We show that $P$ is $\mu$-almost $G$-invariant. Passing to a probability
measure equivalent to $\mu$ if necessary, we may assume that $\mu$ is a
probability measure. Let $g\in G$ be arbitrary. On one hand we have

$\mu\left(g.P\cap P\right)=\int_{g.P}1_{P}\cdot
1/\phi\left(x\right)d\nu=\int_{g.P}1/\phi\left(x\right)d\nu=\mu\left(g.P\right),$

implying that $\mu\left(g.P\backslash P\right)=0$. On other other hand, since
$\nu$ is $G$-invariant we have

$\nu\left(P\backslash g.P\right)=\nu\left(g^{-1}.P\backslash P\right)=0,$

and using that $\mu\mid_{P}\sim\nu\mid_{P}$ we deduce that
$\mu\left(P\backslash g.P\right)=0$. All together we deduce that
$\mu\left(g.P\triangle P\right)=0$, and by Theorem 2.3 and using that
$\nu\ll\mu$ we may assume that $P$ is $G$-invariant. ∎

######  Lemma 9.6.

Unless a.c.i.m does not exist, there exists an a.c.i.m (which is possibly an
a.c.i.p) with a maximal compatible support.
Similarly, unless a.c.i.p does not exist, there exists an a.c.i.p with a
$1$-maximal compatible support.

The following is a proof of the first statement of Lemma 9.6, and the proof of
the second statement follows with minor modifications.

###### Proof.

Passing to a probability measure equivalent to $\mu$ if necessary, we may
assume that $\mu$ is a probability measure. Set

$\alpha:=\sup\left\\{\mu\left(P\right):P\text{ is a compatible support of an
{\sc a.c.i.m} of }\left(X,\mu\right)\right\\}.$

Choose a sequence of a.c.i.m’s $\nu_{1},\nu_{2},\dotsc$ with corresponding
compatible supports $P_{1},P_{2},\dotsc$ such that
$\mu\left(P_{n}\right)\to\alpha$ as $n\to+\infty$. Observe that finite convex
combinations of a.c.i.m’s is again an a.c.i.m, and the union of the underlying
compatible supports is a compatible support for the sum, so we may assume that
$P_{1}\subset P_{2}\subset\dotsm$. We then define $\nu_{0}$ by

$\nu_{0}:=\nu_{1}/2\text{ on }P_{1}\text{ and }\nu_{0}:=\nu_{n}/2^{n+1}\text{
on }P_{n}\backslash P_{n-1}\text{ for }n\geq 2.$

Is is easy to see that $\nu_{0}$ is $\sigma$-finite (possibly finite) and,
since $\nu_{n}$ is $G$-invariant and $P_{n}$ is a $G$-invariant set for every
$n\in\mathbb{N}$, we also see that $\nu_{0}$ is $G$-invariant. Thus, $\nu_{0}$
is an a.c.i.m of $\left(X,\mu\right)$. Letting

$P_{0}:=P_{1}\cup P_{2}\cup\dotsm,$

it is clearly a compatible support for $\nu_{0}$, and we claim that it is also
maximal; indeed, since $\mu\left(P_{0}\right)\geq\mu\left(P_{n}\right)$ for
every $n$ necessarily $\mu\left(P_{0}\right)=\alpha$, so for any compatible
support $P$ of any a.c.i.m $\nu$ of $\left(X,\mu\right)$, by taking a convex
combination of $\nu_{0}$ and $\nu$ we obtain a new a.c.i.m with compatible
support $P_{0}\cup P$. Then
$\alpha=\mu\left(P_{0}\right)\leq\mu\left(P_{0}\cup P\right)\leq\alpha$, hence
$\mu\left(P\backslash P_{0}\right)=0$. ∎

###### Proof of Theorem 9.2.

If there is no a.c.i.m of $\left(X,\mu\right)$ we put $\mathcal{N}=X$.
Otherwise, using the first statement of Lemma 9.6, fix an a.c.i.m $\upsilon$
with a maximal compatible support $\mathcal{P}$, let
$\mathcal{N}:=X\backslash\mathcal{P}$ and look at the $G$-invariant Borel
decomposition

$X=\mathcal{P}\sqcup\mathcal{N}.$

Since $\mathcal{P}$ is a compatible support,
$\upsilon\mid_{\mathcal{P}}\sim\mu\mid_{\mathcal{P}}$. From the maximality of
$\mathcal{P}$, the compatible support of every a.c.i.m of
$\left(X,\mu\right)$, which always exists by Lemma 9.5, must be contained in
$\mathcal{P}$ modulo $\mu$, hence $\mathcal{N}$ supports no a.c.i.m of
$\left(X,\mu\right)$. This establishes the property (3) in the theorem for the
set $\mathcal{N}$.

