# Twisted convolution algebras with coefficients on a differential subalgebra

Felipe I. Flores  Felipe I. Flores 111 2020 Mathematics Subject
Classification: Primary 43A20, Secondary 47L65, 47L30.
Key Words: symmetric Banach ∗-algebra, differential subalgebra,
$C^{*}$-dynamical system, crossed product, spectral invariance, inverse
closed.
Acknowledgements: The author has been partially supported by the NSF grant
DMS-2000105.

###### Abstract

Let $(\alpha,\omega)$ be a measurable twisted action of the locally compact
group ${\sf G}$ on a Banach ∗-algebra $\mathfrak{B}$ and $\mathfrak{A}$ a
differential ∗-subalgebra of $\mathfrak{B}$, invariant under
$(\alpha,\omega)$. We observe that $L^{1}_{\alpha,\omega}({\sf
G},\mathfrak{A})$ is a differential subalgebra of $L^{1}_{\alpha,\omega}({\sf
G},\mathfrak{B})$. We use this fact to provide new examples of groups with
symmetric Banach ∗-algebras. In particular, we prove that discrete rigidly
symmetric extensions of compact groups are symmetric or that semidirect
products ${\sf K}\rtimes{\sf H}$, with ${\sf H}$ symmetric and ${\sf K}$
compact, are symmetric.

Dedicated to the memory of Marius Laurenţiu Măntoiu (1961-2023).

##  1 Introduction

The problem of determining which groups have a symmetric $L^{1}$-algebra (also
called symmetric, or hermitian groups) has a long history, most of which can
be found in [14, Chapters 10,11,12]. Besides being important in the general
theory of Banach ∗-algebras and spectral theory, this property has
applications in K-theory and representation theory. We will avoid discussing
both the history and implications of this property here in order to keep this
exposition short, however, we highly recommend [1, 4, 8, 9, 10, 13, 14, 15,
12, 11, 18] and the references therein. Here are the precise definitions that
will interest us

######  Definition 1.1.

A Banach ∗-algebra $\mathfrak{B}$ is called symmetric if the spectrum of
$b^{*}b$ is positive for every $b\in\mathfrak{B}$ (this happens if and only if
the spectrum of any self-adjoint element is real).

######  Definition 1.2.

A locally compact group ${\sf G}$ is called symmetric (or hermitian) if the
convolution algebra $L^{1}({\sf G})$ is symmetric.

For example, the following classes of groups are known to have symmetric
algebras: compactly generated and of polynomial growth [11, Corollary 1],
groups where all conjugacy classes are relatively compact [8, Theorem 16],
compact extensions of nilpotent groups or connected groups of polynomial
growth (without compact generation) [12].

However, many authors have studied have also studied this property in the
context of ’generalized $L^{1}$-algebras’ or even further generalizations (see
for example [9, 10, 13, 15, 4]) and our main goal in this small article is to
show new examples of symmetric groups by taking advantage of their
investigations and the decomposition

$L^{1}({\sf G})\cong L^{1}_{\alpha,\omega}({\sf H},L^{1}({\sf K})),$ (1.1)

that ocurrs when ${\sf G}$ is an extension of ${\sf K}$ by ${\sf H}$ (using
the kernel-by-quotient convention) [3]. Our main theorem is the following

######  Theorem 1.3.

Let ${\sf G}$ is an extension of ${\sf K}$ by ${\sf H}$ and with ${\sf K}$
compact. Let $(\alpha,\omega)$ be the twisted action of ${\sf H}$ on
$L^{1}({\sf K})$ associated to this extension. Then
$L^{1}_{\alpha,\omega}({\sf H},C^{*}({\sf K}))$ is symmetric if and only if
${\sf G}$ is symmetric.

We rely heavily in the norm estimates that differential subalgebras provide
-particularly $L^{2}({\sf K})$ inside $C^{*}({\sf K})$\- and hence Section 2
is dedicated to a brief discussion on them, focusing on their properties
related to spectral theory. The most interesting fact is that their spectral
invariance passes to convolution algebras (Lemma 2.7), something that does not
occur in general (Example 2.8). Section 3 is entirely devoted to prove the
main theorem and shows how to use the properties of differential subalgebras
in an indirect way. We finish the article by analizing the particular case of
splitting extensions in Section 4. In this section we completely characterize
the symmetry of the extension in terms of the symmetry of the group ${\sf H}$.
Along the last sections we also exemplify how to use the results, together
with pre-existent results in the literature, to get both new examples of
symemtric groups and generalized $L^{1}$-algebras.

##  2 Differential subalgebras

Let $(\mathfrak{A},\lVert\cdot\rVert_{\mathfrak{A}})$,
$(\mathfrak{B},\lVert\cdot\rVert_{\mathfrak{B}})$ be Banach ∗-algebras, such
that $\mathfrak{A}$ is a Banach ∗-subalgebra of $\mathfrak{B}$, we assume that
if $\mathfrak{A}$ is unital, then $\mathfrak{B}$ is also unital and they share
the same unit. We say that $\mathfrak{A}$ is a differential subalgebra of
$\mathfrak{B}$ if there is a constant $C>0$ such that

$\lVert ab\rVert_{\mathfrak{A}}\leq C(\lVert a\rVert_{\mathfrak{A}}\lVert
b\rVert_{\mathfrak{B}}+\lVert a\rVert_{\mathfrak{B}}\lVert
b\rVert_{\mathfrak{A}}),\quad\textup{ for all }a,b\in\mathfrak{A}.$ (2.1)

The concept of differential subalgebras appeared first in [7, 16, 2]. One of
the main applications of this property was found in [5, 6] to derive estimates
on the holomorphic functional calculus. Let us pose the two examples that will
interest us the most.

######  Example 2.1.

Given a Hilbert space $\mathcal{H}$ and $p\in[1,\infty)$, the class of
Schatten $p$-operators $S_{p}(\mathcal{H})$ is a dense ∗-ideal of the
$C^{*}$-algebra of compact operators $\mathbb{K}(\mathcal{H})$, moreover, one
has

$\lVert ST\rVert_{S_{p}(\mathcal{H})}\leq\lVert
S\rVert_{\mathbb{K}(\mathcal{H})}\lVert T\rVert_{S_{p}(\mathcal{H})},\textup{
for all }S,T\in S_{p}(\mathcal{H}).$

######  Example 2.2.

A proper $H^{*}$-algebra $\mathfrak{A}$ is a Banach ∗-algebra which is also a
Hilbert space under the same norm and where $a^{*}$ is the unique element in
$\mathfrak{A}$ satisfying

$\langle ab,c\rangle=\langle b,a^{*}c\rangle\textup{ and }\langle
ba,c\rangle=\langle b,ca^{*}\rangle\textup{, for all }b,c\in\mathfrak{A}.$

Then the left regular representation $\mathfrak{A}\ni a\mapsto
L_{a}\in\mathbb{B}(\mathfrak{A})$ is a contractive ∗-monomorphism. We then
have

$\lVert ab\rVert_{\mathfrak{A}}\leq\lVert
L_{a}\rVert_{\mathbb{B}(\mathfrak{A})}\lVert b\rVert_{\mathfrak{A}},\textup{
for all }a,b\in\mathfrak{A}.$

So that $\mathfrak{A}$ is a differential subalgebra of
$\mathfrak{B}=\overline{L(A)}^{\lVert\cdot\rVert_{\mathbb{B}(\mathfrak{A})}}\cong
C^{*}(\mathfrak{A})$.

######  Remark 2.3.

More examples include $\mathcal{C}^{1}([a,b])$ on $\mathcal{C}([a,b])$ [17,
page 470], some weighted subalgebras of $L^{1}({\sf G})$ [15, Lemma 1], any
full left Hilbert algebra on its $C^{*}$-completion [14, Theorem 11.7.10], or
the algebras of Lemma 2.7.

