# There is no “Weil-”cohomology theory with real coefficients for arithmetic
curves

Christopher Deninger111Funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy EXC
2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure and the CRC
1442 Geometry: Deformations and Rigidity

( )

##  1 Weil cohomologies over $\overline{\mathbb{F}}_{p}$

A well known argument by Serre shows that there is no Weil cohomology theory
with real coefficients for smooth projective varieties over
$\overline{\mathbb{F}}_{p}$. For such a cohomology theory and every elliptic
curve $E$ over $\overline{\mathbb{F}}_{p}$ there would have to be an
${\mathbb{R}}$-algebra anti-homomorphism

######  1.1.

$(\mathrm{End}\,E)\otimes{\mathbb{R}}\longrightarrow\mathrm{End}\,H^{1}(E,{\mathbb{R}})\;,$

However, if $E$ is supersingular, then $(\mathrm{End}\,E)\otimes{\mathbb{Q}}$
is a quaternion algebra over ${\mathbb{Q}}$ which is non-split precisely at
$p$ and $\infty$. Thus $(\mathrm{End}\,E)\otimes{\mathbb{R}}$ would be the
Hamilton quaternions $\mathbb{H}$ whereas
$\mathrm{End}\,H^{1}(E,{\mathbb{R}})=M_{2}({\mathbb{R}})$, and there is no
anti-homomorphism of ${\mathbb{R}}$-algebras $\mathbb{H}\to
M_{2}({\mathbb{R}})$. The algebras $(\mathrm{End}\,E)\otimes{\mathbb{Q}}_{l}$
for $l\neq p$ and $(\mathrm{End}\,E)\otimes Q_{p}$ are split, where $Q_{p}$ is
the quotient field of $W(\overline{\mathbb{F}}_{p})$, the ($p$-typical) Witt
vector ring of $\overline{\mathbb{F}}_{p}$. Hence Serre’s argument does not
contradict the existence of Weil cohomology theories with values in
${\mathbb{Q}}_{l}$ resp. $Q_{p}$. Indeed $l$-adic and crystalline cohomology
are such theories. In particular, embedding ${\mathbb{Q}}_{l}$ or $Q_{p}$ into
${\mathbb{C}}$ one sees that there is a Weil cohomology with values in
${\mathbb{C}}$-vector spaces. Of course such embeddings are highly
uncanonical. However, if the standard conjectures on algebraic cycles hold,
there is a Weil-cohomology theory with values in $\overline{{\mathbb{Q}}}$. It
would give a theory with complex coefficients after choosing an embedding
$\overline{{\mathbb{Q}}}\hookrightarrow{\mathbb{C}}$, which is a more
reasonable process than embedding ${\mathbb{Q}}_{l}$ or $Q_{p}$ into
${\mathbb{C}}$.

In Scholze [15, Conjecture 9.5 ff] the existence of a Weil cohomology theory
$H^{n}_{Kt_{{\mathbb{Q}}}}(X)$ for varieties $X$ over
$\overline{\mathbb{F}}_{p}$ with values in Kottwitz’ category
$\mathrm{Kt}_{{\mathbb{Q}}}$ is conjectured, and assuming well known
conjectures also proved. As yet, we have no indication that such a refined
category as (the $\mathrm{ind}$-category of $\mathrm{Kt}_{{\mathbb{Q}}}$) may
be the target of our conjectural “Weil-”cohomology
$H^{n}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ for
arithmetic curves in the next section. However, the following much simpler
${\mathbb{R}}$-linear abelian $\otimes$-category $K_{{\mathbb{R}}}$ may be a
target category. Its objects are ${\mathbb{Z}}/2$-graded ${\mathbb{C}}$-vector
spaces $V=V_{0}\oplus V_{1}$ with an antilinear isomorphism $\sigma$
respecting the grading and $\sigma^{2}=(-1)^{i}$ on $V_{i}$. Morphisms are
${\mathbb{C}}$-linear maps which respect the grading and commute with
$\sigma$. On $V_{0}$ the involution $\sigma$ defines the real structure
$V^{\sigma=\mathrm{id}}_{0}$, i.e.
$V_{0}=V^{\sigma=\mathrm{id}}_{0}\otimes_{{\mathbb{R}}}{\mathbb{C}}$. On
$V_{1}$ we obtain a quaternionic structure: The quaternions
$\mathbb{H}={\mathbb{R}}\oplus{\mathbb{R}}i\oplus{\mathbb{R}}j\oplus{\mathbb{R}}k$
act on $V_{1}$ by $1,i,j=\sigma$ and $k=i\sigma$, turning $V_{1}$ into an
$\mathbb{H}$-vector space. Thus Serre’s argument would not exclude a Weil
cohomology theory $H^{n}$ with values in $K_{{\mathbb{R}}}$ if
$H^{1}(E)=H^{1}(E)_{1}$ for supersingular elliptic curves. In fact, as Milne
pointed out to me, it follows directly from Grothendieck’s standard
conjectures that there exists a Weil cohomology with values in
$K_{{\mathbb{R}}}$ \- the conjectures imply that the motives over
$\overline{\mathbb{F}}_{p}$ form a polarized Tannakian category, to which one
can apply [1, Theorem 4.29]. He also remarked that this was likely known to
Grothendieck in the 1960s. Note that there are canonical $\otimes$-functors
$\mathrm{Kt}_{{\mathbb{Q}}}\to\mathrm{Kt}_{{\mathbb{R}}}\to K_{{\mathbb{R}}}$,
where Kottwitz’ category $\mathrm{Kt}_{{\mathbb{R}}}$ is the same as the
category $V_{\infty}$ in [11], Example 1.7.

##  2 “Weil-”cohomologies over $\mathrm{spec}\,{\mathbb{Z}}$

In several articles we discussed the possibility of a cohomology theory
$H^{n}(\ ,{\mathcal{C}})$ with values in ${\mathbb{C}}$-vector spaces which is
defined for all arithmetic schemes over $\mathrm{spec}\,{\mathbb{Z}}$ and
their arithmetic compactifications. The theory should be equipped with
functorial endomorphisms $\theta$ whose eigenvalues would be the zeroes and
poles of the corresponding zeta functions. In particular the cohomology groups
can be infinite dimensional. For arithmetic schemes over
$\mathrm{spec}\,{\mathbb{F}}_{p}$ such a theory can be constructed by a formal
process starting with $l$-adic cohomology, c.f. [2, section 4]. The question
arises whether a theory of this type can have real coefficients. For
arithmetic schemes which are not flat over spec ${\mathbb{Z}}$ the arguments
in [5, 4.7] make this improbable but for flat schemes over
$\mathrm{spec}\,{\mathbb{Z}}$ real coefficients remained a theoretical
possibility. In the following, we will explain why no real-valued theory of
the type conjectured in [4], [6] can exist, even for spectra of number rings.
On the other hand, a theory with values in $K_{{\mathbb{R}}}$ refining the
conjectural ${\mathbb{C}}$-theory seems plausible. We begin by recalling some
of the discussion from [4], [6]. Let $k/{\mathbb{Q}}$ be a number field and
let $\overline{\mathrm{spec}\,\mathfrak{o}_{k}}$ be the arithmetic
compactification of $\mathrm{spec}\,\mathfrak{o}_{k}$. The basic expected
properties of
$H^{n}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ and its
operator $\theta$ are the following:

######  2.1.

$\theta$ is a derivation with respect to cup-product.

######  2.2.

$H^{0}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})={\mathbb{C}}$
with $\theta=0$.

######  2.3.

$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ is infinite
dimensional with the eigenvalues of $\theta$ being the non-trivial zeroes of
$\zeta_{k}(s)$ with their multiplicities. Here the eigenvalues $\alpha$ are
counted with their algebraic multiplicities $\dim
H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim\alpha}$.
Here

$H^{\theta\sim\alpha}=\\{h\in H\mid(\theta-\alpha)^{n}(h)=0\quad\text{for
some}\;n\geq 1\\}$

is the generalized $\alpha$-eigenspace of $\theta$.

######  2.4.

There is a $\theta$-equivariant trace isomorphism

$\mathrm{tr}:H^{2}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})\xrightarrow{\sim}{\mathbb{C}}(-1)\quad\text{where}\;{\mathbb{C}}(-1)={\mathbb{C}}\;\text{with}\;\theta=\mathrm{id}\;.$

By 2.1 and 2.4, for $h_{1},h_{2}\in
H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ we would have

######  2.5.

$h_{1}\cup h_{2}=\theta(h_{1}\cup h_{2})=\theta h_{1}\cup
h_{2}+h_{1}\cup\theta h_{2}\;.$

For $h_{1}\in
H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim\alpha}$
and $h_{2}\in
H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim\beta}$
we have

$h_{1}\cup h_{2}\in
H^{2}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim\alpha+\beta}\;.$

Hence $h_{1}\cup h_{2}\neq 0$ can only happen by 2.4 if $\alpha+\beta=1$ and
we expect that:

######  2.6.

the pairing

$\cup:H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim\alpha}\times
H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim
1-\alpha}\xrightarrow{\cup}H^{2}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})\overset{\mathrm{tr}}{\xrightarrow{\sim}}{\mathbb{C}}$

is perfect. In particular, $\alpha$ is an eigenvalue of $\theta$ on $H^{1}$ if
and only if $1-\alpha$ is an eigenvalue and the (algebraic) multiplicities are
equal. Note that via 2.3 this is compatible with the functional equation of
$\zeta_{k}(s)$.

Much more optimistically, one could hope for the following property.

######  2.7.

There is a canononical anti-linear automorphism $\ast$ of
$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ with
$\ast^{2}=-\mathrm{id},\ast\mbox{\scriptsize$\,\circ\,$}\theta=\theta\mbox{\scriptsize$\,\circ\,$}\ast$
and such that we get a scalar product by setting

$\langle h,h^{\prime}\rangle=\mathrm{tr}(h\cup\ast
h^{\prime})\quad\text{on}\;H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})\;.$

In particular $\alpha$ is then an eigenvalue of $\theta$ on $H^{1}$ if and
only if $\overline{\alpha}$ is an eigenvalue. Note that 2.5 and 2.7 imply the
relation

$\langle h,h^{\prime}\rangle=\langle\theta h,h^{\prime}\rangle+\langle
h,\theta h^{\prime}\rangle\;.$

Hence $\theta-\frac{1}{2}$ would be skew-symmetric on
$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ and its
eigenvalues therefore purely imaginary. Using 2.3 the Riemann hypotheses would
follow. Of course this argument follows the lead of Serre [16]. Also note that
if 2.7 is true, skew-symmetry of $\theta-\frac{1}{2}$ would imply that the
algebraic eigenspaces $H^{\theta\sim\alpha}$ of $\theta$ are the usual ones
$H^{\theta=\alpha}$.

The following heuristic argument suggests that property 2.7 may be more than
whishful thinking. In [8] we showed that higher dimensional analogues of 2.7
together with certain Abel-Jacobi isomorphisms would imply Beilinson’s
conjectures on the positivity of height pairings. In particular, suitably
normalized the Néron-Tate height pairing on $E({\mathbb{Q}})$ for an elliptic
curve $E$ over ${\mathbb{Q}}$ would be the scalar product as in 2.7 restricted
to the $1$-eigenspace of $\theta$ on the relevant cohomology group. We expect
an Abel-Jacobi isomorphism to identify $E({\mathbb{Q}})\otimes{\mathbb{C}}$
with this $1$-eigenspace. Thus the same scalar product on cohomology whose
positivity would imply that all zeroes of $L(E,s)$ lie on the line
$\mathrm{Re}\,s=1$ would also imply that the Néron-Tate height pairing on
$E({\mathbb{Q}})$ is positive definite, c.f. [8] for the details.

For $\overline{\mathrm{spec}\,\mathfrak{o}_{k}}$ there is no natural known
candidate of a ${\mathbb{C}}$-vector space which might be isomorphic to
$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}})^{\theta\sim\frac{1}{2}}$
via an Abel-Jacobi map. We will discuss this intriguing problem below.

