When reading the proof, the following perspective may help. Recall the notation in equations (<ref>) and (<ref>). The knowledge of the (simple max-stable) distribution of $X_U$ implies the knowledge of its angular measure $H_U$ and thus of the unordered collection of pairs of atoms and masses $(\omega_r, \mu_r)$ for $r = 1, \ldots, s$. The vector $X_U$ can itself be represented as a max-linear model with $s$ independent factors and coefficient vectors $\beta_r = \mu_r \omega_r$ for $r = 1,\ldots,s$. We first need to ensure that we can match those vectors $\beta_r$ in a unique way to the max-linear coefficient vectors $(b_{vi})_{v \in U}$ for $i \in V$; note that the coordinates $v$ of those vectors are restricted to $U$. Next, from the latter vectors, we need to recover the edge coefficients $\theta = (c_e)_{e \in E}$.

We assume <ref> and <ref>. In the first step of the proof we show that the angular measure of $X_U$ in (<ref>) is composed of $|V|$ distinct atoms and that we can associate every atom in $\{\omega_r : r=1, \ldots, |V|\}$ to some node $v \in V$ and accordingly be able to associate it to one of the atoms $a_{i,U}=(b_{vi}/m_{i,U})_{v\in U}$ for $i\in V$.
For this, we focus on the nature of the atoms $\{a_{i,U}\}$, given the conditions <ref> and <ref>.
As a consequence, the max-linear coefficient matrix $b_{U \times V} = (b_{vi})_{v \in U, i \in V}$ can be recovered from the distribution of $X_U$. In Step 2, we show how  to recover from this matrix the edge parameters $\theta = (c_e)_{e \in E}$.

* Step 1. Recall the representation $H_U = \sum_{i \in V} m_{i,U} \delta_{a_{i,U}}$  in (<ref>) of the angular measure of $X_U$. We shall show that all $|V|$ masses $m_{i,U}$ are positive and that all $|V|$ atoms $a_{i,U}$ are distinct. Moreover, we will show how to match the atoms to the nodes, that is, given an atom $\omega \in \{ \omega_r : r = 1, \ldots, |V|\}$ how to identify the node $i \in V$ such that $\omega = a_{i,U}$.

* All $|V|$ vectors $\{a_{i,U}\}$ have positive  masses $\{m_{i,U}\}$.

Recall $m_{i,U} = \sum_{v \in U} b_{vi}$ and recall from (<ref>) that $b_{vi} > 0$ if and only if $v \in \Desc(i)$.
It follows that $m_{i,U} = 0$ if and only if $\Desc(i) \cap U = \varnothing$ or, in other words, $\Desc(i) \subseteq \bar{U}$.
But this is impossible
since $\Desc(i)$ contains at least one leaf node, that is, a node without children, and such a node belongs to $U$ by <ref>.
We conclude that $m_{i,U} > 0$ for all $i \in V$.

* All $|V|$ vectors $\{a_{i,U}\}$ are distinct.

By (<ref>) it follows that whenever for two different nodes $i,j\in V$ we have $\Desc(i)\cap U\neq \Desc(j)\cap U$ then we can find two atoms, say $\omega'$ and $\omega''$, within the set $\{\omega_r\}$ such that $\omega'=a_{i,U}$ and $\omega''=a_{j,U}$. Because $\Desc(i)\cap U\neq \Desc(j)\cap U$ then necessarily $a_{i,U}\neq a_{j,U}$. Suppose however for two different nodes $i,j\in V$ we have $a_{i,U}=a_{j,U}$. This means that for the $u$-th and $j$-th elements of these vectors we have

\[
\Longleftrightarrow
\frac{b_{ui}}{m_i}=\frac{b_{uj}}{m_j}
\qquad \text{and} \qquad
\Longleftrightarrow
\frac{b_{ji}}{m_i}=\frac{b_{jj}}{m_j}
\]

Considering the ratios above, we should also have

\begin{equation} \label{eqn:aiuU}
\frac{a_{i,u;U}}{a_{i,j;U}}=\frac{a_{j,u;U}}{a_{j,j;U}}
\qquad \Longleftrightarrow \qquad
\frac{b_{ui}}{b_{ji}}=\frac{b_{uj}}{b_{jj}}.
\end{equation}

Because $a_{i,U}=a_{j,U}$ necessarily
$		\Desc(i) \cap U
%	= \{v\in U : b_{vi} > 0\}
%	= \{v\in U : b_{vj} > 0\}
= \Desc(j) \cap U.
By Lemma <ref> there exists a node $u$
such that one of the edge sets $\{(i,j), (i,u), (j,u)\}$ or $\{(j,i), (i,u), (j,u)\}$ is contained in $E$. Without loss of generality, suppose this holds for the first triple. Also, by Lemma <ref> there cannot be another node $k$ with $\Desc(k)\cap U=\Desc(i)\cap U=\Desc(j)\cap U$.

Suppose first $j, u \in U$. From the identities

\begin{align} \label{eqn:identities}
b_{ji} &= c_{ij} b_{ii}, &
b_{ui} &= c_{iu} b_{ii}, &
b_{uj} &= c_{ju} b_{jj},
\end{align}

and the criticality requirement

\[
c_{iu} > c_{ij} c_{ju}
\]

we have

\begin{equation} \label{eqn:bui}
\frac{b_{ui}}{b_{ji}}>\frac{b_{uj}}{b_{jj}}.
\end{equation}

This inequality shows that (<ref>) cannot happen, hence we cannot have $a_{i,U}=a_{j,U}$. This means that all $|V|$ atoms are distinct.

Next suppose $j, u \in \bar{U}$. By Lemma <ref>, there exist nodes $j', u'\in U$ such that $j' \in \desc(j)$ and $u' \in \desc(u)$ and the paths $p(j,j')$ and $p(u,u')$ satisfy the properties in the said lemma. Because all nodes on the path $p(j,j')$ except possibly for $j$ have a unique parent, the path $p(j,j')$ cannot pass through $u$ (which has parents $i$ and $j$), and thus $j' \neq u'$. As $i$ is a parent of $j$, the shortest (and in fact the only) path from $i$ to $j'$ is the one that concatenates the edge $(i, j)$ with $p(j, j')$ (Lemma <ref>). It follows that

\begin{equation} \label{eqn:bjprimei}
b_{j'i} = c_{p(j,j')} c_{ij} b_{ii}
\qquad \text{and} \qquad
b_{j'j} = c_{p(j,j')} b_{jj}.
\end{equation}

By a similar argument, the path that concatenates the edge $(i,u)$ with the path $p(u,u')$ is the unique shortest path from $i$ to $u'$, while the path that concatenates $(j,u)$ with $p(u, u')$ is the unique shortest path from $j$ to $u'$. It follows that

\begin{equation} \label{eqn:buprimei}
b_{u'i} = c_{p(u,u')} c_{iu} b_{ii}
\qquad \text{and} \qquad
b_{u'j} = c_{p(u,u')} c_{ju} b_{jj}.
\end{equation}

Combining these equalities, $c_{iu}>c_{ij}c_{ju}$ implies that we should have

\begin{equation} \label{eqn:buprime}
\frac{b_{u'i}}{b_{u'j}}>\frac{b_{j'i}}{b_{j'j}}.
\end{equation}

However from $a_{i,U}=a_{j,U}$ we have for the $u'$-th and $j'$-th elements of these vectors

\[
\Longleftrightarrow
\frac{b_{u'i}}{m_i}=\frac{b_{u'j}}{m_j}
\qquad \text{and} \qquad
\Longleftrightarrow
\frac{b_{j'i}}{m_i}=\frac{b_{j'j}}{m_j}
\]

Considering the ratios above, we should also have

\begin{equation} \label{eqn:aiuprime}
\frac{a_{i,u';U}}{a_{i,j';U}}=\frac{a_{j,u';U}}{a_{j,j';U}}
\qquad \Longleftrightarrow \qquad
\frac{b_{u'i}}{b_{u'j}}=\frac{b_{j'i}}{b_{j'j}}.
\end{equation}

Because of (<ref>) the equalities in (<ref>) cannot happen, hence we cannot have $a_{i,U}=a_{j,U}$.

The analysis of the cases $(j, s) \in U \times \bar{U}$ and $(j, s) \in \bar{U} \times U$ is similar.
This shows that all $|V|$ vectors $a_{i,U}$ for $i \in V$ are different and because of (<ref>), all vectors $\omega_r$ for $r \in \{1,\ldots,|V|\}$ are different too.

* Distinguishing all atoms $a_{i,U}$ with zeroes on the same positions.

For two different nodes $i, j \in V$, the atoms $a_{i,U}$ and $a_{j,U}$ have the same supports $\{v\in U: a_{vi}>0\}=\{v\in U: a_{vj}>0\}$ when $\Desc(i) \cap U = \Desc(j)\cap U$. By Lemma <ref>-5 there cannot be any other node $k \in V \setminus \{i, j\}$ with the same descendants in $U$. In the representation

\[
= \sum_{t \in V} m_{t,U} \delta_{a_{t,U}}
= \sum_{r = 1}^{|V|} \mu_r \delta_{\omega_r}
\]

there are thus exactly two atoms, $\omega$ and $\omega'$, say, with the same indices of non-zero coordinates as $a_{i,U} = (b_{vi} / m_{i,U})_{v \in V}$ and $a_{j,U} = (b_{vj} / m_{j,U})_{v \in V}$. The question is then how to know whether $\omega = a_{i,U}$ and $\omega' = a_{j,U}$ or vice versa, $\omega = a_{j,U}$ and $\omega' = a_{i,U}$. Let $\mu$ and $\mu'$ be the masses of $\omega$ and $\omega'$, respectively, and consider the vectors $\beta = \mu \omega$ and $\beta' = \mu' \omega'$. An equivalent question is then how to identify $\beta$ and $\beta'$ with the two max-linear coefficient vectors $(b_{vi})_{v \in U}$ and $(b_{vj})_{v \in U}$.