Let us now discuss one of the more important properties of differential
subalgebras and the main reason why they interest us here. As usual, ${\rm
Spec}_{\mathfrak{A}}(a)$ will denote the spectrum of an element
$a\in\mathfrak{A}$, while
$\rho_{\mathfrak{A}}(a)=\sup\\{|\lambda|\mid\lambda\in{\rm
Spec}_{\mathfrak{A}}(a)\\}$ denote its spectral radius.

######  Definition 2.4.

Let $\mathfrak{A}\subset\mathfrak{B}$ be a continuous inclusion of Banach
∗-algebras. We say that:

  * (i)

$\mathfrak{A}$ is inverse-closed or spectrally invariant in $\mathfrak{B}$ if
${\rm Spec}_{\mathfrak{A}}(a)={\rm Spec}_{\mathfrak{B}}(a)$, for all
$a\in\mathfrak{A}$.

  * (ii)

$\mathfrak{A}$ is spectral radius preserving in $\mathfrak{B}$ if
$\rho_{\mathfrak{A}}(a)=\rho_{\mathfrak{B}}(a)$, for all $a\in\mathfrak{A}$.

It is clear that inverse-closedness can be rephrased as ’if $a\in\mathfrak{A}$
is invertible inside of $\mathfrak{B}$, then $a^{-1}\in\mathfrak{A}$’ which
justifies the name.

######  Lemma 2.5.

If $\mathfrak{A}$ is a differential subalgebra of $\mathfrak{B}$, then
$\mathfrak{A}$ is spectral radius preserving in $\mathfrak{B}$. If
$\mathfrak{A}$ is also dense in $\mathfrak{B}$, then it is spectrally
invariant.

###### Proof.

Observe that the inequality 2.1 implies that for all $a\in\mathfrak{A}$, one
has

$\lVert a^{2n}\rVert_{\mathfrak{A}}\leq 2C\lVert
a^{n}\rVert_{\mathfrak{A}}\lVert a^{n}\rVert_{\mathfrak{B}}.$

Since $\mathfrak{A}$ and $\mathfrak{B}$ are Banach algebras, Gelfand’s formula
for the spectral radius holds, so

$\rho_{\mathfrak{A}}(a)=\lim_{n\to\infty}\lVert
a^{2n}\rVert_{\mathfrak{A}}^{1/2n}\leq\lim_{n\to\infty}(2C)^{1/2n}\lVert
a^{n}\rVert_{\mathfrak{A}}^{1/2n}\lVert
a^{n}\rVert_{\mathfrak{B}}^{1/2n}=\rho_{\mathfrak{A}}(a)^{1/2}\rho_{\mathfrak{B}}(a)^{1/2},$

hence $\rho_{\mathfrak{A}}(a)=\rho_{\mathfrak{B}}(a)$. For the second part, we
follow the argument given in [1, Proposition 2]: Suppose $a\in\mathfrak{A}$
has an inverse $a^{-1}\in\mathfrak{B}$. By density, there is a sequence
$\\{a_{n}\\}_{n\in\mathbb{N}}\subset\mathfrak{A}$ converging to $a^{-1}$ in
$\mathfrak{B}$. So $\lim_{n}\lVert 1-aa_{n}\rVert_{\mathfrak{B}}=0$. In
particular, for big enough $n$ one has

$\rho_{\mathfrak{A}}(1-aa_{n})=\rho_{\mathfrak{B}}(1-aa_{n})\leq\lVert
1-aa_{n}\rVert_{\mathfrak{B}}<1.$

Therefore, $aa_{n}$ is invertible in $\mathfrak{A}$ and letting
$c_{n}=(aa_{n})^{-1}\in\mathfrak{A}$ we see that $aa_{n}c_{n}=1$ and hence
$a^{-1}=a_{n}c_{n}\in\mathfrak{A}$. ∎

Let us now turn to the class of differential subalgebras we plan to study.
First it becomes necessary to introduce twisted actions on Banach ∗-algebras.
If ${\mathfrak{B}}$ is a Banach ∗-algebra with isometric involution, we denote
by ${\rm Aut}_{1}({\mathfrak{B}})$ the group of its ∗-automorphisms $L$
satisfying $\lVert L\rVert=\lVert L^{-1}\rVert=1$, by
$\mathcal{M}({\mathfrak{B}})$ its multiplier algebra and by
$\mathcal{UM}({\mathfrak{B}})$, the corresponding unitary group. The following
definition is taken from [3].

######  Definition 2.6.

A (measurable) twisted action is a $4$-tuple $({\sf
G},\alpha,\omega,\mathfrak{A})$, where ${\sf G}$ is a locally compact group,
$\mathfrak{A}$ a Banach ∗-algebra and maps $\alpha:{\sf G}\to{\rm
Aut}_{1}({\mathfrak{A}})$, $\omega:{\sf G}\times{\sf
G}\to\mathcal{UM}({\mathfrak{A}})$, such that $\omega$ and
$x\mapsto\alpha_{x}(a)$ are strongly Borel and satisfy

  * (i)

$\alpha_{x}(\omega(y,z))\omega(x,yz)=\omega(x,y)\omega(xy,z)$,

  * (ii)

$\alpha_{x}\big{(}\alpha_{y}(a)\big{)}\omega(x,y)=\omega(x,y)\alpha_{xy}(a)$,

  * (iii)

$\omega(x,{\sf e})=\omega({\sf e},y)=1,\alpha_{\sf e}={\rm
id}_{{\mathfrak{A}}}$,

for all $x,y,z\in{\sf G}$ and $a\in\mathfrak{A}$. ${\sf e}$ denotes the
identity in ${\sf G}$. When $\mathfrak{A}$ is a $C^{*}$-algebra, $({\sf
G},\alpha,\omega,\mathfrak{A})$ is also called a twisted $C^{*}$-dynamical
system or just a $C^{*}$-dynamical system if the twist $\omega$ is trivial.

Given such a tuple, one usually forms the twisted convolution algebra
$L^{1}_{\alpha,\omega}({\sf G},\mathfrak{A})$, consisting of all Bochner
integrable functions $\Phi:{\sf G}\to\mathfrak{A}$, endowed with the twisted
convolution product

$\Phi*\Psi(x)=\int_{\sf
G}\Phi(y)\alpha_{y}[\Psi(y^{-1}x)]\omega(y,y^{-1}x){\rm d}y$ (2.2)

and the involution

$\Phi^{*}(x)=\Delta(x^{-1})\omega(x,x^{-1})^{*}\alpha_{x}[\Phi(x^{-1})^{*}].$
(2.3)

Making $L^{1}_{\alpha,\omega}({\sf G},\mathfrak{A})$ a Banach ∗-algebra under
the norm $\lVert\Phi\rVert_{L^{1}_{\alpha,\omega}({\sf
G},\mathfrak{A})}=\int_{\sf G}\lVert\Phi(x)\rVert_{\mathfrak{A}}{\rm d}x$.
When the twisted action is trivial (namely $\alpha\equiv{\rm
id}_{\mathfrak{A}}$ and $\omega\equiv 1$), one recovers the projective tensor
product: $L^{1}({\sf G},\mathfrak{A}):=L^{1}_{{\rm id}_{\mathfrak{A}},1}({\sf
G},\mathfrak{A})\cong L^{1}({\sf G})\hat{\otimes}\mathfrak{A}$.

######  Lemma 2.7.

If $\,\mathfrak{A}$ is a dense differential subalgebra of $\mathfrak{B}$ and
stable under the twisted action $({\sf G},\alpha,\omega,\mathfrak{B})$, then
$L^{1}_{\alpha,\omega}({\sf G},\mathfrak{A})$ is a dense differential
subalgebra of $L^{1}_{\alpha,\omega}({\sf G},\mathfrak{B})$. In particular,
$L^{1}_{\alpha,\omega}({\sf G},\mathfrak{A})$ is inverse-closed in
$L^{1}_{\alpha,\omega}({\sf G},\mathfrak{B})$.

