text:
\section{Introduction}

The universal approximation theorem in machine learning states that any continuous function can be approximated by a deep neural network, but in a practical situation, there are many cases where learning fails.
It is believed that this is because the number of parameters used is too large to learn an appropriate model from a hypotheses set.
Therefore, researchers developed task-specific models, such as convolutional networks for image input, and obtained better results.
On the other hand, \citet{zaheer2017deep} developed a model using permutation invariance and permutation equivariance, and obtained good experimental results as well as theoretical development.
Such a model was generalized by  \citet{maron2019universality}  for subgroups of permutation groups.
In this paper, we develope an equivariant and invariant deep neural network model for the action of permutation groups on higher-order tensor spaces.
An important application of this formulation is the task of using graphs and hypergraphs as input.

The central part of our idea is to use the equivariant Reynolds operator.
The equivariant Reynolds operator has the property of transforming a function into an equivariant function, and by adapting this operator to a deep neural nets, we obtain a class of equivariant neural networks.
Such attempts have been made in the past, for example by \citet{yarotsky2021universal,kicki2021new,van2020mdp,mouli2020neural}, but the use of Reynolds operator causes computational problems when the order of the groups is large in the part that computes the equivariant Reynolds operator.

Therefore, we introduce the concept of Reynolds design, which is an analog of spherical design.
Here, a $t$-spherical design is a finite subset $H\subset \mathbb{S}^{d-1}$ such that any polynomial $f$ with degree at most $t$ satisfies
\begin{align}
    \frac{1}{|\mathbb{S}^{d-1}|}\int_{\mathbb{S}^{d-1}} f(x) dx
    = \frac{1}{|H|} \sum_{x \in H} f(x).
    \label{eq:spherical-design}
\end{align}
Using the spherical design,
the computation over the whole of $\mathbb{S}^{d-1}$ can be reduced to that over the small subset $H$.
Similarly, a Reynolds design is a subset $H$ of a group $G$ that realizes
\begin{align*}
    \frac{1}{|G|}\sum_{g\in G} f(g\cdot x)
    = \frac{1}{|H|}\sum_{g\in H} f(g\cdot x)
\end{align*}
for a target function $f$, where $g\cdot x$ is the action of $g$ to $x$.
We call the operator defined on the right-hand side the reductive Reynolds operator, and use this operator instead of the Reynolds operator to construct equivariant and invariant networks.
The advantage of this construction is that it preserves the universal approximation property of the construction using Reynolds operators.
Also, in many cases, the Reynolds design can be much smaller than the original $G$.
To formulate this fact, we prove a representation theorem for equivariant functions using the relation between Young diagrams and the representation of symmetric groups on higher-order tensor spaces (Theorem \ref{represention}).
For example, in the case of graphs with $n$ nodes, the order of the group is $n!$ but we have shown that there exist Reynolds designs of order $n(n-1)$.
Thus, by using Reynolds designs, we can avoid the problem of computational complexity.

As for the invariant network, as in the case of CNN, the input is first transformed by a equivariant network, and after taking the sum of the orbits, it is transformed by a deep neural network.
Note that we do not know how to take the Reynolds design, because the above representation theorem is not proved with invariant functions.
To solve this problem we introduce an invariant called Reynolds dimension. Briefly, the Reynolds dimension is the number of variables required for the input of the functions before they are converted by the Reynolds operator, when the model had universality in the above construction. It is not clear what the Reynolds dimension is when taking a higher order tensor space as input, but our expectation is that it is independent of the dimension $n$. We prove that we can construct a Reynolds design when the Reynolds dimension is fixed. We also find that all input variables are not necessary when the Reynolds dimension is low, therefore we construct networks with reduced inputs (reduced ReyNets).

Using these models, we conduct experiments on synthetic data.
In addition, as an implementation, we use the above representation theorem to develop the Corner MSE, which restricts the loss function for an equivariant function to two components.
As a result, our method significantly outperform the baseline of FNN and Maron's model, confirming its consistency with the theoretical part.

\section{Previous Work}

\textbf{Equivariance and Invariance}.
Various machine learning tasks aims to approximate a certain target map such as the labeling function in classification and regression.
When symmetries exist behind data,
the target map often have invariance or equivariance to the symmetries.
In such cases, invariant or equivariant networks are effective and efficient to approximate the target map because the model complexity can be significantly reduced than neural networks without specific structure for the symmetries.
Convolutional neural networks (CNNs) are well-known as a seminal equivariant model to translation symmetry (\citet{lecun1989backpropagation}; \citet{krizhevsky2012imagenet}).
Inspired by the success of CNNs,
various equivariant models have been proposed.
Besides continuous symmetries such as translation,
symmetries of finite groups often appear in many machine learning tasks.
In the case where sets or point clouds are inputs,
the target functions are typically invariant to the order of data points.
Then, this function has invariance to the permutation group on data points (\citet{qi2017pointnet,zaheer2017deep}).
In the case where graphs or hyper-graphs are inputs,
the symmetry is represented by permutation on a tensor product space.
\citet{gori2005new,scarselli2008graph} introduced graph neural networks.
Various researchers generalized convolution to the setting of graphs motivated by CNNs (\citet{bruna2013spectral,henaff2015deep,kipf2016semi,defferrard2016convolutional,levie2018cayleynets}).
Recently, \citet{kondor2018covariant,maron2019provably,maron2019universality, chen2019equivalence} investigate more efficient graph neural networks.
\citet{ravanbakhsh2017equivariance} treated equivariant and invariant models to subgroups of the symmetric group.
\citet{hartford2018deep} consider interaction between sets.
\citet{graham2019deep} consider relational databases as generalization of graphs and provide equivariant models to handle relational databases.
\cite{maron2020learning} consider sets of symmetric elements as input data.

\textbf{Universality}.
The expressive power of learning models is mathematically validated by universal approximation theorems.
Many universal approximation theorems have been proved for different conditions.
Equivariant and invariant models with universal approximation property are provided for pointclouds networks and sets network (\citet{qi2017pointnet,zaheer2017deep}),
tabular and multi-set networks (\citet{hartford2018deep}), graph and hyper-graph networks (\citet{kondor2018covariant,maron2018invariant,nguyen2020graph}),
and networks invariant to finite translations with rotations and/or reflections \cite{maron2019universality}.
On the other hand, \cite{oono2019graph} proved that graph neural networks of aggregation type are over smoothing when many layers are stacked.
There are also learning models with universality in other settings (\citet{yarotsky2021universal,keriven2019universal,maehara2019simple,segol2019universal}).

\section{Reynolds Operator and Reynolds Designs}

In this section, we organize the mathematical framework for dealing with symmetric networks. The first objects to be considered are the following equivariant and invariant functions.

\begin{df}[Invariant / Equivariant Function] For a group $G$ acting on $\mathbb{R}^{N}$ and $\mathbb{R}^{M}$, a function
$f: \mathbb{R}^{N} \rightarrow \mathbb{R}^{M}$ is invariant if $f(g \cdot x)=f(x)$ holds for any $g \in G$ and any $x \in \mathbb{R}^{N}$, and  equivariant if $f(g \cdot x)=g \cdot f(x)$ holds for any $g \in G$ and any $x \in \mathbb{R}^{N}$.
\end{df}

The next operator, called the Reynolds operator, plays a central role in this paper.

\begin{df}[Reynolds Operator (cf. \cite{mumford1994geometric}, Definition 1.5)]
For a group $G$,
the followings are called the equivariant and invariant Reynolds operator respectively:
\begin{align}
\tau_{G}(f(-))=\frac{1}{|G|} \sum_{g\in G} g^{-1} \cdot f(g \cdot-),
\quad\gamma_{G}(f(-))=\frac{1}{|G|} \sum_{g\in G} f(g \cdot-).
\label{eq:equiv}
\end{align}
\end{df}

More generally,
for a subset $H$ in $G$\footnote{The subset $H$ in $G$ may not be a subgroup of $G$.},
we define $\tau_H$ and $\gamma_H$ by replacing $G$ by $H$ in (\ref{eq:equiv}).

The equivariant Reynolds operator converts an arbitrary map to a equivariant map. Our idea is to use this operator to convert a fully connected deep neural network into an equivariant function.
However, as we see in a moment, the computational complexity of Reynolds operators increases when the order of $G$ is large, for example, for symmetric groups of order $n!$.

Design theory in mathematics provides a suitable subset that represents some property about the whole.
Here, we explain spherical design.
Let $\mathcal{P}_t$ be the set of all polynomials of at most degree $t\in\mathbb{N}$ on $\mathbb{R}^d$.
Then, there exists a finite subset $H$ of the sphere $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ such that an arbitrary polynomial $p\in \mathcal{P}_t$ satisfies (\ref{eq:spherical-design}).
Then, such a subset $H$ is called the $t$-spherical design.

Therefore, we define the following notion of Reynolds design to reduce the computational complexity.
The concept of Reynolds design is first proposed in this paper.

\begin{df}[Reynolds Design]
The Reynolds design $H$ of a function $f$ is a subset $H$ of $G$ that satisfies the following equation.

\begin{align*}
\tau_G(f(-))=\frac{1}{|G|} \sum_{g \in G} g^{-1} \cdot f(g \cdot-)=\frac{1}{|H|} \sum_{g \in H} g^{-1} \cdot f(g \cdot-).
\end{align*}
In this case, $\tau_H$ is called the reductive Reynolds operator.Furthermore, the Reynolds design $H$ of a set of functions $\mathcal{F}$ is defined by the property that the above equation for $H$ holds for all $f \in \mathcal{F}$.

\end{df}

As an example, consider a power of one variable as $f$. Since $\tau_{S_{n}}\left(x_{1}^{i}\right)=x_{1}^{i}+\cdots+x_{n}^{i}$, the number of terms in the summation is only $n$, which can be written as an orbit by the cyclic group $C_n$ of order $n$ as
$\tau_{S_{n}}\left(x_{1}^{i}\right)=x_{1}^{i}+\cdots+x_{n}^{i}=\tau_{C_{n}}\left(x_{1}^{i}\right)$.
Hence $C_n$ is the Reynolds design of $x_1^i$.

\section{Representation Theorem for Equivariant Maps}

In this section, we consider the case where the symmetric group $S_n$ acts on the higher order tensor space $\mathbb{R}^{n^{\ell} \times a}$.
The action of $S_n$ on tensors $\mathbf{X} \in \mathbb{R}^{n^{l} \times a}$ (the last index, denoted $\alpha$ represents feature depth) is defined by  $(g \cdot \mathbf{X})_{i_{1} \ldots i_{l}, \alpha}=\mathbf{X}_{g^{-1}\left(i_{1}\right) \ldots g^{-1}\left(i_{l}\right), \alpha}$.
Note that this is an important class for applications because it deals with the case where the input is a set, graph, etc.
\begin{figure*}[!t]
\centering
    \includegraphics[bb=0 0 1150 450,
        scale=0.25]{figs/table-all2.png}
\caption{
Young diagrams and basis tableaux for $m=3$, and the corresponding basis tableau vectors in $\mathbb{R}^{n^3}= \mathbb{R}^{n}\otimes\mathbb{R}^{n}\otimes\mathbb{R}^{n}$.
}
\label{fig:equivariance}
\end{figure*}

A Young diagram is a way to represent the division of a natural number $m$.
Here, division means to express a natural number $m$ as a sum of several non-negative integers.

\begin{df}[Young Diagram]
Let $D\in[m]$ be fixed. A vector $\mathbf{k}=[k_1,\ldots,k_D]$ of natural numbers $k_1,\ldots,k_D$ is called a Young diagram if it satisfies $m = k_1+\ldots+k_D$ and $k_1\ge \ldots\ge k_D$.
\end{df}

This partition can be represented by a total of $m$ boxes, consisting of $D$ rows with $k_i$ boxes in the $d$-th row.
Here, each row is left-justified.
We formally provide the definition of basis tableaux in order to use the notation in later discussion.

\begin{df}[Basis Tableau]
Let $n\ge m$  and $\mathbf{k}=[k_1,\ldots,k_D]$ a Young diagram.
A vector $\mathbf{T}= \left[\mathbf{t}_1, \ldots, \mathbf{t}_D \right]$ of vectors $\mathbf{t}_d= [t_{d,1}, \ldots, t_{d,k_d}]\in [n]^{k_d}$ is called basis tableau of depth $D$ if it satisfies the following conditions:
\begin{enumerate}
    \item (Different components) $t_{d,w} \ne t_{d',w'}$ for $(d,w)\ne (d',w')$.
    \item (Row monotonicity) $t_{d,1}<t_{d,2}<\ldots< t_{d,k_d}$ for each $d\in[D]$.
    \item (Partial column monotonicity) If $k_d= k_{d+1}$, then $t_{d,1}<t_{d+1,1}$ for $d=1,\ldots, D-1$.

