for all the four STT APIs.

When the work is time-constraint, Google Cloud will be the first choice with
fastest response time and a reasonable word error rate. Amazon Transcribe can
be used when the framework of the project is on the platform of Amazon Web
Services. The parallel job can help to reduce the total transcription time,
however, per job time is too longer than any other STT service. In average,
one transcription job on Amazon Transcribe is 15 times longer than the same
job on Google Cloud. Otherwise, the general suggestion will be Microsoft
Azure, lowest WER and high robustness to noise. It’s more suitable for
precision-constraint applications.

## References

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# Two-color pulse compounds in waveguides with a zero-nonlinearity point

O. Melchert<EMAIL_ADDRESS>Leibniz Universität Hannover,
Institute of Quantum Optics (IQO), Welfengarten 1, 30167 Hannover, Germany
Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering -
Innovation Across Disciplines), Welfengarten 1A, 30167 Hannover, Germany  S.
Bose  Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering -
Innovation Across Disciplines), Welfengarten 1A, 30167 Hannover, Germany
Leibniz Universität Hannover, Institue of Photonics (IOP), Nienburger Str. 17,
30167 Hannover  S. Willms  Leibniz Universität Hannover, Institute of Quantum
Optics (IQO), Welfengarten 1, 30167 Hannover, Germany  Cluster of Excellence
PhoenixD (Photonics, Optics, and Engineering - Innovation Across Disciplines),
Welfengarten 1A, 30167 Hannover, Germany  I. Babushkin  Leibniz Universität
Hannover, Institute of Quantum Optics (IQO), Welfengarten 1, 30167 Hannover,
Germany  Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering -
Innovation Across Disciplines), Welfengarten 1A, 30167 Hannover, Germany  U.
Morgner  Leibniz Universität Hannover, Institute of Quantum Optics (IQO),
Welfengarten 1, 30167 Hannover, Germany  Cluster of Excellence PhoenixD
(Photonics, Optics, and Engineering - Innovation Across Disciplines),
Welfengarten 1A, 30167 Hannover, Germany  A. Demircan  Leibniz Universität
Hannover, Institute of Quantum Optics (IQO), Welfengarten 1, 30167 Hannover,
Germany  Cluster of Excellence PhoenixD (Photonics, Optics, and Engineering -
Innovation Across Disciplines), Welfengarten 1A, 30167 Hannover, Germany

###### Abstract

We study incoherently coupled two-frequency pulse compounds in waveguides with
single zero-dispersion and zero-nonlinearity points. In such waveguides,
supported by a negative nonlinearity, soliton dynamics can be obtained even in
domains of normal dispersion. We demonstrate trapping of weak pulses by
solitary-wave wells, forming nonlinear-photonics meta-atoms, and molecule-like
bound-states of pulses. We study the impact of Raman effect on these pulse
compounds, finding that, depending on the precise subpulse configuration, they
decelerate, accelerate, or are completely unaffected. Our results extend the
range of systems in which two-frequency pulse compounds can be expected to
exist and demonstrate further unique and unexpected behavior.