As a particular case of 2.3 we would get the formula

######  2.8.

$\mathrm{ord}_{s=1/2}\,\zeta_{k}(s)=\dim_{{\mathbb{C}}}H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim\frac{1}{2}}\;.$

Note that if an antilinear involution $\ast$ on
$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ exists with
$\ast^{2}=-\mathrm{id}$ and
$\ast\mbox{\scriptsize$\,\circ\,$}\theta=\theta\mbox{\scriptsize$\,\circ\,$}\ast$
then
$V_{k}=H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim\frac{1}{2}}$
carries a quaternionic structure and hence it has even
${\mathbb{C}}$-dimension. Since the order of $\zeta_{k}(s)$ at $s=1/2$ is even
by the functional equation, this is compatible with (2.8). Alternatively, the
existence of the perfect alternating pairing 2.6 for $\alpha=1/2$

######  2.9.

$\cup:V_{k}\times V_{k}\longrightarrow{\mathbb{C}}$

would also imply that $\dim V_{k}$ is even.

Let us now assume that $k/{\mathbb{Q}}$ is Galois with group $G$. By
contravariant functoriality of the cohomology ring with derivation
$(H^{{\scriptscriptstyle\bullet\,}}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}}),\cup,\theta)$
the group $G$ would act on
$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ and in
particular on $V_{k}$. By the following argument kindly pointed out to me by
Baptiste Morin, it is natural to assume that $G$ acts trivially on
$H^{2}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})\cong{\mathbb{C}}$,
so that the pairings 2.6 and in particular (2.9) would be $G$-invariant.
Namely one may expect a natural continuous $G$-equivariant homomorphism from
the Arakelov Chow group:

(1) $cl:CH^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}})\longrightarrow
H^{2}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})\;.$

The map $cl$ would send the compact kernel of the Arakelov degree map to zero
since ${\mathbb{C}}$ has no non-trivial compact subgroups. Hence $cl$ would
factor over the Arakelov degree map and this would lead to a copy of
${\mathbb{R}}$ with trivial $G$-action inside of
$H^{2}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$. Hence $G$
would act trivially on the entire group.

For an equivalence class $\pi$ of irreducible representations of $G$ over
${\mathbb{C}}$ let
$e_{\pi}\in{\mathbb{C}}[G]^{\mathrm{opp}}=\mathrm{End}\,_{G}{\mathbb{C}}[G]$
be the corresponding idempotent projecting to the $\pi$-isotypical component
of ${\mathbb{C}}[G]$. For any complex vector space $H$ on which $G$ acts,
$e_{\pi}$ defines a projector and we set $H(\pi)=e_{\pi}H$. For a realization
$V_{\pi}$ of $\pi$ consider the $G$-equivariant isomorphism:

$\mathrm{Hom}_{G}(V_{\pi},H)\otimes
V_{\pi}\xrightarrow{\sim}H(\pi)\quad\text{with}\;f\otimes v\mapsto f(v)\;.$

Since $\theta$ commutes with the $G$-action, it respects both
$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})(\pi)$ and the
decomposition

$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})=\bigoplus_{\pi}H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})(\pi)\;.$

Consider the $G$-isomorphism:

$\mathrm{Hom}_{G}(V_{\pi},H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}}))\otimes
V_{\pi}\xrightarrow{\sim}H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})(\pi)\;.$

It is compatible with $\theta$ if $\theta$ acts on the left as
$\theta(f\otimes v)=(\theta\mbox{\scriptsize$\,\circ\,$}f)\otimes v$. In
accordance with the formula

$\zeta_{k}(s)=\prod_{\pi}L({\mathbb{C}}[G](\pi),s)=\prod_{\pi}L(V_{\pi},s)^{\dim
V_{\pi}}$

we expect:

######  2.10.

The eigenvalues $\alpha$ of $\theta$ on
$\mathrm{Hom}_{G}(V_{\pi},H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}}))$
counted with their algebraic multiplicities are the non-trivial zeroes of the
Artin $L$-function $L(V_{\pi},s)$ counted with multiplicity.

A corresponding assertion in the function field case is true, c.f. [10, § 9].

In particular, we would get

######  2.11.

$\dim_{{\mathbb{C}}}\mathrm{Hom}_{G}(V_{\pi},H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim\alpha})=\mathrm{ord}_{s=\alpha}L(V_{\pi},s)\;.$

For $\alpha=1/2$ this amounts to the formula

######  2.12.

$\dim_{{\mathbb{C}}}\mathrm{Hom}_{G}(V_{\pi},V_{k})=\mathrm{ord}_{s=1/2}L(V_{\pi},s)\;.$

######  Theorem 2.13.

For a number field $k$ there is no functorial real valued cohomology theory
$H^{{\scriptscriptstyle\bullet\,}}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{R}})$
with endomorphism $\theta_{{\mathbb{R}}}$ such that
$H^{{\scriptscriptstyle\bullet\,}}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}}):=H^{{\scriptscriptstyle\bullet\,}}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{R}})\otimes_{{\mathbb{R}}}{\mathbb{C}}$
with $\theta:=\theta_{{\mathbb{R}}}\otimes\mathrm{id}$ satisfies property
2.12.

###### Proof.

Let $k$ be a finite Galois extension of ${\mathbb{Q}}$ whose Galois group $G$
has an irreducible symplectic representation $V_{\pi}$ with root number
$W(\pi)=-1$. For example, there exist quaternion fields i.e. normal extensions
of ${\mathbb{Q}}$ with Galois group the quaternion group $H_{8}$ with $8$
elements whose Artin $L$-function for the unique non-abelian representation
(it is symplectic) has root number $-1$. See [12] for examples of such fields.
From the functional equation of $L(V_{\pi},s)$ it follows that
$\mathrm{ord}_{s=1/2}L(V_{\pi},s)\geq 1$ is odd. By property 2.12 the
representation $V_{\pi}$ occurs with odd multiplicity
$\mathrm{Hom}_{G}(V_{\pi},V_{k})$ in $V_{k}$. If a theory
$H^{{\scriptscriptstyle\bullet\,}}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{R}})$
as in the theorem exists set

$V^{{\mathbb{R}}}_{k}=H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{R}})^{\theta\sim\frac{1}{2}}\;.$

Then $V^{{\mathbb{R}}}_{k}$ would be a $G$-module with
$V_{k}=V^{{\mathbb{R}}}_{k}\otimes_{{\mathbb{R}}}{\mathbb{C}}$. and we could
write $V^{{\mathbb{R}}}_{k}$ as a sum of irreducible representations $W$ of
$G$ over ${\mathbb{R}}$. According to [17, 13.2] there are three possibilities
for $W$. If $W$ has the form $W=\mathrm{Res}V$ for an irreducible non-self
dual irreducible representation $V$ over ${\mathbb{C}}$, then we would have
$W\otimes{\mathbb{C}}=V\oplus\overline{V}$ and since both $V$ and
$\overline{V}$ are not self-dual they cannot be symplectic. If $W$ is
orthogonal then $W\otimes{\mathbb{C}}$ is an irreducible orthogonal
${\mathbb{C}}$-representation and hence not symplectic. If $W$ is symplectic
i.e. $W=\mathrm{Res}V$ for a symplectic irreducible representation $V$ over
${\mathbb{C}}$ then $W\otimes{\mathbb{C}}=V\oplus V$. It follows that the
symplectic representation $V_{\pi}$ occurs with even multiplicity in $V_{k}$.
This is a contradiction. ∎

I would like to thank one of the referees for suggesting this simplification
of my original argument.

There is thus no real valued “Weil-”cohomology for arithmetic curves. On the
other hand, the operator $\ast$ in 2.7 would give a canonical quaternionic
structure on $H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$
and hence we could view
$(H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}}),\ast)$ as an
object of $K_{{\mathbb{R}}}$, equal to its degree $1$ part. The groups $H^{0}$
and $H^{2}$ being isomorphic to ${\mathbb{C}}$ by 2.2 and 2.4 must be equal to
their degree zero parts if they actually take values in $K_{{\mathbb{R}}}$. In
other words, they must have natural real structures. For $H^{2}$ the real
structure would be the image of the cycle class map (1) in Morin’s heuristic
argument above. We noted that $cl$ would have to factorize over the Arakelov
degree map onto ${\mathbb{R}}$.

Hence, a “Weil-”cohomology theory with values in $K_{{\mathbb{R}}}$ is a
possibility for arithmetic curves.

In [6], [7] we discussed analogies between the leafwise (or tangential)
cohomology of certain foliated dynamical systems and cohomologies such as the
hoped for $H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$.
The operator $\theta$ corresponds to the infinitesimal generator of the
induced ${\mathbb{R}}$-action by the flow on leafwise cohomology. Complex
leafwise cohomology always has the real structure given by the cohomology of
the real forms. Since such a real structure cannot exist on the cohomologies
$H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})$ as we have
seen, it follows that the analogy is too simple. There has to be a twisted
version of leafwise cohomology with complex coefficients that does not have a
natural real structure. This can become relevant since in [9] we did construct
foliated dynamical systems for arithmetic schemes which have some (but not
all) the expected properties.

We will now state the expected properties of the spaces
$V_{k}=H^{1}(\overline{\mathrm{spec}\,\mathfrak{o}_{k}},{\mathcal{C}})^{\theta\sim
1/2}$ and suggest the problem to find a direct number theoretical construction
of the $V_{k}$. This refines the discussion in [3, § 7]. We need the following
conjecture which is due to Serre (unpublished). Sarnak and Rubinstein also
arrived at this conjecture in a wider context on p. 195 of [14]. Previously
Chowla had conjectured that the Dirichlet $L$-series of quadratic characters
do not vanish at $s=1/2$.

Vanishing Conjecture For an irreducible complex representation of the Galois
group of ${\mathbb{Q}}$ the corresponding Artin $L$-function has a zero at
$s=1/2$ if and only if the representation is symplectic with root number $-1$.
In this case the order of vanishing is one.

If the root number is $-1$, the functional equation implies that the Artin
$L$-function vanishes at $s=1/2$. For orthogonal irreducible representations
it is known that the root number is $+1$ by [10]. Hence there is no a priori
reason why their $L$-functions should vanish at $s=1/2$. For certain
quaternion fields the vanishing conjecture has been verified numerically in
[12].

If $k/{\mathbb{Q}}$ is Galois, it follows from 2.12 and the vanishing
conjecture that precisely the symplectic $\pi$’s with $W(\pi)=-1$ among the
irreducible representations of $G$ are realized in the $G$-module $V_{k}$ and
each of them with multiplicity one. Thus, as ${\mathbb{C}}[G]$-modules we
should have

######  2.14.

$V_{k}=\bigoplus_{\pi\,\text{\rm sympl}\atop W(\pi)=-1}V_{k}(\pi)$ and
$V_{k}(\pi)\cong V_{\pi}$.

Since the $V_{k}(\pi)$ are irreducible self dual and pairwise non-isomorphic,
it would follow that the cup-product on $V_{k}$ gives the (up to scalar)
unique $G$-invariant symplectic pairing on $V_{k}(\pi)$.

All together we get the following suggestion:

######  2.15.

There should be a covariant functor $k\mapsto(V_{k},\cup)$ from the category
of number fields $k$ to the category of finite dimensional complex vector
spaces $V_{k}$ with a non-degenerate alternating form $\cup:V_{k}\times
V_{k}\to{\mathbb{C}}$. If $k$ is Galois over ${\mathbb{Q}}$ with group $G$,
the ${\mathbb{C}}[G]$-module structure on $V_{k}$ resulting from functoriality
should be the one described in 2.14. In particular, we should have

$\dim_{{\mathbb{C}}}V_{k}=\sum_{\pi\,\text{sympl}\atop W(\pi)=-1}\dim
V_{\pi}\;.$

Moreover, we expect the symplectic vector spaces $(V_{k},\cup)$ to be equipped
with an antilinear isomorphism $\ast:V_{k}\to V_{k}$ with $\ast^{2}=-1$ such
that we obtain a scalar product by setting

$\langle v,w\rangle=v\cup\ast w\quad\text{for}\;v,w\in V_{k}\;.$

Like height pairings, the symplectic form $\cup$ should be a sum of local
contributions.