\end{enumerate}
\end{df}

We denote the set of basis tableaux for $m$ of depth $D$ by $\mathcal{T}_{m,D}$,
and
the set $\bigcup_{1\le D \le m} \mathcal{T}_{m,D}$ of basis tableaux with at most depth $m$ by $\mathcal{T}_m$.

Here, we introduce some notation about basis to state Theorem \ref{represention}.
Let $\mathbf{e}_1, \ldots, \mathbf{e}_n$ the standard basis of $\mathbb{R}^n$.
For $\mathbf{u}=[u_1,\ldots,u_m] \in [n]^m$,
we set
\begin{align}
    \mathbf{e}_{\mathbf{u}}:=\mathbf{e}_{u_1,\ldots,u_m}=\mathbf{e}_{u_1}\otimes \cdots\otimes\mathbf{e}_{u_m} \in \underbrace{\mathbb{R}^n\otimes \cdots \otimes \mathbb{R}^n}_{m}=\mathbb{R}^{n^m}.
    \label{eq:basis}
\end{align}
For any basis tableau $\textbf{T}\in \mathcal{T}_{m,D}$,
natural numbers $\mathbf{u}=[u_1,\ldots, u_m]\in [n]^m$ are given as
$
    u_{\ell}:= d\in [n] \quad \textit{if} \quad \ell \in \{\mathbf{t}_{d}\}.
$\footnote{
For a vector $\mathbf{t}=[t_{1},\ldots,t_{k}]$,
we set $\{\mathbf{t}\}:= \{t_{1},\ldots,t_{k}\}$.
For example, when $\mathbf{t}=[1,1]$, $\{\mathbf{t}\}=\{1\}$.
}
Then, we define the map $\phi: \mathcal{T}_m \to [n]^m$ by $\phi(\textbf{T}):= (u_1,..,u_m) \in [n]^m$.

\begin{df}[Basis Tableau Vector]
Let $n\ge m$.
For $D\in[m]$
and
$\mathbf{T}= \left[\mathbf{t}_1, \ldots, \mathbf{t}_D \right]\in \mathcal{T}_{m,D}$,
the basis tableau vector is defined by $\mathbf{e}_{\mathbf{T}}:= \mathbf{e}_{\phi(\mathbf{T})}\in \mathbb{R}^{n^m}$, where the right hand side is defined by (\ref{eq:basis}).
\end{df}
Figure \ref{fig:equivariance} shows an example of Young diagrams, basis tableaux, and basis tableaux vectors when $m = 3$.
For $\mathbf{X} \in \mathbb{R}^{n^m}$,
we define the linear map by
\begin{align}
    \hat{\mathbf{X}}_{b}&:\mathbb{R}^b\ni (a_1, \ldots, a_b) \mapsto  (a_1\mathbf{X}, \ldots, a_b\mathbf{X}) \in \mathbb{R}^{n^m\times b}.
    \label{eq:linear}
\end{align}
In the following,
we mainly treat the case where
$\mathbf{X}=\mathbf{e}_{\mathbf{T}}$ (i.e., $\hat{\mathbf{X}}_{b}=\hat{\mathbf{e}}_{\mathbf{T},b}$).
Then, we have the following theorem.
\begin{thm}[Representation Theorem]\label{represention}
Let $n\ge m$ and $G=S_n$.
For any equivariant continuous map  $F: \mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}^{n^m \times b}$, there exist continuous maps $F_{\mathbf{T}}: \mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}^b$ indexed by basis tableaux $\mathbf{T} \in \mathcal{T}_m$ such that
\begin{align*}
F= \sum_{D=1}^m \sum_{\mathbf{T} \in \mathcal{T}_{m,D}}
 \tau_{H_D}(F_{\textbf{T}}\circ \hat{\mathbf{e}}_{\mathbf{T}, b}),
  \end{align*}where $C_{n-i}$ denote the cyclic group of order $n-i$ on the set $\{i+1,..,n\}$ and
  \begin{align*}
      H_{D} &:= C_n \circ \cdots \circ C_{n-D+1}
      = \{\sigma_D \cdot \sigma_{D-1} \cdots \sigma_{1}  \mid \sigma_i \in C_{n-D+i}~ (i=n,n-1,\ldots,n-D+1)\}.
  \end{align*}
Furthermore,  $ H_{D} $ is a Reynolds design of $F_{\textbf{T}
}\circ \hat{\mathbf{e}}_{\mathbf{T}, b}$ for any $\textbf{T}\in \mathcal{T}_{m,D}$.
\end{thm}

\subsection{Proof of Theorem \ref{represention}}

We introduce the following notion to obtain tableau-based representation of elements in $[n]^m$.

\begin{df}[Extended Tableau]
Let $n\ge D$.
Let $[n]^D_{\#}$ be the set of $D$ different natural numbers at most $n$ defined by
$
    [n]^D_{\#}:= \{[j_1, \ldots, j_D] \in [n]^D \mid j_a \ne j_b ~\text{for}~ a\ne b \in [D]\}.
$
Then,
we call elements in $\tilde{\mathcal{T}}_{m,D}:=[n]^D_{\#}\times \mathcal{T}_{m,D}$ extended tableaux with depth $D$.
\end{df}

We denote the set $\bigcup_{1\le D \le m} \tilde{\mathcal{T}}_{m,D}$ of extended tableaux with at most depth $m$  by $\tilde{\mathcal{T}}_m$.
The action $g\cdot [j_1, \ldots, j_D]:=[g\cdot j_1, \ldots, g\cdot j_D]$ of $G$ on $[n]^D_{\#}$ is well-defined.
Then, we define the action of $G$ on $\tilde{\mathcal{T}}_{m,D}$ by $g\cdot (\mathbf{j}, \mathbf{T}) := (g\cdot \mathbf{j}, \mathbf{T})$.
In the following,
we identify an extended tableau $([1,2,\ldots, D],\mathbf{T})$ with the basis tableau $\mathbf{T}$.

Next, we introduce some notation to represent elements in $[n]^m$ by extended tableaux.
We first define the partial order $\succ$ of vectors of natural numbers that can have different dimensions.
For $\mathbf{t} =[t_1,\ldots,t_k]\in \mathbb{N}^k$ and $\mathbf{t}' =[t'_1,\ldots,t'_k]\in \mathbb{N}^{k'}$,
we denote as
$\mathbf{t} \succ \mathbf{t}'$ if either (i) $k>k'$ or (ii) $k=k'$ and $t_1>t'_1$.
We note that $\mathbf{t}_1\succ \ldots \succ \mathbf{t}_D$ holds for a basis tableau $\mathbf{T}=[\mathbf{t}_1,\ldots,\mathbf{t}_D]$ by definition.
Let $\mathbf{u}=[u_1,\ldots,u_k]\in [n]^{m}$.
We set $D(\mathbf{u}):=|\{\mathbf{u}\}|$.
We define the multiplicity map $\mathrm{mult}_{\mathbf{u}}:\{\mathbf{u}\} \to [m]$ by
$
    \mathrm{mult}_{\mathbf{u}}(u)
    := |\{\ell \in [m] \mid u_\ell = u\}|.
$
For $u\in \{\mathbf{u}\}$,
we define $\mathbf{t}_u:= [t_1,\ldots,t_{k_u}]$,
where $k_u:=\mathrm{mult}_{\mathbf{u}}(u)$,
$u_{t_1}=\cdots={\bm u}_{t_{k_u}}=\mathbf{u}$,
and $t_{1}<\cdots<t_{k_u}$.
Here, we note that either $\mathbf{t}_{u}\succ \mathbf{t}_{u'}$ or $\mathbf{t}_{u}\prec \mathbf{t}_{u'}$ holds for $u\ne u' \in \{\mathbf{u}\}$ by definition.
Thus,
there exist different natural numbers $j_1,\ldots,j_D \subset [n]$ such that $\{j_1,\ldots,j_D\} = \{\mathbf{u}\}$ and $\mathbf{t}_{j_1} \succ \ldots \succ \mathbf{t}_{j_D}$.

\begin{df}[Tableau Representation]
Let $n\ge m$ .
We define the map $\psi: [n]^m \ni \mathbf{u} \mapsto (\mathbf{j},\mathbf{T}) \in \tilde{\mathcal{T}}_m$ called tableau representation by
\begin{align*}
    \psi(\mathbf{u})
    := ([j_1, \ldots, j_D], [\mathbf{t}_{j_1}, \ldots, \mathbf{t}_{j_D}])
    \in \tilde{\mathcal{T}}_{m,D}.
\end{align*}
\end{df}

We define the map $\phi: \tilde{\mathcal{T}}_m \to [n]^m$ as follows:
For any extended tableau $(\mathbf{j}, \textbf{T})\in \tilde{\mathcal{T}}_{m,D}$,
natural numbers $\mathbf{u}=[u_1,\ldots, u_m]\in [n]^m$ are given as
$
    u_{\ell}:= j_d\in [n] \quad \textit{if} \quad \ell \in \{\mathbf{t}_{d}\},
$
and $\phi(\mathbf{j}, \textbf{T}):= (u_1,..,u_m) \in [n]^m$.

\begin{lem}\label{lem:bijective1}
    When $n\ge m$,
    the tableau representation $\psi: [n]^m \to \tilde{\mathcal{T}}_m$ is bijective and $\psi(g\cdot \mathbf{u}) = g\cdot \psi(\mathbf{u})$ for $g \in H_D$.
\end{lem}
\begin{proof}
First, from the construction, $\psi$ is injective and $\phi\circ \psi = \mathrm{id}_{[n]^m}$.
Since $[n]^m$ and $\mathcal{T}$ are finite sets, if we show that $\phi$ is injective, then $\psi$ is bijective.
For the extended tableau $\mathbf{S},\mathbf{T}$, assume that $\phi(\mathbf{S})=\phi(\mathbf{T})$.
Note that the set of row vectors $\{\mathbf{s}_1,...,\mathbf{s}_d\}$ of $\mathbf{S}$ is uniquely determined from $\phi(\mathbf{S})$.
Then, since the extended tableaux satisfy the order $\succ$ in which this goes between the row vectors, the extended tableau $\textbf{S}$ having row vectors $\{\mathbf{s}_1,...,\mathbf{s}_d\}$ is unique. Hence, we have $\textbf{S}=\textbf{T}$.
\end{proof}

\begin{df}[Extended Tableau Vector]
Let $n\ge m$.
The extended tableau vector $\mathbf{e}_{\mathbf{j},\mathbf{T}}$ is defined by $\mathbf{e}_{\mathbf{j},\mathbf{T}}:=\mathbf{e}_{\phi(\mathbf{j},\mathbf{T})}$.
\end{df}
For an extended tableau $(\mathbf{j},\mathbf{T})\in \tilde{\mathcal{T}}_m$,
the linear map $\hat{\mathbf{e}}_{\mathbf{j},\mathbf{T},b}: \mathbb{R}^b \rightarrow \mathbb{R}^{n^m\times b}$
is defined by $\mathbf{X}=\mathbf{e}_{\mathbf{j},\mathbf{T}}$ in
(\ref{eq:linear}).