By Lemma <ref>-2 to 4, we can suppose that $i$ is the unique parent of $j$ and that $i$ and $j$ have a common child $u$. The analysis is now to be split up into different cases, according to whether $j$ and $u$ belong to $U$ or not. Recall that $b_{vi} = c_{p(i,v)} b_{ii}$ and $b_{vj} = c_{p(j,v)} b_{jj}$ for $v \in \Desc(j) \subset \Desc(i)$.

Suppose first that $j, u \in U$. From (<ref>) we deduce (<ref>) thanks to the criticality assumption.

In order to make the correct assignment of the two vectors $\beta=(\beta_v)_{v\in U}$ and $\beta'=(\beta'_v)_{v\in U}$ to the nodes $i$ and $j$, we need to check the inequality (<ref>). If $\beta_u / \beta_j > \beta'_u / \beta'_j$ then we assign the vector $\beta$ to the node $i$ and the vector $\beta'$ to the node $j$. If the equality is reversed, we do the assignment the other way around.

Next suppose that $j, u \in \bar{U}$. According to Lemma <ref>, there exist nodes $j', u'\in U$ so that there is a unique path from $j$ to $j'$ and from $u$ to $u'$.
By Lemma <ref>, the paths from $i$ to $u'$ and $j'$ and from $j$ to $j$ to $u'$ are

\begin{align*}
p(i,u') &= \{(i,u)\} \cup p(u,u'), &
p(i,j') &= \{(i,j)\} \cup p(j,j'), &
p(j,u') &= \{(j,u)\} \cup p(u,u').
\end{align*}

We have the same identities in (<ref>) and (<ref>) which, together with the criticality assumption, lead to the inequality (<ref>).

In order to make the correct assignment of the two vectors $\beta=(\beta_v)_{v\in U}$ and $\beta'=(\beta'_v)_{v\in U}$ we do as above for the case $j,u\in U$.

For the cases $(j,u) \in U \times \bar{U}$ and $(j, u) \in \bar{U} \times U$, we combine methods from the cases $(j,u) \in U \times U$ and $(j,u) \in \bar{U} \times \bar{U}$.

With this we finish the proof that we can learn the structure of every atom $\{\omega_r : r = 1, \ldots, |V|\}$, i.e., for every $r=1, \ldots, |V|$ we can identify the unique node $i\in V$ such that $\omega_r=a_{i,U}=(b_{vi}/m_{i,U})_{v\in U}$. This means that we can also match every element $\beta$ in the collection of vectors $\{\beta_r : r = 1, \ldots, |V|\}$ to the correct node $i \in V$ such that $\beta = (b_{vi})_{v \in U}$.

* Step 2.
In the previous step, we have shown that the distribution of $X_U$ (together with the knowledge of the graph structure) determines the max-linear coefficient matrix $b_{U \times V} = (b_{vi})_{v \in U, i \in V}$. Here, we show that this matrix suffices to reconstruct the vector of edge coefficients $\theta = (c_e)_{e \in E}$.

If $v$ is a child of $i$, then $p(i,v) = \{(i,v)\}$ and thus $b_{vi} = c_{iv} b_{ii}$. If both $i$ and $v$ belong to $U$, then, clearly, we can identify $c_{iv} = b_{vi} / b_{ii}$.

Let $i\in U$ with child $v \in \ch(i) \cap \bar{U}$. By Lemma <ref> there exists a node $v'\in U$ such that there is a unique path from $v$ to $v'$. Rewrite $v=v_1$ and $v'=v_n$ and consider the node set $\{v_1, v_2, \ldots, v_n\}$ on that  unique path. Using the fact that if a node $\ell$ has a single parent $k$, then $b_{\ell\ell} = 1 - c_{k \ell}$ (see (<ref>)), we find the following identities for the max-linear coefficients $b_{v_nj}$ for $j \in \{v_n,\ldots,v_2,i\}$:

\begin{equation} \label{eqn:binin}
\begin{split}
b_{v_nv_n} &= 1 - c_{v_{n-1}v_n},\\
b_{v_nv_{n-1}} &= c_{p(v_{n-1},v_n)} b_{v_{n-1}v_{n-1}}
= c_{v_{n-1} v_n} \left( 1-c_{v_{n-2}v_{n-1}} \right),\\
b_{v_nv_{2}} &= c_{p(v_{2},v_n)} b_{v_2v_2}
= c_{v_2 v_3} \cdots c_{v_{n-1}v_n} \left( 1-c_{v_{1}v_{2}} \right),\\
%		b_{i_n,i_{1}} &= c_{p(i_{1},i_n)} b_{i_1,i_2}
%		= c_{i_1, i_2} c_{i_2, i_3} \cdots c_{i_{n-1}, i_n } \left( 1-c_{v,i_1} \right).
=c_{iv_1}c_{v_1v_2}c_{v_2v_3}\cdots c_{v_{n-1}v_n}b_{ii}.\\
\end{split}
\end{equation}

From the first equation we identify $c_{v_{n-1}v_n}$, from the second $c_{v_{n-2}v_{n-1}}$ and so on until we identify $c_{v_1v_2}$ from the penultimate equation. From the last equation we can identify $c_{iv_1}$ because $b_{ii}$ is available from $b_{U \times V}$ in view of $i \in U$.

The next step of the proof is to extract the edge parameters between a node with latent variable and its children.

Let $i\in \bar{U}$. We will show that we can identify all edge weights $c_{ij}$ for $j \in \ch(i)$. Because $i$ belongs to $\bar{U}$, it should have at least two children, say $v$ and $\bar{v}$. Take $v$ to be a node whose only parent is $i$. Note that we can always find such a node by Lemma <ref>.
Let us first assume $v,\bar{v}\in U$. Because $v$ has only one parent, we have $b_{vv} = 1-c_{iv}$ and thus $c_{iv}=1-b_{vv}$. We also know $b_{vi}=c_{iv}b_{ii}$ and from here $b_{ii}=b_{vi}/c_{iv}=b_{vi}/(1-b_{vv})$. From $b_{\bar{v} i}=c_{i\bar{v}}b_{ii}$ we deduce $c_{i\bar{v}}=b_{\bar{v} i}(1-b_{vv})/b_{vi}$. Hence we have identified all the edge parameters related to children of $i$ which are observable, provided $i$ has two or more children in $U$.

Next assume that both $v,\bar{v}\in \bar{U}$. By Lemma <ref>, there exists a node $v'\in \desc(v) \cap U$ such that there is a unique path from $v$ to $v'$ and which has the properties in the cited statement. Let the sequence of nodes along which the path passes be denoted by $\{v_1=v, v_2, \ldots, v_n=v'\}$.
Using again that for a node $\ell$ with single parent $k$, $b_{\ell\ell} = 1 - c_{k \ell}$ (see (<ref>)), we find the same identities as in (<ref>) for the max-linear coefficients $b_{v_nj}$ for $j \in \{v_n,\ldots,v_2\}$. From the first equation we obtain $c_{v_{n-1} v_n} = 1 - b_{v_nv_n}$, from the second equation $c_{v_{n-2}v_{n-1}} = 1 - b_{v_nv_{n-1}}/(1-b_{v_nv_n})$ and so on until we obtain $c_{v_1 v_2}$ from the penultimate equation in (<ref>). Because we assumed $\pa(v_1)=\{i\}$ we have $b_{v_1v_1}=1-c_{iv_1}$ and thus

\begin{equation*}
b_{v_nv_{1}} = c_{p(v_{1},v_n)} b_{v_1v_1}
= c_{v_1 v_2} c_{v_2 v_3} \cdots c_{v_{n-1} v_n } \left( 1-c_{iv_1} \right),
\end{equation*}

from where we identify $c_{iv_1}$. Since $v_n \in U$, all coefficients $b_{v_nj}$ for $j \in V$ are contained in the max-linear coefficient matrix $b_{U \times V}$. The procedure just described thus allows us to compute all edge coefficients $c_{iv_1}, c_{v_1v_2}, \ldots, c_{v_{n-1}v_n}$.

Because the path $p(v_1, v_n)$ satisfies the properties in Lemma <ref>, and in view of the same lemma, plus the fact that the only parent of $v_1=v$ is $i$, it follows that the path $\{(i,v_1)\} \cup p(v_1,v_n)$ is the unique path between $i$ and $v_n$. Hence we have also

\begin{equation} \label{eqn:bvv}
b_{v_ni} = c_{p(i,{v_n})}b_{ii}
=c_{iv_1}c_{v_1 v_2}c_{v_2 v_3}\cdots c_{v_{n-1} v_n}b_{ii}.
\end{equation}

As $v_n \in U$, the value of $b_{v_n i}$ is known from $b_{U \times V}$. The edge coefficients on the right-hand side were expressed in terms of $b_{U \times V}$ in the previous paragraph. From there, we obtain the value of $b_{ii}$, which will be used next.

Now consider the node $\bar{v}$, renamed to $\bar{v}_1$, which was an arbitrary child of $i$. For $\bar{v}=\bar{v}_1$ too, we can find a node $\bar{v}_m \in U$ and a sequence of nodes $\{\bar{v}_2, \ldots, \bar{v}_m\}$ according to Lemma <ref> satisfying the properties stated there.
For all $r \in \{2,\ldots,m\}$, the node $\bar{v}_r$ has a unique parent, $\bar{v}_{r-1}$. We have the following equalities, using again by $b_{\ell\ell} = 1 - c_{k\ell}$ for a node $\ell$ with unique parent $k$

\begin{equation*}
\begin{split}
&= 1-c_{\bar{v}_{m-1}\bar{v}_m}, \\
&= c_{p(\bar{v}_{m-1},\bar{v}_m)} b_{\bar{v}_{m-1}\bar{v}_{m-1}}
= c_{\bar{v}_{m-1} \bar{v}_m} \, \left( 1-c_{\bar{v}_{m-2}\bar{v}_{m-1}} \right),\\
&= c_{p(\bar{v}_{2},\bar{v}_m)} b_{\bar{v}_2 \bar{v}_2}
= c_{\bar{v}_2 \bar{v}_3} \cdots c_{\bar{v}_{m-1} \bar{v}_m} \left( 1-c_{\bar{v}_{1}\bar{v}_{2}} \right),\\
=c_{i\bar{v}_1}c_{\bar{v}_1 \bar{v}_2}c_{\bar{v}_2 \bar{v}_3}\cdots c_{\bar{v}_{m-1} \bar{v}_m}b_{ii} .
\end{split}
\end{equation*}

In the last equality we used that the path $\{(i,\bar{v}_1)\}\cup p(\bar{v}_1, \bar{v}_m)$ is the unique shortest one between $i$ and $\bar{v}_m$, because of Lemma <ref> and because $p(\bar{v}_1, \bar{v}_m)$ satisfies the properties 1-2 of the same lemma.
Since $\bar{v}_m \in U$, the values of the left-hand sides in the previous equations are contained in the given matrix $b_{U \times V}$.
From the first equality we obtain $c_{\bar{v}_{m-1}\bar{v}_m}$, from the second one $c_{\bar{v}_{m-2}\bar{v}_{m-1}}$ and so on until we identify $c_{\bar{v}_{1}\bar{v}_{2}}$ from the penultimate equality, i.e., all edge parameters linked to $p(\bar{v}_1,\bar{v}_m)$. In the last equation above we replace $b_{ii}$ with the expression derived from (<ref>) and we obtain the parameter $c_{i\bar{v}_1}$.