###### Proof.

It is clear that $L^{1}_{\alpha,\omega}({\sf G},\mathfrak{A})\hookrightarrow
L^{1}_{\alpha,\omega}({\sf G},\mathfrak{B})$ in a natural way. This inclusion
is dense, as $\mathfrak{B}$-valued step functions can be approximated by
$\mathfrak{A}$-valued step functions. Finally, we have

$\displaystyle\lVert\Phi*\Psi\rVert_{L^{1}_{\alpha,\omega}({\sf
G},\mathfrak{A})}$ $\displaystyle\leq\int_{\sf G}\int_{\sf
G}\lVert\Phi(y)\alpha_{y}\big{[}\Psi(y^{-1}x)\big{]}\rVert_{\mathfrak{A}}{\rm
d}y{\rm d}x$ $\displaystyle\leq C\int_{\sf G}\int_{\sf
G}\lVert\Phi(y)\rVert_{\mathfrak{A}}\lVert\Psi(y^{-1}x)\rVert_{\mathfrak{B}}+\lVert\Phi(y)\rVert_{\mathfrak{B}}\lVert\Psi(y^{-1}x)\rVert_{\mathfrak{A}}{\rm
d}y{\rm d}x$ $\displaystyle\leq C(\lVert\Phi\rVert_{L^{1}_{\alpha,\omega}({\sf
G},\mathfrak{A})}\lVert\Psi\rVert_{L^{1}_{\alpha,\omega}({\sf
G},\mathfrak{B})}+\lVert\Phi\rVert_{L^{1}_{\alpha,\omega}({\sf
G},\mathfrak{B})}\lVert\Psi\rVert_{L^{1}_{\alpha,\omega}({\sf
G},\mathfrak{A})}).$

Applying Lemma 2.5 concludes the proof.∎

######  Remark 2.8.

Lemma 2.7 does not hold under the weaker assumption that $\mathfrak{A}$ is a
(dense) inverse-closed ∗-subalgebra of $\mathfrak{B}$. See, for instance [10,
Theorems 4,5]. This shows that, in general, $\mathfrak{A}=L^{1}_{\rm lt}({\sf
G},\mathcal{C}_{0}({\sf G}))$ is an inverse-closed subalgebra but not a
differential subalgebra of $\mathfrak{B}=\mathbb{K}(L^{2}({\sf G}))$.

In contrast with the previous remark, let us apply our results to some honest
differential subalgebras of $\mathbb{K}(\mathcal{H})$.

######  Corollary 2.9.

Let ${\sf G}$ be a symemtric locally compact group. Then $L^{1}({\sf
G},S_{p}(\mathcal{H}))\cong L^{1}({\sf G})\hat{\otimes}S_{p}(\mathcal{H})$ is
a symmetric Banach ∗-algebra.

###### Proof.

$S_{p}(\mathcal{H})$ is a dense differential subalgebra of
$\mathbb{K}(\mathcal{H})$, so Lemmas 2.7 and 2.5 apply and we get

${\rm Spec}_{L^{1}({\sf G},S_{p}(\mathcal{H}))}(\Phi)={\rm Spec}_{L^{1}({\sf
G},\mathbb{K}(\mathcal{H}))}(\Phi),\textup{ for all }\Phi\in L^{1}({\sf
G},S_{p}(\mathcal{H})).$

But $L^{1}({\sf G},\mathbb{K}(\mathcal{H}))$ is symmetric when $L^{1}({\sf
G})$ is symmetric [8, Theorem 1]. ∎

######  Remark 2.10.

If $\mathfrak{D}$ is a symmetric Banach ∗-algebra, then the projective tensor
product $\mathfrak{D}\,\hat{\otimes}{M}_{n}(\mathbb{C})$ gives a symmetric
Banach ∗-algebra [14, Theorem 11.4.2]. We see Corollary 2.9 (together with the
symmetry of $L^{1}({\sf G})\hat{\otimes}\mathbb{K}(\mathcal{H})$) as an
infinite-dimensional version of this fact.

######  Remark 2.11.

Its easy to replicate the techinque used in Corollary 2.9 in any of the cases
mentioned in Remark 2.3, under possibly different assumptions.

##  3 On the extension of a compact group

Let ${\sf G}$ be a (Hausdorff) locally compact group, ${\sf K}$ a normal,
compact subgroup and set ${\sf H}={\sf G}/{\sf K}$. The goal of this section
is to use the above mentioned methods to find conditions on ${\sf H}$ that
guarantee the symmetry of $L^{1}({\sf G})$. Following [3], we will describe
how to decompose

$L^{1}({\sf G})\cong L^{1}_{\alpha,\omega}({\sf H},L^{1}({\sf K})),$ (3.1)

for an appropiate twisted action $({\sf H},\alpha,\omega,L^{1}({\sf K}))$.
Indeed, suppose that $\eta$ is a Borel measurable right inverse of $\pi:{\sf
G}\to{\sf H}={\sf G}/{\sf K}$, with $\eta({\sf K})=1$ and define
$\tau(x,y)=\eta(x)\eta(y)\eta(xy)^{-1}\in{\sf K}$, for $x,y\in{\sf H}$. Then
for $f\in L^{1}({\sf K})$ the relevant formulas are

$\displaystyle\alpha_{x}(f)(t)$
$\displaystyle=f\big{(}\eta(x)^{-1}t\eta(x)\big{)},$ (3.2)
$\displaystyle[\omega(x,y)f](t)=f(\tau(x,y)^{-1}t)$ $\displaystyle\textup{ and
}[f\omega(x,y)](t)=f(t\tau(x,y)^{-1}).$ (3.3)

It is fairly elementary to show that this twisted action extends to
$C^{*}({\sf K})$ and also restricts to $L^{2}({\sf K})$. Let us now prove the
main theorem of the section.

######  Theorem 3.1.

Let $({\sf H},\alpha,\omega,L^{1}({\sf K}))$ defined as above. If
$L^{1}_{\alpha,\omega}({\sf H},C^{*}({\sf K}))$ is a symmetric Banach
∗-algebra, then $L^{1}({\sf G})\cong L^{1}_{\alpha,\omega}({\sf H},L^{1}({\sf
K}))$ is also symmmetric.

###### Proof.

We first consider the (smaller) algebra $L^{1}_{\alpha,\omega}({\sf
H},L^{2}({\sf K}))$. $\mathfrak{A}=L^{2}({\sf K})$ is a proper
$H^{*}$-algebra, and hence a dense differential subalgebra of $C^{*}({\sf
K})$. But $\mathfrak{A}$ is also a dense differential subalgebra of
$L^{1}({\sf K})$ and it then follows from Lemma 2.7 that
$L^{1}_{\alpha,\omega}({\sf H},L^{2}({\sf K}))$ is inverse-closed in both
$L^{1}_{\alpha,\omega}({\sf H},C^{*}({\sf K}))$ and
$L^{1}_{\alpha,\omega}({\sf H},L^{1}({\sf K}))$, so we have

${\rm Spec}_{L^{1}_{\alpha,\omega}({\sf H},L^{1}({\sf K}))}(\Phi)={\rm
Spec}_{L^{1}_{\alpha,\omega}({\sf H},C^{*}({\sf K}))}(\Phi)={\rm
Spec}_{L^{1}_{\alpha,\omega}({\sf H},L^{2}({\sf K}))}(\Phi),$

for all $\Phi\in L^{1}_{\alpha,\omega}({\sf H},L^{2}({\sf K}))$. In
particular, $L^{1}_{\alpha,\omega}({\sf H},L^{2}({\sf K}))$ is a symmetric
dense ∗-ideal of $L^{1}_{\alpha,\omega}({\sf H},L^{1}({\sf K}))$ and hence the
latter has to be a symmetric Banach ∗-algebra, due to [18], or [9, Satz 1]. ∎

The usefulness of Theorem 3.1 relies on the fact that the study of the
symmetry of $L^{1}$-algebras is far more developed -and easier to handle- in
the case where the algebra of coefficients is a $C^{*}$-algebra. Let us
mention some cases of interest.