In case a natural candidate for $V_{k}$ can be found, one may also expect a
formula for the leading coefficient of $\zeta_{k}(s)$ at $s=1/2$ in the style
of the Birch Swinnerton-Dyer or Lichtenbaum Weil-étale conjectures. From the
point of view of the Beilinson conjectures, $V_{k}$ would be the
$1/2$-eigenspace of the Adams operators on $K_{0}(k)$, or the $CH^{1/2}$ group
of $\mathrm{spec}\,k$ or the group of extensions
$\mathrm{Ext}^{1}({\mathbb{C}}(0),{\mathbb{C}}(1/2))$ in some
${\mathbb{C}}$-linear category of “motives” over $k$ where a $1/2$-twist of
the Tate motive exists. Classically all these objects do not make sense in the
number field case. In the function field case the $H^{1}$ of a supersingular
elliptic curve over a finite field can be used as a kind of $1/2$-Tate twist,
c.f.[13]. In the Galois module theory of rings of integers in tamely ramified
extensions $k/{\mathbb{Q}}$, symplectic representations with root number $-1$
play a vital role. Also, exponential motives come to mind because they allow
non-integral twists.

Acknowledgements. I would like to thank the referees for their suggestions to
simplify and improve this note. I am also grateful to Luca Barbieri Viale,
Werner Bley, James Milne, Baptiste Morin, Peter Sarnak and Jean-Pierre Serre
for valuable mathematical and historical comments. Many thanks go to Umberto
Zannier for the invitation to the SNS in Pisa, where the first version of this
note was written.

## References

  * [1] P. Deligne and J. S. Milne.  Tannakian Categories, pages 101–228.  Springer Berlin Heidelberg, Berlin, Heidelberg, 1982.
  * [2] Christopher Deninger.  Motivic $L$-functions and regularized determinants.  In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos. Pure Math., pages 707–743. Amer. Math. Soc., Providence, RI, 1994.
  * [3] Christopher Deninger.  Motivic $L$-functions and regularized determinants. II.  In Arithmetic geometry (Cortona, 1994), Sympos. Math., XXXVII, pages 138–156. Cambridge Univ. Press, Cambridge, 1997.
  * [4] Christopher Deninger.  Some analogies between number theory and dynamical systems on foliated spaces.  In Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), number Extra Vol. I, pages 163–186, 1998.
  * [5] Christopher Deninger.  On dynamical systems and their possible significance for arithmetic geometry.  In Regulators in analysis, geometry and number theory, volume 171 of Progr. Math., pages 29–87. Birkhäuser Boston, Boston, MA, 2000\.
  * [6] Christopher Deninger.  Number theory and dynamical systems on foliated spaces.  Jahresber. Deutsch. Math.-Verein., 103(3):79–100, 2001.
  * [7] Christopher Deninger.  Analogies between analysis on foliated spaces and arithmetic geometry.  In Groups and analysis, volume 354 of London Math. Soc. Lecture Note Ser., pages 174–190. Cambridge Univ. Press, Cambridge, 2008.
  * [8] Christopher Deninger.  The Hilbert-Polya strategy and height pairings.  In Casimir force, Casimir operators and the Riemann hypothesis, pages 275–283. Walter de Gruyter, Berlin, 2010.
  * [9] Christopher Deninger.  Dynamical systems for arithmetic schemes, 2022.  arXiv:1807.06400.
  * [10] A. Fröhlich and J. Queyrut.  On the functional equation of the Artin $L$-function for characters of real representations.  Invent. Math., 20:125–138, 1973.
  * [11] J. S. Milne.  Motives over finite fields.  In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos. Pure Math., pages 401–459. Amer. Math. Soc., Providence, RI, 1994.
  * [12] Sami Omar.  On Artin $L$-functions for octic quaternion fields.  Experiment. Math., 10(2):237–245, 2001.
  * [13] Niranjan Ramachandran.  Values of zeta functions at $s=1/2$.  Int. Math. Res. Not., (25):1519–1541, 2005.
  * [14] Michael Rubinstein and Peter Sarnak.  Chebyshev’s bias.  Experiment. Math., 3(3):173–197, 1994.
  * [15] Peter Scholze.  $p$-adic geometry.  In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, pages 899–933. World Sci. Publ., Hackensack, NJ, 2018.
  * [16] Jean-Pierre Serre.  Analogues kählériens de certaines conjectures de Weil.  Ann. of Math. (2), 71:392–394, 1960.
  * [17] Jean-Pierre Serre.  Linear representations of finite groups.  Graduate Texts in Mathematics, Vol. 42. Springer-Verlag, New York-Heidelberg, 1977.  Translated from the second French edition by Leonard L. Scott.


# A Neophyte With AutoML: Evaluating the Promises of Automatic Machine
Learning Tools

Oleg Bezrukavnikov<EMAIL_ADDRESS>Letovo SchoolMoscow and  Rhema
Linder<EMAIL_ADDRESS>University of TennesseeKnoxvilleTennessee

###### Abstract.

This paper discusses modern Auto Machine Learning (AutoML) tools from the
perspective of a person with little prior experience in Machine Learning (ML).
There are many AutoML tools both ready-to-use and under development, which are
created to simplify and democratize usage of ML technologies in everyday life.
Our position is that ML should be easy to use and available to a greater
number of people. Prior research has identified the need for intuitive AutoML
tools. This work seeks to understand how well AutoML tools have achieved that
goal in practice. We evaluate three AutoML Tools to evaluate the end-user
experience and system performance. We evaluate the tools by having them create
models from a competition dataset on banking data. We report on their
performance and the details of our experience. This process provides a unique
understanding of the state of the art of AutoML tools. Finally, we use these
experiences to inform a discussion on how future AutoML tools can improve the
user experience for neophytes of Machine Learning.

AutoML, user experience, Machine Learning, visualization

††conference: ; ; ††copyright: none††ccs: Human-centered computing Human
computer interaction (HCI)††ccs: Computing methodologies Machine Learning

##  1\. Introduction

While smaller datasets can be directly manipulated and analyzed by a real
person in order to discover some insights (Zgraggen et al., 2016), AutoML
tools are designed for very large datasets, which are very difficult to
understand and interpret and often require indirect analysis. As datasets
become bigger in size and more complex in terms of data stored, Machine
Learning techniques get more useful and even necessary for analysis. These
techniques help to better interpret big data without the need of detailed
analysis of all data points by transmitting most of the intense work to the
computer. Some kinds of datasets require ML techniques. For example, datasets
with large number of images over the web (Deng et al., 2009; Rerabek and
Ebrahimi, 2016), information on behaviour of users of different apps (Granka
et al., 2004), and information on people’s preferences (Agichtein et al.,
2006).

Ideally, a neophyte developer could create Machine Learning models fast and
easily. However, creating Machine Learning models is difficult, often
requiring a lot of math expertise and hours of intense labor. As well, there
are many datasets with a very large amount of data. Although new techniques
(Belabbas and Wolfe, 2009) help ML-engineers to deal with such type of data
faster, it still requires a lot of work. To reduce the cost of ML analysis,
prior research has developed a variety of AutoML tools. These new tools are
created to simplify and democratize (Shang et al., 2019) the usage of ML
technologies in everyday life.

In this paper we discuss some of the modern AutoML tools from the perspective
of a person with little prior experience, known as a “neophyte”, in Machine
Learning. These tools attempt the lofty goal of giving users an experience
that “automatically composes a Machine Learning pipeline (Shang et al.,
2019).” However, the level of automation depends on the User Experience of
AutoML tools. We recognise User Experience (UX) as “the subjective
relationship between user and application (Calvillo-Gámez et al., 2015;
McCarthy and Wright, 2004).” Using specialized algorithms, AutoML tools
compare different Machine Learning techniques while performing hyperparameter
optimization (Feurer and Hutter, 2019). This process of testing ranks
technique-parameter combinations, giving the user a choice of well-performing
Machine Learning models fitted to the particular dataset.

Our overall aim is to understand how AutoML tools perform in terms of User
Experience and accuracy. We will discuss their performance, ease of use, and
some other parameters through experimentation, which we ran on a number of
datasets. As well, we provide our recommendations which are informed by our
experiences using the tools. This is an empirical approach. The three AutoML
tools, which we decided to test are TPOP, AutoKeras and AutoGluon. These tools
are not the only ones in AutoML field, but we have chosen them as they are
quite well known, affordable and high-level. The present research evaluates
the degree AutoML tools achieve their goal of automation and suitability for
widespread use. We contextualize AutoML related work, describe out methods for
evaluation, show results, and finally end with a discussion. Our discussion
summarized our recommendations for researchers to address AutoML tools’
functionality to best improve their user experience for neophytes of Machine
Learning.

##  2\. Related Work

Recent books (Hutter et al., 2019) and surveys (Zöller and Huber, [n.d.]) of
the whole of AutoML research provide a more complete understanding than we
will address. Our related work section highlights three major types of
research. Like our research, related papers also compare AutoML systems. Prior
research typically either introduces a new benchmark for AutoML systems
comparison, introduces a new AutoML system, or shares results of an AutoML
challenge. While both our paper and the prior work compares numeric scores
among AutoML systems, our research also focuses on UX.

Papers introducing new benchmarks. A large body of research work focuses on
inventing benchmarks (Gijsbers et al., 2019; Balaji and Allen, 2018). This
approach helps compare different AutoML systems with quantified accuracy
relative to state-of-the-art and baseline methods. This research introduces a
new benchmark, explains it, and also quantifies performance of selected AutoML
models on this benchmark. In their paper, Gijsbers et al. share a benchmark
for comparison of some of the most modern AutoML tools like TPOT (Olson and
Moore, 2019a), Auto-WEKA (Thornton et al., 2013), auto-sklearn (Feurer et al.,
2019), and H20 AutoML (Cook, 2016). Their benchmarks are open source and
designed to be updated as AutoML tools change. Gijsbers et al. have worked
with AutoML communities to maintain interoperability and fairness. Balaji and
Allen focus on the benchmark methodology, explaining how their benchmark was
created and proving its high quality. They also refer to the OpenML datasets
(Gijsbers et al., 2019), which are presented in the earlier paper, and discuss
quality issues, such as missing target features. These problems often make
programmatic comparisons among AutoML tools difficult to impossible. In this
paper, while we are not introducing new benchmarks, we are also performing
tests on data to compare User Experience.

Papers introducing new AutoML systems. Despite the fact that automated methods
for generating models is a long-standing problem, dating from before 1990 (as
for example (Tenorio and Lee, 1989)), it remains a popular and unsolved
problem in many contexts. There are many AutoML systems currently in use, as
well those under development, that try to solve these long-standing problems.
For example, AutoGluon (Erickson et al., 2020) is an Amazon developed project.
It is used for Tabular Prediction, Image Classification, Object Detection, and
Text Classification.

Northstar (Kraska, 2018) is in active development, but unreleased for public
use, by professors from MIT and Brown University. Kraska et al.’s Northstar is
an AutoML tool that goes beyond creating models inspired by previous systems
(Crotty et al., 2015). Northstar integrates visualization systems that show
connections in data, helping analysts manipulate presentations to better
understand their data. This solution seeks to make Machine Learning more
accessible to an average person. We agree with Kraska et al. that UX and
simplicity are very important in Machine Learning field. Therefore, in our
paper, we focus on the UX of contemporary AutoML tools.

AutoML challenges. Competitions and challenges are a popular activity for
people in the Machine Learning community. Popular examples include the
following platforms:

  * •

https://www.kaggle.com/competitions

  * •

https://www.drivendata.org/competitions/

  * •

https://ods.ai/competitions

  * •

https://mlcontests.com/.

Many of the challenges are sponsored by companies such as Google, Amazon and
etc. Competitions often offer winners money or free software access as a
reward. The main point of these challenges is that they help to boost research
and awareness in certain fields of Machine Learning (Goodfellow et al., 2013).
This results in finding better solutions and developing new technologies for
big companies, but not always smaller companies and individual programmers.

The concept of an AutoML challenge is less mature than established Machine
Learning challenge venues. One of the biggest AutoML challenges is ChaLearn
(Guyon et al., 2019). It is mostly popular for competitions in the field of
image processing and analysis. In their paper, Guyon et al. analyses data from
different ChaLearn challenges, identifying possible strategies, scoring
metrics, and results. Despite this effort, the participants creating AutoML
systems for the challenge tended to pay little attention to User Experience.
This demonstrates that there is a lack of work comparing UX across AutoML
tools. Our paper address this deficiency by concentrating on the User
Experience rather than system performance.