If some of the children of $i$ are in $U$ and some others are in $\bar{U}$, we apply a combination of the techniques used in the two cases described above – when two children are in $U$ or when two children are in $\bar{U}$.

This concludes the proof of the sufficiency (if) part.

§§ before inverting the roles of i and v. Here v is a parent of i.
Let $v\in U$ and its child $i\in \bar{U}$. By Lemma <ref> we have a node $i'\in U$ and a unique path from $i$ to $i'$. Rewrite $i=i_1$ and $i'=i_n$ and consider the unique path $\{i_1, i_2, \ldots, i_n\}$. 	Using the fact that if a node $\ell$ has a single parent $k$, then $b_{\ell\ell} = 1 - c_{k \ell}$ (see (<ref>)),
we find the following identities for the max-linear coefficients $b_{i_n,j}$ for $j \in \{i_n,\ldots,i_2,v\}$:

\begin{equation} \label{eqn:binin}
\begin{split}
b_{i_n,i_n} &= 1 - c_{i_{n-1},i_n},\\
b_{i_n,i_{n-1}} &= c_{p(i_{n-1},i_n)} b_{i_{n-1},i_{n-1}}
= c_{i_{n-1}, i_n} \left( 1-c_{i_{n-2},i_{n-1}} \right),\\
b_{i_n,i_{2}} &= c_{p(i_{2},i_n)} b_{i_2,i_2}
= c_{i_2, i_3} \cdots c_{i_{n-1}, i_n} \left( 1-c_{i_{1},i_{2}} \right),\\
%		b_{i_n,i_{1}} &= c_{p(i_{1},i_n)} b_{i_1,i_2}
%		= c_{i_1, i_2} c_{i_2, i_3} \cdots c_{i_{n-1}, i_n } \left( 1-c_{v,i_1} \right).
=c_{v,i_1}c_{i_1,i_2}c_{i_2,i_3}\cdots c_{i_{n-1},i_n}b_{vv}.\\
\end{split}
\end{equation}

From the first equation we identify $c_{i_{n-1},i_n}$, from the second $c_{i_{n-2},i_{n-1}}$ and so on until we identify $c_{i_1,i_2}$ from the penultimate equation. From the last equation we can identify $c_{v,i_1}$ because $b_{vv}$ is available.

The next step of the proof is to extract the edge parameters between a node with latent variable and its children.

Let $v\in \bar{U}$. In the light of earlier notation, it would be more convenient to denote the parent by $i$. In what earlier notation? In the previous paragraph? To me it looks easier to change the previous paragraph than all what follows next. I don't understand your considerations. Please see my reply above. We will show that we can identify all edge weights $c_{vj}$ for $j \in \ch(v)$. Because $v \in \bar{U}$, it should have at least two children, say $i$ and $\bari$.

Take $i$ to be a node whose only parent is $v$. Note that we can always find such a node by Lemma <ref>.

Let us first assume $i,\bari\in U$. Because $i$ has only one parent we have $b_{ii} = 1-c_{vi}$

and from here $c_{vi}=1-b_{ii}$. We know also $b_{iv}=c_{vi}b_{vv}$ and from here $b_{vv}=b_{iv}/c_{vi}=b_{iv}/(1-b_{ii})$. From $b_{\bari v}=c_{v\bari}b_{vv}$ we have $c_{v\bari}=b_{\bari v}(1-b_{ii})/b_{iv}$. Hence we have identified all the edge parameters related to children of $v$ which are observable, provided $v$ has two or more children in $U$.

Next assume that both $i,\bari\in \bar{U}$. By Lemma <ref>, there exists a node $s\in \desc(i) \cap U$ such that there is a unique path from $i$ to $s$ and which has the properties in the cited statement. Let the sequence of nodes along which the path passes be denoted by $\{i_1=i, i_2, \ldots, i_n=s\}$.

Using again that for a node $\ell$ with single parent $k$, $b_{\ell\ell} = 1 - c_{k \ell}$, It's the same identity as the one used above; to be mentioned somewhere around (<ref>) perhaps? see earlier comment. we find the same identities as in (<ref>) for the max-linear coefficients $b_{i_n,j}$ for $j \in \{i_n,\ldots,i_2\}$. From the first equation we obtain $c_{i_{n-1}, i_n} = 1 - b_{i_n,i_n}$, from the second equation $c_{i_{n-2},i_{n-1}} = 1 - b_{i_n,i_{n-1}}/(1-b_{i_n,i_n})$ and so on until we obtain $c_{i_1, i_2}$ from the prelast equation in (<ref>). Because we assumed $\pa(i_1)=\{v\}$ we have $b_{i_1,i_1}=1-c_{v,i_1}$ and thus

\begin{equation*}
%	\begin{split}
%		b_{i_n,i_n} &= 1 - c_{i_{n-1},i_n},\\
%		b_{i_n,i_{n-1}} &= c_{p(i_{n-1},i_n)} b_{i_{n-1},i_{n-1}}
%		= c_{i_{n-1}, i_n} \left( 1-c_{i_{n-2},i_{n-1}} \right),\\
%		&\vdots\\
%		b_{i_n,i_{2}} &= c_{p(i_{2},i_n)} b_{i_2,i_2}
%		= c_{i_2, i_3} \cdots c_{i_{n-1}, i_n} \left( 1-c_{i_{1},i_{2}} \right),\\
b_{i_n,i_{1}} = c_{p(i_{1},i_n)} b_{i_1,i_1}
= c_{i_1, i_2} c_{i_2, i_3} \cdots c_{i_{n-1}, i_n } \left( 1-c_{v,i_1} \right),
%	b_{i_n,v}&=p\pth{v}{i_n}_*b_{vv}.\\
%	\end{split}
\end{equation*}

from where we identify $c_{v,i_1}$. Since $i_n \in U$, all coefficients $b_{i_n,j}$ for $j \in V$ are contained in the max-linear coefficient matrix $b_{U \times V}$. The procedure just described thus allows us to compute all edge coefficients $c_{v,i_1}, c_{i_1,i_2}, \ldots, c_{i_{n-1},i_n}$.

Because the path $p(i_1, i_n)$ satisfies the properties in Lemma <ref>, and in view of the same lemma, plus the fact that the only parent of $i_1=i$ is $v$, it follows that the path $\{(v,i_1)\} \cup p(i_1,i_n)$ is the unique path between $v$ and $i_n$. Hence we have also

\begin{equation} \label{eqn:bvv}
b_{i_n,v} = c_{p(v,{i_n})}b_{vv}=c_{v,i_1}c_{i_1, i_2}c_{i_2, i_3}\cdots c_{i_{n-1}, i_n}b_{vv}.
\end{equation}

As $i_n \in U$, the value of $b_{i_n, v}$ is known from $b_{U \times V}$. The edge coefficients on the right-hand side were expressed in terms of $b_{U \times V}$ in the previous paragraph. From there, we obtain the value of $b_{vv}$, which will be used next.

Now consider the node $\bari$, renamed to $\bari_1$, which was an arbitrary child of $v$. For $\bari=\bari_1$ too we can find a node $\bari_m \in U$ and a sequence of nodes $\{\bari_2, \ldots, \bari_m\}$ according to Lemma <ref> satisfying the properties stated there.

For all $r \in \{2,\ldots,m\}$, the node $\bari_r$ has a unique parent, $\bari_{r-1}$. We have the following equalities, using again by $b_{\ell\ell} = 1 - c_{k\ell}$ for a node $\ell$ with unique parent $k$

\begin{equation*}
\begin{split}
&= 1-c_{\bari_{m-1},\bari_m}, \\
&= c_{p(\bari_{m-1},\bari_m)} b_{\bari_{m-1},\bari_{m-1}}
= c_{\bari_{m-1}, i_m} \, \left( 1-c_{\bari_{m-2},\bari_{m-1}} \right),\\
&= c_{p(\bari_{2},\bari_m)} b_{\bari_2, \bari_2}
= c_{\bari_2, \bari_3} \cdots c_{\bari_{m-1}, \bari_m} \left( 1-c_{\bari_{1},\bari_{2}} \right),\\
b_{\bari_m,v}&=c_{p(v,\bari_m)}b_{vv}=c_{v,\bari_1}c_{\bari_1, \bari_2}c_{\bari_2, \bari_3}\cdots c_{\bari_{m-1}, \bari_m}b_{vv} .
%	b_{i_n,v}&=p\pth{v}{i_n}_*b_{vv}.\\
\end{split}
\end{equation*}

In the last equality we used that the path $\{(v,\bari_1)\}\cup p(\bari_1, \bari_m)$ is the unique shortest one between $v$ and $\bari_m$, because of Lemma <ref> and because $p(\bari_1, \bari_m)$ satisfies the properties 1-2 of the same lemma.
Since $\bari_m \in U$, the values of the left-hand sides in the previous equations are contained in the given matrix $b_{U \times V}$.
From the first equality we obtain $c_{\bari_{m-1},\bari_m}$, from the second one $c_{\bari_{m-2},\bari_{m-1}}$ and so on until we identify $c_{\bari_{1},\bari_{2}}$ from the penultimate equality, i.e., all edge parameters linked to $p(\bari_1,\bari_m)$. In the last equation above we replace $b_{vv}$ with the expression derived from (<ref>) and we obtain the parameter $c_{v,\bari_1}$.