\begin{lem}[Normalization]\label{lem:bijective2}
    Let $n\ge D$.
    For $\mathbf{j}\in[n]^D_{\#}$, there uniquely exists $g\in H_D$ such that $\mathbf{j}= g^{-1}\cdot [1,2,\ldots, D]\in[n]^D_{\#}$.
    Hence, this correspondence $[n]^D_{\#} \ni \mathbf{j} \to g \in H_D$ is bijective.
\end{lem}

\begin{proof}
There uniquely exists $\sigma_{1} \in C_{n}$ such that $\sigma_{1}\left(j_{1}\right)=1$.
Inductively, there uniquely exists $\sigma_{d} \in C_{n-d+1}$ such that $\sigma_{d}\left(\sigma_{d-1}\cdot \sigma_{d-2}  \cdots  \sigma_1 ( j_{d})\right)=d$ for $d=2,\ldots,D$.
Then $g := \sigma_{D} \cdots \sigma_{1}\in H_D$ satisfies $g\cdot \mathbf{j}=  [1,2,\ldots, D]$ by definition.
\end{proof}

From Lemma \ref{lem:bijective2},
an extended tableau $(\mathbf{j},\mathbf{T})\in \tilde{\mathcal{T}}_{m,D}$ is uniquely represented by
$
    (\mathbf{j},\mathbf{T}) = g^{-1}\cdot([1,\ldots,D], \mathbf{T})
    =g^{-1}\cdot\mathbf{T}
$
as in Figure \ref{fig:reduction},
where $g\in H_D$ and we identified $\mathbf{T}\in \mathcal{T}_{m,D}$ with $([1,\ldots,D], \mathbf{T})\in \tilde{\mathcal{T}}_{m,D}$ in the last equation.
Thus, we have $\tilde{\mathcal{T}}_{m,D}= \bigcup_{g\in H_D} g^{-1} \mathcal{T}_{m,D}$,
and
$\tilde{\mathcal{T}}_{m}= \bigcup_{1\le D \le m} \bigcup_{g\in H_D} g^{-1} \mathcal{T}_{m,D}$.
From Lemma \ref{lem:bijective1}, for each $\mathbf{u} \in [n]^m$, there uniquely exists $g\in H_D$ and $\mathbf{T}\in\mathcal{T}_{m}$ such that $\psi (\mathbf{u}) = g^{-1}\cdot \mathbf{T}$ (or equivalently $\mathbf{u} = \psi^{-1}(g^{-1}\cdot \mathbf{T})$).
In the following, we omit the bijective $\psi$ for notational simplicity.

In the following,
we prove Theorem \ref{represention}.
We can write $F =  \sum_{\mathbf{u} \in\left[n\right]^{m}}  f_{\mathbf{u}} \cdot \hat{e}_{\mathbf{u}, b}$ by maps $f_{\mathbf{u}}: \mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}^b$.
Then since $F$ is equivariant, we have
$
\sum_{\mathbf{u} \in\left[n\right]^{m}} f_{\mathbf{u}}\cdot \hat{e}_{\mathbf{u}, b}(g\cdot x)
=F(g \cdot x)
=g\cdot F(x)
=\sum_{\mathbf{u} \in\left[n\right]^{m}} f_{\mathbf{u}} \cdot \hat{e}_{g\cdot \mathbf{u}, b}(x).
$
This implies that
\begin{align}
    f_{\mathbf{u}} \cdot \hat{e}_{\mathbf{u}, b}(g\cdot x)
=f_{g^{-1}\cdot \mathbf{u}}\cdot \hat{e}_{\mathbf{u}, b}(x). \label{eq:component}
\end{align}

\begin{figure*}[!t]
\centering
    \includegraphics[bb=0 0 1590 300,
    scale=0.24]{figs/permutation2.png}
\caption{
Reduction from an extended tableau to a basis tableau by permutations.
An extended tableau $(\mathbf{j},\mathbf{T})$ is converted to a basis tableau by multiplying $g=\sigma_D \cdot \sigma_{D-1}\cdots \sigma_{1}$.
}
\label{fig:reduction}
\end{figure*}

Then,
we obtain the following equations:
\begin{align}
F(x)
=\sum_{1\leq D\leq m}\sum_{\textbf{T}  \in \mathcal{T}_{m, D}} \tau_{H_{D}}\left( F_{\textbf{T} } \cdot \hat{e}_{\textbf{T} , b}\right)(x).
\label{eq:representation}
\end{align}
We provide the detailed calculation of (\ref{eq:representation}) in the appendix.

\section{Equivariant ReyNets and Universality}

In this section, we define equivariant models using Theorem \ref{represention}.

\begin{df}[Equivariant Reynolds Nets]
We assume that $n\ge m$.
For any basis tableaux $\mathbf{T} \in \mathcal{T}_m$, Let $\mathcal{N}_{\mathbf{T}}: \mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}^b$ be a Multi layer perceptron.
The map $\mathcal{E}: \mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}^{n^m \times b}$
\begin{align*}
\mathcal{E}= \sum_{D=1}^m \sum_{\mathbf{T} \in \mathcal{T}_{m,D}} \tau_{H_D}(\mathcal{N}_{\textbf{T}
}\circ \hat{\mathbf{e}}_{\mathbf{T}, b}),
  \end{align*} is called reduced Equivariant Reynolds Nets (equivariant ReyNets).
\end{df}

Theorem \ref{represention} guarantees the universal approximation property of this model.
\begin{thm}[Universality]\label{thm:universality-equiv}
    We assume that $n\ge m$ and $G=S_n$.
  Let $F:  \mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}^{n^m \times b}$ be a continuous equivariant function. For any compact set $K \subset \mathbb{R}^{n^{\ell}\times a}$ , there exists a equivariant Reynolds network that approximates $F$ to an arbitrary precision on $K$.
Namely, equivariant Reynolds nets are a universal approximator for equivariant functions.
\end{thm}

The most important application, the case of the graph, corresponds to the $m = 2$ case. In this case, the Young diagrams are $\textbf{k}=(2)$ and $(1,1)$, and the basis tableaux is also uniquely numbered in the above two. In other words, the number of MLP $\mathcal{N}_T$ required to construct the model is two. For the Reynolds design, $H_1$ is of order $n$ and $H_2$ is of order $n(n-1)$, so the total computational complexity is $O(n^2)$. The number of summations that were originally required $n!$ times has been reduced to $n^2$ times, resulting in a significant reduction in the amount of calculation. We can see some experiments on these in the experiments section.

\section{Invariant ReyNets and Universality}

In this section, we define the invariant models using the equivariant models defined in the previous section. First, we will define invariant Reynolds nets. Then, we review the generators of invariant polynomials and define Reynolds dimension using them. Finally, we explain the universal approximation of invariant networks using Reynolds dimension. We will deal with the case where a general finite group $G$ acts on an $n$-dimensional vector space or a set of $n$ variables. Note that such formulations include the case of higher order tensor space as $G=S_n$.
\begin{df}[Invariant Reynolds Nets]
A invariant Reynolds network (invariant ReyNet) is a function $\mathcal{I}$ :
$\mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}$ defined as
$$
\mathcal{I} = \mathcal{M} \circ \Sigma \circ \mathcal{E},
$$
where $\mathcal{E}: \mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}^{n^m \times b}$ is a equivariant Reynolds network,
$\Sigma$ is the orbit sum\footnote{
The orbit sum $\Sigma: \mathbb{R}^{[n]^m\times b} \to \mathbb{R}^{[n]^m/G\times b}$ is defined by $\Sigma(\mathbf{X})_{G\cdot \mathbf{u},\beta}:= \sum_{g\in G} x_{g\cdot \mathbf{u},\beta}$ for $\mathbf{X}=[x_{\mathbf{u},\beta}] \in \mathbb{R}^{[n]^m\times b}$, $\mathbf{u}\in [n]^m$, $\beta \in [b]$ and $G\cdot \mathbf{u} \in [n]^m/G$.},
and $\mathcal{M}$ is a Multi-Layer Perceptron (MLP).
\end{df}

We define the notion of generator of invariant polynomials, which is necessary to define Reynolds dimension.
\begin{df}[Generator of invariant polynomials]
Let $G$ be a finite group.
A set of invariant polynomials $r_1(x_1,..,x_n),..,r_s(x_1,..,x_n)$ are called {\it a generator of invariant polynomials} if for any invariant polynomial $f(x_1,..,x_n)$, there is a polynomial $p(y_1,..,y_s)$ such that $f(x_1,..,x_n)=p( r_1(x_1,..,x_n),..,r_s(x_1,..,x_n))$ holds.
\end{df}

The existence of such a generator is non-trivial for general group actions, but has been shown by Hilbert for finite groups or more generally linearly reductive groups \cite{hilbert1890}.
Under the preparation so far, we define the following Reynolds dimension.
\begin{df}[Reynolds dimension]
Let $G$ be a finite group.
The smallest natural number $d$ satisfying the following property is the Reynolds dimension of the group $G$.
There exist polynomial $h_1,..,h_s$ of $d$-variable and an index subset $\{j_1,\ldots,j_d\}\subset [n]$  such that$\gamma_{G}(h_{1}(x_{j_1}, \ldots, x_{j_d})), \ldots, \gamma_{G}(h_{s}(x_{j_1}, \ldots, x_{j_d}))$ is a generator of the invariant polynomials.
\end{df}

Before discussing universality for invariant functions, let us review some notation. We define $\mathrm{Stab}_G([d])$ to be the set of elements of $G$ for which $x_{j_1}, \ldots, x_{j_d}$ are fixed.
In addition, $[G/G']$ a complete system of representatives of $G/G'$ for a subgroup $G'\subset G$ is a set of order $|G/G'|$ that satisfies $G = \bigcup_{a \in [G/G']} a G'$.

\begin{prp}\label{inv design}
In the same situation as Definition 18, $[G/\mathrm{Stab}_G([d])]$ is a Reynolds design of  $h_i$.
\end{prp}
\begin{thm}[Universality]\label{thm:universality-inv}
We assume that $G=S_n$.
Let $d$ be the Reynolds dimension of G.
Then, Reyonlds invariant nets constructed above for
$\mathcal{E}: \mathbb{R}^{n^l \times a} \rightarrow \mathbb{R}^{n^d \times b}$ is a universal approximator for invariant functions $f: \mathbb{R}^{n^{\ell} \times a} \rightarrow \mathbb{R}^b$.
More strongly, the input space of $\mathcal{E}$  can be replaced by a composite $ \mathcal{E}\circ \mathcal{Z}$ with the the zero padding map\footnote{
The zero padding map $\mathcal{Z}: \mathbb{R}^{d\times a} = \mathbb{R}^d \otimes \mathbb{R}^a \to \mathbb{R}^{n^{\ell}\times a} = \mathbb{R}^{n^{\ell}}\otimes \mathbb{R}^a$ is the linear map defined by
$
    \mathcal{Z}((x_1, \ldots, x_d) \otimes \mathbf{e}_{\alpha})
    := (x_1, \ldots, x_d, \underbrace{0,\ldots, 0}_{n^\ell -d}) \otimes \mathbf{e}_{\alpha}
$
for $\alpha=1,2,\ldots,a$.
} $\mathcal{Z}: \mathbb{R}^{d\times a} \to \mathbb{R}^{n^{\ell}\times a}$.
\end{thm}