##  3\. Methods

This methods section outlines how we selected AutoML tools, our dataset,
metrics for performance comparison, and how we evaluated user experience.

###  3.1. Tools Selection

Our reasoning for evaluating AutoML tools was to better understand their
potential for helping people who are not ML specialist to benefit from modern
techniques. We selected AutoML tools that were both popular and recently
developed. The three AutoML tools, which we decided to test are TPOT (Olson
and Moore, 2019b), AutoKeras (Jin et al., 2019) and AutoGluon (Erickson et
al., 2020). These tools are not the only ones in the AutoML field, but we have
chosen them because they are well known, affordable, and functional. AutoGluon
is the most recent tool we are testing. However, there are many AutoML tools
under development. For example, an unreleased system developed by Kraska et
al. supports automated data processing, data visualisation, and ML models
(Kraska, 2018).

###  3.2. Selected Tools

The first public version of TPOT was released in 2015. It is one of the
earliest AutoML tools. TPOT was developed at the Computational Genetics
Laboratory of the University of Pennsylvania with the goal of maximizing
classification accuracy on supervised tasks (Schrider and Kern, 2018). It is
based on the scikit-learn Machine Learning Python library (Hackeling, 2017).
TPOT (Koza and Koza, 1992) uses a combination of genetic programming (Koza and
Koza, 1992) and scikit-learn to provide effective models to users, based on
performance with their data. TPOT’s primary features are solving
Classification and Regression problems, but may also provide support for
Neural Networks.

AutoKeras was published in 2017 by DATA Lab at Texas A&M University (Jin et
al., 2019). Jin et al. released AutoKeras with novel search strategies that
find neural network configurations with the help of Bayesian optimization.
Further, AutoKeras is designed to adapt to different hardware to optimize its
performance. It is based on a Deep Learning library called Keras. When
creating a solution model for user, AutoKeras provides different Neural
Network approaches from the Keras library. AutoKeras provides users with
Classification and Regression tools for structured data as well as images and
text.

Amazon released the public version of AutoGluon in 2019. AutoGluon uses both
Machine Learning and Deep Learning algorithms to generate an effective model
for users. Its main goal is to democratize ML by decreasing the amount of time
and resources required for training. Unlike most other AutoML frameworks that
primarily focus on model hyperparameter selection, AutoGluon Tabular combines
multiple models as ensembles. Experiments ran by AutoGluon developers show
that such an approach offers better use of allocated training time than the
search of the best model. While AutoKeras highlights support for Image and
Text Regression, AutoGluon provides tools for Object detection that identify
and locate targets embedded in larger pictures.

###  3.3. Dataset Selection

Competitions in ML can help early career data scientists learn to work with
realistic data. Our dataset is from a competition in which the lead author of
this paper recently participated (https://onti.ai-academy.ru/competition). We
picked this because competition datasets are designed to present simulated
real-world problems.

For the competition we were given generated data, presenting expenses of bank
clients through 3 years. For each client expense transaction we were given the
time and date when it was performed, the category of products bought and the
sum of the transaction. For the training data we were also given the age group
of the clients. The data placed each client in one of four age groups. The
task was to train a ML model to predict the age group of a client. The score
was counted as the number of correct answers divided by the number of
questions.

We modified the dataset for the purposes of our study to limit computation
time to be appropriate for a consumer grade computer. This dataset, after some
basic operations on data, provided us with a dataset (30000 rows and 1015
columns). We decided to cut the data to test the AutoML, because the
approximate time of work on full data was too big. It predicted us around 10h
of work with all data, but we wanted to test under conditions of a single
workstation. We tested the tools on a dataset size of 5000x1015. We expected
the tools to complete model selection on this data in around an hour. However,
in practice the computation took more time than we expected. Therefore, we
performed evaluation of tools on three training data sizes: 250, 500, 1,000
rows.

###  3.4. Performance Evaluation

In order to compare AutoML tools quantitatively, we established a procedure
for testing performance on the selected data. This section describes
operations we applied to the data before testing.

####  3.4.1. Procedure

In order to compare the performance of the tools, we used a consumer-grade
computer with the following hardware:

  * •

NVIDIA GeForce GTX 1050 Ti

  * •

Core i7, 7700HQ

  * •

RAM 24GB DDR4

All the tools were used under the same conditions, having only the Jupyter
Notebook and command line running on the computer. We close non-essential
windows programs in order to prioritize computing power for training.

We performed the following for each test:

  * •

Data Clearing

  * •

Features Addition

  * •

Model Creation

  * •

Train-test Splitting

  * •

Model Fitting

  * •

Score Evaluation

For _Data Clearing_ , we removed rows with NaN values. For _Features Addition_
, we added new features based on aggregate functions on existing numerical
values. For _Model Creation_ we used TPOT and AutoGluon with default
parameters. We used AutoKeras with specified max_trials parameter
(max_trials=5, max_trials=20) as these greatly impact training time. This
parameter defines how many Neural Network models from Keras library will be
used for model creation on the data. For _Train-test Splitting_ , we used the
first N rows of data as training data and last M rows of data as test data.
This selection helped us to choose the same data for all the tools so that
their performance evaluation is more fair. Mirroring the programming
competition, we included data with approximately equal portions of each class
for training and testing. For _Model Fitting_ , we used the fit function
corresponding to each tool with its default parameters. This function starts
the various model training processes, which are approached differently for
different AutoML tools. For _Score Evaluation_ , we used default evaluation
functions that measure accuracy as the total correctly predicted target values
over the total number of examples.

We limited our training data size to reflect the constraint of limited
computational resources of a general developer with a consumer-grade computer.
We performed evaluation of tools on three training data sizes: 250, 500, 1,000
rows. Each training data size had 1,015 columns. We did not compare results
for bigger number of rows due to time requirements. The slowest tool required
our computer to perform over two hours of training on 1000 rows, which we
believe is a considerable time for an independent developer, who might have
lower computational resources than we had.

####  3.4.2. Metrics

We compared the performance of different tools using the same metric as in the
competition - accuracy (number of right guesses divided by number of answers).
For example, a random guess would yield an accuracy of around 0.25 as there
are 4 classes of similar size in the data. Our best result during the
competition was 0.6438.

###  3.5. User Experience Evaluation

We decided to compare AutoML tools by qualitatively analyzing aspects of the
user experience they provide.

####  3.5.1. Documentation.

We sought to evaluate the documentation of the selected AutoML tools. We found
methods for evaluating documentation that focus on preserving continued open-
sourced development (Lethbridge et al., 2003; McCauley et al., 1996). However,
we did not find methods for evaluating documentation that focuses on early
learners of Machine Learning. Based on our own experiences, we decided to
evaluate three characteristics of documentation, which we see as important
features for AutoML tool users. We developed these characteristics to focus on
User Experience of end-users, rather than potential contributors. They are:

  * •

_Quick start tutorials_ \- help new users to start working with the library.

  * •

_Documented integrations_ \- integrations that help users to find new
potentially helpful usecases of the library.

  * •

_Detailed documentation_ \- helps advanced users utilize the library at its
full potential.

The results section shows our analysis of each AutoML tool according to the
characteristics we define above.

####  3.5.2. Simplicity of the First Use

We believe that the simplicity of the start of usage is a crucial part of user
experience. Many tools and technologies do not receive their deserved
attention because users find them hard to initially configure. We compared the
chosen AutoML tools by the differences in the first setup and number of lines
of code required to start. As ML models are often intended to be used in
different projects, we also compared the simplicity of model export and import
of the chosen AutoML tools as it is crucial for integration.

####  3.5.3. Logs

Logs help developers understand what programs are doing. They are especially
useful when anomalies occur (Fu et al., 2014). As ML principles of work are
sometimes quite confusing to understand, good logs play a big role in making
ML tools democratized(Shang et al., 2019) and more accessible. We compared how
good are logs of the AutoML tools at presenting information and how
customizable is this information.

##  4\. Results

We present numeric results from performance tests of the chosen AutoML tools
and our analysis of their user experience.

###  4.1. Performance Results

In Table 1, we present a comparison of performance of AutoGluon, AutoKeras,
and TPOT on our data from ML competition.

Our results show AutoML tool have higher accuracy than random (0.25) chance.
TPOT is being quite accurate while taking more time to train, AutoKeras can be
very fast with properly set max_trials parameters (less the 2000 seconds to
train on 1000 rows of data), and AutoGluon has the best results in accuracy
and takes reasonable training time (0.597 accuracy trained on 1000 rows). The
baseline Catboost model outperforms all AutoML tools for 250 and 500 rows of
training data in accuracy and requires a much less training time. Overall,
AutoGluon has similar accuracy results to baseline. AutoGluon outperforms the
baseline model for 1000 rows of training data (accuracy 0.597 vs 0.586). Our
baseline model used parameters selected by professional data scientists.
AutoGluon performed with a high accuracy without the benefit of specialist’s
tuning.

Table 1. Each row of this table lists an AutoML tool’s accuracy and training time performance on a subset of the ML competition data in our study. Overall, only AutoGluon performed better than our manually-programmed baseline.  Tool | Train Size (rows) | Work time (seconds) | Test Size (rows) | Accuracy
---|---|---|---|---
Catboost baseline | 250 | 64 | 29000 | 0.536
Catboost baseline | 500 | 107 | 29000 | 0.565
Catboost baseline | 1000 | 136 | 29000 | 0.586
TPOT | 250 | 2038 | 29000 | 0.512
TPOT | 500 | 5280 | 29000 | 0.550
TPOT | 1000 | 9814 | 29000 | 0.568
AutoKeras (max_trials = 5) | 250 | 375 | 29000 | 0.417
AutoKeras (max_trials = 20) | 500 | 1229 | 29000 | 0.450
AutoKeras (max_trials = 20) | 1000 | 1857 | 29000 | 0.513
AutoGluon | 250 | 169 | 29000 | 0.509
AutoGluon | 500 | 892 | 29000 | 0.552
AutoGluon | 1000 | 2482 | 29000 | 0.597

###  4.2. User Experience Results

####  4.2.1. Documentation

In Table 2, we show the results of our analysis of documentation. It shows
that AutoKeras pays very close attention to details, AutoGluon focuses mainly
on quick start tutorials, and all the tools have different side services’
integrations described in their documentation.

Table 2. This figure depicts AutoML tools’ documentation analysis. Overall, TPOT has the least number of use cases and AutoKeras has the most detailed documentation. Tool |  Simplicity of start |  Integrations featured |  Details in documentation
---|---|---|---
TPOT |  Has a quick start tutorial for Classification and Regression on its GitHub page and in github.io documentation. Also features specific examples of usage. |  Has a tutorial on work with Pytorch and Dask on github.io. |  Has detailed documentation on Classification and Regression.
AutoGluon |  Has a quick start tutorial for all main usecases on the official website. Has tabular quick start tutorial on the GitHub page. |  Presents guidilines for pytorch users and explains work with H2O algorithms |  Although it has documentation on some additional features, AutoGluon does not have detailed documentation on the main AutoML tools on its website. Has docs catalog on github, which might be used to generate documentation.
AutoKeras |  Has a quick start tutorial for all main usecades on the official website. Has Image Classification quick start on the GitHub page. |  Explains usage of TensorFlow Cloud with autokeras. Has a tutorial on Trains integration. |  Has detailed documentation on all main classes and functions of the core features.

####  4.2.2. Simplicity of the First Use

One way to make tools more approachable for neophyte developers is to reduce
the barrier to entry, making the first use as easy as possible. When compared,
all of the three AutoML tools had decent documentation and examples for first
usage and required a small amount of code to repeat a basic example. Yet,
AutoGluon required the least amount of code, featuring different time saving
functions.