If some of the children of $v$ are in $U$ and some are in $\bar{U}$, we apply a combination of the techniques used in the two cases described above – when two children are in $U$, and when all two children are in $\bar{U}$.

§ PROOF NECESSITY OF IDENTIFIABILITY PROBLEM

To be merged with the previous section.

to be checked once again! In particular - does the proof for a single tournament needs to be by induction? The part Necessity of the theorem.

For the case of a single tournament, I don't think induction is needed. See the alternative proof, Lemma <ref>.

Nice corollary, but perhaps no longer needed in the new necessity proof.

Let $H_{\theta}^{(\tau)}$ be the angular measure with atoms $a_{i}^{\tau}= (b^{\tau}_{ji}/m_i^{\tau}, j\in V_{\tau})$ and masses $m_i^{\tau}$ for $i\in V_{\tau}$ related to a max-linear model as in Assumption <ref> with respect to a single tournament $\tau=(V_{\tau}, E_{\tau})$. Let $H_{\theta}$ be the angular measure as in (<ref>) related to a max-linear model with respect to a ttt, $\T$, which contains at least $\tau$. Then

\begin{align*}
b_{ij}^{\tau}&=b_{ij},  && i\in V_{\tau}, j\in V_{\tau}\setminus s,\\
b_{is}^{\tau}b_{ss}&=b_{is},  && i\in V_{\tau},
\end{align*}

where $s$ is the source of $\tau$.

For every $j\in V_{\tau}\setminus s$ we have $b_{jj}$ depending only on $(c_e, e\in E_{\tau})$ by Lemma <ref> so $b_{jj}=b_{jj}^{\tau}$. Then for all $i\in V_{\tau}$ and $j\in V_{\tau}\setminus s$ with $i \ne j$ we have  $b_{ij}=c_{p(j,i)}b_{jj}=c_{ji}b_{jj}=c_{ji}b_{jj}^{\tau}=b_{ij}^{\tau}$  if $i \in \desc(j)$ and $b_{ij} = b_{ij}^\tau = 0$ if $i \not\in \desc{j}$. However, if $s$ has parents, they are necessarily in a different tournament than $\tau$, so $b_{ss}\neq 1=b_{ss}^{\tau}$. But then $b_{is}=c_{p(s,i)}b_{ss}=b_{is}^{\tau}b_{ss}$ for all $i\in V_{\tau}$.

I'm not yet 100% sure about the proof of the next lemma; see the questions in the proof. Lemma <ref> and its proof give an alternative approach.

Let $X$ follow a max-linear model as in Assumption <ref> with respect to a ttt $\T$ consisting of a single tournament $\tau=(V,E)$. If the node $v\in V$ has at least one parent and at least one child, then the parameter vector $\theta=(c_e,e\in E)$ is not identifiable from the distribution of $X_{V \setminus v}$.

Consider a node $v$ with at least one parent and at least one child. We will proceed by induction on the number of children for a fixed number of parents. So in the initial step let $|\pa(v)|\geq 1$ and $\ch(v)=\{w\}$. Note that because of the in-/out-degree ordering in a tournament, we have $\Pa(v)=\pa(w)=V\setminus w$. Also put $U=V\setminus v$.

The angular measure $H_{\theta,U}$ is determined by atoms $\omega_r$ with masses $\mu_r$ as in (<ref>). Recall also the representation of $H_{\theta,U}$ in terms of $a_{i,U}$ and $m_{i,U}$ in (<ref>).

Since $\Desc(v)\cap U=\Desc(w)\cap U=\{w\}$, we have $a_{v,U}=a_{w,U}=(0,\ldots, 0,1)$, a vector of length $|V|-1$. Hence one of the atoms in $\{\omega_r\}$, say $\omega_0$, equals $(0,\ldots, 0,1)$. The corresponding masses, $m_{v,U}=b_{wv}$ and $m_{w,U}=b_{ww}$, should be summed up, giving $\mu_0=b_{wv}+b_{ww}$, which is the mass of $\omega_0$.
Because for all $i\neq j\in V\setminus \{v,w\}$ we have $\Desc(j)\cap U\neq \Desc(i)\cap U$, all atoms $\{a_{i,U}\}_{i\in V\setminus\{v,w\}}$ have distinct support sets (sets of zero coordinates) and hence the atoms $\{\omega_r\}\setminus \omega_0$ can be uniquely associated to those in $\{a_{i,U}\}_{i\in V\setminus\{v,w\}}$.

Therefore, the angular measure $H_{\theta,U}$ has $|V|-1$ atoms of dimension $|V|-1$: we have

\begin{equation} \label{eqn:atoms}
\{\omega_1, \ldots, \omega_{|V|-2}\} \cup \{\omega_0\}
%		\left(	\Big\{\Big(a_{j,U}=\frac{b_{ij}}{m_{j,U}}, i\in U\Big)\Big\}_{j\in }\cup \Big(0,\ldots, 0,\frac{b'_{n,q}}{b'_{n,q}}\Big)\right)
\bigl\{a_{i,U} : i\in V\setminus \{v,w\}\bigr\} \cup \{(0,\ldots, 0,1)\big\},
\end{equation}

where $a_{i,U}=(b_{ji}/m_{i,U}, j\in U)$ and with masses

\begin{equation} \label{eqn:mass}
\{\mu_1, \ldots, \mu_{|V|-2}\} \cup \{\mu_0\}
\bigl\{m_{i,U} : i\in V\setminus \{v,w\}\bigr\} \cup \big\{b_{wv}+b_{ww}\big\}.
\end{equation}

Let $V':= V\setminus \{v,w\} = \pa(V)$. The only coefficients that contain the edge weights $(c_{iv}, i\in V')$ and $(c_{iw}, i\in \pa(w)=V\setminus w)$ are $(b_{wj}, j\in V', b_{wv}+b_{ww})$. This means that we have $|V|-2$ independent equations to identify $|V|-2+|V|-1$ parameters which is impossible. We note that $b_{wv}+b_{ww}$ is a linear combination of $(b_{wj}, j\in V')$. Why? We have $b_{wv}+b_{ww} = \sum_{j \in \{v,w\}} b_{wj}$, but $v, w \not\in V'$? For all $i\in \pa(w)\setminus v=V\setminus \{v,w\}=V'$ we have $b_{wi}=c_{iw}b_{ii}$ and hence $c_{iw}=b_{wi}/b_{ii}$, which shows that we can compute the values  $(c_{iw}, i\in \pa(w)\setminus v)$.

However, the edge weights $c_{vw}, \big(c_{iv}, i\in \pa(v)\big)$ are free This is what we need to show. How have we shown it now? Note also that the edge weights are not completely free. There is only one “degree of freedom” (one more unknown than there are equations). and whatever value they may take, the angular measure with atoms in (<ref>) and masses in (<ref>) is the same. This concludes the initial step of the induction.

The induction hypothesis is that for $|\ch(v)|=n-1\ge 1$ the edge weights $\big(c_{iv}, i\in \pa(v)\big)$
are free parameters in the angular measure $H_{\theta, U}$.

Let's add one more node, say $z$, and connect it with edges $(i,z), i\in V\setminus z$. We have $|\ch(v)|=n$. Let the set of parents of $v$ be $\pa(v)=\{u_1, \ldots, u_r\}$ with $|\pi(u_1,v)|=1, |\pi(u_2,v)|=2$, etc. We will show that $c_{vz}$ is dependent on $(c_{iv}, i\in \pa(v))$, hence the latter remain free parameters.

Let $w\in V$ be such that $\pi(v,w)=\{(v,w)\}$. For all $i\neq j\in V\setminus \{v,w\} $ we have $\Desc(i)\cap U\neq \Desc(j)\cap U$ and hence also $\{k\in U: b_{ki}>0\}\neq \{k\in U: b_{kj}>0\}$. This means that except for two atoms, $a_{v,U}$ and $a_{w,U}$, the other atoms $\{a_{i,U}\}_{i\in V\setminus \{v,w\}}$ have unique zero coordinates and hence there are $|V|-2$ atoms from the set  $\{\omega_r\}$ that can be associated to the atoms $\{a_{i,U}\}_{i\in V\setminus \{v,w\}}$.

We have however $\Desc(v)\cap U=\Desc(w)\cap U$ which means that we have $\{i\in U: b_{iv}>0\}=\{i\in U:b_{iw}>0\}$. Take $\bar{w}\in \ch(v)$. For $i=w$
we have $b_{wv}=c_{vw}b_{vv}$ and for $i=\bar{w}$ we have $b_{\bar{w}v}=c_{v\bar{w}}b_{vv}$ and $b_{\bar{w}w}=c_{w\bar{w}}b_{ww}$. Then the constraint $c_{v\bar{w}}>c_{vw}c_{w\bar{w}}$ arising from the requirement $\theta\in \mathring{\Theta}_*$ implies that we should have

\[
\]

For two vectors $\beta=(\beta_i, i\in U)$ and $\beta'=(\beta'_i, i\in U)$ from the angular measure $H_{\theta, U}$ we should check: if $\beta_{\bar{w}}/\beta_w>\beta'_{\bar{w}}/\beta_w'$ then $\beta=(b_{iv}, i\in U)$ and $\beta'=(b_{iw}, i\in U)$; if $\beta_{\bar{w}}/\beta_w<\beta'_{\bar{w}}/\beta_w'$ then $\beta=(b_{iw}, i\in U)$ and $\beta'=(b_{iv}, i\in U)$.

Therefore we can uniquely associate the remaining two atoms of $\{\omega_r\}$ to $a_{v,U}, a_{w,U}$.