### Introduction

The incoherent interaction of optical pulses is a central concern in nonlinear
optics. For instance, strong and efficient control of light pulses has been
shown for a soliton, which induces a strong refractive index barrier that
cannot be surpassed by quasi group-velocity matched waves located in a domain
of normal dispersion Demircan et al. (2013); Demircan et al. (2014a),
resulting in mutual repulsion. This mechanism is naturally supported by the
supercontinuum generation process Driben et al. (2010); Demircan et al.
(2014b). A transfer of this concept to waveguides supporting group-velocity
matched copropagation of pulses in separate domains of anomalous dispersion
yields an entirely attractive interaction Melchert et al. (2019a). In this
case, cross-phase modulation (XPM) induced potential wells provide a binding
mechanism that enable molecule-like bound states of pulses. They form a single
compound pulse, consisting of two subpulses at vastly different frequencies.
These objects were previously studied by putting emphasis on their frequency-
domain representation, showing that a soliton can act as localized trapping
potential with discrete level spectrum Melchert et al. (2019a), supporting the
formation of two-frequency pulse compounds in cases where both subpulse-
amplitudes are of similar size Melchert et al. (2019a); Melchert and Demircan
(2021). Perturbations of various type where studied in this context Melchert
et al. (2021); Willms et al. (2022); Oreshnikov et al. (2022). A complementing
approach in terms of a multi-scales analysis, putting emphasis on the
representation in the time domain, showed that they form a class of
generalized dispersion Kerr solitons which can be described using the concept
of a meta-envelope Tam et al. (2019). Such two-color solitons were recently
verified experimentally in mode-locked laser cavities Lourdesamy et al.
(2021); Mao et al. (2021). Here, we extend the range of systems in which such
pulse compounds can be observed. We consider waveguides with a single zero-
dispersion point and a single zero-nonlinearity point, where the nonlinear
coefficient is negative in the domain of normal dispersion. This setup allows
for group-velocity matching within a large range of frequencies, and allows
insight into the complex interplay of sign changing nonlinear and dispersive
effects. Photonic-crystal fibers with frequency dependent nonlinearity with
the above properties can be obtained by doping with nanoparticles Driben et
al. (2009); Bose et al. (2016a, b); Arteaga-Sierra et al. (2018); Linale et
al. (2020); Hernandez et al. (2022). Noble gas filled hollow-core waveguides
also offer the possibility to have a negative refractive index within a domain
of normal dispersion Junnarkar and Uesugi (2000). For a model system with the
above properties, we demonstrate the existence of trapped states in solitary-
wave wells, show that two-frequency pulse compounds with mutually bound
subpulses of similar amplitudes are supported, and discuss the dynamics of
such pulse complexes in presence of the Raman effect. The latter leads to the
surprising finding that, when the center frequency of the solitary wave-well
shifts, a trapped state of higher order can transit into the ground-state. For
our analysis we consider two-frequency pulse compounds for which the subpulses
can be well distinguished in the frequency domain, so that their mutual
interaction can be described by an incoherent interaction stemming from XPM
alone.

### Generalized nonlinear Schrödinger equation.

Subsequently, we model pulse propagation in waveguides with frequency-
dependent nonlinearity in terms of the generalized nonlinear Schrödinger
equation (GNSE) Agrawal (2019); Bose et al. (2016a); Zhao et al. (2022)

$\displaystyle i\partial_{z}A=$ $\displaystyle-\sum_{n\geq
2}\frac{\beta_{n}}{n!}(i\partial_{t})^{n}A-(1-f_{R})\gamma_{\mathrm{eff}}|A|^{2}A$
$\displaystyle-f_{R}\gamma
A\int_{0}^{\infty}h_{R}(t^{\prime})|A(z,t-t^{\prime})|^{2}~{}{\mathrm{d}}t^{\prime},$
(1)