######  Example 3.2.

The case of continuous twisted actions (namely, when $\alpha$ and $\omega$ are
continuous maps) has received the most attention, so we can cite relevant
results. Note that this is always the case if ${\sf H}$ is discrete. In this
case, the condition of symmetry in theorem 3.1 is always fulfilled when ${\sf
H}$ is hypersymmetric, see [4].

Another important case covered here could be the one of semidirect products
${\sf G}={\sf K}\rtimes{\sf H}$. However, this is example is easier and will
be treated directly in Section 4. In Theorem 4.1 we proved that is enough to
only require ${\sf H}$ to be symmetric. It is known that there are symmetric
but not hypersymmetric groups.

As a corollary to the previous theorem, we explore different types of decay on
the coefficient algebra. For $p\in[1,\infty]$, the Banach space $L^{p}({\sf
K})$, of $p$-integrable functions, is a Banach ∗-algebra when gifted with the
operations (2.2), (2.3) and stable under the twisted action determined by
(3.2) and (3.3).

######  Corollary 3.3.

Let $p\in[1,\infty]$, then $L^{1}_{\alpha,\omega}({\sf H},L^{p}({\sf K}))$ is
symmetric if and only if $L^{1}_{\alpha,\omega}({\sf H},C^{*}(K))$ is
symmetric.

###### Proof.

Note that if $p\geq q$, then $L^{p}({\sf K})$ is a differential subalgebra of
$L^{q}({\sf K})$, so by Lemma 2.7, one has

${\rm Spec}_{L^{1}_{\alpha,\omega}({\sf H},L^{p}({\sf K}))}(\Phi)={\rm
Spec}_{L^{1}_{\alpha,\omega}({\sf H},L^{q}({\sf K}))}(\Phi),\textup{ for all
}\Phi\in L^{1}_{\alpha,\omega}({\sf H},L^{p}({\sf K})).$

If $L^{1}_{\alpha,\omega}({\sf H},C^{*}(K))$ is symmetric, then
$L^{1}_{\alpha,\omega}({\sf H},L^{1}({\sf K}))$ is symmetric because of
Theorem 3.1, hence $L^{1}_{\alpha,\omega}({\sf H},L^{p}({\sf K}))$ is
symmetric for all $p$.

On the other hand, if $L^{1}_{\alpha,\omega}({\sf H},L^{p}({\sf K}))$ is
symmetric and $p\geq 2$, then by the reasoning above
$L^{1}_{\alpha,\omega}({\sf H},L^{p}({\sf K}))$ is a symmetric dense ∗-ideal
of $L^{1}_{\alpha,\omega}({\sf H},L^{2}({\sf K}))$, which makes the latter
symmetric. But the latter algebra is also a symmetric dense ∗-ideal of
$L^{1}_{\alpha,\omega}({\sf H},C^{*}({\sf K}))$, hence
$L^{1}_{\alpha,\omega}({\sf H},C^{*}({\sf K}))$ is symmetric. If $p<2$, then
it follows immediately that $L^{1}_{\alpha,\omega}({\sf H},L^{2}({\sf K}))$ is
symmetric and we can repeat the last argument. ∎

Corollary 3.3 effectively gives us a converse to Theorem 3.1. As such, it
provides a way to construct (new) symmetric algebras associated to
$C^{*}$-dynamical systems from symmetric groups. Let us briefly illustrate
how.

######  Example 3.4.

Suppose that ${\sf H}$ is either (i) compactly generated and of polynomial
growth or (ii) has relatively compact conjugacy classes. Suppose further that
${\sf K}$ is an abelian compact group such that ${\sf H}$ acts continuously by
group homomorphisms. In each case, the group ${\sf K}\rtimes{\sf H}$ satisfies
the same condition, so it is symmetric. Then by Pontryagin duality we obtain a
$C^{*}$-dynamical system $({\sf G},\alpha,\mathcal{C}_{0}(\hat{{\sf K}}))$ and
Corollary 3.3 implies that the algebra $L^{1}_{\alpha}({\sf H},C_{0}(\hat{{\sf
K}}))$ is symmetric and inverse closed in $C_{0}(\hat{{\sf
K}})\rtimes_{\alpha}{\sf H}$.

##  4 Splitting extensions

As mentioned in the section title, we will finish the paper by exploring a
little bit further the case of splitting extensions and provide a sufficient
condition for Theorem 3.1 to apply. In fact, the condition provided is also
necessary, yielding a very satisfactory answer. Note that we kept the
notations from the previous section.

######  Theorem 4.1.

Suppose that $\eta:{\sf H}\to{\sf G}$ is a group homomorphism. Then
$L^{1}({\sf G})$ is symmetric whenever $L^{1}({\sf H})$ is symmetric.

###### Proof.

It is clear that in this case, $\tau\equiv{\sf e}$ and consequently
$\omega\equiv 1$. Let $\lambda:C^{*}({\sf K})\to\mathbb{B}(L^{2}({\sf K}))$ be
the embedding obtained from the left regular representation. Then we define
the unitary representation $U:{\sf H}\to\mathbb{U}(L^{2}({\sf K}))$ given by

$U_{x}(f)(t)=f(\eta(x)^{-1}t\eta(x)),\textup{ for all }f\in L^{2}({\sf K}).$
(4.1)

Now observe that, for $f,g\in L^{2}({\sf K})$, one has

$\displaystyle\big{(}U_{x}\lambda(f)U_{x}^{*}g\big{)}(t)$
$\displaystyle=\big{(}\lambda(f)U_{x}^{*}g\big{)}(\eta(x)^{-1}t\eta(x))$
$\displaystyle=\int_{\sf K}f(s)U_{x}^{*}g(s^{-1}\eta(x)^{-1}t\eta(x)){\rm d}s$
$\displaystyle=\int_{\sf K}f(s)g(\eta(x)s^{-1}\eta(x)^{-1}t){\rm d}s$
$\displaystyle=\int_{\sf K}\alpha_{x}(f)(s)g(s^{-1}t){\rm d}s$
$\displaystyle=\lambda(\alpha_{x}(f))(t)$

Hence the pair $(U,\lambda)$ is a covariant representation of $({\sf
H},\alpha,C^{*}({\sf K}))$. We now define an isometric ∗-monomorphism

$\varphi:L_{\alpha}^{1}({\sf H},C^{*}({\sf K}))\to L^{1}({\sf
H},\mathbb{K}(L^{2}({\sf
K})))\quad\textup{by}\quad\varphi(\Phi)(x)=\lambda(\Phi(x))U_{x}.$

Note that this map is well defined as $\lambda(\Phi(x))\in C^{*}({\sf
K})\subset\mathbb{K}(L^{2}({\sf K}))$, making $\varphi(\Phi)$ Bochner
measurable. Is clear that $\varphi$ is isometric, let us verify its algebraic
properties. We see that

$\displaystyle\varphi(\Phi)*\varphi(\Psi)$ $\displaystyle=\int_{\,{\sf
H}}\lambda(\Phi(y))U_{y}\lambda(\Phi(y^{-1}x))U_{y^{-1}x}{\rm d}y$
$\displaystyle=\int_{\,{\sf
H}}\lambda(\Phi(y))\lambda(\alpha_{x}(\Phi(y^{-1}x)))U_{x}{\rm d}y$
$\displaystyle=\lambda\Big{(}\int_{\,{\sf
H}}\Phi(y)\alpha_{x}(\Phi(y^{-1}x)){\rm d}y\Big{)}U_{x}$
$\displaystyle=\varphi(\Phi*\Psi)$

and that

$\varphi(\Phi^{*})(x)=\Delta(x^{-1})U_{x}\lambda(\Phi(x^{-1}))^{*}=\Delta(x^{-1})\varphi(\Phi)(x^{-1})^{*}=\varphi(\Phi)^{*}(x).$

Therefore we can identify $L_{\alpha}^{1}({\sf H},C^{*}({\sf K}))$ with a
closed ∗-subalgebra of $L^{1}({\sf H},\mathbb{K}(L^{2}({\sf K})))$. The latter
is symemtric by [8, Theorem 1], so the former has to be symmetric. Then
Theorem 3.1 implies that $L^{1}({\sf G})$ is symmetric. ∎

######  Remark 4.2.

As ${\sf H}$ is a quotient of ${\sf G}$, we obtain as a corollary that ${\sf
G}$ is symmetric if and only if ${\sf H}$ is symmetric.