We now restrict our attention to the nonsingular $G$-space
$\left(\mathcal{P},\mu\mid_{\mathcal{P}}\right)$. If there is no a.c.i.p of
$\left(\mathcal{P},\mu\mid_{\mathcal{P}}\right)$ we put
$\mathcal{P}_{\infty}=\mathcal{P}$. Otherwise, using the second statement of
Lemma 9.6, fix an a.c.i.p $\varrho$ with a $1$-maximal compatible support
$\mathcal{P}_{1}$, let
$\mathcal{P}_{\infty}=\mathcal{P}\backslash\mathcal{P}_{1}$ and look at the
$G$-invariant Borel decomposition

$\mathcal{P}=\mathcal{P}_{1}\sqcup\mathcal{P}_{\infty}.$

The very same reasoning as in the first part of this proof establishes
properties (1) and (2) in the theorem for the sets $\mathcal{P}_{1}$ and
$\mathcal{P}_{\infty}$.

Finally, the maximality of $\mathcal{P}$ as the compatible support of the
a.c.i.m $\upsilon$ together with the $1$-maximality of $\mathcal{P}_{1}$ as
the compatible support of the a.c.i.p $\varrho$, imply the uniqueness of the
decomposition. ∎

##  10\. Concluding Remarks and Further Questions

####  10.0.1. Transience from the Right

Following Definition 1.2, one may consider another notion of a transient set,
namely a Borel set $A\subset X$ with
$\lambda\big{(}R_{A}\left(x\right)^{-1}\big{)}<+\infty$ for every $x\in X$.
When $G$ is not unimodular it does not coincide with our working definition of
a transient set.

######  Question 10.1.

Suppose $G$ is a non-unimodular lcsc group and $\left(X,\mu\right)$ a
nonsingular $G$-space. If there is a $\mu$-positive set $A\subset X$ with
$\lambda\big{(}R_{A}\left(x\right)^{-1}\big{)}<+\infty$ for every $x\in X$, is
$\left(X,\mu\right)$ dissipative?

Note that the converse is always true: when $(X,\mu)$ is dissipative it
contains even a $\mu$-positive _strongly transient_ set, which obviously has
this property.

####  10.0.2. Shelah–Weiss Pointwise Characterization of Transience

In [35], Shelah & Weiss showed that for an aperiodic Borel automorphism $f$ of
a standard Borel space $X$, every Borel set $A\subseteq X$ that satisfies
$\mu\left(A\right)=0$ for every quasi-invariant ergodic probability measure
$\mu$ on $X$, is contained in $\bigcup_{n\in\mathbb{Z}}f^{n}\left(W\right)$
for some wandering set $W\subset X$ (note that the conclusion is measure-
free). It is natural to ask for the following continuous analog:

######  Question 10.2.

Let $G$ be an lcsc non-compact group with a left Haar measure $\lambda$ and
$X$ a free Borel G-space. Suppose that $A\subset X$ is a Borel set with
$\lambda\left(R_{A}\left(x\right)\right)>0$ for every $x\in G.A$. Is it true
that $A\subset G.T$ for some transient set $T\subset X$ provided that
$\mu\left(G.A\right)=0$ for every quasi-invariant ergodic probability measure
$\mu$ on $X$?

Note that since every Borel $G$-space admits a cross-section $C$, that is
$G.C=X$ and there exists an identity neighborhood $U\subset G$ such that
$U.x\cap U.y=\emptyset$ for all distinct $x,y\in C$ (see e.g. [37, §2] and the
references therein), the assumption
$\lambda\left(R_{A}\left(x\right)\right)>0$ whenever $x\in G.A$ is necessary.