for a complex-valued envelope $A=A(z,t)$. Therein, time $t$ is measured in a
reference frame moving with the group velocity at $\omega_{0}\approx
2.2559\,\mathrm{rad/fs}$, and $z$ is the propagation distance. Following Ref.
Zhao et al. (2022), the dispersion coefficients are taken as
$\beta_{2}=-1.183\times 10^{-2}\,\mathrm{fs^{2}/\upmu m}$,
$\beta_{3}=8.10383\times 10^{-2}\,\mathrm{fs^{3}/\upmu m}$,
$\beta_{4}=-9.5205\times 10^{-2}\,\mathrm{fs^{4}/\upmu m}$,
$\beta_{5}=0.20737\,\mathrm{fs^{5}/\upmu m}$,
$\beta_{6}=-0.53943\,\mathrm{fs^{6}/\upmu m}$,
$\beta_{7}=1.3486\,\mathrm{fs^{7}/\upmu m}$,
$\beta_{8}=-2.5495\,\mathrm{fs^{8}/\upmu m}$,
$\beta_{9}=3.0524\,\mathrm{fs^{9}/\upmu m}$, and
$\beta_{10}=-1.7140\,\mathrm{fs^{10}/\upmu m}$. As function of the angular
frequency detuning $\Omega=\omega-\omega_{0}$, they define the propagation
constant $\beta(\Omega)=\sum_{n=2}^{10}\beta_{n}\Omega^{n}/n!$, with relative
group delay $\beta_{1}(\Omega)=\partial_{\Omega}\beta(\Omega)$ [Fig. 1(a)] and
group-velocity dispersion
$\beta_{2}(\Omega)=\partial_{\Omega}^{2}\beta(\Omega)$ [Fig. 1(b)]. The
nonlinear coefficients are modeled as
$\gamma(\Omega)=\gamma_{0}+\gamma_{1}\Omega$, with
$\gamma_{0}=0.11\,\mathrm{W^{-1}/m}$ and $\gamma_{1}=4.8728\times
10^{-5}\,\mathrm{ps\,W^{-1}/m}$, and as
$\gamma_{\rm{eff}}(\Omega)=\gamma_{0,\mathrm{eff}}+\gamma_{1,\mathrm{eff}}\Omega$,
with $\gamma_{0,\mathrm{eff}}=0.7453\,\mathrm{W^{-1}/m}$, and
$\gamma_{1,\mathrm{eff}}=-4.6822\times 10^{-3}\,\mathrm{ps\,W^{-1}/m}$ [Fig.
1(c)]. For the considered parameters, the zero-disperion point, defined by
$\beta_{2}(\Omega_{\rm{ZDP}})=0$, and the zero-nonlinearity point, defined by
$\gamma_{\rm{eff}}(\Omega_{\rm{ZNP}})=0$, are at
$\Omega_{\rm{ZDP}}\approx\Omega_{\rm{ZNP}}\approx 0.16\,\mathrm{rad/fs}$. The
Raman effect is included as
$h_{R}(t)=(\tau_{1}^{2}+\tau_{2}^{2})\tau_{1}^{-1}\tau_{2}^{-2}\,\exp(-t/\tau_{2})\,\sin(t/\tau_{1})$
with $f_{R}=0.18$, $\tau_{1}=12.2\,\mathrm{fs}$, and
$\tau_{2}=32\,\mathrm{fs}$ Blow and Wood (1989). For the solution of Eq. (1)
with $f_{R}=0.18$ we use a split-step Fourier method Agrawal (2019). When
neglecting the Raman effect, i.e. for $f_{R}=0$, we use the conservation
quantity error method Heidt (2009); Melchert and Demircan (2022). To assess
time-frequency interrelations within $A(z,t)$, we use the spectrogram
$P_{S}(t,\Omega)=\left|\int
A(z,t^{\prime})\exp\left[-(t^{\prime}-t)^{2}/2\sigma^{2}-i\Omega
t^{\prime}\right]~{}{\rm d}t^{\prime}\right|^{2}$ Melchert et al. (2019b),
employing a Gaussian window function with root-mean-square width
$\sigma=50\,\mathrm{fs}$.

Figure 1:  Specifics of the model. (a) Group-delay, (b) group-velocity
dispersion, and, (c) effective nonlinear coefficient. Dot and circle indicate
a pair of group-velocity matched pulses. Domain of normal dispersion is shaded
gray. (d) Potential strength as function of soliton center frequency
$\Omega_{\rm{S}}$. Labels on top indicate $\Omega_{\rm{TR}}$, i.e. group-
velocity matched frequencies at which trapped states exist. (e) Wavenumber
eigenvalues $\kappa^{\prime\prime}_{n}$ and potential depth $V_{0}=\min(V)$ as
function of $\Omega_{\rm{S}}$. Vertical dashed line indicates the pair of
frequencies in (a-c).

### Coupled nonlinear Schrödinger equations.

In search of incoherently coupled two-frequency pulse compounds, we
intentionally neglect the Raman effect and consider complex-valued envelopes
$A_{1}$, and $A_{2}$, of two group-velocity (GV) matched pulses with a vast
frequency gap [Fig. 1(a)], described by the two coupled nonlinear Schrödinger
equations (NSEs)

$\displaystyle
i\partial_{z}\,A_{1}-\frac{\beta_{2}^{\prime}}{2}\partial_{t}^{2}\,A_{1}+\gamma^{\prime}\left(|A_{1}|^{2}+2|A_{2}|^{2}\right)A_{1}=0,$
(2a) $\displaystyle
i\partial_{z}\,A_{2}-\frac{\beta_{2}^{\prime\prime}}{2}\partial_{t}^{2}\,A_{2}+\gamma^{\prime\prime}\left(|A_{2}|^{2}+2|A_{1}|^{2}\right)A_{2}=0.$
(2b)