Considering integration of models in different projects, AutoGluon and
AutoKeras offered standard ML export options. AutoGluon models could be
exported and imported with the help of Joblib Python library ”dump” and ”load”
functions or by using AutoGluon models’ inbuilt ”save” and ”load” functions.
AutoKeras models could be exported and imported with a call of ”save” and
”load_model” inbuilt functions. TPOT did not present a simple export and
import solution. TPOT allowed users to save a model’s training pipeline and
use it in future model trainings, but there was no option to export and import
the trained model itself.

####  4.2.3. Logs

When we used the selected tools, we found their logging to be quite different,
see Figure 4. TPOT and AutoGluon both incorporated a standard verbosity
parameter (Li et al., 2017), which enables its users to choose an integer
verbosity level that limits the level of details they see about ML training.
Popular programming tools like Hadoop often offer various levels of detail in
their logging through _verbosity parameters_ (Li et al., 2017). Some standard
verbosity settings for software would be ”ERROR”, ”WARNING”, and ”INFO”. These
help system designers analyse program logic.

TPOT had 4 levels of verbosity and AutoGluon had 5. In contrast, AutoKeras has
some basic logs out of the box, but offers no verbosity parameter to customize
the level of details. Instead, it provides programming hooks where its users
can make custom logs with tensorflow functions. While this feature might be
particularly useful for advanced users, it is more demanding for newcomers. In
Table 3, we describe the output of TPOT and AutoGluon, given the range of
verbosity level settings. At the lowest setting of verbosity, tools will not
print out anything unless there are warnings. At the highest verbosity
setting, tools log detailed information on each training step. The log
information shows the current accuracy metrics and an overall progress
indicator.

In Figure 4 we present examples of logs provided by AutoKeras, TPOT, and
AutoGluon. While all of these tools have the same goal of providing a model,
their logs are different because the nature of algorithms used are different
for each tool. As we mention in the _Tools Description_ section, AutoKeras
uses neural networks for predictions, TPOT uses genetic programming, and
AutoGluon uses model stacking. Correspondingly, the tools present log
information on epochs, generations, and specific model training. The _Epoch_
is a part of neural network training during which the learning algorithm works
through the entire training dataset one time and updates its parameters
(Brownlee, 2018). In the logs of AutoKeras we can see how model prediction
metrics vary from epoch to epoch. The _Generation_ is a part of genetic
programming, which includes a random mutation of the training ML model (Koza
and Koza, 1992). Each output about generation by TPOT provides us with
information on data analysis and evolution progress. AutoGluon performs
training of different models on the given dataset. In its logs we can see
outputs of each specific model. In this case, the models are the Gradient
Boosting model and Catboost model.

Table 3. This figure presents verbosity levels description for AutoGluon and TPOT. AutoKeras does not have standard logs with verbosity levels so we omit it from the table. Verbosity level |  AutoGluon |  TPOT
---|---|---
0 |  Only prints exceptions. |  TPOT prints nothing.
1 |  Provides top level AutoGluon info (accuracy, directory, models). |  Prints only warnings.
2 |  Inclueds basic info about all models that are being trained. |  TPOT will print training results and provide a progress bar.
3 |  Prints information about info storage in addition to basic info. |  Includes detailed information about training and a progress bar.
4 |  Presents time spent on various model trainings, their result, and accuracy. Details how models are trained and how data was interpreted. For example, it prints parameters used in classifiers, splits of train and test, and final scores. |  No such option in TPOT.

Overall, AutoGluon seemed to present the most useful logs. In contrast,
AutoKeras and TPOT logs were mixed up, confusing, and hard to interpret,
AutoGluon keeps its logs more minimalistic. It carefully utilizes line breaks,
spaces, and colors to highlight the most important information, see Figure 1.
In addition, AutoGluon provides clear summaries of each new model training
after it is done.

Figure 1. This figure depicts example logging by AutoGluon. It shows both
specific AutoGluon logs and logs of the model being trained by AutoGluon, in
our case LightGBMClassifier.

AutoKeras and TPOT tend to print out all information without convenient line
breaks. These two tools also lack clear summaries of training, making it hard
to understand how the model is developing. TPOT also outputs hard to interpret
information on how the data features are analysed and modified. Because TPOT
combines information across mutations, features, and iterations, the user can
become easily overwhelmed. The difficulty with showing too much log
information is that it can hide more important parts, such as how much
accuracy is increasing over time. While TPOT shows all of its information to
the user, the way it is presented without order or summaries makes it
confusing.

##  5\. Discussion and Implications for Design

In this section we present our assessment on the current state of AutoML. We
also share some tips which we learned through work with different AutoML
tools.

###  5.1. AutoML Tools’ Limitations

This section outlines some flaws of AutoML tools we managed to discover.

####  5.1.1. Error Handling

As was mentioned in the Performance Evaluation section, once while using
AutoKeras we received a Runtime error. This error was not connected with our
code, but only with AutoKeras algorithms. Sadly, this type of error is
commonly faced by many developers. We believe that AutoML tools should make
their inside algorithms error resistant. This lack of visibility of the
internal state is highly disruptive to the effort to democratize Machine
Learning. Instead, designers and researchers should strive to make tools more
accessible for those without specific knowledge to interpret the errors and
apply needed modifications to input data.

####  5.1.2. Indecipherable Logs

Out of all the tools we have evaluated, considering logs, AutoGluon seemed to
provide the least confusing experience. AutoGluon displayed minimalistic
output with carefully utilized line breaks and colors to highlight the most
important information. However, its approach to logs is not ideal. We believe
that logs of future AutoML tools should not only utilize colors and be very
clean as AutoGluon’s logs, but should also refer to model training. Logs would
be more beneficial if they emphasized how models develop over the course of
training.

We highlight our experience that shows AutoML tools lack basic visualization
support. There are several Python visualisation libraries, which might be used
to improve visual attractiveness and readability of logs including and not
limited to matplotlib, plotly, pytables, ggplot (Hunter, 2007; Stančin and
Jović, 2019; Fahad and Yahya, 2018). Several AutoML integrations for model
training visualization are now available. Some of such integrations are
mentioned in our section on _Auxiliary AutoML Integrations_. However, all
external tools require additional time to learn and might require a usage fee.
Out-of-the-box support that makes use of a Python visualisation library would
make it easier to analyse model training and performance. An example of
possible helpful visualization is given in the Figure 2. This visualization
was created with Tensor Dash, an app and Python library that enables remote
monitoring of Machine Learning model training developed by Harshit Maheshari
and Arun Kumar.

Figure 2. This figure presents an example of training process illustration. It
was generated with TensorDash. The charts present loss and accuracy changes
during the model training.

Figure 3. This figure presents an example of training process illustration. It
was generated with Weights&Biases. The charts present loss and accuracy
changes during the model training and number of epochs of current training
trial for each training step.

Figure 4. This figure depicts example logging output from the three selected
AutoML tools. One difference in logging is that AutoKeras reports on epochs,
while TPOT displays evolving generations.

####  5.1.3. Time Estimation Problem

One of the biggest barriers to entry to the ML field is the large amount of
computational resources and machine time required to perform training on big
data. Another source of friction into the ML field is the difficulty of
knowing how many hours it will take to train. For many companies and
specialists it is necessary to know how long training will take. Training time
differs based on computational resources, the nature of the data itself, the
size of the data, and the tools chosen by the developer. Better estimations on
training time could help users schedule training and dedicate available
resources at the most convenient time. This would provide a better allocation
of resources. Some steps towards solution of this problem have been already
offered by prior research (Justus et al., 2018). However, there is still no
readily available implementation. Very large companies can absorb the cost of
a lack of estimates because of their resources. But, this is not the case for
small companies. As more small companies and individuals enter the ML field,
we predict that there will be a growing demand for training time prediction
functionality.

###  5.2. Building Towards Reduced Friction

In this section we offer suggestions for creating a better user experience for
future development of tools of AutoML tools.

####  5.2.1. Auxiliary AutoML Integrations

AutoML tools should continue to utilize auxiliary tools that simplify the
user’s needs. There are several ML integrations tools which help tracking,
automating, and analysing ML experiments. Integrations include TensorDash,
Weights&Biases, Atlas, MLflow, TensorBoard, Comet, and Valohai. These tools
are designed to improve the experiences of developers during AutoML training.

In addition to the Performance Evaluation, we tested ML training visualization
dashboards from Weights&Biases and TensorDash. TensorDash is a Python library
and a mobile application, while Weights&Biases is a Python library and a web-
application. TensorDash allows users to track their ML experiments from their
phone. Weights&Biases provides users with an online dashboard for tracking,
comparing, and analyzing their ML experiments. We used both these tools
together with AutoKeras with the help of the ”callbacks” model’s parameter. An
example of charts generated by TensorDash is given in the Figure 2. One of the
drawbacks of using TensorDash is that each TensorDash project contains charts
for only one training. As AutoKeras performs several trainings with different
parameters, our projects’ charts were constantly overwritten. This left us
without the results of training results across trials. Weights&Biases had a
solution to this problem. Weights&Biases plots all data on the same chart,
which allows you to examine all the trials together. Some of the other
advantages of Weights&Biases over TensorDash are advanced chart scoping
instruments and GPU usage charts. An example of charts generated by
Weights&Biases is given in the Figure 3.

Overall, we found TensorDash and Weights&Biases easy to use. We especially see
value in the speed with which its charts update during the training process.

####  5.2.2. The Visual Future of AutoML

The possible implementations of AutoML by developers is growing quickly. While
a skilled developer can produce good results, the tools are too complex for
non-specialists. Moreover, as the amount of data stored by different companies
grows rapidly, the programmers and analysts are faced with challenges that
grow in both size and complexity. As Herbert Simon states (Simon, 1974),
humans are best at operating on information precisely when it can fit in their
cognitive memory. However, human memory has limitations which are revealed as
the amount of information for analysis increases. When a Data Science related
task involves an enormous amount of data, limitations of the human condition
prevent analysts from keeping the data in their memory. Therefore, systems
which would offload their memory are necessary.

We highlight new efforts that utilize visual interfaces that intend to make
AutoML more accessible for non-developers. The Northstar project by Kraska et.
al. (Kraska, 2018) suggests a solution, which would enable anyone to create
prediction models and find hidden dependencies between different data just by
performing interactions with a simple user interface. Built on the base of
Vizdom, Northstar aims to provide a flexible interface that is friendly to
both mouse and hand interaction. This no code future of AutoML would highly
democratize ML making its benefits accessible for everyone. Our own
experiences show that using AutoML tools is difficult, requiring expertise in
Python, AutoML libraries knowledge, and expertise in tools’ internal error
handling. Northstar’s approach begins to remove these requirements. The
Northstar project intends to create a tool, which would allow general users to
perform ML training with the help of simple table data and graph manipulation.
After model training, Northstar intends to provide users with generated Python
code for prediction. This code can be given to developers and applied in
various pipelines. By using the visual interface, Northstar fully manages code
that allows users to perform various ML tasks without writing code directly.

As described by its creators, the main goal of the Northstar is ”truly
democratizing Data Science” (Shang et al., 2019). We agree with this goal. The
Northstar approach of reducing the amount of code though interactive
applications is very promising. While workers with various specializations
lack programming skills, they are becoming more accustomed to using
interactive applications. As more tools that combine visual interfaces with ML
backends are released, it will result in further democratization of ML
development. In the end, we believe this will make ML more ready for use by
the general public.

##  6\. Conclusion

We have analyzed three AutoML tools from the perspective of performance and
user experience. While autoML tools attempt “automatically composes a Machine
Learning pipeline (Shang et al., 2019)”, our goal was to explore whether this
is indeed the case. Instead of focusing on algorithm design and performance,
we described our experiences utilizing AutoML tools features. During this
process, we encountered confusing log output, low-support for training time
estimates, and a lack of support for visualizing internal state of training
processes and results. This work both details our experiences, documenting our
experiences of Auto-ML state-of-the-art tools, and discusses recommendations
for further improvement. As democratization of ML continues, we hope that soon
every interested developer will be enabled to take advantage of this amazing
technology.