Consider $(b_{zi}, i\in V)$. These are the only coefficients that contain edge weights $(c_{iz}, i\in \pa(z))$. Let's divide $V$ in disjoint subsets $\{\pa(z)\setminus v\}\cup \{v\}\cup \{z\}$. For all $i\in \pa(z)\setminus v$ we have $b_{zi}=c_{iz}b_{ii}$ and because $\{\pa(z)\setminus v\}\cap U=\pa(z)\setminus v$ we can obtain the values $c_{iz}=b_{zi}/b_{ii}$.
For $i=v$ we have $b_{zv}=c_{vz}b_{vv}$ and therefore $c_{vz}=b_{zv}/b_{vv}$,  but $b_{vv}$ is not available from the set of atoms $\{a_{i,U}\}_{i\in V}$. We know from Lemma <ref> that $b_{vv}$ depends on the edge weights on all paths to $v$, hence it is a function of $(c_{iv}, i\in \pa(v))$, which by the induction hypothesis are free parameters. Finally, note that the coefficient $b_{zz}$ is a linear combination of $(b_{zi}, i\in\pa(z))$, which have been already used, so $b_{zz}$ does not contain further information.

Alternative statement of the lemma with shorter but more abstract proof:

Somehow I don't feel convinced yet by the proof. We have a latent variable at a node $u \in \bar{U}$ such that (I1)–(I2) are not both satisfied. What are the possible cases? Are all cases covered? Also, notions of “free parameter” and “no new information is brought in”, “being available from” are not always very solid. But I guess the analysis can be made rigorous and that all the ideas are there.

New proof attempt: the two identifiability conditions (I1)–(I2) are first reformulated in Lemma <ref>, and next, the necessity proof follows.

Let $\T = (V, E)$ be a ttt with unique source. For $\bar{U} \subset V$, conditions <ref>–<ref> are equivalent to the following ones:

*  any $u \in \bar{U}$ has at least two children;

*  any $u \in \bar{U}$ is the source of some tournament in $\T$.

Conditions <ref> and <ref> are the same. The claim of Lemma <ref> is thus that if <ref> is fulfilled, then <ref> and <ref> are equivalent.

On the one hand, suppose <ref> and <ref> are satisfied; we will show that <ref> holds too. Let $u \in \bar{U}$. If $u$ belongs to only a single tournament, then $u$ is equal to the source of $\T$ and thus certainly the source of the tournament to which it belongs. If $u$ belongs to two or more tournaments, then it is the source of at least one of them, because otherwise it would have parents from two different tournaments, yielding a forbidden v-structure.

On the other hand, suppose <ref> and <ref> are satisfied; we will show that <ref> holds too. If $u \in \bar{U}$ belongs to only a single tournament, then it must be the source of that tournament. But then $u$ has no parents at all and is therefore equal to the source of $\T$.

Let $\bar{U} \subset V$ be such that at least one of the two conditions <ref> and <ref> is not satisfied and let $\theta = (c_e)_{e \in E} \in \mathring{\Theta}_*$. We will show that there exists another parameter $\theta' = (c_e')_{e \in E} \in \mathring{\Theta}_*$ such that $\theta' \ne \theta$ but the distribution of $X_{V \setminus u}$ under $\theta'$ is the same as the one under $\theta$.

We consider two cases: case (1) <ref> does not hold, i.e., there exists $u \in \bar{U}$ which is not the source of any tournament in $\T$; case (2) <ref> holds but not <ref>, i.e., every $u \in \bar{U}$ is the source of some tournament in $\T$ but there exists $u \in \bar{U}$ with less than two children.
Case (1): there exists $u \in \bar{U}$ which is not the source of any tournament in $\T$.
Then $u$ belongs to only a single tournament, say $\tau = (V_\tau, E_\tau)$; indeed, a node that belongs to two different tournaments must be the source of at least one of them, because otherwise it would have parents from two different tournaments, yielding a forbidden v-structure.

Let $X_{V_{\tau}}$ be a max-linear model restricted to a single tournament, $\tau$. The coefficients associated to $X_{V_{\tau}}$, denote by $b_{vi}^\tau$ for $v,i \in V_\tau$, are determined by the weights of the edges $e \in E_\tau$.
By Lemma <ref>, we can modify the edge weights $c_e$ for $e \in E_\tau$ in such a way that the max-linear coefficients $b_{vi}^\tau$ for $v \in V_\tau \setminus u$ and $i \in V_\tau$ remain unaffected.
Let $\tilde{c}_e$ for $e \in E_\tau$ denote such a modified vector of edge weights. Define $\theta' = (c_e')_{e \in E} \in \mathring{\Theta}_*$ by

\begin{align*}
c_e' &=
\begin{cases}
\tilde{c}_e & \text{if $e \in E_\tau$,} \\
c_e & \text{if $e \in E \setminus E_\tau$.}
\end{cases}
\end{align*}

Then $\theta' \in \mathring{\Theta}_*$ too: indeed, $c_{ii}' > 0$ for all $i \in V$ by assumption on $\theta$

and the fact the vector $(\tilde{c}_e : e \in E_\tau)$ satisfies $\tilde{c}_{ii} > 0$ for $i \in V_\tau$. By construction, the distribution of $X_{V_{\tau} \setminus u}$ is the same under $\theta'$ as under $\theta$.

We show that the distribution of $X_{V \setminus u}$ under $\theta'$ is the same as the one under $\theta$. We proceed by induction on the number of tournaments.

If $\T$ consists of a single tournament, then $\T = \tau$ and there is nothing more to show.

So suppose $\T$ consists of $m \ge 2$ tournaments. The skeleton graph of $\T$ is a block graph and thus a decomposable graph. By the running intersection property, we can order the tournaments $\tau_1,\ldots,\tau_m$ with node sets $V_1,\ldots,V_m$ in such a way that $\tau_1 = \tau$, the tournament containing $u$, and such that $V_m \cap (V_1 \cup \ldots \cup V_{m-1})$ is a a singleton, say $\{s\}$. Then $s \ne u$ since $u$ belongs to only a single tournament.

Write $W = V_1 \cup \ldots \cup V_{m-1} \setminus u$. The joint distribution of $X_{V \setminus u}$ can be factorized into two parts: first, the distribution of $X_{W}$ and second, the conditional distribution of $X_{V_m \setminus s}$ given $X_{W}$. It is sufficient to show that both parts remain the same when $\theta$ is replaced by $\theta'$.

* By the induction hypothesis, the distribution of $X_{W}$ is the same under $\theta'$ as under $\theta$.

* By the global Markov property (Proposition <ref>), the conditional distribution of $X_{V_m \setminus s}$ given $X_{W}$ is the same as the conditional distribution of $X_{V_m \setminus s}$ given $X_s$. But the latter is determined by the joint distribution of $X_{V_m}$, which, in turn, only depends on the weights of the edges $e$ in $\tau_m$. By construction, these edge weights are the same under $\theta$ as under $\theta'$. It follows that the conditional distribution of $X_{V_m \setminus u}$ given $X_W$ is the same under $\theta'$ as under $\theta$.

We conclude that the distribution of $X_{V \setminus u}$ is the same under $\theta'$ as under $\theta$. Since $\theta \ne \theta'$, the parameter is not identifiable.

Earlier attempt:
Let $s$ be the source node of $\tau$. Then $s \ne u$ by assumption, so that $s \in \pa(u) \subset V_\tau$.

For $v \in V \setminus u$, we have

\[
X_{v} = \bigvee_{i \in \An(v)} b_{vi}' Z_i,
\]

where $b'$ is the coefficient vector associated to $\theta'$. We claim that $b_{vi}' = b_{vi}$ for $v \in V \setminus u$ and $i \in \An(v)$. We consider several cases, depending on whether $v$ is a descendant of $s$ (the source of $\tau$) or not, $v$ belongs to $V_\tau$ or not, and is a descendant of any other node in $V_\tau$ or not.

* First, suppose $v \in V \setminus \desc(s)$. Then $b_{vi}'$ is a function of edge weights $c_{e}'$ for edges $e$ not in $E_\tau$. But $c_{e}' = c_{e}$ for such edges, so $b_{vi}' = b_{vi}$.

* Second, suppose $v \in \desc(s) \setminus V_\tau$ is not the descendant of any other node in $V_\tau$ but $s$. Then again, $b_{vi}$ is a function of edge weights $c_{e}'$ for edges not in $E_\tau$ and thus $b_{vi}' = b_{vi}$.

* Third, suppose $v \in V_\tau \setminus \{u, s\}$. Let $i \in \An(v)$. Then either $i \in V_\tau$ or $i \not\in V_\tau$. If $i \in V_\tau$, then, in the notation of Corollary <ref> referring to the parameter vector $\theta'$, we have

\[
b_{iv}' = (b_{iv}')^\tau = b_{iv}^\tau = b_{iv}
\]

by the choice of $(\tilde{c}_e)_{e \in E_\tau}$.
Is this correct? Can we identify the coefficients $b_{iv}$ from the distribution of $X_{V_\tau \setminus u}$?
If $i \not\in V_\tau$, then $i \in \an(s)$ and

\[
b_{iv}' = b_{ii}' p(i,s)' c_{sv}'.
\]

But $b_{ii}'$ and $p(i,s)'$ are the same under $\theta'$ as under $\theta$ since they do not involve edges in $E_\tau$, while $c_{sv}' = c_{sv}$ by Lemma <ref>. Hence, $b_{iv}' = b_{iv}$.