The incoherently coupled NSEs (2) further neglect higher orders of dispersion
as well as four-wave mixing contributions between the two pulses. Their mutual
interaction is included via XPM alone. Considering the pair of GV matched
frequencies $\Omega_{1}=-0.20\,\mathrm{rad/fs}$, and
$\Omega_{2}=0.57\,\mathrm{rad/fs}$ [Fig. 1(a)], yields
$\beta_{2}^{\prime}=-0.0303\,\mathrm{fs^{2}/\upmu m}$,
$\gamma^{\prime}=1.68\,\mathrm{W^{-1}/m}$,
$\beta_{2}^{\prime\prime}=0.0234\,\mathrm{fs^{2}/\upmu m}$, and
$\gamma^{\prime\prime}=-1.91\,\mathrm{W^{-1}/m}$. This distinguishes the
present setup from earlier ones where
$\beta_{2}^{\prime},\,\beta_{2}^{\prime\prime}<0$, and
$\gamma^{\prime},\,\gamma^{\prime\prime}>0$ Melchert et al. (2019a). Below we
look for solutions to Eqs. (2) in the form

$\displaystyle A_{1}(z,t)=U_{1}(t)e^{i\kappa^{\prime}z},\quad\text{and}\quad
A_{2}(z,t)=U_{2}(t)e^{i\kappa^{\prime\prime}z},$ (3)

wherein $U_{1}$, $U_{2}$ are real-valued envelopes, and $\kappa^{\prime}$,
$\kappa^{\prime\prime}$ are the corresponding wave numbers. Substituting Eqs.
(3) into Eqs. (2) yields the two coupled ordinary differential equations
(ODEs)

$\displaystyle\ddot{U}_{1}-\frac{2}{\beta_{2}^{\prime}}\left[\gamma^{\prime}\left(|U_{1}|^{2}+2|U_{2}|^{2}\right)-\kappa^{\prime}\right]U_{1}=0,$
(4a)
$\displaystyle\ddot{U}_{2}-\frac{2}{\beta_{2}^{\prime\prime}}\left[\gamma^{\prime\prime}\left(|U_{2}|^{2}+2|U_{1}|^{2}\right)-\kappa^{\prime\prime}\right]U_{2}=0,$
(4b)

where the dots denote derivatives with respect to time.

Figure 2: Solitary-wave well with two trapped states. (a) Trapping potential
$V$, wavenumber eigenvalues $\kappa^{\prime\prime}_{n}$, and eigenfunctions
$\phi_{n}$, $n=0,1$. (b) Time-domain propagation dynamics of the soliton and
its trapped state $n=0$. (c) Corresponding spectrum. Filtered view in (b)
details the time-domain view of the frequency range enclosed by the dashed box
in (c). (d,e) Same as (b,c) for $n=1$.

### Trapped states.

Imposing the condition $\max(U_{2})\ll\max(U_{1})$ decouples Eqs. (4):
assuming Eq. (4a) to describe a freely propagating soliton
$U_{1}(t)=\sqrt{P_{0}}\,{\mathrm{sech}}(t/t_{0})$ with
$P_{0}=|\beta_{2}^{\prime}|(\gamma^{\prime}\,t_{0}^{2})^{-1}$ and
$\kappa^{\prime}=\gamma^{\prime}P_{0}/2$, Eq. (4b) takes the form of a
Schrödinger-type eigenvalue problem

$\displaystyle-(\beta_{2}^{\prime\prime}/2)\ddot{\phi}_{n}+V(t)\phi_{n}=\kappa_{n}^{\prime\prime}~{}\phi_{n},$
(5)