\section{Reduction of input variables}
So far, we have taken the input of $\mathcal{N}$ as an $n^{\ell}$-dimensional space. However, when $n$ or $\ell$ are large, this can be an obstacle to the calculation. Therefore, we take $d$-dimensional subspace as an input by Theorem \ref{thm:universality-inv}. This is called the $d$-reduced equivariant (invariant) Reynolds network. For example, consider the case $\ell=2$ and assume that the set of invariant polynomials $\mathbb{R}[\textbf{x}]^{S_n}$ has Reynolds dimension $4$ by the valuables $x_{1,1},x_{1,2},x_{2,1}, x_{2,2}$, then  by Theorem \ref{thm:universality-inv}, we can restrict the input space of $f_\textbf{T}$ in our model to $x_{1,1},x_{1,2},x_{2,1}, x_{2,2}$ and still preserve universality for invariant functions.
The reduced Reynolds networks give us the great benefit of generalization over $n$.
In other words, by transferring $f_\textbf{T}$ learned for small $n$ to $f_\textbf{T}$ for large $n$, we can obtain a model for large $n$. See Figure 3 for more details.
When we consider a practical task, for example, the problem of classifying graphs, it is not a simple matter to determine how many Reynolds dimensions it has. However, this problem has been formulated mathematically as a problem in higher-order tensor spaces, and we expect that the Reynolds dimension will be independent of $n$ in this case.
\begin{conjecture}
Let $G$ be an $S_n$ acting on $\mathbb{R}^{n^{\ell}}$ as in Section 4. For arbitrary enough large $n$, the Reynolds dimension $d(n)$ of G is independent of $n$.
\end{conjecture}

\section{Experiments}

\begin{table}[]
\centering
\caption{Results of comparison to a baseline method}
\fontsize{8pt}{9pt}\selectfont
\begin{tabular}{@{}lllllllll@{}}
\toprule
Task                & \multicolumn{4}{l}{Symmetry}                                                  & \multicolumn{4}{l}{Diagonal}                                                  \\
\textit{\textbf{n}} & 3                 & 5                 & 10                & 20                & 3                 & 5                 & 10                & 20                \\ \midrule
FNN               & 1.730e-4          & 9.180e-4          & 1.454e-3          & 3.0583          & 1.295e-4          & 2.655e-4          & 1.148e-4          & 1.081e-1          \\
\citet{maron2018invariant}               & 6.600e-3          & 3.786e-3          & 9.294e-4          & 4.471e-3          & 2.065e-3          & 2.266e-3          & 4.098e-3          & 4.743e-4          \\
ReyNet (ours)     & 2.147e-4          & 3.960e-4          & 1.408e-3          & 3.151e-3          & 1.007e-4          & 2.472e-4          & 6.635e-4          & 1.112e-4          \\
4red-ReyNet (ours)         & \textbf{8.544e-5} & \textbf{4.889e-5} & \textbf{7.529e-5} & \textbf{6.554e-5} & \textbf{6.947e-5} & \textbf{1.932e-5} & \textbf{5.568e-5} & \textbf{3.566e-5} \\ \bottomrule
\toprule
Task                & \multicolumn{4}{l}{Power}                                                     & \multicolumn{4}{l}{Trace}                                                     \\
\textit{\textbf{n}} & 3                 & 5                 & 10                & 20                & 3                 & 5                 & 10                & 20                \\ \midrule
FNN               & 2.586          & 3.091e+1          &    5.756e+1       & 6.268e+2          & 2.821e-4          & 1.135e-3          & 9.529e-3          & 5.149e-2          \\
\citet{maron2018invariant}               & 4.036e-1          & 3.462e-1          & 7.062e-1          & 4.735e-1          & 1.241e-3          & 6.696e-3          & 3.663e-2          & 5.527e-2          \\
ReyNet (ours)     & 3.798e-1          & 1.257             & 3.620             & 3.065             & 1.884e-3          & 2.949e-3          & 4.275e-2          & 5.338e-2          \\
4red-ReyNet (ours)         & \textbf{1.204e-1} & \textbf{1.330e-1} & \textbf{1.217e-1} & \textbf{1.165e-1} & \textbf{3.491e-4} & \textbf{1.914e-3} & \textbf{6.758e-3} & \textbf{1.220e-2} \\ \bottomrule
\end{tabular}
\label{tab:exp1}
\end{table}

We evaluated our models compared to fully-connected neural nets (FNNs) and \citet{maron2018invariant} using synthetic datasets for equivariant and invariant tasks.
In the experiments, we used two models; Reynolds Networks (ReyNets) and $4$-reduced Reynolds Networks (4red-ReyNets).

\subsection{Datasets}
We created four types of synthetic datasets for the comparison.
Given an input matrics data $\mathbf{A} \in \mathbb{R}^{n \times n}$, each task is defined by:
\begin{itemize}
    \item \textbf{Symmetry}: projection onto the symmetric matrices $F(\mathbf{A})=\frac{1}{2}(\mathbf{A} + \mathbf{A}^\top)$,
    \item \textbf{Diagonal}: diagonal extraction $F(\mathbf{A})=diag(\mathbf{A})$,
    \item \textbf{Power}: computing each squared element $F(\mathbf{A})=[A_{i,j}^2]$,
    \item \textbf{Trace}: computing the trace $F(\mathbf{A})=tr(\mathbf{A})$,
\end{itemize}
where the task function $F$ is equivariant in \textit{symmetry}, \textit{diagonal}, and \textit{power}, and is invariant in \textit{trace}.
Following \citet{maron2018invariant}, we sampled i.i.d. random matrices $\mathbf{A}$ with uniform distribution in $[0, 10]$; then we transformed it by the task function $F$.
In our experiments, we provided $n \in \{ 3, 5, 10, 20\}$, and the size of training dataset and test dataset is 1000 respectively.

\subsection{Implementation Details}
For equivariant tasks, we applied equivariant ReyNets.
For invariant tasks, we applied invariant ReyNets following the architecture of the invariant model proposed by \citet{maron2018invariant};
an equivariant ReyNet is followed by max pooling\footnote{This operation outputs the max value of diagonal and non-diagonal elements of an input matrix.}, and fully-connected layers.
We adopted the ReLU function as activation.
We used Adam optimizer and set learning rate as 1e-3 and weight decay as 1e-5.
Batch size is 100.
Note that the models of \citet{maron2018invariant} are re-implemented by PyTorch in reference to the author's implementation in Tensorflow.

In terms of an objective function, we modified mean squared error (MSE) loss based on Theorem \ref{represention}.
In a straightforward manner, equivariant tasks are regression task so that MSE loss is selected to decrease the gap of all elements between an output matrix and ground truth matrix: $\ell_{std}\colon \mathbb{R}^{n\times n} \times \mathbb{R}^{n\times n} \rightarrow \mathbb{R}$.
However, thanks to Theorem \ref{represention}, ReyNet does not require calculating whole elements but the gap of 1st row of 1st column element and 1st row of 2nd column element: $\ell_{corner}\colon \mathbb{R}^{1\times 2} \times \mathbb{R}^{1\times 2} \rightarrow \mathbb{R}$.
In this paper, we call the latter objective function as corner MSE loss.

\subsection{Results}
\begin{wraptable}[10]{r}[0mm]{50mm}
 \caption{Comparison of Objective Function}
 \label{tab:exp2}
 \centering
 \small
    \begin{tabular}{l|ll}
    \toprule
        $n$    & mse & corner mse \\ \midrule
        3 & 1.438e-4 & \textbf{1.249e-4} \\
        5 & \textbf{4.912e-5} & 9.375e-5 \\
        10 & 1.157e-4 & \textbf{6.487e-5} \\
        20 & 1.608e-4 & \textbf{8.537e-5} \\ \bottomrule
    \end{tabular}
\end{wraptable}
Table \ref{tab:exp1} shows the result of synthetic datasets, which is the average of MSE of five different seeds.
As a result, our proposed models outperform \citet{maron2018invariant} overall.
Interestingly, with regards to the \textit{power} task, FNN and ReyNet are worse than other models.
The function $F$ of \textit{power} task amplifies the output error to large value.
Simultaneously, when the number of parameters of the model is very large along with the size of $n$, the output of the model becomes sensitive.
Therefore, we considered that the error became large due to the combination with large neural networks and the \textit{power} function.

Moreover, we validated the effect of an objective function.
We trained 4r-ReyNets with standard MSE loss and Corner MSE loss respectively.
Table \ref{tab:exp2} shows the result of using each objective function.
Accordingly, Corner MSE loss achieved lower error than standard MSE loss except for $n=5$ case.
Therefore, using Corner MSE loss is a good choice for equivariant tasks.

\subsection{Generalization Error}
\begin{figure}[t]
    \centering
  \begin{minipage}[b]{0.45\hsize}
    \centering
    \includegraphics[width=0.95\columnwidth]{figs/symmetry-compare.pdf}
    \subcaption{Symmetry}\label{gen-sym}
  \end{minipage}
  \begin{minipage}[b]{0.45\hsize}
    \centering
    \includegraphics[width=0.95\columnwidth]{figs/power-compare.pdf}
    \subcaption{Power}\label{gen-diag}
  \end{minipage}
  \caption{Generalization Error}\label{fig:generalization}
\end{figure}
Notably, our 4red-ReyNet does not depend on the size of inputs $n$.
In order to test the generalization performance, we trained 4red-ReyNet with $n_{train}=3$ dataset and then validated the MSE on $n_{test}\in \{3, 4, \dots, 20\}$ datasets.
The results are depicted in Figure \ref{fig:generalization}.
We can see that our 4-reduced ReyNet is generalized to the input size.
Note that with regard to invariant tasks, we confirmed the model is not generalized to the tasks as \citet{maron2018invariant} has reported.

\section{Conclusion}
We considered equivariant and invariant neural networks over higher order tensor spaces.
The method of converting deep neural nets to equivariant neural nets using Reynolds operators had some difficulties in terms of computational complexity, which can be successfully avoided by using Reynolds designs.
Then, we constructed equivariant Reynolds networks (equivariant ReyNets) based on the Reynolds designs and proved the universality.
Similarly, we also introduced Reynolds designs induced by Reynolds dimension in the invariant case.
Then, we also constructed invariant Reynolds networks (invariant ReyNets) and proved the universality as well.
Furthermore, we showed that input variables of ReyNets can be reduced based on the Reynolds dimension.
In the section of numerical experiments,
we showed that equivariant and invariant ReyNets performs better than or comparable to existing models.
Moreover, we observed that ReyNets with a few input variables can generalize well to the cases with more input variables.