* Fourth, suppose $v \in \desc(s) \setminus V_\tau$ is the descendant of some node $w \in V_\tau \setminus s$. If $i \not\in \an(w)$, then proceed as in the previous step, then clearly $b_{vi}' = b_{vi}$ by arguments already made while if $i \in \an(w)$, then proceed as in the previous bullet point but with an extra factor $p(w,v)$ at appropriate places

Case (2): any $u \in \bar{U}$ is the source of some tournament in $\T$ but there exists $u \in \bar{U}$ with less than two children.
Any $u \in \bar{U}$ must have at least one child (a node without children cannot be the source of a tournament). But then there exists $u \in \bar{U}$ with exactly one child: $\ch(u) = \{w\}$. The tournament of which $u$ is the source can only consist of the nodes $u$ and $w$ and the edge $(u, w)$. Now there are two subcases, depending on whether $u$ has any parents or not.
Case (2).a: $u$ has no parents.
Then $u$ is the source node of $\T$ with a single child $w$.
Removing the node $u$ yields the ttt $\T_{\setminus u} := (V \setminus u, E \setminus \{(u, w)\})$ with single source $w$. The random vector $X_{V \setminus u}$ follows the recursive max-linear model (<ref>) with respect to $\T_{\setminus u}$. Its distribution is determined by the coefficients $c_e$ for $e \in E \setminus \{(u, w)\}$. The value of $c_{uw}$ can thus be chosen arbitrarily in $(0, 1)$ without affecting the distribution of $X_{V \setminus u}$.
Case (2).b: $u$ has parents.
Any ancestor of $w$ different from $u$ must be an ancestor of $u$ too, since otherwise there would be a v-structure at $w$
; therefore,

\begin{equation*}
%		\label{eq:Anw}
\An(w) = \an(u) \cup \{u, w\}.
\end{equation*}

Let $\lambda > 0$ and close enough to $1$ (as specified below). Define $\theta' = (c_e')_{e \in E}$ by modifying the weights of edges adjacent to $u$: specifically,

\begin{align*}
c_{ju}' &= \lambda c_{ju}, \qquad j \in \pa(u); \\
c_{uw}' &= \lambda^{-1} c_{uw}; \\
c_{e}' &= c_e, \qquad e \in E \setminus [\{(j,u) : j \in \pa(u)\} \cup \{(u, w)\}].
\end{align*}

In words, $\theta'$ coincides with $\theta$ for edges $e$ that do not involve $u$, and $\theta' = \theta$ if and only if $\lambda = 1$. Since the parameter space $\mathring{\Theta}_*$ is open, $\theta'$ belongs to $\mathring{\Theta}_*$ for $\lambda$ sufficiently close to $1$. We claim that the distribution of $X_{V \setminus u}$ is invariant under $\lambda$. Hence, for $\lambda$ different from but sufficiently close to $1$, we have found a parameter $\theta' \neq \theta$ producing the same distribution of $X_{V \setminus u}$ as $\theta$.

Under $\theta'$, the random vector $X_{V \setminus u}$ follows the max-linear model

\[
X_v = \bigvee_{i \in \An(v)} b_{vi}' Z_i, \qquad v \in V \setminus u,
\]

where $(Z_i)_{i \in V}$ is a vector of independent unit-Fréchet random variables and where the coefficients $b_{vi}'$ are given by equations (<ref>), (<ref>) and (<ref>) with $c_e$ replaced by $c_e'$.

First, suppose $v \in V \setminus u$ is not a descendant of $u$. Then for any $i \in \An(v)$, the coefficient $b_{vi}'$ is a function of edge weights $c_e'$ for edges $e \in E$ different from $(u, w)$ and from $(j, u)$ for $j \in \pa(u)$. It follows that $c_e' = c_e$ for such edges and thus $b_{vi}' = b_{vi}$ for $v \in V \setminus \Desc(u)$ and $i \in \An(v)$.

Second, suppose $v \in \desc(u)$. Then necessarily $v \in \Desc(w)$ too and for any $i \in \An(u)$, the path $p(i, v)$ passes by (or arrives in) $w$. Furthermore, any ancestor of $v$ not in $\An(w)$ is a descendant of $w$:

\begin{equation}
\label{eq:Anv}
\An(v) = \an(u) \cup \{u, w\} \cup [\desc(w) \cap \An(v)].
\end{equation}

It follows that

\[
X_v = \bigvee_{i \in \an(u)} b_{vi}' Z_i
\vee \left( b_{vu}' Z_u \vee b_{vw}' Z_w \right)
\vee \bigvee_{i \in \desc(w) \cap \An(v)} b_{vi}' Z_i,
\]

where the last term on the right-hand side is to be omitted if $v = w$. We treat each of the three terms on the right-hand side separately.

* For $i \in \an(u)$, we have

\[
b_{vi}' = c_{ii}' c_{p(i,v)}' = c_{ii}' c_{p(i,u)}' c_{uw}' c_{p(w,v)}'
\]

where $c_{p(w,v)}' = 1$ if $v = w$. The coefficients $c_{ii}'$ and $c_{p(w, v)}'$ only involve weights $c_e'$ for edges $e \in E$ different from $(u, w)$ and $(j, u)$ for $j \in \pa(u)$; it follows that $c_{ii}' = c_{ii}$ and $c_{p(w,v)}' = c_{p(w,v)}$. Further, given $i \in \an(u)$ there exists $j \in \pa(u)$ such that $p(i, u)$ passes by $j$ right before reaching $u$ (with $i = j$ if $i \in \pa(u)$), and then

\[
c_{p(i,u)}' c_{uw}'
= c_{p(i,j)}' c_{ju}' c_{uw}'
= c_{p(i,j)}' (\lambda c_{ju}) (\lambda^{-1} c_{uw}).
\]

Since $p(i,j)$ does not involve edges meeting $u$, we find that $c_{p(i,j)}' = c_{p(i,j)}$, so that the above expression does not depend on $\lambda$. We conclude that $b_{vi}' = b_{vi}$ for $i \in \an(u)$.

* If $v \in \desc(w)$ and $i \in \desc(w) \cap \An(v)$, the coefficient $b_{vi}'$ is

\[
b_{vi}' = c_{ii}' c_{p(i,v)}'.
\]

The path $p(i,v)$ does not involve edges touching $u$, so $c_{p(i,v)'} = c_{p(i,v)}$.
By Lemma <ref>, the coefficient $c_{ii}' = b_{ii}'$ is a function of the edge weights $c_e'$ for edges $e$ in the tournament shared by $i$ and its parents. Since $i \in \desc(w)$ and since $w$ is the only child of $u$, none of these edges touches $u$, and thus $c_{e}' = c_{e}$ for all such edges.

It follows that $c_{ii}' = c_{ii}$ too. We conclude that $b_{vi}' = b_{vi}$ for $v \in \desc(w)$ and $i \in \desc(w) \cap \An(v)$.

* The random variable $b_{vu}' Z_u \vee b_{vw}' Z_w$ is independent of all other variables $Z_i$ for $i \in V \setminus \{u,w\}$ and its distribution is equal to $(b_{vu}' + b_{vw}') Z$ for $Z$ a unit-Fréchet variable.
We will show that $b_{vu}' + b_{vw}'$ does not depend on $\lambda$.
Since $1 = \sum_{i \in \An(v)} b_{vi}'$, the partition (<ref>) yields

\[
b_{vu}' + b_{vw}'
= 1 - \sum_{i \in \an(u)} b_{vi}' - \sum_{i \in \desc(w) \cap \An(v)} b_{vi}'.
\]

(The last sum on the right-hand side is zero if $v$ is not a descendant of $w$.)
In the two previous bullet points, we have already shown that the coefficients $b_{vi}'$ for $i$ in $\an(u)$ or $\desc(w) \cap \An(v)$ do not depend on $\lambda$.
By the stated identity, the sum $b_{vu}' + b_{vw}'$ then does not depend on $\lambda$ either.

We have thus shown that if $\bar{U}$ does not satisfy <ref>–<ref>, then we can find $u \in \bar{U}$ such that the distribution of $X_{V \setminus u}$ is the same under $\theta'$ as under $\theta$. As $\theta' \ne \theta$ by construction, this means that the parameter $\theta$ is not identifiable from the distribution of $X_{V \setminus u}$. But as $U = V \setminus \bar{U} \subseteq V \setminus \{u\}$, the parameter $\theta$ is not identifiable from the distribution of $X_U$ either. This confirms the necessity of <ref>–<ref> for the identifiability of $\theta$ from the distribution of $X_U$.

Older attempt, probably to be discarded:

Let $u \in \bar{U}$ be such that at least one of the two conditions <ref> and <ref> is not satisfied. We will show that there exists two different parameters $\theta$ and $\theta'$ in $\mathring{\Theta}_*$ such that the distribution of $X_{V \setminus u}$ is the same under $\theta$ and under $\theta'$.

We consider two cases: first, when <ref> is not satisfied, and second, when <ref> is satisfied but <ref> is not. In all cases, it is useful to remember that in view of (<ref>)–(<ref>), the distribution of $X_{V \setminus u}$ is determined by the coefficients $b_{vi}$ for $v \in V \setminus u$ and $i \in \An(v) = \an(v) \cup \{v\}$. But since $b_{vv} = 1 - \sum_{i \in \an(v)} b_{vi}$, see (<ref>), the distribution of $X_{V \setminus u}$ is determined by the coefficients $b_{vi}$ for $v \in V \setminus u$ and $i \in \an(v)$. The latter coefficients are a function of the edge weights $c_{ab}$ for edges $(a, b) \in E$ such that $b \in \An(v)$ for some $v \in V \setminus u$; see (<ref>) and (<ref>). We conclude that the distribution of $X_{V \setminus u}$ is determined by the edge weights $c_{ab}$ for edges $(a, b) \in E$ such that $b \in \An(v)$ for some $v \in V \setminus u$.

Case 1: <ref> is not satisfied.
This means that there exists $u \in \bar{U}$ with less than two children. We consider two subcases: $u$ has no children at all or $u$ has exactly one child.
Case 1.a: there exists $u \in \bar{U}$ without children.
Since $u$ has no children, it does not belong to $\An(v)$, and this for all $v \in V \setminus u$. By the argument made at the start of the proof, the coefficients $b_{vi}$ for $v \in V \setminus u$ and $i \in \an(v)$ do not depend at all on the edge weights $c_{iu}$ for $i \in \an(u)$. We may thus alter the values of those edge weights without changing the distribution of $X_{V \setminus u}$.
Case 1.b: there exists $u \in \bar{U}$ with exactly one child.
Let $\ch(u) = \{w\}$. Any parent of $w$ different from $u$ must be a parent of $u$ too, since otherwise there would be a v-structure at $w$; therefore,

\begin{equation}
\label{eq:Anw}
\An(w) = \an(u) \cup \{u, w\}.
\end{equation}

We consider two further subcases, depending on whether $w$ has any children or not.