######  Example 4.3.

Let ${\sf K}$ be a compact group and ${\sf H}$ be a discrete group acting on a
set $\Omega$, we denote by ${\sf K}^{\Omega}$ the directed product of ${\sf
K}$ with itself, indexed by $\Omega$. We then can form the group ${\sf G}={\sf
K}^{\Omega}\rtimes{\sf H}$ and Theorem 4.1 implies that this group is
symmetric exactly when ${\sf H}$ is symmetric.

## References

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ADDRESS

Felipe I. Flores

Department of Mathematics, University of Virginia,

114 Kerchof Hall. 141 Cabell Dr,

Charlottesville, Virginia, United States

E-mail<EMAIL_ADDRESS>


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#  Improved Graph- based semi-supervised learning Schemes

Farid Bozorgnia  CAMGSD, Department of Mathematics, Instituto Superior
Técnico, Lisbon, Portugal<EMAIL_ADDRESS>
###### Abstract.

In this work, we improve the accuracy of several known algorithms to address
the classification of large datasets when few labels are available. Our
framework lies in the realm of graph-based semi-supervised learning. With
novel modifications on Gaussian Random Fields Learning and Poisson Learning
algorithms, we increase the accuracy and create more robust algorithms.
Experimental results demonstrate the efficiency and superiority of the
proposed methods over conventional graph-based semi-supervised techniques,
especially in the context of imbalanced datasets.

###### Key words and phrases:

Graph-based algorithms, Semi-supervised learning, Laplace learning

F. Bozorgnia was supported by the FCT fellowship SFRH/BPD/33962/2009 and by
Marie Skłodowska-Curie grant agreement No. 777826 (NoMADS)

##  1\. Introduction

In this paper, we consider semi-supervised learning approaches for classifying
many unlabeled data when very few labels are given. Our objective is to refine
existing classification algorithms to perform effectively in the cases with
limited labeled data. We introduce innovative modifications to established
algorithms to enhance accuracy and robustness.

When there is an insufficient amount of labeled data available, models may
struggle to learn effectively and to predict accurately. One way to improve
this is by incorporating lots of unlabeled data along with those few labeled
samples. This approach helps the model to learn about the structure of data,
enhancing its performance especially when there aren’t many examples to learn
from. Considering mentioned advantage, semi-supervised learning has received
much attention in the past few decades [18, 11, 17, 22]. A common approach for
doing this is graph-based semi-supervised learning. We use graphs to show how
different samples of data are connected, helping to understand both labeled
and unlabeled data better. We refer to recent surveys [19, 12] as a
comprehensive review of existing algorithms on graph-based learning.

Although existing semi-supervised algorithms show good performance on balanced
data, they frequently struggle in complex real-world applications,
particularly with highly imbalanced datasets. One of the issues is overfilling
to the limited labeled data available and not making full use of the
structural information present in the unlabeled data. The Imbalance Ratio (IR)
is an important factor and represents the ratio of samples in each of Minority
and Majority classes. The bigger IR, the more difficult the problem becomes,
to see more about techniques to handle imbalanced data see [20]. Our objective
is to introduce innovative modifications to established algorithms, enhancing
both accuracy and robustness against these challenges.

A common approach to use unlabeled data in semi-supervised learning is to
build a graph over the data. This means geometric or topological properties of
the unlabeled data have been used to improve algorithms. In [13] a general
framework for constructing a graph is proposed and they named it Graph-based
Semi-supervised Learning via Improving the Quality of the Graph Dynamically.

First, we construct a weight matrix $W$ or similarity matrix over the data
set, which encodes the similarities between pairs of data nodes. If our data
set consists of $n$ points
${\mathcal{X}}=\\{x_{1},x_{2},\cdots,x_{n}\\}\subset\mathbb{R}^{d}$, then the
weight matrix $W$ is an $n\times n$ symmetric matrix, where the element
$w_{ij}$ represents the similarity between two data points $x_{i}$ and
$x_{j}$. When constructing a KNN graph, the weight between node $x_{i}$ and
node $x_{j}$ typically depends on the distance or similarity between the
nodes. A common choice for the weight function is the Gaussian kernel or the
inverse of the distance between nodes.

The similarity is always nonnegative and should be large when $x_{i}$ and
$x_{j}$ are close together spatially, and small (or zero), when $x_{i}$ and
$x_{j}$ are far apart. Subsequently, by leveraging information from the
constructed graph, labels are propagated from the given labeled instances to
the unlabeled data.

The main contribution of the current work is as follows.

  * •

Modified GRF and Improved GRF Learning Algorithms: To speed up convergence and
handle imbalances more effectively by incorporating the stationary
distribution $\pi$ of the random walk on the graph.

  * •

Improved Poisson Learning Algorithm: Introduces regularization terms to
enhance performance on imbalanced data.

##  2\. Problem setting and previous works

This section is devoted as an introduction to PDEs on graph and setting our
problem. Let ${\mathcal{X}}=\\{x_{1},\cdots,x_{n}\\}$ denote the vertices of a
graph with symmetric edge weight $w_{ij}$ between
$x_{i},x_{j}\in{\mathcal{X}}$ and let $W=[w_{ij}]$. We assume there is a
subset of the nodes
${\mathcal{X}}_{l}={\\{x_{1},\cdots,x_{l}}\\}\subset{\mathcal{X}}$ for which
labels are provided by a labeling function
$g:{\mathcal{X}}_{l}\rightarrow{\mathbb{R}}^{k}$, forming our training set
$(x_{i},y_{i})_{i=1}^{l}$. The degree of a vertex $x_{i}$ is given by
$d_{i}=\sum_{j}w_{ij}$ and let $d=\sum_{i}d_{i}$. Matrix $D$ denotes the
diagonal matrix which has $d_{i}$ on main the diagonal.

Let $\ell^{2}({\mathcal{X}})$ denote the set of functions
$u:{\mathcal{X}}\rightarrow\mathbb{R}$ equipped with the inner product

$(u,v)=\sum\limits_{x\in{\mathcal{X}}}u(x)v(x),$

for functions $u,v:{\mathcal{X}}\rightarrow\mathbb{R}$. We also define a
vector field on the graph to be an antisymmetric function
$V:{\mathcal{X}}\times{\mathcal{X}}\rightarrow\mathbb{R}^{2}$, i.e.
$V(x,y)=-V(y,x)$ and denote the space of all vector fields by
$\ell^{2}({\mathcal{X}}^{2})$.

The gradient of a function $u\in\ell^{2}({\mathcal{X}})$ is the vector field

$\nabla u(x,y)=u(y)-u(x).$

The unnormalized graph laplacian ${\mathcal{L}}$ of a function
$u\in\ell^{2}({\mathcal{X}})$ is defined as

${\mathcal{L}}u(x):=-{\rm div\,}(\nabla
u)(x)=\sum_{y\in{\mathcal{X}}}w_{xy}(u(x)-u(y)).$

Most graph-based semi-supervised learning aims to find a function on the graph
that closely matches the given labels while also maintaining smoothness. It is
shown in [23] that the graph Laplacian regularization is effective because it
forces the labels to be consistent with the graph structure.

To model label propagation in semi-supervised learning, it is assumed that the
learned labels vary smoothly and do not change rapidly within high-density
regions of the graph (smoothness assumptions) [4]. Based on this assumption
different approaches have been proposed, we refer to the pioneer methods
Laplace learning, [21] see also [3, 4].

Let $U$ stand for the label matrix $U$ that shows the class information of
each element. A general form of a graph-based semi-supervised learning for
data classification can be formulated as a minimization of an energy:

$R(U)+\mu F(U),$

where $R(U)$ is a regularization term incorporating the graph weights, and
$F(U)$ is a forcing term, which usually incorporates the labeled points and
their class information.