####  10.0.3. Kaimanovich-Type Characterization

Theorem C is formulated for unimodular groups. We do not know whether it is
true generally:

######  Question 10.3.

Suppose $G$ is an lcsc non-unimodular group and $\left(X,\mu\right)$ a
dissipative nonsingular probability $G$-space. Is it true that for $\mu$-a.e.
$x\in X$ there exists $r_{x}>0$ such that $\lambda\left(g\in
G:\nabla_{g}(x)\geq r_{x}\right)=+\infty$?

####  10.0.4. First Cohomology, Skew Products and Maharam Extensions

Schmidt’s monograph [33] provides a thorough study of the cohomology of
nonsingular $G$-spaces for countable group $G$. See also [1, Chapter 8]. While
we did not include this theory in this work, one naturally expects to adapt it
to an lcsc acting group $G$, thus defining cocycles [33, §2], essential values
[33, §3] and recurrence of cocycles [33, Definition 3.13]. It is then expected
that a cocycle is recurrent iff the skew-product it forms is conservative
(following [33, §5]), and that the Radon–Nikodym cocycle of a conservative
nonsingular $G$-spaces is recurrent (following [34, §4]), hereby showing that
the Maharam Extension of a conservative nonsingular $G$-space is conservative.

##  Appendix A Measures on Homogeneous Spaces

The construction of measures on homogeneous spaces was established by Mackey
[25, §1] and extended later by many others. We recall its basics here in a way
that suits our needs. We will follow Mackey’s seminal works [25, 26] and the
presentation in [15, §2.6] (see also [39, Chapter V, §4]).

A group pair $C\lessdot G$ is a pair of an lcsc group $G$ and a closed
subgroup $C$ of $G$. We denote by $G/C$ the coset space $\left\\{gC:g\in
G\right\\}$ with the quotient topology it inherits from $G$. The group $G$
acts continuously and transitively on $G/C$ via

$g.\left(hC\right)=ghC,\quad g\in G,\,hC\in G/C.$

We will refer to measures on $G/C$ simply as invariant or quasi-invariant,
with the understanding that this always means invariance or quasi-invariance
with respect to the natural action of $G$ on $G/C$. Since this action is
transitive, it becomes ergodic with respect to whatever invariant or quasi-
invariant measure we may put on $G/C$.

######  Theorem A.1 (Mackey).

For every group pair $C\lessdot G$, the coset space $G/C$ becomes a standard
Borel space. Equivalently, with the natural action of $C$ on $G$, the orbit
equivalent relation $E_{C}^{G}$ is smooth.

This theorem was proved by Mackey with two different formulations in [25,
Lemma 1.1] and [26, Theorem 7.2], which are equivalent according to Theorem
5.2. See also [41, Corollary A.8]. In fact, it is true also beyond lcsc groups
(see [22, Theorem (12.17)] and [16, Theorem 2.2.10]; cf. [16, Proposition
3.4.6]). A useful fact for the proof of the following extension of Mackey’s
theorem, is that the quotient topology itself is Polish, making $G/C$ a Polish
$G$-space.

######  Theorem A.2.

Suppose $C\lessdot G$ is a group pair with compact $C$, and let $H$ be a
closed subgroup of $G$. Then, with the natural action of $H$ on $G/C$, the
orbit equivalence relation $E_{H}^{G/C}$ is smooth.

###### Proof.

Since $H$ is closed its action on $G/C$ is continuous in the quotient
topology, which is a Polish topology by [16, Theorem 2.2.10]. The orbits in
this action are

$H_{gC}=HgC,\quad gC\in G/C,$

and since $H$ is closed and $C$ is compact we deduce that all orbits are
closed. We finally recall that a Borel equivalence relation with closed
equivalence classes is smooth (see [22, Theorem (12.16)] and [16, Theorem
6.4.4(iv)]). ∎

The construction of measures on homogeneous spaces yields a quasi-invariant
Radon measure on $G/C$. In the following we describe it as well as _Weil’s
formula_ it satisfies. Given a group pair $C\lessdot G$, treating each of $G$
and $C$ as an lcsc group on its own right, we denote by $\lambda_{G}$ and
$\lambda_{C}$ their respective left Haar measures and by $\varDelta_{G}$ and
$\varDelta_{C}$ the corresponding modular functions. Every group pair
$C\lessdot G$ admits a rho-function, i.e. a continuous function
$\rho:G\to\mathbb{R}_{+}$ with

$\rho\left(gc\right)=\frac{\varDelta_{C}\left(c\right)}{\varDelta_{G}\left(c\right)}\cdot\rho\left(g\right),\quad\left(g,c\right)\in
G\times C.$

(See [15, Proposition 2.54]). The following theorem is classical and in many
occasions is formulated for locally compact groups that may not be second
countable. We focus solely on lcsc groups.