Or rather depending on whether $w$ has other parents but $u$.
Case 1.b.i: there exists $u \in \bar{U}$ with $\ch(u) = \{w\}$ and $\ch(w) = \varnothing$.
For all $v \in V \setminus \{u, w\}$, the nodes $u$ and $w$ do not belong to $\an(v)$.
By the argument at the start of the proof, the coefficients $b_{vi}$ for $v \in V \setminus \{u, w\}$ and $i \in \an(v)$ are not a function of the edge weights $c_{iu}$ for $i \in \an(u)$ nor of $c_{uw}$.

Further, by (<ref>), the formula for $X_w$ is

\begin{equation}
\label{eq:Xw}
X_w =
\bigvee_{i \in \an(u)} b_{wi} Z_i
\vee b_{wu} Z_u
\vee b_{ww} Z_w
\end{equation}

with coefficients ERROR: $p(i, w)$ need not pass by $u$; only true if $\pa(w) = \{u\}$, not otherwise.

\begin{align*}
b_{wi} &= b_{ii} c_{p(i, w)} = b_{ii} c_{p(i, u)} c_{uw}, \qquad i \in \an(u), \\
b_{wu} &= c_{uu} c_{uw}, \\
b_{ww} &= 1 - \sum_{i \in \an(u)} b_{wi} - b_{wu}.
\end{align*}

The unit-Fréchet variables $Z_u$ and $Z_w$ are independent and they do not appear in the expression for $X_v$ for $v \in V \setminus \{u, w\}$ (since $u$ nor $w$ are ancestors of such $v$). Hence, in (<ref>), we may replace the maximum $b_{wu} Z_u \vee b_{ww} Z_w$ by $(b_{wu} + b_{ww}) Z_{uw}$ for a single unit-Fréchet variable $Z_{uw}$ independent of all variables $Z_v$ for $v \in V \setminus \{u, w\}$, and this without changing the distribution of $X_{V \setminus u}$. But $b_{wu} + b_{ww} = 1 - \sum_{i \in \an(u)} b_{wi}$, yielding the representation

\[
X_w = \bigvee_{i \in \an(u)} b_{wi} Z_i
\vee \left(1 - \sum_{i \in \an(u)} b_{wi}\right) Z_{uw}.
\]

Let $\theta = (c_e)_{e \in E} \in \mathring{\Theta}_*$ denote the original parameter vector. For $\lambda > 0$ close to but different from $1$, let $\theta'$ be the parameter vector that results from replacing $c_{uw}$ by $\lambda c_{uw}$ and by replacing $c_{iu}$ for $i \in \pa(u)$ by $\lambda^{-1} c_{iu}$. Since $\mathring{\Theta}_*$ is open, we have $\theta' \in \mathring{\Theta}_*$ for $\lambda$ close enough to $1$. Obviously, the modification does not alter the products $c_{p(i, u)} c_{uw}$ for $i \in \an(u)$. The diagonal coefficients $b_{ii}$ for $i \in \an(u)$ have not been affected either. It follows that the distribution of $X_{V \setminus u}$ is the same under $\theta$ and under $\theta'$.
Case 1.b.ii: there exists $u \in \bar{U}$ with $\ch(u) = \{w\}$ and $\ch(w) \ne \varnothing$.
We proceed in the same way as in Case 1.b.i. This time, we also need to consider $X_v$ for $v \in \desc(w)$. By Lemma <ref>, item 3, any ancestor of $v \in \desc(w)$ that does not belong to $\An(w)$ must be a descendant of $w$ too. The ancestor set of any node $v \in \desc(w)$ can thus be partitioned into

\[
\An(v) = \an(u) \cup \{u, w\} \cup [\desc(w) \cap \An(v)].
\]

By (<ref>), we find, for $v \in \desc(w)$,

\[
X_v = \bigvee_{i \in \an(u)} b_{vi} Z_i \vee b_{vu} Z_u \vee b_{vw} Z_w \vee \bigvee_{i \in \desc(w) \cap \An(v)} b_{vi} Z_i.
\]

For $i \in \an(u)$ and $v \in \desc(w)$, the shortest path $p(i, v)$ must pass by ERROR: $u$ and $w$; similarly, the shortest path $p(u, v)$ must pass by $w$.

Case 2: <ref> is satisfied but <ref> is not.
Then there exists $u \in \bar{U}$ that belongs to a single tournament, say $\tau = (V_\tau, E_\tau)$, but is not equal to the source of $\T$. Let $s$ be the source node of $\tau$. Then $s \in \pa(u) \subset V_\tau$: if $u$ would have no parents, it would be equal to the source of $\T$, and moreover all parents of $u$ are in $V_\tau$, since $\tau$ is the only tournament to which $u$ belongs.

The idea is now to combine Lemma <ref> and Corollary <ref>: modify the edge weights $c_e$ for $e \in E_\tau$ without changing the coefficients $b_{ij}^\tau$ for $i \in V_\tau \setminus u$ and $j \in V_\tau$. This change then does affect the coefficients $b_{vi}$ for $v \in V \setminus u$ and $i \in \an(v)$ either.

§ ACKNOWLEDGEMENTS

The comments and suggestions by two anonymous Reviewers have been greatly helpful in the preparation of the final version of the manuscript.
Stefka Asenova is particularly grateful to Eugen Pircalabelu and Ngoc Tran for their availability and precious help.

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# Regularization of Persistent Homology Gradient Computation

Padraig Corcoran & Bailin Deng
School of Computer Science & Informatics
Cardiff University
Wales

###### Abstract

Persistent homology is a method for computing the topological features present
in a given data. Recently, there has been much interest in the integration of
persistent homology as a computational step in neural networks or deep
learning. In order for a given computation to be integrated in such a way, the
computation in question must be differentiable. Computing the gradients of
persistent homology is an ill-posed inverse problem with infinitely many
solutions. Consequently, it is important to perform regularization so that the
solution obtained agrees with known priors. In this work we propose a novel
method for regularizing persistent homology gradient computation through the
addition of a grouping term. This has the effect of helping to ensure
gradients are defined with respect to larger entities and not individual
points.

##  1 Introduction

Persistent homology is a computational method from the field of applied
topology for computing the topological features present in a given data [1].
Informally, the features in question relate to the number and scale of
connected components and holes of different dimensions in the data. The direct
application of persistent homology has proven to be a useful method in the
analysis of many different types of data including image [2], health [3] and
network data [4].

Given the recent advances and interest in neural networks or deep learning,
there exists a trend of attempting to integrate existing computational methods
as computational steps in this framework. This includes, for example, the
integration of integer programming [5] and shortest path methods [6] with deep
learning. In these works, deep learning usually acts as a preprocessing step
to the method in question where it performs representation learning. Deep
learning requires all computational steps to be differentiable such that
gradients can be back-propagated through each step and used to update the
method parameters. Therefore, for a given computational method to be
integrated with deep learning the method in question must be differentiable.
Many useful computational methods, such as integer programming, in their
native form are not differentiable. Consequently much work has been investing
in making such computational methods differentiable so that they may be
integrated with deep learning. That is, developing methods for computing the
gradients of the input with respect to the output of the methods in question.

As discussed above, persistent homology has proven to be a useful
computational method. Therefore integrating it with deep learning has much
potential. However, in its native form, persistent homology is not
differentiable. Therefore recently there has been much interest in attempting
to make it differentiable so that it may be integrated with deep learning [7,
8, 9, 10]. Computing gradients of persistent homology outputs with respect to
inputs is an inverse problem [11, 12]. Like many inverse problems, it is ill-
posed with infinitely many solutions. This point is obvious when one considers
that infinitely many different datasets can have the same number and scale of
connected components and holes of different dimensions. When attempting to
solve an arbitrary ill-posed inverse problem, if one does not consider the
ill-posed nature of the problem, the solution obtained may not agree with
known priors. A common approach to overcome this issue is to perform
regularization which biases the solution toward known priors. This approach is
commonly used to solve inverse problems in the field of image processing. In
this field two commonly used regularization approaches are total variation
(TV) and total generalized variation (TGV) which lead to piecewise constant
and piecewise linear images respectively [13].

To the authors knowledge, the use of regularization when computing gradients
of persistent homology has yet to be considered. Consequently, the use of
current methods for computing such gradients can lead to solutions which do
not agree with known priors. For example, consider the two dimensional point
dataset in Figure 1 which contains two compact clusters. If we compute the
gradients of persistent homology using current methods and in turn minimize a
loss function measuring the distance between the clusters (see Appendix A), we
obtain the result in Figure 1. Although this result approaches the
minimization of the loss function in question, it does not agree with a
reasonable prior that changes to topology features should be made at the level
of larger entities and not individual points. In this case the entities in
question are the two clusters. As a second example, consider the two
dimensional point dataset in Figure 1 which contains a single horseshoe shaped
cluster. If we compute the gradients of persistent homology using current
methods and in turn minimize a loss function measuring the width of the
horseshoe opening (see Appendix A), we obtain the result in Figure 1. Again,
although this result approaches the minimization of the loss function in
question, it does not agree with a reasonable prior that changes to topology
features should be made at the level of larger entities. In this case the
entities in question are the two ends of the horseshoe.

Figure 1: Two sets of points in $\mathbb{R}^{2}$ containing 100 and 300
elements are displayed in (a) and (d) respectively. Each set of points has an
associated loss function measuring the distance between the two clusters and
two horseshoe ends respectively. The results of minimizing these loss
functions without regularization are displayed in (b) and (e) respectively.
The results of minimizing these loss functions with the proposed
regularization are displayed in (c) and (f) respectively.

The artefacts in the above two example results are a consequence of the fact
that persistent homology gradients are defined with respect to individual
points and not larger entities. This motivates the following insight. When
computing persistent homology gradients, this computation should be
regularized through the addition of a grouping term such that gradients are
defined with respect to larger entities and not individual points. In this
article we propose a novel method for regularizing the computation of
persistent homology gradients which achieves this goal. The result of applying
this method to the two dimensional datasets in Figures 1 and 1 with the
respective loss functions described above are displayed in Figures 1 and 1
respectively. It is evident that in the case of these two examples, the
proposed method does not exhibit the artefacts encountered above.