In Laplace learning algorithm the labels are extended by finding the minimizer
$U:{\mathcal{X}}\rightarrow\mathbb{R}^{k}$ for the following problem

(1) $\begin{cases}&\min
J_{n}(U):=\frac{1}{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}w_{ij}|U(x_{i})-U(x_{j})|^{2}\\\
&\text{subject to }U(x_{i})=y_{i},\quad\text{ for }i=1,2,\cdots,l.\end{cases}$

The minimizer will be a harmonic function satisfying

$\left\\{\begin{array}[]{ll}{\mathcal{L}}U(x)=0,&x\in{\mathcal{X}}\setminus{\mathcal{X}}_{l},\\\
U=g,&\text{ on }{\mathcal{X}}_{l},\\\ \end{array}\right.$

where ${\mathcal{L}}$ is the unnormalized graph Laplacian given by

$\mathcal{L}U(x_{i})=\sum\limits_{j=1}^{n}w_{ij}\,(U(x_{i})-U(x_{j})).$

Let $U=(u_{1},\cdots,u_{k})$ be a solution of (1), the label of node
$x_{i}\in{\mathcal{X}}\setminus{\mathcal{X}}_{l}$ is dictated by

$\underset{j\in{\\{1,\cdots,k}\\}}{\textrm{arg max}}{u_{j}(x_{i})}.$

This means that Laplace learning uses harmonic extension on a graph to
propagate labels. If the number of labeled data samples is finite while the
number of unlabeled data tends to infinity, then Laplace learning becomes
degenerate and the solutions become roughly constant with a spike at each
labeled data point.

Later it has been observed that the Laplace learning can give poor results in
classification [15]. The results are often poor because the solutions have
localized spikes near the labeled points, while being almost constant far from
them. To overcome this problem several versions of the Laplace learning
algorithm have been proposed, for instance, Laplacian regularization, [2],
weighted Laplacian, [10] and $p$-Laplace learning, [8, 16]. Also, the limiting
case in $p$-Laplacian when $p$ tends to infinity is so-called Lipschitz
learning is studied in [14] and similar to continuum PDEs is related to
finding the absolute minimal Lipschitz extension of the training data.
Recently, in [7] another approach to increase accuracy of Laplace learning is
given and called Poisson learning.

In [9] the authors consider the case that the number of labeled data points
grows to infinity also when the total number of data points grows. Let $\beta$
denote the labeling rate. They show that for a random geometric graph with
length scale $\varepsilon$ if $\beta\geq\varepsilon^{2}$, then the solution
becomes degenerate and spikes occurs, while for the case
$\beta\leq\varepsilon^{2}$, Laplacian learning is well-posed and consistent
with a continuum Laplace equation.

The authors in [7] have proposed a scheme, called Poisson learning that
replaces the label values at training points as sources and sinks, and solves
the Poisson equation on the graph as follows:

(2)
$\left\\{\begin{array}[]{ll}\mathcal{L}\mathbf{u}(x_{i})=y_{i}-\overline{y}&1\leq
i\leq l,\\\ \mathcal{L}\mathbf{u}(x_{i})=0&l+1\leq i\leq n,\\\
\end{array}\right.$

with further condition $\sum_{i=1}^{n}d(x_{i})\mathbf{u}(x_{i})=0$, where
$\overline{y}=\frac{1}{l}\sum_{i=1}^{l}y_{i}$ is the average label vector. In
the next section, we review this scheme in detail and will consider some
modification to this algorithm.

In [6, 5] a graph-based semi-supervised learning scheme based on the theory of
the spatial segregation of competitive systems, is given. The scheme is based
on the minimizing (1) with constraint

$u_{i}(x)\cdot u_{j}(x)=0,\qquad\text{ for all }x\in{\mathcal{X}},\;1\leq
i\neq j\leq k.$

##  3\. Proposed Algorithms

###  3.1. Modified Gaussian Random Fields (MGRF)

Let, as before ${\mathcal{X}}={\\{x_{1},\cdots,x_{n}}\\}$ denote the data
points or vertices in a graph. We assume there is a subset of the nodes
${\mathcal{X}}_{l}={\\{x_{1},\cdots,x_{l}}\\}\subset{\mathcal{X}}$ that their
labels are given with a label function
$g:{\mathcal{X}}_{l}\rightarrow{\mathbb{R}}^{k}$. It is further assumed that
$y_{i}=g(x_{i})\in\\{e_{1},\cdots,e_{k}\\}$ where $e_{i}$ is the standard
basis in $\mathbb{R}^{k}$ and represents the $i\textsuperscript{th}$ class. In
graph-based semi-supervised learning, we aim to extend labels to the rest of
the vertices ${\mathcal{X}}_{u}=\\{x_{l+1},\cdots,x_{n}\\}$.

The scheme described below is known as Gaussian Random Fields (GRF) [21]. For
any matrix $U=(U_{ij})\in\mathbb{R}^{n\times k}$, $U_{i\cdot}$ represents the
$i^{\text{th}}$ row of $U$. They consider the following minimization problem

(3) $\underset{U_{l\cdot}=Y_{l\cdot}}{\text{arg min}}\,Tr(U^{T}LU),\quad
U\in\mathbb{R}^{n\times k}$

Here $L=D-W$. Let us split the weight matrix $W$ into four blocks as

$\mathbf{W}=\left(\begin{array}[]{cc}W_{ll}&W_{lu}\\\ W_{ul}&W_{uu}\\\
\end{array}\right).$

Also let $P=D^{-1}W$ and decompose $P$ and $U$ to $U_{l}$ and $U_{u}$, then
the solution of minimization problem (3) is given by:

(4) $U_{u}=(I-P_{uu})^{-1}P_{ul}U_{l}.$

The following iterative scheme was proposed in [21].

(5) $U^{(m+1)}=\alpha SU^{(m)}+(1-\alpha)Y,$

where $S=D^{\frac{-1}{2}}WD^{\frac{-1}{2}}$ and $Y$ denotes the vector of
initial labels.

We modify the scheme (5) to achieve better accuracy. In scheme (5), $U^{(m)}$
is a vector where the sign of the $i^{\textrm{th}}$ element of $U^{m}$
indicates the class of $x_{i}$. For a sample $x_{i}\in{\mathcal{X}}_{l}$ its
label belongs ${\\{\pm 1}\\}$.

Let $l_{1},l_{2}$ denote the number of labels in Class 1 and Class 2,
respectively with $l=1_{1}+l_{2}$. Then the vector $B\in\mathbb{R}^{n}$ is
defined as follows

$B=\left\\{\begin{array}[]{ll}\frac{l_{2}}{l},&x_{i}\in{\mathcal{X}}_{l}\,\textrm{and}\,y_{i}=1,\\\
\frac{-l_{1}}{l},&x_{i}\in{\mathcal{X}}_{l}\,\textrm{and}\,y_{i}=-1,\\\
0&x_{i}\in{\mathcal{X}}\setminus{\mathcal{X}}_{l}.\\\ \end{array}\right.$

Algorithm 1, named ”MGRF”, extends the GRF scheme by modifying the initial
labels vector $Y$ and iterating over all nodes.

Algorithm 1 (MGRF)

Input:Matrix $W$, initial label $Y$, parameter $\alpha$, tolerance
$\varepsilon$.
Output: Label $y_{i}$ for each point $x_{i}$ or equivalently
$U\in\mathbb{R}^{n}$

  * •

Calculate matrices: $D$, $S$ and $B$

  * •

Initialize $U^{0}=D^{-1}Y$

  * •

while $\|U^{(m+1)}-U^{(m)}\|>\varepsilon$ do

  * •

$U^{(m+1)}=\alpha SU^{(m)}+(1-\alpha)D^{\frac{-1}{2}}B^{T}$

  * •

end while

  * •

Assign each point $x_{i}$ the sign of $U^{*}_{i}$

  * •

$U^{*}=\underset{m\rightarrow\infty}{Lim}U^{m}.$

Next, we explain how to improve the efficiency of the scheme MGRF. Let $\pi$
be the unique stationary distribution of the random walk on the graph
represented by the normalized affinity matrix $S$, then $\pi$ satisfies the
equation:

$\pi S=\pi,$

$\pi$ is a row vector representing the stationary distribution.