\newpage

\subsection{Datasets}
We created four types of synthetic datasets for the comparison.
Given an input matrics data $\mathbf{A} \in \mathbb{R}^{n \times n}$, each task is defined by:
\begin{itemize}
    \item \textbf{Symmetry}: projection onto the symmetric matrices $F(\mathbf{A})=\frac{1}{2}(\mathbf{A} + \mathbf{A}^\top)$,
    \item \textbf{Diagonal}: diagonal extraction $F(\mathbf{A})=diag(\mathbf{A})$,
    \item \textbf{Power}: computing each squared element $F(\mathbf{A})=[A_{i,j}^2]$,
    \item \textbf{Trace}: computing the trace $F(\mathbf{A})=tr(\mathbf{A})$,
\end{itemize}
where the task function $F$ is equivariant in \textit{symmetry}, \textit{diagonal}, and \textit{power}, and is invariant in \textit{trace}.
Following \citet{maron2018invariant}, we sampled i.i.d. random matrices $\mathbf{A}$ with uniform distribution in $[0, 10]$; then we transformed it by the task function $F$.
In our experiments, we provided $n \in \{ 3, 5, 10, 20\}$, and the size of training dataset and test dataset is 1000 respectively.

\subsection{Implementation Details}
For equivariant tasks, we applied equivariant ReyNets.
For invariant tasks, we applied invariant ReyNets following the architecture of the invariant model proposed by \citet{maron2018invariant};
an equivariant ReyNet is followed by max pooling\footnote{This operation outputs the max value of diagonal and non-diagonal elements of an input matrix.}, and fully-connected layers.
We adopted the ReLU function as activation.
We used Adam optimizer and set learning rate as 1e-3 and weight decay as 1e-5.
Batch size is 100.
Note that the models of \citet{maron2018invariant} are re-implemented by PyTorch in reference to the author's implementation in Tensorflow.

In terms of an objective function, we modified mean squared error (MSE) loss based on Theorem \ref{represention}.
In a straightforward manner, equivariant tasks are regression task so that MSE loss is selected to decrease the gap of all elements between an output matrix and ground truth matrix: $\ell_{std}\colon \mathbb{R}^{n\times n} \times \mathbb{R}^{n\times n} \rightarrow \mathbb{R}$.
However, thanks to Theorem \ref{represention}, ReyNet does not require calculating whole elements but the gap of 1st row of 1st column element and 1st row of 2nd column element: $\ell_{corner}\colon \mathbb{R}^{1\times 2} \times \mathbb{R}^{1\times 2} \rightarrow \mathbb{R}$.
In this paper, we call the latter objective function as corner MSE loss.

\subsection{Results}
\begin{wraptable}[10]{r}[0mm]{50mm}
 \caption{Comparison of Objective Function}
 \label{tab:exp2}
 \centering
 \small
    \begin{tabular}{l|ll}
    \toprule
        $n$    & mse & corner mse \\ \midrule
        3 & 1.438e-4 & \textbf{1.249e-4} \\
        5 & \textbf{4.912e-5} & 9.375e-5 \\
        10 & 1.157e-4 & \textbf{6.487e-5} \\
        20 & 1.608e-4 & \textbf{8.537e-5} \\ \bottomrule
    \end{tabular}
\end{wraptable}
Table \ref{tab:exp1} shows the result of synthetic datasets, which is the average of MSE of five different seeds.
As a result, our proposed models outperform \citet{maron2018invariant} overall.
Interestingly, with regards to the \textit{power} task, FNN and ReyNet are worse than other models.
The function $F$ of \textit{power} task amplifies the output error to large value.
Simultaneously, when the number of parameters of the model is very large along with the size of $n$, the output of the model becomes sensitive.
Therefore, we considered that the error became large due to the combination with large neural networks and the \textit{power} function.

Moreover, we validated the effect of an objective function.
We trained 4r-ReyNets with standard MSE loss and Corner MSE loss respectively.
Table \ref{tab:exp2} shows the result of using each objective function.
Accordingly, Corner MSE loss achieved lower error than standard MSE loss except for $n=5$ case.
Therefore, using Corner MSE loss is a good choice for equivariant tasks.

\subsection{Generalization Error}
\begin{figure}[t]
    \centering
  \begin{minipage}[b]{0.45\hsize}
    \centering
    \includegraphics[width=0.95\columnwidth]{iclr2022/figs/symmetry-compare.pdf}
    \subcaption{Symmetry}\label{gen-sym}
  \end{minipage}
  \begin{minipage}[b]{0.45\hsize}
    \centering
    \includegraphics[width=0.95\columnwidth]{iclr2022/figs/power-compare.pdf}
    \subcaption{Power}\label{gen-diag}
  \end{minipage}
  \caption{Generalization Error}\label{fig:generalization}
\end{figure}
Notably, our 4red-ReyNet does not depend on the size of inputs $n$.
In order to test the generalization performance, we trained 4red-ReyNet with $n_{train}=3$ dataset and then validated the MSE on $n_{test}\in \{3, 4, \dots, 20\}$ datasets.
The results are depicted in Figure \ref{fig:generalization}.
We can see that our 4-reduced ReyNet is generalized to the input size.
Note that with regard to invariant tasks, we confirmed the model is not generalized to the tasks as \citet{maron2018invariant} has reported.