The layout of this paper is as follows. In Section 2 we describe the proposed
method for regularizing the computation of persistent homology gradients. In
Section 3 we briefly draw some conclusions from this work.

##  2 Regularized Gradient Computation

This section is structured as follows. In Section 2.1 we briefly review
necessary background material on persistent homology and describe a current
method for computing gradients with respect to persistent homology. In Section
2.2 we describe the proposed method for performing regularization of this
gradient computation.

###  2.1 Persistent Homology & Gradient Computation

An (abstract) simplicial complex $\mathcal{K}$ is a finite collection of sets
such that for each $\sigma\in\mathcal{K}$ all subsets of $\sigma$ are also
contained in $\mathcal{K}$. Each element $\sigma\in\mathcal{K}$ is called a
$k$-simplex where $k=\left|{\sigma}\right|-1$ is the dimension of the simplex.
Given a finite set of points $X=\\{x_{1},\dots,x_{m}\\}$ in $\mathbb{R}^{n}$,
the corresponding Rips complex $\mathcal{R}_{r}$ for a specified $r\geq 0$
radius value is defined as follows:

$\mathcal{R}_{r}(X)=\\{\sigma\subseteq X:\forall x,y\in\sigma,\|x-y\|\leq
r\\}$ (1)

The $k$-simplices in this simplicial complex equal unordered ($k+1$)-tuples of
points which are pairwise within distance $r$ [14]. Computing the homology of
$\mathcal{R}_{r}(X)$ returns the homology groups $H_{i}$ for each natural
number $i$. An element of $H_{i}$ represents the existence of an
$i$-dimensional hole in $\mathcal{R}_{r}(X)$. That is, an element of $H_{0}$
represents the existence of a path-component in $\mathcal{R}_{r}(X)$ while an
element of $H_{1}$ represents the existence of a one dimensional hole in
$\mathcal{R}_{r}(X)$ [15]. A Rips filtration of $X$ is a finite sequence of
$N$ Rips complexes $\mathcal{R}_{r_{1}}(X),\dots,\mathcal{R}_{r_{N}}(X)$
associated with an increasing sequence $r_{1},\dots,r_{N}$ of radius values. A
Rips filtration induces a sequence of inclusion maps defined as:

$\mathcal{R}_{r_{1}}(X)\xhookrightarrow{}\mathcal{R}_{r_{2}}(X)\xhookrightarrow{}\dots\xhookrightarrow{}\mathcal{R}_{r_{N}}(X)$
(2)

Given a Rips filtration, instead of computing the homology of each Rips
complex in the sequence independently, persistent homology computes the
homology of the inclusions
$\mathcal{R}_{p}(X)\xhookrightarrow{}\mathcal{R}_{q}(X)$ for all $p<q$ [15].
The result of this computation is a set of persistence diagrams
$\\{D_{0},\dots,D_{h}\\}$ where $D_{i}$ corresponds to the homology group
$H_{i}$. Each persistence diagram is a multiset of points
$\\{(p,q)\in\mathbb{R}^{2},p\leq q\\}$ where $(p,q)\in D_{i}$ represents the
existence of an element of $H_{i}$ appearing in $\mathcal{R}_{p}(X)$ and
subsequently disappearing in $\mathcal{R}_{q}(X)$. The value $q-p$ is called
the persistence of the element in question.

Let $(p,q)$ be an element in a given persistence diagram $D_{i}$. There exists
a map $\gamma:\mathbb{R}\rightarrow\mathcal{R}_{r}(X)$ which maps $p$ and $q$
to simplices in the Rips filtration whose addition results in the appearance
and disappearance respectively of the corresponding element in $H_{i}$ [16,
9]. Therefore, one can adjust the values $p$ and $q$ by adjusting the radius
values at which the simplices $\gamma(p)$ and $\gamma(q)$ respectively are
added in the Rips filtration. Specifically, the radius value at which a
simplex is added in the Rips filtration is defined by the following map
$\delta:\mathcal{R}_{r}\rightarrow\mathbb{R}$:

$\delta(\sigma)=\max_{x,y\in\sigma}\|x-y\|$ (3)

Both the maps $\gamma$ and $\delta$ are differentiable and in turn the map
$\gamma\circ\delta$ is differentiable [16]. Let $l$ be a real valued
differentiable loss function which is a function of $D_{i}$. The map
$l\circ\gamma\circ\delta$ is in turn differentiable and can be minimized using
any gradient based optimization technique.

###  2.2 Regularization

As described in the previous section, the map $l\circ\gamma\circ\delta$ is
differentiable. If the map $l$ is a function of a single element $(p,q)$ in a
given persistence diagram $D_{i}$, the map $l\circ\gamma\circ\delta$ is
locally a function of at most four elements of $X$. The elements in question
are the two pairs of points maximum distance apart in the simplices whose
introduction resulted in the appearance and disappearance of the topological
feature in question. In this case, taking a single in the direction of the
gradient of $l\circ\gamma\circ\delta$ will alter the positions of at most
these four elements of $X$.

The above approach to minimizing a loss function changes topological features
at the level of individual points. This does not agree with our prior that
changes to topological features should be made at the level of larger entities
consisting of sets of points. To overcome this challenge we propose a novel
method for regularizing persistent homology gradient computation through the
addition of a grouping term. This has the effect of helping to ensure
gradients are defined with respect to larger entities and not individual
points.

Let Set be the space of finite sets of points in $\mathbb{R}^{n}$. Let
$X\in\text{Set}$ be the set of input points which we wish to optimize with
respect to the loss function $l$. Let $X^{\prime}$ be the initial value of $X$
before optimization and $\rho:\text{Set}\rightarrow\text{Set}$ be the
corresponding bijection from $X^{\prime}$ to $X$. Let $G$ be the set of all
unordered pairs of elements in $X^{\prime}$ and $k:[0,\infty]\rightarrow[0,1]$
be a given kernel. Broadly speaking, a kernel maps smaller values to a value
approaching $1$ and larger values to a value approaching $0$. An example of a
kernel is the uniform kernel defined as follows where $s$ is a specified scale
parameter:

$k(x)=\begin{cases}1&\|x\|\leq s\\\ 0&\|x\|>s\\\ \end{cases}$ (4)

Given a specified kernel, we define the proposed regularization term
$\tau:\text{Set}\rightarrow\mathbb{R}$ as follows:

$\tau(X)=\sum_{(a,b)\in G}k(\|a-b\|)(\|a-b\|-\|\rho(a)-\rho(b)\|)^{2}$ (5)

This term measures the discrepancy between pairwise distances in $X^{\prime}$
and the corresponding pairwise distances in $X$. Minimizing this term helps to
ensure the structure of local groups of points in $X^{\prime}$ is preserved in
$X$. That is, if a change is made to a single point this term will help ensure
that a similar change is made to all points in a corresponding group where
this group is a local neighbourhood defined by the kernel in question.

Let $\varrho:\text{Set}\rightarrow\mathbb{R}$ be a specified loss function
which is a function of the persistent homology of the corresponding input. We
define a regularized version of this loss $l:\text{Set}\rightarrow\mathbb{R}$,
which integrates the regularization term in Equation 5, as follows:

$l(X)=\varrho(X)+\lambda\tau(X)$ (6)

In this equation $\lambda$ is a specified real valued weighting parameter.
This regularized version of the loss will help ensure that changes to
topological features will not be made at the level of individual points, but
will instead be made at the level of larger entities consisting of sets of
points.

The result of applying the proposed regularization method to the sets of
points in Figures 1 and 1 are displayed in Figures 1 and 1 respectively. In
both cases a uniform kernel with a $s$ parameter value of $1.0$ and a
$\lambda$ parameter value of $1.0$ was used. The loss function terms $\varrho$
in question equal the squared persistence of the element in $H_{0}$
corresponding to the merging of the two clusters and the squared persistence
of the element in $H_{1}$ corresponding to the merging of the two ends of the
horseshoe respectively. These functions are formally defined in the Appendix A
section of this article. Optimization was performed using the Adam optimizer
with a learning rate of $0.01$ [17].

From these figures, we see that the distance between the clusters and the
distance between the ends of the horseshoe are reduced and this is achieved in
a manner which agrees with our prior that changes to topological features are
made at the level of larger entities. This contrasts with the results
displayed in Figures 1 and 1 respectively where no regularization is applied
and consequently changes to topological features are made at the level of
individual points.

##  3 Conclusions & Future Work

To the authors knowledge, this work presents the first attempt to perform
regularization of persistent homology gradient computation. Given the
increasing usefulness of integrating persistent homology and deep learning
plus the need to perform such regularization, we believe this topic has the
potential to develop into an active area of research in the field of applied
topology.

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##  Appendix A Loss Functions for Figure 1

In this section we formally define the loss function terms $\varrho$ in
Equation 6 applied to the datasets in Figures 1 and 1 to obtain the results in
Figures 1, 1, 1 and 1.

Let $\mathcal{D}$ be the space of persistence diagrams. Let
$\varrho_{0}:\mathcal{D}\rightarrow\mathbb{R}$ be a loss function defined as:

$\varrho_{0}(D)=\sum_{\begin{subarray}{c}\\{(p,q)\in D:\\\ q-p>0.10,\\\
q-p<\infty\\}\end{subarray}}(q-p)^{2}.$ (7)

This loss function was applied to the persistence diagram $D_{0}$
corresponding to the dataset in Figure 1 to compute the results in Figures 1
and 1.

Let $\varrho_{0}:\mathcal{D}\rightarrow\mathbb{R}$ be the loss function
defined as:

$\varrho_{1}(D)=\sum_{\begin{subarray}{c}\\{(p,q)\in D:\\\
q-p>0.25\\}\end{subarray}}(q-p)^{2}.$ (8)

This loss function was applied to the persistence diagram $D_{1}$
corresponding to the dataset in Figure 1 to compute the results in Figures 1
and 1.


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