Consider matrix $G$, where all rows of $G$ are equal and composed of $\pi$. By
subtracting $G$ from $S$ in each iteration, we effectively bias the random
walk towards the stationary distribution consequently, accelerates the
convergence towards the stationary distribution of the graph. This strategy
makes the algorithm robust to initial conditions and parameter choices.
Furthermore, it prevents the dominance of majority classes, thus improving the
overall classification performance, particularly in imbalanced data sets.

The details of the scheme are given in algorithm 2 which we call it Improved
Gaussian Random Fields (IGRF).

Algorithm 2 (IGRF)

Input:Matrix $W$, $Y$, parameters $\alpha_{1},\alpha_{2},\alpha_{3},$
tolerance $\varepsilon$
Output: Label $y_{i}$ for each point $x_{i}$ or equivalently
$U\in\mathbb{R}^{n}$

  * •

Calculate matrices: $S$, $B$ and $\pi$

  * •

Initialize $U^{0}=D^{-1}Y$

  * •

while $\|U^{(m+1)}-U^{(m)}\|>\varepsilon$ do

  * •

$U^{(m+1)}=(\alpha_{1}S-\alpha_{2}G+\alpha_{3}I)U^{(m)}+(1-\alpha_{1})D^{\frac{-1}{2}}B^{T}$

  * •

end while

  * •

Assign each point $x_{i}$ the sign of $U^{*}_{i}$

  * •

$U^{*}=\underset{m\rightarrow\infty}{Lim}U^{m}.$

###  3.2. Improved Poisson Learning(IPL)

In this section, we aim to improve the efficiency of Poisson Learning given by
Algorithm 3 (see [7]). Let $b\in\mathbb{R}^{k}$ be the vector whose
$i^{\textrm{th}}$ entry $b_{i}$ is the fraction of data points belonging to
class $i$.

Algorithm 3 Poisson Learning

Input: ${\mathcal{X}},W$, $Y\in\mathbb{R}^{k\times l}$ (given labels), $b$,
$T$:(number of iterations)
Output:$U\in\mathbb{R}^{n\times k}$ predicted class labels, and classification
results

  * •

$D\leftarrow W\mathds{1}$

  * •

$L\leftarrow D-W$

  * •

$\overline{y}\leftarrow\frac{1}{l}Y\mathds{1}$

  * •

$B\leftarrow[Y-\overline{y},zeros(k,n-l)]$

  * •

$U\leftarrow\textrm{zeros}(n,k)$

  * •

For $m=1:T$ do

  * •

$U^{(m)}=U^{(m-1)}+D^{-1}(B^{T}-LU^{m-1})$

  * •

end for

  * •

$U^{m}=U^{m}\cdot\text{diag}(b/\bar{y})$.

The main step of the Poisson Algorithm can be rewritten as

(6) $\begin{split}U^{(m)}&=U^{(m-1)}+D^{-1}(B^{T}-(D-W)U^{(m-1)})\\\
&=U^{(m-1)}+D^{-1}B^{T}-U^{(m-1)}+D^{-1}WU^{(m-1)}),\\\
&=PU^{(m-1)}+D^{-1}B^{T}\end{split}$

where $P=D^{-1}W.$ Choosing $U^{0}=D^{-1}B^{T}$ then (6) implies

$U^{(m)}=\left(\sum_{i=0}^{m-1}P^{i}\right)D^{-1}B^{T}.$

Next, we determine the limit of $P^{m}$ as $m$ tends to infinity. From Markov
chains theory, for an irreducible and aperiodic Markov chain, $P^{m}$
converges to a matrix where all rows are identical and represent the
stationary distribution of the Markov chain. It represents the long-term
relative frequencies of being in each state of the Markov chain.

This stationary distribution can be computed by finding the left eigenvector
corresponding to the eigenvalue one of the transition matrix $P$, i.e,

$\pi P=\pi.$

It is handy to verify that
$\pi=(\frac{d_{1}}{d},\frac{d_{2}}{d}\cdots,\frac{d_{n}}{d}).$ We make ansatz
that the limit of matrix $P$ is $Q$ with matrix $Q$ being rank one matrix with
rows $\pi$. Thus

$\underset{m\rightarrow\infty}{Lim}P^{m}=\underset{m\rightarrow\infty}{Lim}P^{m-1}P=QP=Q.$

It worths to see $P$ has an eigenvalue equal $one,$ means $I-P$ is not
invertible means one can not extract the fixed point for iterative scheme
given by (6). This shows the sequence in (6) is not convergent.

Our change in the Poisson algorithm is to subtract matrix $Q$ from $P$. Our
iterative scheme is as follows

$U^{(m+1)}=(P-Q)U^{m}+D^{-1}B^{T}.$

Keeping in mind the facts $QP=Q$, and $Q^{J}=Q(J=1,2,\cdots$) then it is easy
to check

$\underset{m\rightarrow\infty}{\text{lim}}\,(P-Q)^{m}=0.$

Moreover, with a slightly change, our iterative scheme called Improved Poisson
Learning is

(7) $U^{(m+1)}=(P-\alpha_{1}Q)U^{m}+\alpha_{2}D^{-1}B^{T},$

where parameters $0<\alpha_{1}\leq 1$ and $\alpha_{2}>0$.

The scheme updates the label matrix $U$ using a combination of transition
probabilities, regularization terms, and initial label information. It
converges faster as it focuses entirely on the graph’s intrinsic structure and
connectivity. Furthermore, we might add the term $\alpha I$ to iterative
scheme (7).

Algorithm 4 (IPL)

Input:Matrix $W$, initial label matrix $Y$, parameters
$\alpha_{1},\alpha_{2},\alpha_{3}$.
Output:Label $y_{i}$ for each point $x_{i}$ or equivalently
$U\in\mathbb{R}^{n\times k}$

  * •

Calculate matrices: $D$ transition matrix $P$, $Q$

  * •

Initialize $U^{0}=D^{-1}Y$

  * •

while not convergent do

  * •

$U^{(m+1)}=(P-\alpha_{1}Q+\alpha_{2}I)U^{m}+\alpha_{3}D^{-1}B^{T}$

  * •

end while

  * •

Assign each point $x_{i}$ class $y_{i}=\underset{j}{\text{arg max}}\,U^{*}$
where

  * •

$U^{*}=\underset{m\rightarrow\infty}{Lim}U^{m}$

Adding $\alpha_{2}I$ to the transition matrix $P$ effectively increases the
self-transition probabilities of nodes in the graph by $\alpha_{2}$. This can
be interpreted as giving more weight to the existing labels in the updating
process. It may help stabilize the iterative process by ensuring that each
node retains some influence from its current label in the next iteration. The
addition of $\alpha_{2}I$ can also help prevent numerical instability issues,
particularly since $P$ and $Q$ have eigenvalues close to one.

##  4\. Experimental results

In this section, we compare the proposed algorithms with some existing semi-
supervised learning algorithms. For imbalanced data, we use evaluation metrics
that are robust, such as precision, recall, and ${\rm F}_{1}$-score. These
metrics provide a more comprehensive assessment of model performance on
imbalanced data compared to accuracy.

For simplicity reason, we name the minor class as positive class
interchangeably. Precision measures the accuracy of the positive predictions.
It is the ratio of true positive predictions to the total predicted positives.
Recall measures the ability to capture all positive samples. It is the ratio
of true positive predictions to the total actual positives. $F_{1}$ Score is
the harmonic mean of precision and recall. It balances the two metrics,
especially useful in the case of imbalanced data sets. The term of $F_{1}$
score for each class is defined as follows:

(8) ${\rm
F}_{1}(j)=2\times\dfrac{{\rm{Recall}}(j)\times{\rm{Precision}}(j)}{{\rm{Recall}}(j)+{\rm{Precision}}(j)}\quad
j=1,2.$

Accuracy measures the overall correctness of the model. It is the ratio of all
correct predictions to the total number of samples.