######  Theorem A.3 (Measures on homogeneous spaces).

For a group pair $C\lessdot G$:

  1. (1)

There is an essentially unique (up to a positive constant multiple) invariant
$\sigma$-finite Radon measure $\kappa$ on $G/C$ iff
$\varDelta_{G}\mid_{C}\equiv\varDelta_{C}$.

  2. (2)

There is always an essentially unique (up to measure class) quasi-invariant
$\sigma$-finite Radon measure $\kappa$ on $G/C$, whose Radon–Nikodym cocycle
is jointly continuous.

  3. (3)

_Weil’s formula_ : For every $\kappa$ as in (2), there is a rho-function
$\rho$ with

(A.0.1)
$\displaystyle\int_{G}\varphi\left(g\right)d\lambda_{G}\left(g\right)=\int_{G/C}\left[\int_{C}\varphi\left(gc\right){\textstyle\frac{\varDelta_{G}\left(c\right)}{\varDelta_{C}\left(c\right)}}d\lambda_{C}\left(c\right)\right]\rho\left(g\right)^{-1}d\kappa\left(gC\right),$
$\text{ for every Borel function }\varphi:G\to\left[0,\infty\right).$

###### Proof.

Since $G$ is assumed to be $\sigma$-compact, so is its continuous image $G/C$.
Thus, every Radon measure on $G/C$ would be automatically $\sigma$-finite.
Part (1) can be found in [15, Theorem (2.49)]. Part (2) can be found in [15,
Theorem (2.59)]. As for part (3), in [15, Theorem (2.56)] there can be found
the formula

$\int_{G}\varphi\left(g\right)\rho\left(g\right)d\lambda_{G}\left(g\right)=\int_{G/C}\left[\int_{C}\varphi\left(gc\right)d\lambda_{C}\left(c\right)\right]d\kappa\left(gC\right)$

for every compactly supported continuous $\varphi:G\to\left[0,+\infty\right)$.
Since we deal with second countable groups it applies to every Borel function
$\varphi:G\to\left[0,+\infty\right)$ (see [15, §2.7, pp. 64–65]). Finally, for
every Borel function $\varphi:G\to\left[0,+\infty\right)$, substituting
$\varphi/\rho$ for $\varphi$ in this formula and using the rho-function
property, we obtain the formula appears in part (3). ∎

The uniqueness of $\kappa$ as stated in Theorem A.3(2) is restricted to the
class of Radon measures whose Radon–Nikodym cocycle is jointly continuous.
Also the famous uniqueness of the Haar measure is formulated in the common
literature under further regularity property (e.g. Radon). We will need to
strengthen this uniqueness which is due to Mackey.

It was shown by Mackey that in fact quasi-invariance is sufficient to
completely determine the measure class in the construction of measures on
homogeneous spaces (a fact that is already significant when $C$ is the trivial
subgroup). Given that every $\sigma$-finite measure is equivalent to a
probability measure, which is always Radon, Mackey’s theorem renders any
additional regularity assumptions redundant. We formulate this for compact
$C$, in which case $\varDelta_{G}\mid_{C}\equiv\varDelta\mid_{C}$ hence
$\kappa$ is invariant according to Theorem A.3(1).

######  Theorem A.4 (Mackey).

For a group pair $C\lessdot G$ with compact $C$, with the invariant measure
$\kappa$ constructed in Theorem A.3(1), the following hold:

  1. (1)

Every quasi-invariant $\sigma$-finite Borel measure is equivalent to $\kappa$.

  2. (2)

Every invariant $\sigma$-finite Borel measure is a positive constant multiple
of $\kappa$.

  3. (3)

The pushforward of $\lambda_{G}$ along the canonical projection $G\to G/C$ is
precisely $\kappa$ up to a positive constant multiple.