redpajama_set_name: RedPajamaArXiv


Introduction. According to the Digital News Report 2017, Spain is the European country with the highest penetration of WhatsApp for news consumption. Hypothesis and objectives. This tool of immediate communication allows media to increase from a qualitative perspective sources, traffic and even involve active audiences in fact-checking tasks. Our goals are to describe the process of creating B of Bulo feature in the newspaper SUR, analyze its operation and impact on social networks. Methodology. A mixed methodology is applied, by combining quantitative and qualitative techniques: content analysis, interviews with those responsible for the management of the service, participant observation and monitoring of visits. Results. The publication of content on hoaxes is done every 48 or 72 hours, and those related to children or politics are avoided. Despite these editorial limitations, news pieces in this section are part of the list of the most read contents. Conclusions. The results confirm the success of the proximity relationship established between newsroom and audiences via WhatsApp.
Fake news; post-truth; fact-checking; local press; Telegram; WhatsApp.
1. Introduction. 2. Objectives and methodology. 2.1. Methodological strategies. 3. Results. 3.1. Productive routines. 3.2. Fact-checking strategies. 3.3. Key to success. 4. Conclusion and discussion. 5. List of References.
Fake news have become one of the main problems of the Digital Era today. Its relevance forced the Dictionary of the Royal Academy of Spanish to incorporate the term post-truth in 2017, which refers to "the deliberate distortion of a reality, that manipulates beliefs and emotions in order to influence on public opinion and social attitudes." The intensive circulation of this kind of contents requires the journalists' reaction to protect and advocate the quality and veracity of professional information.
The fact that a hoax gets to the cover of a relevant media reflects in most occasions, a lack of fact-checking by journalists involved. On January 24, 2013, El País published in the cover an exclusive photograph of the president Hugo Chávez intubated while receiving medical treatment in Cuba. Other media also received the proposal by the agency Gtres Online, the only source having that material, but only El País decided to publish it. The image was a fake. Exclusive news offer distinction but when they are not fact-checked they might seriously damage the medium's credibility, and these failures promote the audience desertion. In this sense, editorials of digital news media must face a double pressure: be the first in disseminating a story and publish it with the complete correction and care of a written edition (Torres, 2007). Many media are tempted to prioritize speed and impact over fact-checking, without understanding that time becomes an ethical problem when rapidness is preferred over correctness (CdPeriodismo, 2013). Some experts, such as the journalist Delia Rodríguez, have warned about the dangers of these practices: "everything has an author, a responsible ordered the subject, hands that executed everything even though they did not sign it. If you copy, if you lie, if you manipulate, if you write trash, there is always someone who will notice, because we are on the Internet and we are all experts in something" (Rodríguez, 2016).
Despite these advises, which comply with the basic principles of the profession, El Mundo published a story on November 26, 2016, about a brave dad and his sick daughter Nadia, for whom he couldn't afford a cure. The newspaper helped collect almost 150,000 Euros, although only the father's version was published. Everything turned out to be a charade. Other journalists (Josu Mezo of the blog Mala Prensa, Ángela Bernardo of Hipertextual, and Manuel Ansede and Elena Sevillano of El País) dismantled the story and did their homework: contrasted sources. Therefore, this is one of the main weaknesses of the Spanish journalism, the lack or even the absence of sources (Chicote, 2010).
The journalist Craig Silverman confirms in 'Verification Handbook'(2014)that something as basic as confirming data, is being lost in editorial offices, and the technical easiness of rectifying any mistaken information, shouldn't reduce the publication process. In this sense, the Era of Information has changed the journalistic paradigm. Media have lost their leading role and are not the only senders of information (Paniagua, 2017). Social networks, which have multiplied in the last ten years, now occupy that space, they have already favored a communicative hyperactivity based on an excess of contents that generates 'noise', allowing rumors to filter and fake news to emerge, which both journalists and consumers are not always able to discern.
The strategies implanted on media, derived from this context of informational uncertainty have been varied. The agency France Presstried to stop this situation in 2009 when forbidding the use of Facebook or Wikipedia to its writers as source of information. Others, such as Steve Buffy, editor of 'Gazzete Communications', recommended using Mark Zuckerberg's social network instead as starting point for research, of course, with great caution.
Even so, Facebook is working on improving its system after what happened in 2016, when some former employees of the social network ensured having 'camouflaged' news that exalted the most conservative policies among Facebook Trending Topics, or in other words, the system that the platform had to show the most outstanding news to users. Zuckerberg did, back then, what some months later turned into one of the social network's greatest problems: news automation. At that moment, the algorithm did not distinguish between real or false. This way, the social network started filling with fake news that users deemed to be true. All this potentiated by the election period in the United States. Even Barack Obama was concerned about the boom of this kind of news and the impossibility of users to discriminate them. The II Report of the Spanish Internet Users Association confirms that 70% of users have trouble distinguishing a hoax from a reliable news (Asociación de Internautas, 2009).
Due to these backgrounds, in the last five-year period several platforms dedicated to detecting this kind of fake information have started. For instance, in Spain, three of them were born in a year: Maldito Bulo in November 2016, El Tragabulos in January 2017 and B de Bulo in March 2017.
Although journalists are more connected than ever, the lack of resources, making decisions based on sentences, abbreviations, emojis, or a brief communication that may or may not be accurate representations of truth, make difficult to maintain quality standards that must characterize journalism. A situation that has caused that only a third of the Spanish population trusts media (Reuters Institute Digital News Report, 2015).
This devastating landscape coexists with another reality, and it is the expansion of mobile communication. The Asociación para la Investigación de los Medios de Comunicación (AIMC) indicated in 2017 that 94.3% of Internet users browse the web from their smartphones and 62.7% of users do so out of home. WhatsApp has turned the Spanish population into the greatest users of its application in Europe, being present in 98% of smartphones in the country (CNMC Report).
In the labor aspect, based on a CIS survey, 25% of Spanish use WhatsApp for work-related issues. In this context, the use of electronic communication has surpassed face to face and voice to voice communication by far. Studies demonstrate that Generation Y and Millennials prefer using the instant messaging platforms or other social networks instead of attending an office and communicate in-person.
For years, Twitter has established the informative agenda, but now media face a post-Twitter era, because platforms like WhatsApp or Telegram also allowed to transform immediate messages into news, in such a way that the audience can condition the agenda (Batra, 2016). To actively involve and engage the audience into a collaborative journalism based on immediate communication, experiencing and introducing it into fact-checking and contents filtering processes represents a challenge but also an opportunity to increase audience and its loyalty.
Since 2015, media like La Sexta Noticias, El Periódico de Cataluña or Sur have incorporated these new consumers' habits into their routines, by applying data-supported strategies such as the fact that 69% of users that comment television content, send their messages through WhatsApp (Suñé, 2016). In the international field, initiatives of Reddit, Propublica, The Guardian, BBC or The Washington Post have also demonstrated that media can benefit from active audiences obtaining more sources and comments with links to evidences.
Despite this professional adaptation to the context of immediate communication, the circulation of fake news is quite alarming. The problem is not the new networks, but confirming who signs the content, thus seeking, comparing and always going to reliable brands, just like with food (Casal, 2016), otherwise, journalism can undergo irreparable damage. The actor Denzel Washington recorded a video to refute the rumor about his support to Trump by saying: "When you don't read the newspaper you are uninformed, but if you read it, you are misinformed." (Mèdia.cat, 2016).
In this same trend, fake news published on Facebook generated a great debate about whether Trump achieved the presidency thanks to them because, according to a study of the Pew Research Center, 60% of United States citizens obtained information about elections through this social network (Gottfried and Shearer, 2016). Zuckerberg's response consisted of creating an alert system, similar to Google's Fact-Check, to indicate users whether a news was fake or not. In the browser's case, the option must be used by media when they are completely sure that they have done their job in a responsible manner. On the other hand, Facebook is studying the possibility of keep trusting artificial intelligence when using an algorithm that identifies fake news. Although this can entail a new problem long term: overflowing the thin line between filtering and censuring. At the same time, a group of four young individuals have solved the fake news problem for now by implementing an extension for Chrome, theweb browser (Itkowitz, 2016).
Media in collaboration with research centers have also adopted an active role to protect themselves against the circulation of fake news. Initiatives like 'The Trust Project', internationally supported by media such as La Repubblica, The Economist, La Stampa, The Washington Post or Trinity Mirror promote the creation of protocols and tools to identify and certify the content on Internet. Duke Reporter's Lab has also elaborated a directory where he located 126 active sites dedicated to contents' fact-checking. PolitiFact is one of the most known because it was awarded with a Pulitzer prize.
In Spain the situation is disturbing. Reuters Institute Digital News Report 2015 indicated that Spanish media were the least credible of Europe and the second ones in the world, after North American media. These data demand an active reaction and response from media and researchers, in order to redirect a discredited communicative activity. Developing new tools or strategies in a responsible manner that facilitate contents' fact-checking and detecting rumors is essential to safeguard the credibility of media. In Spain, there have started fact-checking initiatives such as El Objetivo, Polétika, La Chistera, Maldito Bulo or El Tragabulos. This article is focused on analyzing one of these experiences, B de Bulo.
B de Bulo represents a special model because this section has a mobile phone to receive messages from the audience through WhatsApp about those contents they want to know whether to be true or false. Many fake messages turn into viral contents that circulate through WhatsApp, and this section selects the fake contents that the audience might receive, analyses them and turns to different sources to explain why the message is a fraud, offering a useful service for the citizen.
Despite the relevance of the phenomenon, from the academy the literature about it is rather scarce and recent. There is no thesis focused on the use of instant messaging applications for fact-checking by the media. Up until now, studies have mainly dealt with the analysis of certain specific coverages (Ramalho, 2014), international reference cases like The Washington Post or The Guardian (López, Rodríguez and Álvarez, 2016), methodological proposals (Sedano and Palomo, 2018) or the use that Spanish media make of WhatsApp (Fares, 2018; Negreira, López and Lozano, 2017).
Many research articles have explored citizen participation in digital news media. It is more difficult to find studies focused on the internal professional view, describing and/ or analyzing how editorials manage direct communication, what strategies are being developed and how journalists manage these new routines. These limitations are part of our objectives.
This study has a triple goal: to describe the creation process of B de Bulo, a section included in the Malaga newspaper Sur (Grupo Vocento) which online edition was founded in March 2017; secondly, to analyze its impact on social networks: how many followers has reached (Twitter and Facebook), what news have been most shared, and which had a greater impact among users. And, in third place, we will study the productive routines that surround said section, in order to know how themes are selected, how they perform fact-checking of published information and the type of feedback established with users.
To approach these objectives, a mixed methodology is applied, which combines quantitative and qualitative techniques: content analysis, interviews with the staff responsible for maintaining said section, participant observation and statistical data studies.
In-depth interview offers evidences about the nature of the phenomenon to be studied, facilitating the context and its emerging origin, and it enriches from the internal view of whom has a direct experience with the situation to be analyzed (Miller and Barry, 2016: 51). Therefore, in a first stage we conducted an interview with Elena De Miguel, chief editor of the digital department of Diario Sur, who allowed to identify the reasons underlying the creation of the B de Bulo section, how it was managed, how long did it take to prepare its onset, what are the main problems faced on a regular basis, what are the sources used, fact-checking techniques used and how are contents selected.
Participant observation was another ethnographic method implemented for data collection, understood as the active view that enables the researcher to learn about an activity in the natural scenario where it takes place, while becoming involved in the routine (Kawulich, 2006). In this case, the researcher was, at the same time, the professional responsible for implementing the section, which favored the members of the environment to act naturally, avoiding an intruder's look, which could have caused an uncomfortable situation in the editorial's office.
This qualitative approach was complemented with the content analysis of all news published for four months in the B de Bulo section since its foundation, from March 10, 2017 to July 10 that same year, reviewing themes, the use of sources and categorizing how news are fact-checked or how are hoaxes refuted.
Regarding social networks, we have studied its growth both on Twitter and Facebook, the two platforms used. Moreover, we analyzed the reach on Facebook and its interactions to identify the news that reached most people, as well as the news shared the most or the number of reactions. A similar analysis was performed on Twitter, by studying the favorites and retweets obtained. To achieve these data, SUR facilitated the access of the researcher to the statistics system available as user on both platforms.
B de Bulo is born thanks to an intern's initiative, with the purpose of implementing a fact-checking section open to the community; as a way of helping citizens to detect rumors.
After conducting the interviews and analyzing 50 news (an average of 12 per month) published from March to July 2017 in B de Bulo, the use of two types of sources was noted: messages coming from interactive communication spaces (WhatsApp, Telegram, Facebook or email) and, on the other hand, the search of contents performed by editors, mainly Jon Sedano, as participant observer of the study. Through the first pathway, during the four months of analysis, alerts were received from 18 different individuals on WhatsApp, one from Telegram and four via Facebook. Some of these users sent several messages with different hoaxes, in order to know almost in real time, the content's veracity or to request further investigation and refute it on the web, if a fake.
The bidirectionality of communication with the user is key for the section's success. Therefore, when readers what to know whether the content they have sent to the media is real, if the hoax is already published on the web they are sent the link so that they read the news and hence, increase the piece's traffic with the certainty that the content is interesting for the user; if it has not been published yet but it is already known that it is a hoax thanks to official notices, it is made known, and lastly, if there is no evidence about the interest it may have for the audience, but is still considered a relevant issue, it is fact-checked on the site. To evaluate whether it is relevant or not, we studied whether the theme had an impact (by looking at related news on the Internet or through social networks) or if it is an isolated case. If the story might interest a broad audience, then the work on it is initiated.
The 75% of content published during the four months of analysis was directly proposed by journalists, and 25% was the audience's initiative: 20% was delivered through WhatsApp by 18 different people, 4% through Facebook, and only one suggestion came via Telegram.
The person contacting from Telegram sent two messages: one of them about an already published new (Heineken beer factory in Seville with suckled pigeons) and another one from a missing girl. In the former's case, the link to the original story was sent, while for the latter we thanked the contact, but it was not published because it affected a minor.
Regarding Facebook, messaging individuals alerted about different events, of which two were published in B de Bulo: the gang attacking those whose headlights illuminate their vehicles and the new blood alcohol level control for copilots.
Professional ethics is one of the parameters influencing in the publication criteria of hoaxes. In this sense, from the news received by users, there were two that were discarded once investigated. The first one was about a gas station in Malaga with water in its deposits that damaged several vehicles. In this case, we concluded it was a hoax, and since the message didn't have much impact, it was considered that its publication would do more damage than benefit to the company, therefore the reader sending the message was told it was only a hoax. The second alert was the case of a girl kidnapped in Mexico, and it was discovered that the girl was already safe. B de Bulo received four messages from WhatsApp about her and one from Facebook, and these users were explained that the girl was safe and that she has only disappeared for two hours. Again, publishing the news was not deemed convenient because she was a minor and was already safe.
The rest of news published in the section were proposed by the journalist. In order to locate said contents the following routine was articulated: Firstly, there is a search on Twitter with hashtags such as #bulo or #stopbulos to know about recent messages where users talk about hoaxes. At the same time, several official accounts are followed which echo in these kinds of publications, such as National Police, Civil Guard, Traffic, VOST (Team of digital volunteers in emergencies) or SOSdesaparecidos, among others.
Finally, to the phone of Diario Sur there also came messages containing hoaxes from the audience punctually.
Considering this content, or the content available searching news on Internet that use the term 'bulo' [hoax], as well as from the analysis of the main viral news, we obtained a daily basis to select the section's potential contents. Despite the volume located sometimes, there is an editorial limitation in Diario Sur which is the fact that each publication must be distanced a minimum of 48 hours from the next one, in order to avoid granting excessive notoriety to originally fake news on the cover of the digital newspaper, which might lead to the reader's confusion. In addition, this measure favors the correct fact-checking of said contents. Another restriction is that children or politics related contents are avoided.
The journalist always offers arguments, based on facts and data, to justify why a message is a fake. The fact-checking process is different in each story, although from a general perspective, the steps to be followed are similar. Thus, first there is the search of the original source to confirm its veracity. Given the case of photographs and videos, images are downloaded, and a native Google search is done in order to see whether there are related news.
In other occasions keywords in several languages are used in order to find any helpful data. For instance, the video about a wind generator that broke due to the wind's force in Cadiz was really filmed in Italy and manipulated afterwards: the beginning was cut off to avoid identifying the language and noise was added in order to hide the original oral intervention. There was a search about wind generators breaking to locate the piece's origin. Since the video couldn't be found, a photogram of the best quality possible was extracted and when it was uploaded to Google, it returned a similar new. Following the trail of this story the video was found, which many media placed in Italy. Once the location and accurate dates were obtained there was a search until finding a relevant media publishing the new. Once everything was fact-checked, the article was written in the section alluding to the original source.
Practically all contents are fact-checked by going to the original source. This allows to confirm the veracity of the fact or the search of further data that helps determine whether it is a hoax or not.
Another way of fact-checking a fake new is to use the message from an official source. In this case, the hoax is explained and confirmed versus the official information about it, adding the link, tweet or the quote in the text, so that readers can confirm for themselves it is a lie.
Lastly, in the case of local news, mainly events, there is contrast versus primary sources that could have access to said data, such as local or national police, the healthcare service, etc. besides people directly involved in the issue, like the case of the gas station. In order to do a fact-checking of the news, in addition to talking to the station and with another employee from the company, contact was established with a competitor gas station nearby, as well as the taxi drivers' union and with Consumer Assistance Centre of Malaga. They all confirmed it was a hoax. On the other hand, the hoaxes talking about the kidnapping of children in a town and that the police already working on the issue, are solved by contacting the delegation of that area in order to confirm the message's veracity.
Regarding the average time, the fact-checking requires, it differs for each news. The faster to produce are those where the official sources confirm or deny a content; in these cases, the news explaining what happened is written and it is accompanied by the original message from the official source. If there are no official data about it, the fact-checking can range between some hours and a day. Exceptionally, in those case where there is the need to wait for the reply of involved parties, that usual range has been exceeded.
Before the question about why users share hoaxes, we found two drivers: the first one is to discredit a company, person or institution; and the second one is to create social alarm.
Fear sells, and the analysis of the 50 news published in B de Bulo during its first months of existence showed that 54% of them are contents about kidnapping, deaths, frauds, terrorist activities, disappeared individuals, accidents, etc. Likewise, it was seen that only 14% of these facts were linked to the local context, which allows the media to elaborate interest contents for other contexts that are not necessarily close (López, 1999). In this sense, the local newspaper generates content of local involvement, connected and open to consumer preferences without a geographic limitation.
Often media experience with innovation without obtaining any benefit in return. However, the section B de Bulo has been widely accepted by readers. The most read article had 214.192 visits. In fact, three of those articles are part of a very relevant list: the five most visited news in the digital section of Diario Sur during 2017.
All the content is published on Facebook and Twitter accounts, also developed by B de Bulo. After monitoring social networks, it was seen that these news were more successful on Facebook, obtaining 1235 followers and 1200 "likes". Regarding the most shared publications, the ones related to kidnapping evoked more clicks and reactions.
Technology is not neutral. This feature obliges journalists to develop mechanisms in order to do fact-checking of all contents received through instant messaging of applications like WhatsApp or Telegram essentially for two reasons: the first one, that media need to create solid bridges with the public in order to try to increase their trust, collaborate in an efficacious manner and create commitment and loyalty. On the other hand, most of media use email, text messaging, instant messaging, blogs and comments in websites, therefore the possibility of a misinterpretation increases. Media must find the perfect balance between both scenarios.
Recently, benefits of low cost communication have reduced to describing the activity of audiences on Twitter, Facebook, Instagram or Snapchat. However, current journalism is affected by an over-stimulated society, constantly open to new experiences. Citizens are interested about the search of novelty, which produces a dopamine shot. At the same time, every WhatsApp alert represents a new stimulus for journalists. Messaging applications have turned into a new and faster information source, and the audience activity confirms its preference by these platforms, therefore they are very efficacious channels to connect with the public. Likewise, this private and direct communication can be an intense experience because it depicts the watchdog of the gatekeeper, and the formula of media to recover the audience's trust.
Citizens have new active roles to help media improve, to reduce misinformation, to warn about the circulation of fake news or rumors. In this context, journalists must remember that obtaining the first news is a challenge, but being rigorous or accurate is a need for survival in quality journalism.
Another evident conclusion after the analysis of B de Bulo in Diario Sur is that WhatsApp is an ideal tool for small or local media because they do not usually have the need to elaborate lists with more than 253 users, which is the application limit, and since it is the application with the greatest audience, with it they have the opportunity to increase their public.
The absence of similar previous studies opens many options for developing comparisons versus other news fact-checking sections on media, platforms completely destined to contrast hoaxes or a qualitative analysis of the population's behavior before fake news in moments of social crises.
Funded research: This article is a result of the research project entitled "Influence of audience on journalistic innovation: risks and opportunities", reference code CSO2015-64955-C4-3-R MINECO/FEDER. Funded by the State Program of Research, Development and Innovation oriented towards Challenges of Society.
Batra, B. (2016). "News Comunication Throught WhatsApp". International Journal of Informative & Futuristic Research, 3(10), 3725-3733.
Kawulich, Barbara B. (2006): "Participant Observation as a Data Collection Method". Forum Qualitative Sozialforschung / Forum: Qualitative Social Research, 6(2). http://nbn-resolving.de/urn:nbn:de:0114-fqs0502430.
López, Xosé; Rodríguez, Ana Isabel; y Álvarez, Lucía (2016): "El fact checking como reclamo y como servicio en los cibermedios. Análisis de las experiencias de 'The Washington Post' y 'The Guardian'". Telos: Cuadernos de comunicación e innovación, 103. 74-83.
Miller, Jody y Barry, Glassner (2016): "The 'Inside' and the 'Outside': Finding Realities in Interviews". En Silverman, David (ed.), Qualitative Research. Londres: Sage. p.51-66.
Negreira, María Cruz; López García, Xosé, y Lozano, Lara. (2017). Instant Messaging Networks as a New Channel to Spread the News: Use of WhatsApp and Telegram in the Spanish Online Media of Proximity. Recent Advances in Information Systems and Technologies. WorldCIST 2017. Advances in Intelligent Systems and Computing, 571, 64-72.
B Palomo, J Sedano (2018): "WhatsApp as a verification tool for fake news. The case of B de Bulo". Revista Latina de Comunicación Social, 73, pp. 1384 to 1397.
Article received on 14 January 2018. Accepted on 16 April.
Published on 25 October 2018.