## References

  * (1)
  * Agichtein et al. (2006) Eugene Agichtein, Eric Brill, Susan Dumais, and Robert Ragno. 2006.  Learning user interaction models for predicting web search result preferences. In _Proceedings of the 29th annual international ACM SIGIR conference on Research and development in information retrieval_. 3–10.
  * Balaji and Allen (2018) Adithya Balaji and Alexander Allen. 2018.  Benchmarking automatic machine learning frameworks.  _arXiv preprint arXiv:1808.06492_ (2018).
  * Belabbas and Wolfe (2009) Mohamed-Ali Belabbas and Patrick J Wolfe. 2009.  Spectral methods in machine learning and new strategies for very large datasets.  _Proceedings of the National Academy of Sciences_ 106, 2 (2009), 369–374.
  * Brownlee (2018) Jason Brownlee. 2018\.  What is the Difference Between a Batch and an Epoch in a Neural Network?  _Machine Learning Mastery_ (2018).
  * Calvillo-Gámez et al. (2015) Eduardo H Calvillo-Gámez, Paul Cairns, and Anna L Cox. 2015\.  Assessing the core elements of the gaming experience.  In _Game user experience evaluation_. Springer, 37–62.
  * Cook (2016) Darren Cook. 2016\.  _Practical machine learning with H2O: powerful, scalable techniques for deep learning and AI_.  ” O’Reilly Media, Inc.”.
  * Crotty et al. (2015) Andrew Crotty, Alex Galakatos, Emanuel Zgraggen, Carsten Binnig, and Tim Kraska. 2015\.  Vizdom: interactive analytics through pen and touch.  _Proceedings of the VLDB Endowment_ 8, 12 (2015), 2024–2027.
  * Deng et al. (2009) Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. 2009\.  Imagenet: A large-scale hierarchical image database. In _2009 IEEE conference on computer vision and pattern recognition_. Ieee, 248–255.
  * Erickson et al. (2020) Nick Erickson, Jonas Mueller, Alexander Shirkov, Hang Zhang, Pedro Larroy, Mu Li, and Alexander Smola. 2020.  AutoGluon-Tabular: Robust and Accurate AutoML for Structured Data.  _arXiv preprint arXiv:2003.06505_ (2020).
  * Fahad and Yahya (2018) SK Ahammad Fahad and Abdulsamad Ebrahim Yahya. 2018.  Big Data Visualization: Allotting by R and Python with GUI Tools. In _2018 International Conference on Smart Computing and Electronic Enterprise (ICSCEE)_. IEEE, 1–8.
  * Feurer and Hutter (2019) Matthias Feurer and Frank Hutter. 2019.  Hyperparameter Optimization, See Hutter et al. (2019), 3–38.
  * Feurer et al. (2019) Matthias Feurer, Aaron Klein, Katharina Eggensperger, Jost Tobias Springenberg, Manuel Blum, and Frank Hutter. 2019.  Auto-sklearn: efficient and robust automated machine learning.  In _Automated Machine Learning_. Springer, Cham, 113–134.
  * Fu et al. (2014) Qiang Fu, Jieming Zhu, Wenlu Hu, Jian-Guang Lou, Rui Ding, Qingwei Lin, Dongmei Zhang, and Tao Xie. 2014\.  Where do developers log? an empirical study on logging practices in industry. In _Companion Proceedings of the 36th International Conference on Software Engineering_. 24–33.
  * Gijsbers et al. (2019) Pieter Gijsbers, Erin LeDell, Janek Thomas, Sébastien Poirier, Bernd Bischl, and Joaquin Vanschoren. 2019.  An open source AutoML benchmark.  _arXiv preprint arXiv:1907.00909_ (2019).
  * Goodfellow et al. (2013) Ian J Goodfellow, Dumitru Erhan, Pierre Luc Carrier, Aaron Courville, Mehdi Mirza, Ben Hamner, Will Cukierski, Yichuan Tang, David Thaler, Dong-Hyun Lee, et al. 2013\.  Challenges in representation learning: A report on three machine learning contests. In _International conference on neural information processing_. Springer, 117–124.
  * Granka et al. (2004) Laura A Granka, Thorsten Joachims, and Geri Gay. 2004\.  Eye-tracking analysis of user behavior in WWW search. In _Proceedings of the 27th annual international ACM SIGIR conference on Research and development in information retrieval_. 478–479.
  * Guyon et al. (2019) Isabelle Guyon, Lisheng Sun-Hosoya, Marc Boullé, Hugo Jair Escalante, Sergio Escalera, Zhengying Liu, Damir Jajetic, Bisakha Ray, Mehreen Saeed, Michèle Sebag, Alexander Statnikov, Wei-Wei Tu, and Evelyne Viegas. 2019.  Analysis of the AutoML Challenge Series 2015-2018, See Hutter et al. (2019), 191–236.
  * Hackeling (2017) Gavin Hackeling. 2017\.  _Mastering Machine Learning with scikit-learn_.  Packt Publishing Ltd.
  * Hunter (2007) J. D. Hunter. 2007\.  Matplotlib: A 2D graphics environment.  _Computing in Science & Engineering_ 9, 3 (2007), 90–95.  https://doi.org/10.1109/MCSE.2007.55
  * Hutter et al. (2019) Frank Hutter, Lars Kotthoff, and Joaquin Vanschoren (Eds.). 2019\.  _Automatic Machine Learning: Methods, Systems, Challenges_.  Springer.
  * Jin et al. (2019) Haifeng Jin, Qingquan Song, and Xia Hu. 2019.  Auto-keras: An efficient neural architecture search system. In _Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_. 1946–1956.
  * Justus et al. (2018) Daniel Justus, John Brennan, Stephen Bonner, and Andrew Stephen McGough. 2018.  Predicting the computational cost of deep learning models. In _2018 IEEE International Conference on Big Data (Big Data)_. IEEE, 3873–3882.
  * Koza and Koza (1992) John R Koza and John R Koza. 1992.  _Genetic programming: on the programming of computers by means of natural selection_. Vol. 1.  MIT press.
  * Kraska (2018) Tim Kraska. 2018\.  Northstar: An interactive data science system.  _Proceedings of the VLDB Endowment_ 11, 12 (2018), 2150–2164.
  * Lethbridge et al. (2003) Timothy C Lethbridge, Janice Singer, and Andrew Forward. 2003\.  How software engineers use documentation: The state of the practice.  _IEEE software_ 20, 6 (2003), 35–39.
  * Li et al. (2017) Heng Li, Weiyi Shang, and Ahmed E Hassan. 2017.  Which log level should developers choose for a new logging statement?  _Empirical Software Engineering_ 22, 4 (2017), 1684–1716.
  * McCarthy and Wright (2004) John McCarthy and Peter Wright. 2004.  Technology as experience.  _interactions_ 11, 5 (2004), 42–43.
  * McCauley et al. (1996) Renée A McCauley, Ursula Jackson, and Bill Manaris. 1996\.  Documentation standards in the undergraduate computer science curriculum.  _ACM SIGCSE Bulletin_ 28, 1 (1996), 242–246.
  * Olson and Moore (2019a) Randal S Olson and Jason H Moore. 2019a.  TPOT: A tree-based pipeline optimization tool for automating machine learning.  In _Automated Machine Learning_. Springer, 151–160.
  * Olson and Moore (2019b) Randal S. Olson and Jason H. Moore. 2019b.  TPOT: A Tree-based Pipeline Optimization Tool for Automating Machine Learning, See Hutter et al. (2019), 163–173.
  * Rerabek and Ebrahimi (2016) Martin Rerabek and Touradj Ebrahimi. 2016.  New light field image dataset. In _8th International Conference on Quality of Multimedia Experience (QoMEX)_.
  * Schrider and Kern (2018) Daniel R Schrider and Andrew D Kern. 2018.  Supervised machine learning for population genetics: a new paradigm.  _Trends in Genetics_ 34, 4 (2018), 301–312.
  * Shang et al. (2019) Zeyuan Shang, Emanuel Zgraggen, Benedetto Buratti, Ferdinand Kossmann, Philipp Eichmann, Yeounoh Chung, Carsten Binnig, Eli Upfal, and Tim Kraska. 2019.  Democratizing data science through interactive curation of ml pipelines. In _Proceedings of the 2019 International Conference on Management of Data_. 1171–1188.
  * Simon (1974) Herbert A Simon. 1974\.  How big is a chunk?: By combining data from several experiments, a basic human memory unit can be identified and measured.  _Science_ 183, 4124 (1974), 482–488.
  * Stančin and Jović (2019) Igor Stančin and Alan Jović. 2019.  An overview and comparison of free Python libraries for data mining and big data analysis. In _2019 42nd International Convention on Information and Communication Technology, Electronics and Microelectronics (MIPRO)_. IEEE, 977–982.
  * Tenorio and Lee (1989) Manoel Fernando Tenorio and Wei-Tsih Lee. 1989.  Self Organizing Neural Networks for the Identification Problem.  In _Advances in Neural Information Processing Systems 1_ , D. S. Touretzky (Ed.). Morgan-Kaufmann, 57–64.  http://papers.nips.cc/paper/149-self-organizing-neural-networks-for-the-identification-problem.pdf
  * Thornton et al. (2013) Chris Thornton, Frank Hutter, Holger H Hoos, and Kevin Leyton-Brown. 2013.  Auto-WEKA: Combined selection and hyperparameter optimization of classification algorithms. In _Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining_. 847–855.
  * Zgraggen et al. (2016) Emanuel Zgraggen, Robert Zeleznik, and Philipp Eichmann. 2016\.  Tableur: Handwritten spreadsheets. In _Proceedings of the 2016 CHI Conference Extended Abstracts on Human Factors in Computing Systems_. 2362–2368.
  * Zöller and Huber ([n.d.]) Marc-André Zöller and Marco F Huber. [n.d.].  Benchmark and Survey of Automated Machine Learning Frameworks.  ([n. d.]).


# Synthesis of a quantum tree Weyl matrix

Sergei A. Avdonin${}^{\text{1}}$, Kira V. Khmelnytskaya${}^{\text{2}}$,
Vladislav V. Kravchenko${}^{\text{3}}$
${}^{\text{1}}$ Department of Mathematics and Statistics, University of
Alaska, Fairbanks, AK 99775, USA
${}^{\text{2}}$ Faculty of Engineering, Autonomous University of Queretaro,
Cerro de las Campanas s/n, col. Las Campanas Querétaro, Qro. C.P. 76010 México
${}^{\text{3}}$ Department of Mathematics, Cinvestav, Campus Querétaro,
Libramiento Norponiente #2000, Fracc. Real de Juriquilla, Querétaro, Qro.,
76230 México
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A method for successive synthesis of a Weyl matrix (or Dirichlet-to-Neumann
map) of an arbitrary quantum tree is proposed. It allows one, starting from
one boundary edge, to compute the Weyl matrix of a whole quantum graph by
adding on new edges and solving elementary systems of linear algebraic
equations in each step.

##  1 Introduction

Quantum graphs or differential equations networks have wide applications in
science and engineering and give rise to challenging problems involving many
areas of modern mathematics, from combinatorics to partial differential
equations and spectral theory. A number of surveys and collections of papers
on quantum graphs appeared last years, including the first books on this topic
by Berkolaiko and Kuchment [8] and Mugnolo [14]. In the present work we
consider tree graphs, that is, finite connected compact graphs without cycles.
The Weyl or Titchmarsh-Weyl matrix of a quantum graph is one of the key
mathematical objects, it naturally appears in direct and inverse spectral
theory, in the control theory of quantum graphs and in numerous applications.
Its importance lies in the fact that the Weyl matrix (or more precisely, its
transposed matrix) is the Dirichlet-to-Neumann map of the quantum graph. For a
fixed value of the spectral parameter, a vector of arbitrary Dirichlet-type
boundary values of a solution multiplied by the transposed Weyl matrix gives
one a vector of the corresponding Neumann-type boundary values, if only the
value of the spectral parameter is not a Dirichlet eigenvalue of the quantum
graph. Moreover, the singularities of the Weyl matrix, considered as a
function of the spectral parameter, determine the Dirichlet spectrum of the
quantum graph. Many papers on inverse problems for quantum graphs exploit the
Weyl matrix or equivalent spectral data, see, e.g., [6], [9], [15], [7], [10],
[4].