###### Proof.

Part (1). It was shown in [25, Theorem 1.1] that the measure class of $\kappa$
is unique among the quasi-invariant $\sigma$-finite Borel measures on $G/C$
which are finite on compact sets. As every $\sigma$-finite measure is
equivalent to a probability measure, which is inherently Radon (see e.g. [22,
Theorem (17.10)]), and equivalence of measures preserves quasi-invariance, the
desired uniqueness hereby follows. (cf. [26, Theorem 7.1 and its Corollary]).

Part (2). If $\kappa^{\prime}$ is any invariant $\sigma$-finite Borel measure
equivalent to $\kappa$, then $d\kappa^{\prime}/d\kappa$ is an invariant
function on $G/C$. Since the action is transitive, $d\kappa^{\prime}/d\kappa$
is necessarily a positive constant.

Part (3). Let $q:G\to G/C$ be the canonical projection, which is continuous
and open map, and consider the pushforward measure $q_{\ast}\lambda_{G}$ on
$G/C$. This is a Borel measure and it is invariant since

$q^{-1}\left(g.B\right)=gq^{-1}\left(B\right)\text{ for every Borel set
}B\subset G/C\text{ and }g\in G.$

Then with Part (2), in order to finish we have left to show that
$q_{\ast}\lambda_{G}$ is $\sigma$-finite (which fails when $C$ is non-compact:
$G=\mathbb{R}^{2}$ and $C=\mathbb{R}$). For an arbitrary compact set $K\subset
G$ we have $q^{-1}\left(K\cdot C\right)\subset K.C$ and then, since $K\cdot C$
is compact in $G$,

$q_{\ast}\lambda_{G}\left(K.C\right)\leq\lambda_{G}\left(K\cdot
C\right)<+\infty.$

As $K$ is arbitrary and $G$ is $\sigma$-compact, it follows that
$q_{\ast}\lambda_{G}$ is $\sigma$-finite. ∎

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Since we constructed $V_C$ from a cycle in the underlying graph of $\TG - X$ we have $V_C \cap Y = \emptyset$.
This is a contradiction.

Now we can use the polynomial-time $8$-approximation of Even et al. [27] for Weighted Subset Feedback Vertex Set
and <ref> to conclude the following.[Here, vertices get weight $\infty$ if there are undeletable, and one otherwise.]

There is a polynomial-time $8$-approximation for .

In the remainder of this section we prove the following.

Given an instance $I=(G,V^\infty,T,k)$ of Subset Feedback Vertex Set with Undeletable Vertices
we can construct in $O(k^2(|V|+|E|))$ time an instance $I'=(G,T',k')$ of
Subset Feedback Vertex Set with $k'\le k$ such that $I$ is a
-instance if and only if $I'$ is a -instance.

Note that the running time of the algorithm behind <ref> depends only linearly on the size of the graph.
The proof of <ref> is deferred to the end of this section.
First, we introduce two data reductions rules and then perform the reduction behind <ref> in two steps.
We use these data reduction rules to get an equivalent instance where $G[T \cap V^\infty]$ is an independent set.
We start by detecting some -instances.

Let $I=(G,V^\infty,T,k)$ be an instance of Subset Feedback Vertex Set with Undeletable Vertices
such that there is a vertex $u$ and a simple cycle $C$ intersecting $T$ where $V(C) \setminus (T \cap V^\infty) \subseteq \{u\} \subseteq V^\infty$.
Then output a trivial -instance.

We now show that we can detect in linear-time whether <ref> is applicable and that it is safe.
The latter means that the application of <ref> does not turn a -instance into a -instance or vice versa.

<ref> is safe and can be applied in linear time.