with trapping potential
$V(t)=2\gamma^{\prime\prime}P_{0}\,{\mathrm{sech}}^{2}(t/t_{0})$. Since
$\beta_{2}^{\prime\prime}>0$ at $\Omega_{2}=0.57\,\mathrm{rad/fs}$, the
attractive nature of $V$ is enabled by $\gamma^{\prime\prime}<0$ [Fig.
1(b,c)]. In Eq. (5), the wavenumber eigenvalues are real-valued and satisfy
$\kappa_{n}^{\prime\prime}<0$. To each eigenvalue corresponds an eigenfunction
$\phi_{n}(t)$ with $n$ zeros. In analogy to the Pöschl-Teller potential in
one-dimensional quantum scattering theory Landau and Lifshitz (1981); Lekner
(2007), which can be solved exactly, we write
$V(t)=-\nu\,(\nu+1)\,\beta_{2}^{\prime\prime}(2t_{0}^{2})^{-1}\,{\mathrm{sech}}^{2}(t/t_{0})$
with strength-parameter
$\nu=-1/2+\left[1/4+4|(\gamma^{\prime\prime}/\gamma^{\prime})(\beta_{2}^{\prime}/\beta_{2}^{\prime\prime})|\right]^{1/2}$.
The number of trapped states is $N_{\rm{TR}}=\lfloor\nu\rfloor+1$, where
$\lfloor\nu\rfloor$ is the integer part of $\nu$, and the wavenumber
eigenvalues are
$\kappa_{n}^{\prime\prime}=-\beta_{2}^{\prime\prime}\,(2t_{0}^{2})^{-1}\,(\nu-n)^{2}$,
with $n=0,\ldots,\lfloor\nu\rfloor$. Equation (5) suggests an analogy to
quantum mechanics, with $\phi_{n}$ assuming the role of the wavefunction of a
fictitious particle of mass $m=1/\beta_{2}^{\prime\prime}$, confined to a
localized trapping potential. The quantized number of trapped states is akin
to an atomic number, and a bare soliton, with none of its trapped states
occupied, resembles the nucleus of an one-dimensional atom. By this analogy,
the soliton along with its trapped states represents a nonlinear-photonics
meta-atom. The variation of the potential-strength $\nu$ and the discrete
level spectrum of the solitary-wave well $V$ as function of the soliton center
frequency $\Omega_{\rm{S}}$ are shown in Figs. 1(d,e): for decreasing
$\Omega_{\mathrm{S}}$, the trapping potential induced by the soliton features
an increasing number of bound states. An example for the choice
$\Omega_{\rm{S}}=-0.20\,\mathrm{rad/fs}$ and $t_{0}=50\,\mathrm{fs}$, with
$\Omega_{\rm{TR}}=0.57\,\mathrm{rad/fs}$ and $\nu\approx 1.98$ [Fig. 1(d)] is
detailed in Fig. 2. There exist $N_{\rm{TR}}=2$ trapped states at
$(\kappa_{0}^{\prime\prime},\,\kappa_{1}^{\prime\prime})=(-18.29,-4.47)\,\mathrm{m^{-1}}$,
given by $\phi_{0}(t)\propto{\mathrm{sech}}^{\nu}(t/t_{0})$, and
$\phi_{1}(t)\propto{\mathrm{sech}}^{\nu-1}(t/t_{0})\,{\mathrm{tanh}}(t/t_{0})$
[Fig. 2(a)]. In the vicinity of $\Omega_{\mathrm{TR}}$, due to
$\kappa_{0}^{\prime\prime},\,\kappa_{1}^{\prime\prime}<0$, a wavenumber-gap
separates the trapped states from linear waves with propagation constant
$\beta^{\prime\prime}=(\beta_{2}^{\prime\prime}/2)(\Omega-\Omega_{\mathrm{TR}})^{2}\geq
0$. The stable propagation of intitial conditions
$A_{0}(t)=U_{1}(t)e^{-i\Omega_{\rm{S}}t}+\phi_{n}(t)e^{-i\Omega_{\rm{TR}}t}$,
with weak trapped states of amplitude $\max(|\phi_{n}|)=0.05\sqrt{P_{0}}$,
$n=0,1$, in terms of Eq. (1) in absence of the Raman effect ($f_{R}=0$) is
demonstrated in Figs. 2(b-e). To account for the change in group-velocity of
the soliton in presence of a linear variation of $\gamma$ Haus and Ippen
(2001), we consider
$v_{0}^{-1}=\beta_{1}(\Omega_{\mathrm{S}})+\gamma_{1,\rm{eff}}P_{0}$.