######  Example 4.1.

We consider balanced Two-Moon pattern, we generate a set of 1000 points, with
noise level $0.15$. Increasing noise will make classes more overlapping. In
Algorithm IPL, the parameters are set as
$\alpha_{1}=0.001,\alpha_{2}=0.02,\alpha_{3}=1$.

Figure 1. The classification on Two-Moon.

In Table 1, we present a comparison between Algorithms IGRF and IPL, utilizing
the two moons dataset consisting of $n=1000$ points with a noise level set at
$0.15$.

For Algorithm IGRF, we construct the affinity matrix using the RBF (Gaussian)
kernel, which offers a smoother affinity measure than the $k$-nearest
neighbors graph. The parameters of IGRF are set as
$\alpha_{1}=.99,\alpha_{2}=0.005,\alpha_{3}=0.05.$

In the IPL algorithm, we employ a grid search over hyperparameters to
determine the optimal parameters for various numbers of initial labels.
However, this process is time-consuming. The table displays the average
accuracy for each number of initial labels per class.

Table 1. Two moon, balanced case Average overall accuracy over 100 trials for
two moon
---
number of labels per class | 1 | 2 | 3 | 4 | 5
IGRF | 94.64 | 97.15 | 98.05 | 98.348 | 98.53
IPL | 86.30 | 89.55 | 91.86 | 92.12 | 93.04
Poisson | 83.23 | 87.985 | 90.406 | 91.953 | 92.70

######  Example 4.2.

Next, we generate imbalance two half moons with 950 samples from Majority
class and 50 samples from Minority. We implement the IGRF scheme with two
labeled points per classes. The average of accuracy over 100 runs is $96.765$.
Also we compute the average of the Confusion Matrix over 100 runs.

$\left(\begin{array}[]{cc}30.27&19.73\\\ 11.62&938.38\\\ \end{array}\right).$

Figure 2. The classification on Imbalanced Two-Moons.

######  Example 4.3.

In this example, we implement Algorithm 1 (MGRF) and analyze a scenario where
the data points are distributed across two circles. The configuration is set
as follows: the major class contains 1,000 samples,while the minor class
contains 100; noise level is set at 0.1; the maximum number of iterations is
1,500; the tolerance is set at $=10^{-8}$, $\alpha=.985$ and there are 3
labeled samples per class. Refer to Figure 3 for more details. The average
confusion matrix, calculated over 100 trials, is presented below. Figure 3
also displays the initial labeled points and the predictions made by the MGRF
algorithm.

$\left(\begin{array}[]{cc}919.72&80.28\\\ 25.47&74.53\\\ \end{array}\right).$

Figure 3. The classification on (two-circles ).

######  Example 4.4.

Table 2 describes the accuracy $(\%)$ of IPL algorithm compared to Poisson and
Segregation algorithms, on Cifar-10 data set with 3 classes and 4500 nodes
(1500 nodes per class) with several labels per class. The results are averages
over 100 trials, which demonstrates superior performance of our scheme IPL for
different label rates.

Table 2. Cifar-10 data set Average accuracy over 30 trials for 3 classes on
Cifar-10 dataset
---
number of labels | 2 | 3 | 4 | 5 | 10 | 20 | 40
Poisson | 37.5 | 39.5 | 40.3 | 41.4 | 44.10 | 48.7 | 50.3
Segregation | 34.9 | 35.6 | 36.2 | 38.6 | 42.4 | 45.3 | 47.7
IPL | 45.226 | 46.546 | 47.382 | 48.341 | 53.152 | 54.966 | 57.031

Dimensionality reduction techniques like PCA or feature transformations
provide better feature separation for the proposed algorithms, however, we did
not involve PCA in algorithm. We just used PCA, in Figure 4 to plot the
classification for 3 classes on the Cifar-10 data set.

Figure 4. The classification of Cifar-10.

######  Example 4.5.

We tested our proposed method on 10 KEEL imbalanced benchmark data sets which
contains different levels of imbalance in data and different sample sizes [1]
as summarized in Table 3. In Table. 3 the column denoted by $n$ stands for the
total number of samples, $p$ is the number of features, and IR indicates the
ratio between majority and Minority class samples in each data set.

Table 3. Information of selected imbalanced benchmark datasets Dataset | IR | $p$ | $n$ | %Minority
---|---|---|---|---
abalone9-18 | 16.40 | 8 | 731 | 5.74
appendicitis | 4.04 | 7 | 106 | 19.81
ecoli2 | 5.46 | 7 | 336 | 15.47
hypo | 15.49 | 6 | 2012 | 6.06
new-thyroid1 | 5.14 | 5 | 215 | 16.27
shuttle-c0-vs-c4 | 13.86 | 9 | 1829 | 6.72
sick | 11.61 | 6 | 2751 | 7.92
vowel0 | 9.98 | 13 | 988 | 9.1
yeast3 | 8.10 | 8 | 1484 | 10.98

To evaluate our Algorithms, we use the metrics ${\rm F}_{1}$-Score, Recall,
Accuracy and Precision for each classes. To ensure consistency for all
experiments, for each benchmark, first, the data set is shuffled.
Subsequently, 1 percent of the samples are randomly chosen in accordance with
the dataset’s IR as the labeled samples. This process is independently
repeated 100 times, then the averages of the previously mentioned metrics are
computed. I should mention that for data set ”shuttle-c0-vs-c4” we choose 4
labels per class which is $0.43\%$ of data. We did not compare with existing
schemas since most schemes with only $0.43\%$ of labeled samples per class
give poor results.

Table 4 displays the comparison between performance of our proposed method,
with Poisson Learning. For each data set, the second column indicates the
number of samples randomly selected from each class as labeled data.

Table 4. Information of selected imbalanced benchmark datasets Names | IGRF
---|---
Dataset | Accuracy | F1 min | F1 maj | Recall min | Recall maj | Precision min | Precision maj
yeast3 | $.864$ | $.524$ | $.919$ | $.693$ | $.882$ | $.421$ | $.959$
appendicitis | $.758$ | $.529$ | $.837$ | $.705$ | $.771$ | $.423$ | $.917$
abalone9-18 | $.9519$ | $.7356$ | $.9749$ | $.7380$ | $.9941$ | $.7333$ | $.9565$
ecoli2 | $.875$ | $.694$ | $.922$ | $.912$ | $.869$ | $.561$ | $.982$
hypo | $.9024$ | $.7257$ | $.5657$ | $.82352$ | $.9169$ | $.6488$ | $.9407$
new-tyroid1 | $.877$ | $.425$ | $.931$ | $.278$ | $.994$ | $.898$ | $.876$
shuttle-c0-vs-c4 | .9737 | .757 | .986 | .6097 | .986 | 1 | .970
sick | $.8146$ | $.4044$ | $.8196$ | $.7833$ | $.8196$ | $.2725$ | $.8197$
vowel0 | $.725$ | $.172$ | $.835$ | $.317$ | $.766$ | $.118$ | $.919$

##  5\. Conclusion

In this work, we present several schemes to enhance the efficacy of
classification algorithms when dealing with large datasets with limited
labeled data. The method leverage the efficiency of Poisson and Gaussian
Random Fields methods while maintains simplicity and convergence.

The Modified Gaussian Random Fields (MGRF) and IGRF algorithms optimize label
propagation across a graph by adjusting the initial label vector and
iteratively refining the classification process. This approach demonstrates a
substantial improvement.

The Improved Poisson Learning (IPL) algorithm, addresses the limitations of
Poisson Learning by incorporating a subtraction of the matrix from the
transition matrix, enhancing the stability and convergence of the label
propagation process. This adjustment allows the algorithm to perform more
effectively the case that data is imbalanced, ensuring that minority classes
are represented and classified.

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