Direct construction of the Weyl matrix for a sufficiently large quantum tree
is quite a challenging problem. It requires solving large systems of equations
involving solutions (and their derivatives) of differential equations on all
edges of the tree. The main result of the present work is a simple procedure
for a progressive synthesis of the Weyl matrix of an arbitrary quantum tree.
Starting from just one leaf edge and adding successively new edges, it allows
one to compute the Weyl matrices for ever larger quantum trees from the Weyl
matrices for the smaller ones. We call this procedure the synthesis of the
Weyl matrix. In a sense, it is inverse with respect to the leaf peeling
method, which was developed in [4], see also [5], [2]. The proposed synthesis
of the Weyl matrix is based on revealed relations between Weyl solutions of a
larger quantum tree graph $\Omega$ with those of a smaller one
$\widetilde{\Omega}$, obtained from $\Omega$ by cutting out all the leaf edges
of an internal vertex.

In Section 2 we recall necessary definitions and express the Weyl solutions in
terms of fundamental systems of solutions of the Sturm–Liouville equation on
each edge. Section 3 presents the main result of the work, the procedure of
the synthesis of a Weyl matrix of an arbitrary quantum tree graph. Finally,
Section 4 contains some concluding remarks.

##  2 Preliminaries

Let $\Omega$ be a finite connected compact graph without cycles (a tree graph)
consisting of $P$ edges, $e_{1},\ldots,e_{P},$ and $P+1$ vertices,
$V=\left\\{v_{1},...,v_{P+1}\right\\}.$ The notation $e_{j}\sim v$ means that
the edge $e_{j}$ is incident to the vertex $v$. Every edge $e_{j}$ is
identified with an interval $(0,L_{j})$ of the real line. The boundary
$\Gamma=\left\\{\gamma_{1},\ldots,\gamma_{m}\right\\}$ of $\Omega$ is the set
of all leaves of the graph (the external vertices). The edge adjacent to some
$\gamma_{j}\ $is called a leaf or boundary edge.

A continuous function $u$ defined on the graph $\Omega$ is a $P$-tuple of
functions $u_{j}\in C\left[0,L_{j}\right]$ satisfying the continuity condition
at the internal vertices $v$: $u_{i}(v)=u_{j}(v)$ for all $e_{i},e_{j}\sim v$.
Then $u\in C(\Omega)$.

Let $q\in\mathcal{L}_{1}(\Omega)$ be real valued, and $\lambda$ a complex
number. Consider the Sturm-Liouville equation on $\Omega$:

$-u^{\prime\prime}(x)+q(x)u(x)=\lambda u(x).$ (2.1)

A function $u$ defined on $\Omega$ is said to be a solution of (2.1) if
besides (2.1) we have that

$u\in C(\Omega),$ (2.2)

and for every internal vertex $v$ the Kirchhoff-Neumann condition is fulfilled

$\sum_{e_{j}\sim v}\partial u_{j}(v)=0,\text{\quad for all }v\in
V\setminus\Gamma.$ (2.3)

Here $u_{j}$ is a restriction of $u$ onto $e_{j}$, $\partial u_{j}(v)$ stands
for the derivative of $u$ at the vertex $v$ taken along the edge $e_{j}$ in
the direction outward the vertex, and the sum is taken over all the edges
incident to the internal vertex $v$.

For simplicity we assume that the considered graph does not have vertices of
degree two, because every such vertex can be regarded as an internal point of
an edge which is a sum of two edges incident at such a vertex, and the
continuity condition (2.2) together with the Kirchhoff-Neumann condition (2.3)
guarantee that any solution of (2.1) on the incident edges keeps satisfying
(2.1) also on the union of them.

###### Definition 2.1

A solution $w_{i}$ of (2.1) on $\Omega$ is called the Weyl solution associated
with the leaf $\gamma_{i}$ if it satisfies the boundary conditions

$w_{i}(\gamma_{i})=1\quad\text{and}\quad w_{i}(\gamma_{j})=0\text{ for all
}j\neq i.$ (2.4)

If $\lambda$ in (2.1) is not a Dirichlet eigenvalue of the quantum tree
$\Omega$, the Weyl solution $w_{i}$ exists and is unique for any
$i=1,\ldots,m$. In particular, since the potential $q$ is real valued, the
Dirichlet spectrum is real and thus the boundary value problem (2.1), (2.4) is
uniquely solvable for all $\lambda\notin\mathbb{R}$.

###### Definition 2.2

The $m\times m$ matrix-function $\mathbf{M}(\lambda)$,
$\lambda\notin\mathbb{R}$, consisting of the elements
$\mathbf{M}_{ij}(\lambda)=\partial w_{i}(\gamma_{j})$, $i,j=1,\ldots,m$ is
called the Weyl matrix.

For a fixed value of $\lambda$, the transposed Weyl matrix represents a
Dirichlet-to-Neumann map of the quantum graph defined by $\Omega$ and
$q\in\mathcal{L}_{1}(\Omega)$. Indeed, if $u$ is a solution of (2.1)
satisfying the Dirichlet condition at the boundary vertices
$u(\lambda,\gamma)=f(\lambda)$, then $\partial
u(\lambda,\gamma)=\mathbf{M}^{T}(\lambda)f(\lambda)$,
$\lambda\notin\mathbb{R}$.

It is clear that the direct computation of the Weyl matrix, which involves
finding Weyl solutions and computing their derivatives at leaves may be a
difficult task, especially, when $\Omega$ consists of a large number of edges.
The corresponding systems of equations which need to be solved in this case
may be too large, because they should combine information on the solutions and
their derivatives on each edge and at all internal vertices.

The main result of the present work is a simple procedure which allows one to
synthesize the Weyl matrix progressively, by adding edges to smaller graphs
and computing the Weyl matrices for the obtained larger graphs from the Weyl
matrices for the smaller ones. We call this procedure synthesis of the Weyl
matrix. It allows one to compute the Weyl matrix of any quantum tree starting
from one leaf edge and adding successively new edges.

By $\varphi_{i}(\rho,x)$ and $S_{i}(\rho,x)$ we denote the so-called
fundamental solutions of the Sturm–Liouville equation on the edge $e_{j}:$

$-y^{\prime\prime}(x)+q_{i}(x)y(x)=\rho^{2}y(x),\quad x\in(0,L_{i}),$ (2.5)

satisfying the initial conditions

$\varphi_{i}(\rho,0)=1,\quad\varphi_{i}^{\prime}(\rho,0)=0,$
$S_{i}(\rho,0)=0,\quad S_{i}^{\prime}(\rho,0)=1.$

Here $q_{i}(x)$ is the component of the potential $q(x)$ on the edge $e_{i}$,
and $\rho=\sqrt{\lambda}$, $\operatorname{Im}\rho\geq 0$. For a leaf edge
$e_{i}$ it is convenient to identify its leaf $\gamma_{i}$ with the left
endpoint $x=0.$ Then the Weyl solution $w_{i}(\rho,x)$ has the form

$w_{ii}(\rho,x)=\varphi_{i}(\rho,x)+\mathbf{M}_{i,i}(\rho^{2})S_{i}(\rho,x)\quad\text{on
the adjacent leaf edge }e_{i}$

and

$w_{ij}(\rho,x)=\mathbf{M}_{i,j}(\rho^{2})S_{j}(\rho,x)\quad\text{on every
other leaf edge }e_{j},\quad j\neq i.$

Hereafter, the notation $w_{ij}(\rho,x)$ means that we consider $j$-th
component of a solution $w_{i}(\rho,x)$, that is, the solution $w_{i}(\rho,x)$
on the edge $e_{j}$.

On internal edges $e_{j}$ we have

$w_{ij}(\rho,x)=a_{ij}(\rho)\varphi_{j}(\rho,x)+b_{ij}(\rho)S_{j}(\rho,x),$

where the choice of which vertex is identified with zero is arbitrary, and in
general the factors $a_{ij}(\rho)$, $b_{ij}(\rho)$ are unknown.

Since in direct and inverse spectral problems involving the Weyl matrix one
deals with solutions on large ranges of the parameter $\rho$, it is convenient
to use the Neumann series of Bessel functions representations for
$\varphi_{j}(\rho,x)$ and $S_{j}(\rho,x)$, introduced in [13] and applied in a
number of direct and inverse problems (see, e.g., [1], [2], [3], [11], [12]).
One of the features of these representations is the existence of estimates for
the remainders of the series independent of $\operatorname{Re}\rho$.

##  3 Synthesis of Weyl matrix

Consider a quantum tree graph $\widetilde{\Omega}$ whose leaves are
$\gamma_{0},\gamma_{1},\ldots,\gamma_{m}$. Assume its Weyl matrix
$\widetilde{\mathbf{M}}(\rho^{2})$ to be known for some value of $\rho$.
Attach a number of edges to the leaf $\gamma_{0}$ (see Fig. 1), so that
$\gamma_{0}$ becomes an internal vertex of a new larger graph $\Omega$, and
$\gamma_{1},\ldots,\gamma_{m},\gamma_{m+1},\ldots,\gamma_{m+m_{1}}$ the leaves
of $\Omega$. Here $m_{1}$ is the number of the new attached edges. Thus,
$\Omega$ is a tree graph obtained from $\widetilde{\Omega}$ by attaching
$m_{1}$ edges to $\gamma_{0}$. For simplicity we call these new edges
$e_{m+1},\ldots,e_{m+m_{1}}$ and denote their respective lengths as
$L_{m+1},\ldots,L_{m+m_{1}}$. We assume that a corresponding potential
$q_{j}\in\mathcal{L}_{1}(0,L_{j})$, $j=m+1,\ldots,m+m_{1},$ is given on each
new edge, and the leaf $\gamma_{j}$ is identified with $x=0.$

Figure 1: Tree graph $\Omega$ is obtained from a subgraph $\widetilde{\Omega}$
(its edges are presented by solid lines) by attaching to the vertex
$\gamma_{0}$ a number of new edges (dashed lines).

Our task is to find the Weyl matrix $\mathbf{M}(\rho^{2})$ of the quantum
graph $\Omega$. The idea is to construct the Weyl solutions $w_{i}(\rho,x)$ of
$\Omega$ in terms of the Weyl solutions $\widetilde{w}_{j}(\rho,x)$ of
$\widetilde{\Omega}$. We start with the Weyl solutions $w_{m+j}(\rho,x)$
associated with the new leaves $\gamma_{m+j}$, $j=1,\ldots,m_{1}$.

Let us look for $w_{m+j}(\rho,x)$ in the form

$w_{m+j}(\rho,x)=c_{j}(\rho)\widetilde{w}_{0}(\rho,x)\quad\text{on
}\widetilde{\Omega}.$ (3.1)

That is, on the subgraph $\widetilde{\Omega}$ the Weyl solution
$w_{m+j}(\rho,x)$ coincides with $\widetilde{w}_{0}(\rho,x)$ up to a
multiplicative constant $c_{j}(\rho)$. In this case, it automatically
satisfies the homogeneous Dirichlet condition at
$\gamma_{1},\ldots,\gamma_{m},$ and we still need to satisfy the conditions
$w_{m+j}(\rho,\gamma_{m+j})=1$ and $w_{m+j}(\rho,\gamma_{m+i})=0$ for $i\neq
j$.

From (3.1) we have

$\varphi_{m+j}(\rho,L_{m+j})+\mathbf{M}_{m+j,m+j}(\rho^{2})S_{m+j}(\rho,L_{m+j})=c_{j}(\rho)$
(3.2)

and

$\varphi_{m+j}^{\prime}(\rho,L_{m+j})+\sum_{k=1}^{m_{1}}\mathbf{M}_{m+j,m+k}(\rho^{2})S_{m+k}^{\prime}(\rho,L_{m+k})=c_{j}(\rho)\widetilde{\mathbf{M}}_{0,0}(\rho^{2}),$
(3.3)

where $\widetilde{\mathbf{M}}_{0,0}(\rho^{2})$ is an element of the Weyl
matrix $\widetilde{\mathbf{M}}(\rho^{2})$:
$\widetilde{\mathbf{M}}_{0,0}(\rho^{2})=\partial\widetilde{w}_{0}(\gamma_{0})$.