Figure 3: Incoherently coupled two-color pulse compounds. (a-c) Paramterized
solution of Eqs. (4) (see text for parameters). (a) Scaled amplitudes
$u_{n}=U_{0,n}/\sqrt{P_{0}}$, (b) pulse duration $t_{n}$, and, (c) shape
exponent $\nu_{n}$, $n=1,2$. (d) Pulse pair for
$\kappa^{\prime\prime}=-7.99\,\mathrm{m^{-1}}$, and, (e) pulse pair for
$\kappa^{\prime\prime}=-4.68\,\mathrm{m^{-1}}$. (f) Time-domain propagation
dynamics of the pulse pair in (d). (g) Corresponding spectrum. Filtered views
in (f) detail the time-domain view of the frequency ranges enclosed by the
dashed boxes in (g).

### Simultaneous solution of the coupled ODEs.

Solitary-wave solutions of the coupled nonlinear Eqs. (4) beyond the above
linear limit yield two-frequency pulse compounds of Eq. (1). Under suitable
conditions, such solutions can be specified analytically Haelterman et al.
(1993); Silberberg and Barad (1995); Afanasyev et al. (1989); Pelinovsky and
Kivshar (2000); Melchert and Demircan (2021). However, in order to obtain
solutions for general parameter settings, Eqs. (4) need to be solved
numerically. This is, e.g., possible via shooting methods Haelterman and
Sheppard (1994); Mitchell et al. (1997), spectral renormalization methods
Ablowitz and Musslimani (2005); Lakoba and Yang (2007), conjugate gradient
methods Lakoba (2009); Yang (2009), or Newton methods Dror and Malomed (2016).
We here employ a Newton method employing a boundary value Runge-Kutta
algorithm Kierzenka and Shampine (2001). So as to systematically study
solutions to Eqs. (4) we set
$\kappa^{\prime}=|\beta_{2}^{\prime}|(2t_{0}^{2})^{-1}$ with
$t_{0}=50\,\mathrm{fs}$, and start at the location
$\kappa^{\prime\prime}=-20\,\mathrm{m^{-1}}$ in parameter space, i.e. below
the lowest eigenvalue obtained from Eq. (5). In this case we expect $U_{2}$ to
vanish, and $U_{1}$ to yield a fundamental soliton
$U_{1}(t)=\sqrt{P_{0}}\,{\mathrm{sech}}(t/t_{0})$ with
$P_{0}=|\beta_{2}^{\prime}|(\gamma^{\prime}\,t_{0}^{2})^{-1}$. We set initial
trial functions with parity similar to the soliton and the lowest lying
trapped state, and continue the obtained solutions to larger values of
$\kappa^{\prime\prime}$. The resulting solutions are of the form
$U_{n}(t)=U_{0,n}\,{\mathrm{sech}}^{\nu_{n}}(t/t_{n})$, $n=1,2$, with
parameters summarized in Figs. 3(a-c). Consistent with our results above, we
find that a weak nonzero solution $U_{2}$ with $t_{2}=t_{0}$ and
$\nu_{2}\approx 1.98$ originates at
$\kappa^{\prime\prime}\approx-18.3\,\mathrm{m^{-1}}$. Let us point out that
the above choice of $\max(\phi_{n})/\sqrt{P_{0}}=0.05$ indeed characterises
weak trapped states [Fig. 3(a)]. For
$\kappa^{\prime\prime}>-18.3\,\mathrm{m^{-1}}$, the amplitude of $U_{1}$
continuously decreases while that for $U_{2}$ increases. Above
$\kappa^{\prime\prime}\approx-4\,\mathrm{m^{-1}}$, $U_{1}$ vanishes and
$U_{2}$ describes a fundamental soliton with wavenumber
$\kappa^{\prime\prime}$. Let us note that at
$\kappa^{\prime\prime}\approx-4.68\,\mathrm{m^{-1}}$ we find a pair of
solutions with hyperbolic-secant shape
$U_{n}=U_{0,n}{\mathrm{sech}}(t/t_{0})$, $n=1,2$ [Fig. 3(e)], i.e. a two-color
soliton pair as in Ref. Melchert and Demircan (2021). The stable propagation
of an initial condition
$A_{0}(t)=U_{1}(t)e^{-i\Omega_{\rm{S}}t}+U_{2}(t)e^{-i\Omega_{\rm{TR}}t}$ with
$U_{0,1}\approx U_{0,2}$ [$\kappa^{\prime\prime}=-8\,\mathrm{m^{-1}}$; Fig.
3(d)] in terms of Eq. (1) with $f_{R}=0$ is demonstrated in Figs. 3(f,g). To
account for the change in group-velocity of the pulse compound in Fig. 3(f),
we consider
$v_{0}^{-1}=\beta_{1}(\Omega_{\mathrm{S}})+\gamma_{1,\rm{eff}}(U_{0,1}^{2}+2U_{0,2}^{2})$,
extending the group-velocity correction of Ref. Haus and Ippen (2001) to two-
color pulse compounds.