Grab your official TBM t-shirt today! Choose from a range of over 20 different colours.
/// Tool Box Media has a highly developed production and shipping system and we make every effort to ship your order in a fast and effecient manner.
/// Tool Box Media printing processes have been developed specifically to address quality, efficiency and cost, using the latest in printing technology to deliver cutting edge printing options.


text:
Teacher of Maths
Whitmore High School, Harrow, HA2 0AD
For September 2023
30 January 2023 at 9am
View all Mathematicsjobs
Main Pay Range 1 £32,407 to Upper Pay Scale 3 £48,055
To be a well-qualified teacher of Maths with excellent subject knowledge. The ability to offer A level Maths would be an advantage
To have a sound knowledge of current issues in language and learning and the development of literacy, including the use of ICT in effective teaching and learning
To be willing to experiment with and apply new approaches to teaching and learning
To help create an environment that values feedback from all in order to improve student learning and motivation
To be interested in children as individuals, in how they learn and in a collaborative approach to learning
To be aware of appropriate strategies for mixed-attainment teaching and adapt to the varying demands of each classroom situation
To be able to build good-humoured working relationships with many colleagues
To believe in the importance of teamwork, both at departmental and whole school levels
To be committed to the comprehensive ideal, to the principles and practice of equal opportunities and of mixed-attainment teaching
A commitment to Equality, Diversity and Inclusion and the principles and practice of equal opportunities
Be committed to the principles and practice of safeguarding all young people
State of the art building with well-equipped classrooms, excellent resources and modern facilities
Fantastic support from your line managers and SLT who will develop you and want to see you succeed in your career
ECF programme with dedicated mentor for 1st and 2nd Year ECTs
You'll be working in an OFSTED rated 'Outstanding' school
Exceptional and wide ranging CPD opportunities
Staff Wellbeing group who promote a healthy work life balance and arrange activities such as weekly staff yoga, on-site massage, weekly fruit, breakfast boxes, well-being garden, book club with free Kindle loan for staff etc.
Free annual flu jab
Free use of the on-site gym
Cycle to Work tax free bike scheme
Whitmore High School is committed to safeguarding and promoting the welfare of young people and all staff are expected to share this commitment. Our CP Policy is available on the school's website www.whitmore.harrow.sch.uk
Maths - Jan 23.pdf - 631 KB (opens in new tab)
Whitmore High School Prospectus.pdf - 1.14 MB (opens in new tab)
About Whitmore High School
Local authority maintained school, ages 11 to 18
Whitmore High School website (opens in new tab)
applications@whitmore.harrow.sch.uk
teaching-vacancies.service.gov.uk/organisations/whitmore-high-school (opens in new tab)
Whitmore High School is a mixed comprehensive school with an excellent record of achievement. Students are expected to achieve the highest standards of work and behaviour in a positive, friendly but disciplined atmosphere. Our staff are highly qualified and passionate about ensuring all students reach their full potential and we work closely with parents to achieve this aim. At Whitmore High School we aim to provide the best possible education for all our students by: * Promoting a broad and balanced curriculum in which students take an active part in their own learning. * Recognising the needs of individuals, placing suitable high demands on them and stimulating them to achieve success. * Encouraging students to aim for the highest standards of work and behaviour at all times. * Creating an ordered and friendly community where relationships are positive and founded on courtesy, respect and shared values. * Acknowledging the importance of links between home, school and the wider community. * Maintaining a spirit of constructive self-criticism, regularly considering improvements to existing practice.
Arranging a visit to Whitmore High School
To arrange a visit and increase the chance of a successful application email applications@whitmore.harrow.sch.uk.
Teacher of Maths / Further Maths
Dame Alice Owen's School
Dugdale Hill Lane, Potters Bar, Hertfordshire, EN6 2DU
Teacher of Mathematics
Glenthorne High School
Sutton Common Road, Sutton, Surrey, SM3 9PS
Bolder Academy
1 Macfarlane Lane, Isleworth, TW7 5DB
Waingels
Waingels Road, Reading, Berkshire, RG5 4RF
Find more school jobs in Harrow