Additionally, the continuity condition gives us the equalities

$\mathbf{M}_{m+j,m+k}(\rho^{2})S_{m+k}(\rho,L_{m+k})=c_{j}(\rho),\quad
k=1,\ldots,m_{1}\text{ and }k\neq j.$ (3.4)

Thus, for each $j$, from (3.2), (3.3) and (3.4) we have $m_{1}+1$ equations
for the $m_{1}+1$ unknowns

$\left\\{\mathbf{M}_{m+j,m+k}(\rho^{2}),\,k=1,\ldots,m_{1};\text{
}c_{j}(\rho)\right\\}.$

Note that in the linear algebraic system (3.2), (3.3), (3.4) the magnitudes
$\varphi_{m+j}(\rho,L_{m+j})$, $S_{m+k}(\rho,L_{m+k})$,
$\varphi_{m+j}^{\prime}(\rho,L_{m+j})$, $S_{m+k}^{\prime}(\rho,L_{m+k})$ are
known, since all the potentials $q_{m+k}(x)$, $k=1,\ldots,m_{1}$ are known.

Thus, from (3.2), (3.3) and (3.4) we find
$\mathbf{M}_{m+j,m+k}(\rho^{2}),\,k=1,\ldots,m_{1}$ and $c_{j}(\rho)$.

From (3.1) we obtain additionally,

$\mathbf{M}_{m+j,i}(\rho^{2})=c_{j}(\rho)\widetilde{\mathbf{M}}_{0,i}(\rho^{2})\text{\quad
for }i=1,\ldots,m,$ (3.5)

and thus we have already completed the rows $m+1,\ldots,m+m_{1}$ of the Weyl
matrix $\mathbf{M}(\rho^{2})$.

Now, choose an $i\in\left\\{1,\ldots,m\right\\}$. We look for $w_{i}(\rho,x)$
such that on $\widetilde{\Omega}$ the equality be valid

$w_{i}(\rho,x)=\widetilde{w}_{i}(\rho,x)+\alpha_{i}(\rho)\widetilde{w}_{0}(\rho,x),$

where $\alpha_{i}(\rho)$ is a constant. This is a natural choice, because

$w_{i}(\rho,\gamma_{i})=1\quad\text{and}\quad
w_{i}(\rho,\gamma_{j})=0\text{\quad for }j=1,\ldots,m\text{ and }j\neq i.$

Moreover, we have

$w_{i}(\rho,\gamma_{0})=\alpha_{i}(\rho)$

and

$\partial
w_{i}(\rho,\gamma_{0})=\widetilde{\mathbf{M}}_{i,0}(\rho^{2})+\alpha_{i}(\rho)\widetilde{\mathbf{M}}_{0,0}(\rho^{2}).$

Thus, for all $j=m+1,\ldots,m+m_{1}$ we have

$w_{ij}(\rho,\gamma_{0})=\alpha_{i}(\rho)$ (3.6)

and

$\sum_{j=m+1}^{m+m_{1}}\partial
w_{ij}(\rho,\gamma_{0})=-\widetilde{\mathbf{M}}_{i,0}(\rho^{2})-\alpha_{i}(\rho)\widetilde{\mathbf{M}}_{0,0}(\rho^{2}).$
(3.7)

Equality (3.6) can be written as

$\mathbf{M}_{i,j}(\rho^{2})S_{j}(\rho,L_{j})=\alpha_{i}(\rho),\quad
j=m+1,\ldots,m+m_{1},$ (3.8)

while (3.7) takes the form

$\sum_{j=m+1}^{m+m_{1}}\mathbf{M}_{i,j}(\rho^{2})S_{j}^{\prime}(\rho,L_{j})=\widetilde{\mathbf{M}}_{i,0}(\rho^{2})+\alpha_{i}(\rho)\widetilde{\mathbf{M}}_{0,0}(\rho^{2}).$
(3.9)

For every $i\in\left\\{1,\ldots,m\right\\}$, equations (3.8) and (3.9) give us
$m_{1}+1$ equations for the $m_{1}+1$ unknowns

$\left\\{\mathbf{M}_{i,j}(\rho^{2}),\,j=m+1,\ldots,m+m_{1};\text{
}\alpha_{i}(\rho)\right\\}.$

The magnitudes $S_{j}(\rho,L_{j})$, $S_{j}^{\prime}(\rho,L_{j})$,
$\widetilde{\mathbf{M}}_{i,0}(\rho^{2})$,
$\widetilde{\mathbf{M}}_{0,0}(\rho^{2})$ in (3.8) and (3.9) are already known.

Finally, to obtain $\mathbf{M}_{i,j}(\rho^{2})$ for $i,j=1,\ldots,m$ we
observe that

$\partial
w_{i}(\rho,\gamma_{j})=\partial\widetilde{w}_{i}(\rho,\gamma_{j})+\alpha_{i}(\rho)\partial\widetilde{w}_{0}(\rho,\gamma_{j})$

and thus

$\mathbf{M}_{i,j}(\rho^{2})=\widetilde{\mathbf{M}}_{i,j}(\rho^{2})+\alpha_{i}(\rho)\widetilde{\mathbf{M}}_{0,j}(\rho^{2})\quad\text{for
}i,j=1,\ldots,m\text{.}$ (3.10)

Let us summarize the procedure of construction of the Weyl matrix
$\mathbf{M}(\rho^{2})$ of the quantum tree $\Omega$ from the Weyl matrix
$\widetilde{\mathbf{M}}(\rho^{2})$ of the quantum subtree
$\widetilde{\Omega}$.

1) For each $j=1,\ldots,m_{1}$ solve the
$\left(m_{1}+1\right)\times\left(m_{1}+1\right)$-system of linear algebraic
equations (3.2)-(3.4) to find the constant $c_{j}(\rho)$ and the Weyl matrix
entries $\mathbf{M}_{m+j,m+k}(\rho^{2}),\,k=1,\ldots,m_{1}$. Compute the
entries $\mathbf{M}_{m+j,i}(\rho^{2}),$ $i=1,\ldots,m$ from (3.5). Thus, the
rows from $m+1$ to $m+m_{1}$ of the Weyl matrix $\mathbf{M}(\rho^{2})$ are
computed.

2) For each $i=1,\ldots,m$ solve the
$\left(m_{1}+1\right)\times\left(m_{1}+1\right)$-system of linear algebraic
equations (3.8), (3.9) to find the constant $\alpha_{i}(\rho)$ and the Weyl
matrix entries $\mathbf{M}_{i,j}(\rho^{2})$, $j=m+1,\ldots,m+m_{1}$. Compute
the entries $\mathbf{M}_{i,j}(\rho^{2}),$ $j=1,\ldots,m$ from (3.10). This
completes the construction of the Weyl matrix $\mathbf{M}(\rho^{2})$.

Finally, let us consider the situation when $\widetilde{\Omega}$ is just a
single segment with the vertices $\gamma_{0}$ and $\gamma_{1}$. Since the
potential on this segment is supposed to be given, we may assume that both
corresponding Weyl solutions on such $\widetilde{\Omega}$ are known:
$\widetilde{w}_{0}(\rho,x)$ and $\widetilde{w}_{1}(\rho,x)$, that satisfy the
boundary conditions

$\widetilde{w}_{0}(\rho,\gamma_{0})=1,\quad\widetilde{w}_{0}(\rho,\gamma_{1})=0$

and

$\widetilde{w}_{1}(\rho,\gamma_{0})=0,\quad\widetilde{w}_{1}(\rho,\gamma_{1})=1.$

Then the entries of the $2\times 2$ \- Weyl matrix
$\widetilde{\mathbf{M}}(\rho^{2})$ have the form

$\widetilde{\mathbf{M}}_{0,0}(\rho^{2})=\partial\widetilde{w}_{0}(\rho,\gamma_{0}),\quad\widetilde{\mathbf{M}}_{0,1}(\rho^{2})=\partial\widetilde{w}_{0}(\rho,\gamma_{1}),$
$\widetilde{\mathbf{M}}_{1,0}(\rho^{2})=\partial\widetilde{w}_{1}(\rho,\gamma_{0}),\quad\widetilde{\mathbf{M}}_{1,1}(\rho^{2})=\partial\widetilde{w}_{1}(\rho,\gamma_{1}).$

Now, as a first step for synthesis of the Weyl matrix of a quantum tree we
attach a number of edges to the vertex $\gamma_{0}$, that gives us a quantum
star graph. The procedure described above gives us its Weyl matrix.
Subsequently, attaching new edges to the leaves and applying the above
procedure leads to the computation of the Weyl matrix of an ever larger
quantum tree. Thus, starting with one edge we synthesize the Weyl matrix of
the whole quantum tree.

###### Remark 3.1

The requirement on the potential $q$ to be real valued is not essential. The
proposed procedure for the synthesis of the Weyl matrix is applicable to
complex valued potentials without modifications. The only constraint that must
be checked at each step is that $\rho\in\mathbb{C}$ does not belong to the
Dirichlet spectrum of either graph $\widetilde{\Omega}$ or $\Omega$.

##  4 Conclusions

A procedure of the synthesis of a Weyl matrix of an arbitrary quantum tree
graph is developed. It allows one to compute the Weyl matrix for a large
quantum tree successively, by adding new edges and computing the Weyl matrices
for ever larger subgraphs. In each step relatively small systems of linear
algebraic equations are solved. Due to the fact that the transposed Weyl
matrix is the Dirichlet-to-Neumann map of the quantum tree graph, the
synthesis procedure will find applications in solving a variety of boundary
value and control problems on quantum tree graphs.

Funding The research of Sergei Avdonin was supported in part by the National
Science Foundation, grant DMS 1909869, and by Moscow Center for Fundamental
and Applied Mathematics. The research of Vladislav Kravchenko was supported by
CONACYT, Mexico, via the project 284470.

Data availability The data that support the findings of this study are
available upon reasonable request.

Declarations

Conflict of interest The authors declare no competing interests.

## References

  * [1] Avdonin S. A., Khmelnytskaya K. V. and Kravchenko V. V. Recovery of a potential on a quantum star graph from Weyl’s matrix. arXiv:2210.15536.
  * [2] Avdonin S. A., Khmelnytskaya K. V. and Kravchenko V. V. Reconstruction techniques for quantum trees. arXiv: 2302.05970.
  * [3] Avdonin S. A. and Kravchenko V. V. (2023), Method for solving inverse spectral problems on quantum star graphs. _Journal of Inverse and Ill-posed Problems_ , 31, no. 1, 31-42.
  * [4] Avdonin S. and Kurasov P. (2008), Inverse problems for quantum trees, Inverse Problems and Imaging, 2, no. 1, 1–21.
  * [5] Avdonin S. and Zhao Yu. (2021), Leaf peeling method for the wave equation on metric tree graphs. _Inverse Probl. Imaging_ 15, no. 2, 185–199.
  * [6] Belishev M. (2004), Boundary spectral inverse problem on a class of graphs (trees) by the BC method,  Inverse Problems, 20, 647–672.
  * [7] Belishev M. and Vakulenko A. (2006), Inverse problems on graphs: recovering the tree of strings by the BC-method, J. Inv. Ill-Posed Problems, 14, 29–46.
  * [8] Berkolaiko G. and Kuchment P. (2013), Introduction to Quantum Graphs, AMS, Providence, R.I.
  * [9] Brown B.M. and Weikard, R. (2005), A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, no. 2062, 3231–3243.
  * [10] Freiling G. and Yurko V. (2007), Inverse problems for differential operators on trees with general matching conditions, Appl. Anal., 86, no. 6, 653–667.
  * [11] Kravchenko V. V. (2020), _Direct and inverse Sturm-Liouville problems: A method of solution_ , Birkhäuser, Cham.
  * [12] Kravchenko V. V., Khmelnytskaya K. V. and Çetinkaya F. A. (2022), Recovery of inhomogeneity from output boundary data, _Mathematics_ , 10, 4349, https://doi.org/10.3390/math10224349.
  * [13] Kravchenko V. V., Navarro L. J. and Torba S. M. (2017), Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions, _Appl. Math. Comput._ 314, 173–192.
  * [14] Mugnolo D. (2014), Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems, Springer, Cham.
  * [15] Yurko V. (2005), Inverse spectral problems for Sturm–Liouville operators on graphs, Inverse Problems, 21, 1075-1086.