Figure 4: Perturbation by the Raman effect. (a) Propagation dynamics of a
soliton and a weak trapped state of order $n=0$
($L_{D}=t_{0}^{2}/|\beta^{\prime}_{2}|$). (b) Same for $n=1$. (c) Propagation
dynamics of the pulse compound of Fig. 3(d). Inset labeled A shows a
spectrogram at $z/L_{D}=600$, with $t_{c}$ indicating the peak-location of the
pulse compound. Further insets are detailed in the text.

### Perturbation by the Raman effect.

We next assess the impact of the Raman effect on the propagation dynamics of
the above pulse compounds. In Fig. 4(a) we show a fundamental soliton and a
weak trapped state of order $n=0$, propagating under Eq. (1). While the
soliton experiences a self-frequency-shift, resulting in a deceleration in the
time-domain, the trapped state remains bound by the trapping potential, see
the spectrogram in Fig. 4(a) (inset A). Let us note that the level-spectrum of
the solitary-wave well is affected by the soliton’s frequency downshift [Fig.
1(e)]. While the soliton decelerates, the trapped state starts to oscillate
within the trapping potential (inset B). This deceleration induced oscillation
within the solitary-wave well bears an unexpected consequence when considering
the trapped state of order $n=1$ [Fig. 4(b)]: upon propagation, the initially
swift oscillations (inset B) grow in size (inset C) until finally, the trapped
state transitions into a trapped state of order $n=0$ (inset D). The shape-
conversion of the mode from $n=1\rightarrow 0$ is also evident in the
spectrogram in Fig. 4(b) (inset A). During this transition, a small amount of
radiation emanates from the localized pulses. When considering instances of
incoherently coupled two-color pulse compounds [Figs. 3(a-c)], the Raman
effect can have different consequences [Fig. 4(c)]: when $U_{0,1}>U_{0,2}$,
the pulse compound decelerates
($\kappa^{\prime\prime}=-14.52\,\mathrm{m^{-1}}$); when $U_{0,1}<U_{0,2}$, the
pulse compound accelerates ($\kappa^{\prime\prime}=-4.84\,\mathrm{m^{-1}}$);
in an intermediate parameter range where $U_{0,1}\approx U_{0,2}$, the pulse
compound is nearly unaffected
($\kappa^{\prime\prime}=-9.68\,\mathrm{m^{-1}}$). The latter is a result of
the deceleration of one subpulse being counterbalanced by an acceleration of
its binding partner.

### Summary and conclusions.

In conclusion, we have demonstrated the existence of two-color pulse compounds
in waveguides with a single zero-dispersion point and adequate frequency-
dependent nonlinearity. A strong mutual binding of two group-velocity matched
pulses at vastly different center frequencies, earlier demonstrated for
waveguides with two domains of anomalous dispersion Melchert et al. (2019a),
can, in the present case, be achieved by having $\gamma<0$ in a domain where
$\beta_{2}>0$. The reported study extends the range of systems that support
two-color pulse compounds, and allows to understand the complex propagation
dynamics reported in a recent study on higher-order soliton evolution in a
photonic crystal fiber with one zero-dispersion point and frequency dependent
nonlinearity Zhao et al. (2022). Instances of such two-color pulse compounds
can readily be identified in the propagation studies reported in Ref. Zhao et
al. (2022).

## Acknowledgements

The authors acknowledge financial support from Deutsche Forschungsgemeinschaft
within the Cluster of Excellence PhoenixD (EXC 2122, projectID 390833453).

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