Since $C$ is a witness that $I$ is a -instance, <ref> is safe.
We check whether there is cycle $C$ such that  $ V(C) \setminus (T \cap V^\infty) = \emptyset$
by simply checking whether $G[T \cap V^\infty]$ is a forest.
Assume that $G[T \cap V^\infty]$ is a forest, otherwise we are done and output a trivial -instance.
First, we partition $V(G[T \cap V^\infty]) = Q_1 \uplus \dots \uplus Q_c$ such that $Q_i$ is a maximal connected component of $G[T \cap V^\infty]$. Clearly, this can be done in linear time.
For each connected component $Q_i$ of $G[T \cap V^\infty]$
we first unmark all vertices in $V\setminus (T \cap V^\infty)$.
Then we iterate over all vertices $v \in Q_i$ and mark all vertices in $N_G(v) \cap (V^\infty \setminus T)$.
If we find a vertex $w \in Q_i$ such that there is a vertex $u \in N_G(w) \cap (V^\infty \setminus T)$ which is already marked,
then the path from $v$ to $w$ in $Q_i$ together with $u$ is a cycle $C$ where $V(C) \setminus (T \cap V^\infty) \subseteq \{u\} \subseteq V^\infty$. Hence, we output a trivial -instance in this case.
Moreover, if there is some simple cycle $C$ where $V(C) \setminus (T \cap V^\infty) \subseteq \{u\} \subseteq V^\infty$,
then all vertices $V(C) \setminus \{u\}$ belong to the same connected component of $G[T \cap V^\infty]$.
Thus, the above described procedure will find $C$.
A simple application of the Handshaking Lemma shows that this procedure ends after linear time.

The purpose of the next data reduction rule is to merge undeletable terminal vertices which do share an edge.

Let $I=(G,V^\infty,T,k)$ be an instance of Subset Feedback Vertex Set with Undeletable Vertices
such that there is $\{v,w\} \in E$ with $\{v,w\} \subseteq T \cap V^\infty$ and $N_G(v) \cap N_G(w) \cap V^\infty = \emptyset$.
Then set $\widehat G = (V(G - w),E(G-w) \cup \{ \{v,u\} \mid \{w,u\} \in E \})$
and $G' :=\widehat G - (N_G(v) \cap N_G(w))$.
Output $I'=(G',V^\infty \cap V(G'), T\cap V(G'),k - |N_G(v)\cap N_G(w)|)$.

Note that for each edge in $G[T \cap V^\infty]$, either <ref> or <ref> is applicable.
While it is easy to apply <ref> exhaustively in polynomial time,
more effort is required to do the same in linear time.
To this end, we will construct in linear time an equivalent instance such that <ref> are not applicable.

Given an instance $I$ of Subset Feedback Vertex Set with Undeletable Vertices, we can compute an equivalent instance $I'$ of Subset Feedback Vertex Set with Undeletable Vertices in linear time such that <ref> are not applicable to $I'$.

We first check in linear time by <ref> whether <ref> is applicable.
Assume that <ref> is not applicable, otherwise we are done.
Hence, $G[T \cap V^\infty]$ is a forest.

We now aim to apply <ref> for all edges in $G[T \cap V^\infty]$ at once.
First, we partition $V(G[T \cap V^\infty]) = Q_1 \uplus \dots \uplus Q_c$
such that $Q_i$ is a maximal connected component of $G[T \cap V^\infty]$.
Then we replace $Q_i$ with a fresh vertex $q_i$.
To this end, we construct the graph $G' := (V', E')$ where

\begin{align*}
V' := &(V \setminus (T \cap V^\infty)) \cup \{ q_i \mid i \in [c] \} \text{ and}\\
E' := &\{ \{a,b\} \in E \mid \{a,b\} \subseteq V' \} \cup \{ \{q_i,w\} \mid w \in V' \text{ and } \exists v \in Q_i \colon \{v,w\} \in E \}.
\end{align*}

Note that $G'$ and hence $V'^\infty = (V^\infty \cap V') \cup \{ q_i \mid i \in [c]\}$
and $T' = (T \cap V') \cup \{ q_i \mid i \in [c] \}$ can be computed in linear time.
To compute the remaining budget $k'$, we set $K = \emptyset$.
Then, for each connected component $Q_i$ of $G[T \cap V^\infty]$,
we first unmark all vertices.
Second, we iterate over all vertices $v \in Q_i$ and mark all vertices in $N_G(v) \setminus (T \cap V^\infty)$.
If we find a vertex $w \in Q_i$ such that a vertex $u \in N_G(w) \setminus (T \cap V^\infty)$ which is already marked,
then we add $u$ to $K$, because $G[Q_i \cup \{ u\}]$ contains a cycle intersecting $T$ where $u$ is the only deletable vertex.
Recall that $u \not\in V^\infty$, since <ref> was not applicable.
A simple application of the Handshaking Lemma show that this procedure ends after linear time.
If $k < |K|$, then we return a trivial -instance,
because for each vertex $v \in K$ there is a simple cycle
intersecting $T$ where $v$ is the unique vertex not in $V^\infty \cap T$.
Otherwise, we output $I' = (G' - K,V'^\infty,T' \setminus K,k' = k - |K|)$.
It is easy to verify that <ref> is not applicable in $I'$.
We now claim that $I$ is a -instance
if and only if
$I'$ is a -instance.