redpajama_set_name: RedPajamaCommonCrawl


text:
Search Results: 415 books from 21 publishers. Learn more ›
Animal Performers
Sara Green (author)
Publisher: Bellwether Media Inc. ISBN: 9781681035864
Did you know that the Labrador retriever that starred in Marley and Me was a rescue dog from a shelter in Florida? His is a true rags-to-riches story! This title explores the history of animals in entertainment while highlighting some of the pet pals we have grown to love on the silver screen!
Creating Thanksgiving Crafts
Dana Meachen Rau (author), Kathleen Petelinsek (illustrator)
Publisher: Cherry Lake Publishing ISBN: 9781624312144
Whether you're looking for something to do on a rainy day or making a personalized gift for a friend, crafts are a great way to pass the time. They also provide an opportunity to learn how to draw meaning from technical texts. The activities in this book push readers to read for comprehension, use information gained from the text and illustrations to follow step-by-step directions, determine relationships among steps in a technical process, and build the skills they need to make the perfect crafts for the Thanksgiving season.
Valerie Bodden (author)
Publisher: The Creative Company ISBN: 9781566607568
A biography of Dutch artist Vincent van Gogh, examining his contributions to the Post-Impressionist style and his early death, as well as many of his greatest paintings.
Adele Richardson (author)
A biography of Italian Renaissance artist Michelangelo, examining his masterful commissions and papal relationships, as well as some of his most enduring works.
What is Modernism?
Kate Riggs (author)
With prompting questions and historical background, an early reader comes face to face with famous works of Modern art and is encouraged to identify images and consider different meanings.
What is Romanticism?
With prompting questions and historical background, an early reader comes face to face with famous works of Romantic art and is encouraged to identify feelings and consider dreamlike subjects.
Charles Dickens (author), Roberto Innocenti (illustrator)
This perennial Dickensian favorite about a miser's startling change of heart is illustratedby Italian artist Roberto Innocenti and reissued for the readers of all the Christmases Yet To Come.
Mark Twain (author) C. F. Payne (illustrator)
Mark Twain's classic American story about a boy with a nose for mischief and a knack for avoiding any sort of work is here presented in vibrant detail, with new illustrations by C. F. Payne.
A Glass
Etienne Delessert (author, illustrator)
Children's author and illustrator Etienne Delessert tells the story of Eglantine Besson, the woman who became his mother, and of the glass that came to represent their relationship.
Ludwig van Beethoven: Great Composer
Anna Carew-Miller (author)
Publisher: Mason Crest ISBN: 9781422289693
A German composer born in the 1700s, Ludwig van Beethoven has given the world some of the most well-known and long-lasting music of all time. From his Fr Elise to his 9th Symphony (in which a choir sings words from the poem Ode to Joy), Beethoven's music is still loved, almost 200 years after his death. Amazingly, Beethoven composed music while being unable to hear almost anything from the time he was 26, a feat which makes his brilliant compositions all the more wonderful. Few musicians have had the incredible impact on music that Beethoven had. Learn the story of one of the most important musical composers of all time in Ludwig van Beethoven: Great Composer.
Ted Staunton (author)
Publisher: Orca Book Publishers ISBN: 9781459812383
When first-year film student Spencer O'Toole is asked to make a music video for a band, he leaps at the chance. But Jerry, Spencer's dad, shows up, and somehow the band assumes he's in charge, despite the fact that he has zero background in film. And then there's Scratch, violent gang member turned sleazy music producer, who keeps making big promises but fails to deliver on a single one. Spencer has no idea how he's going to get this thing made. When the band invites him and his dad up to a cottage for the weekend, Spencer takes the opportunity to ditch Jerry. But one small fib snowballs into dozens of lies, and soon Spencer finds himself in way over his head.
A Sense of Art
Christine Dugan (author)
Publisher: Teacher Created Materials ISBN: 9781433399404
Learn how architects and designers use perimeter and area to make famous works of art! In this vibrant title, readers will continue to develop their geometry skills, reviewing topics like dimensions, irrational numbers, polygons, and measurement. With the help of eye-catching images, easy-to-read text, simple practice problems, STEM topics, and clear mathematical diagrams, readers will feel confident in their ability to measure perimeter and area in useful ways!
Moira Anderson (author)
Find three-dimensional shapes while learning about architecture and historic architects! A good understanding of geometry and three-dimensional shapes is essential for architects building houses because all houses have three-dimensional shapes like pentagons and prisms. Readers can follow the process of building a house from start to finish, mastering geometric concepts through practice problems. These problems, along with clear mathematical diagrams, easy-to-read text, and vibrant images will keep readers engaged and confident in their geometry skills and STEM skills.
Action! Making Movies
Sarah Garza (author)
Action! It's time to enter the world of movie magic! Readers are taken behind the scenes to find out what is needed to make a film. From the director to the actor, the director of photography to the costume designer, children will be fascinated to learn about the various aspects that go into motion pictures. With its vivid images, informational text, and impressive facts, this nonfiction title will have readers engaged through the entire book as they discover amazing facts about their favorite genre--whether it be comedy, drama, action, or horror!
Geoffrey Chaucer: Medieval Writer
Joanne Mattern (author)
Geoffrey Chaucer was a British poet during the Middle ages and is still considered one of the greatest English writers of all time. Through this intriguing biography, readers will learn about Chaucer's life as he took part in the Hundred Years War, worked for King Edward III, and went on to write the well-known collection, "The Canterbury Tales". Featuring lively illustrations, fascinating facts and sidebars, easy to read text, an accommodating glossary, and helpful index, this book will have readers excited and eager to learn more about this influential author.
Fearless! Stunt People
Jessica Cohn (author)
When stunt performers go to work, they make dangerous feats look fun and easy, but there is a lot of risk behind this profession. Explore what stunt people do to prepare and keep themselves safe for a truly unique profession. Readers will be introduced to a brief history of stunt performance as well as the latest tips and tricks, including stunt driving, professional fighting, and even digital effects such as computer-generated imagery and are encourged to explore college majors and areas of study are recommended to become a stunt person. With thrilling language and stunning images in conjunction with informational text, a useful glossary, and an interview with a real-life stunt-man, readers are sure to be fascinated and stimulated from cover to cover!
Behind the Canvas: An Artist's Life
Blanca Apodaca (author)
Step inside a studio and learn what it takes to be an artist! This nonfiction title engages readers through stimulating facts and vivid images and diagrams in conjunction with a brief background on well-known art movements such as Expressionism, Surrealism, and Impressionism, an interview with a real-life artist, a glossary, informational text, an index, and list of useful sources for learning more about art.
Origami Fun: Ocean Animals
Robyn Hardyman (author)
Because the ocean biome is the biggest biome in the world, it's not hard to believe that it is packed with many interesting creatures. Aspiring young paper engineers will enjoy making crafts of some of these aquatic animals to develop their paper-folding skills.
Origami Fun: Holidays
Most people would agree that handmade decorations have more charm than store-bought alternatives, so this origami book teaches kids enough paper crafts to make eight popular holidays more festive! The background information on the holidays makes sure origami artists know how to best celebrate each occasion.
Origami Fun: Farm Animals
How can you raise farm animals without acres of land? This children's title has the answer, showing readers how to mountain-fold and valley-fold a cow, horse, sheep, goat, pig, chicken, and other livestock. Animal herds and flocks can form with additional paper and repeated folding.
Origami Fun: Dinosaurs
In nature, extinction is an irreversible process. Animal species that die off cannot be brought back to life. This origami how-to, however, gives young dinosaur lovers the opportunity to resurrect a stegosaurus, a triceratops, and more with paper. Making each species is as easy as 1-2-3…
Origami Fun: Pets
Taking care of a pet can be a lot of work. A dog, for example, needs to be fed, walked, played with, and more. But the pets made of paper in this origami title are very little work. Young pet lovers will give this craft book their full attention!
Origami Fun: Birds
A flock of many different birds is ready to be assembled in this makerspace title. Origami outlines ensure that young crafters will not have to "wing it" when it comes to making paper birds. Along with the clear folding instructions are bonus bird facts that just might turn kids into birders!
Origami Fun: Aircraft
Sometimes all you need to craft a vehicle is a single sheet of paper. In this origami guide for kids, young flight enthusiasts will find easy-to-understand specs for building airplanes of all shapes and also an invitation to conduct test flights to compare the planes' speeds and flight paths.
Every saga has a beginning. This marketing tagline for Star Wars: Episode I— The Phantom Menace could also be the tagline for this biography of the Stars Wars brand. Young readers will travel back to a long time ago in a galaxy far, far away…


text:
News & Views » Columns
Vicious vigorish
Lansing's red tape is stifling Detroit's ability to service tens of thousands of low-income children — and state Rep. Hansen Clarke, D-Detroit, is trying to do something about it. Currently, Michigan takes 40 cents of every Medicaid dollar slated for the state's schools. The money is taken for "administrative" needs and goes straight into the state's general fund, Clarke says. The Medicaid dollars — about $100 million a year — are intended to reimburse schools for federally mandated programs such as counseling, social work, and speech and physical therapy. Nationwide, states take an average of 10 percent of school Medicaid dollars for administrative costs, says Clarke, who's seeking a state Senate seat. Last week, Clarke proposed amending the state budget to reduce Lansing's take to 20 percent. Detroit especially is crossing its collective fingers in hope the measure passes. "It's desperately needed," says Bruce Elkins, Medicaid coordinator for Detroit Public Schools. "We have all these mandated services going back to the 1970s, without adequate resources to hire people and implement those services." Elkins estimates that the district spends about $100 million a year to service 86,000 Medicaid-eligible special-education children. The city will get $10 million from Medicaid in the 2002-2003 school year for such services, Elkins says.
More than 50 percent of that cost is paid by Detroit taxpayers, he adds. "Quite honestly, it's a very unfortunate situation. The state is denying the schools money that was intended for the schools." Clarke learned of the situation after Detroit Public Schools announced layoffs earlier this year. He began digging around to find ways to generate more money for the district so employees could keep their jobs and improvements could occur. At minimum, the reduction for administrative costs must be made, he says. "The state is actually undermining the schools' efforts in places like Detroit to improve services for kids," says Clarke. "And I don't think anyone can justify a 40 percent administrative fee."
Lisa M. Collins is a Metro Times staff writer. E-mail lcollins@metrotimes.com
Columns Education News Hits


text:
INDIAN PREMIER LEAGUE, 2018
MA Chidambaram Stadium, Chennai
Known as
Chepauk; Madras Cricket Club Ground
Anna Pavilion End, V Pattabhiraman Gate End
Home to
Tamil Nadu, Chennai Super Kings
Indian Premier League, 2018 matches scheduled at this venue
14:30 GMT CSK vs KKR, 5th Match
Venue Description:
The stadium is named after the former President of BCCI and Tamil Nadu Cricket Association, MA Chidambaram. It was formerly known as Madras Cricket Club Ground. The ground was established in 1916, making it the oldest venue in India which is in continuous use. The stadium is more commonly known as Chepauk - the locality in which it is situated.
Since 2008, the venue is home to Chennai Super Kings in the Indian Premier League. It hosted its first ever international game in 1934, a Test match between India and England, which the visitors won by 202 runs. The stadium has gone through several changes since then. The most notable ones being, the addition of floodlights in 1996 and the reconstruction in 2009 to accommodate more people and give state-of-art facility.
The most fascinating thing about the ground is the crowd support. It is common knowledge that the Chepauk crowd is the most unbiased and the most knowledgeable crowd in the country. Saeed Anwar got a standing ovation when he made a record-breaking 194. When Pakistan won the 1999 Test, the team received good appreciation from the crowd. As mark of respect, the Pakistani team also took a lap of honour around the ground.
How does the pitch play?
When the pitches were uncovered and unattended to, it used to help the fast bowlers. However, since India started playing the spin quartet of Bishan Singh Bedi, BS Chandrasekar, S Venkatraghavan and EAS Prasanna, the wickets were made spin-friendly. Taking a look at the stats gives you a clear indication as to what type of bowlers hold the aces on the Chepauk wicket.
Among the top 10 wicket-takers at the ground, nine of them are spinners and the only fast bowler on the list is Kapil Dev. Anil Kumble sits on top of the list with 48 wickets, followed closely by Harbhajan Singh - 42 wickets.
Interesting facts about Chepauk:
The first ever Ranji Trophy game was played at this venue. Madras' AG Ram Singh ran through Mysore with a 11-wicket haul in the match.
For a long time, Chepauk hosted the famous Pongal (Harvest) Test. It used to be played in the month of January.
India registered their first ever Test win at this stadium, when they beat England by an innings and 8 runs.
The second tied Test in the history of cricket was played here, between India and Australia in 1986.
Saeed Anwar recorded the highest individual score in ODIs - 194 - at this ground. The milestone remained for more than a decade, before Sachin Tendulkar scored 200 against South Africa.
Sunil Gavaskar went past Don Bradman's 29 Test hundreds at this very venue.
This ground was also witness to one of Sachin Tendulkar's greatest Test innings. The Indian maestro battled back spasms and almost took his side home against Pakistan in 1999. But his 136 went in vain as the arch-rivals sealed a 12-run victory.
Total matches
Matches won batting first
Matches won bowling first
Average scores
1st inns - 337, 2nd inns - 356
Highest total
759/7 (190.4 Ov) by	IND vs ENG
Lowest total
83/10 (38.5 Ov) by	IND vs ENG
387/4 (98.3 Ov) by	IND vs ENG
171/10 (85.2 Ov) by	IND vs AUS
Explore other venues
Punjab Cricket Association IS Bindra Stadium
Rajiv Gandhi International Stadium
MA Chidambaram Stadium
Sawai Mansingh Stadium
M.Chinnaswamy Stadium
Maharashtra Cricket Association Stadium
Arun Jaitley Stadium
Holkar Cricket Stadium
History Points Table Teams Venues
Home Matches Points Table
News Fixtures This day that year
Editorials Results Poll
Photos Rankings Players
Trivia Cricket on Twitter Series
© 2020 Cricbuzz.com


Hip Replacement Surgery – WebMD – Better information … – Nov 05, 2013 · Total joint replacement involves surgery to replace the ends of both bones in a damaged joint to create new joint surfaces….
Hip replacement – Encyclopedia of Surgery: A Guide for … – Hip replacement is a procedure in which the surgeon removes damaged or diseased parts of the patient's hip joint and replaces them with new artificial parts….
Hip replacement surgery involves replacing components of the hip joint with a synthetic implant, to repair the damaged bearing surfaces that are causing pain….
My Hip Replacement Surgery 2019 4.5 out of 5 based on 412 ratings.