Assume that $X$ is a solution for $I$.
Note that for each vertex $v \in K$, there is a simple cycle
intersecting $T$ where $v$ is the unique vertex not in $V^\infty \cap T$.
Hence, $K \subseteq X$.
Set $X' := X \setminus K$ and observe that $X' \subseteq V(G')$.
The set $X'$ is a solution for $I'$, because it is of size at most $k'$
and for each cycle in $G'-K$ which intersects to $T' \setminus K$,
we can construct a cycle in $G$ which intersects to $T$
by replacing a vertex $q_j \in \{ q_i \mid i \in [c] \}$ with a path in $G[Q_j]$.

Assume that $X'$ is a solution for $I'$.
We set $X := X' \cup K$ and note that $X$ is of size at most $k$.
We may assume towards a contradiction that $X$ is not a solution for $I$.
Hence there is a simple cycle $C$ in $G - X$ which contains a vertex of $T$.
We now construct a closed walk of $C'$ in $G'- T$ from $C$ by replacing a
subpath on vertices in $Q_i$ with $q_i$, for all $i \in [c]$.
Since we only replaced vertices from $T \cap V^\infty$ of $I$ with vertices in $T' \cap V'^\infty$,
the closed walk $C'$ contains a simple cycle in $G'-X'$ containing a vertex from $T'$—a contradiction.

We now show an algorithm to dispose all vertices undeletable vertices in $T$
such that the running time dependence only linearly on the size of the graph.

Let $I=(G,V^\infty,T,k)$ be an instance of Subset Feedback Vertex Set with Undeletable Vertices.
We can construct in $O(k(|V|+|E|))$ time
an instance $I'=(G',V'^\infty,T',k')$ of Subset Feedback Vertex Set with Undeletable Vertices
such that

* $k'\leq k$,

* $V'^\infty \setminus T' = \emptyset$,

* $I$ is a -instance if and only if $I'$ is a -instance.

Let $I=(G,V^\infty,T,k)$ be an instance of Subset Feedback Vertex Set with Undeletable Vertices.
Since we aim for a running time of $O(k(|V|+|E|))$,
we can assume that <ref> are not applicable on $I$, see <ref>.

The goal now is to duplicate each vertex in $V^\infty \setminus T$ $k+1$ times,
such that we cannot delete all of them even if there are not in $V^\infty$.
Note that we might create many copies of an edge doing this naïvely.
To avoid this, we observe that it suffices to replace
each maximal connected component in $G[V^\infty \setminus T]$ with $k+1$ vertices.
We partition $V(G[V^\infty \setminus T]) = Q_1 \uplus \dots \uplus Q_c$
such that $Q_i$ is a maximal connected component of $G[V^\infty \setminus T]$
and we construct $G' := (V',E')$, where

\begin{align*}
V' &:= (V \setminus (V^\infty\setminus T)) \cup \{ q_i^j \mid i \in [c],j \in [k+1] \} \text{ and }\\
E' &:= E(G[V \setminus (V^\infty\setminus T)]) \cup \{ \{q_i^j,w\} \mid \exists v \in Q_i \colon \{v,w\}\in E\}.
\end{align*}

Since $|\{ q_i^j \mid i \in [c],j \in [k+1] \}| \leq |V|(k+1)$ and
we copy each edge at most $k+1$ times, $G'$ is constructed after $O(k(|V|+|E|))$ time.

We now claim that $I$ is a -instance
if and only if $I' := (G',T' := V^\infty \cap V(G'),T,k)$ is a -instance
of Subset Feedback Vertex Set with Undeletable Vertices.

Let $X$ be a solution for $I$.
Then $X$ is also a solution for $I'$,
because $X \cap V^\infty = \emptyset$ and for each cycle $C'$ in $G'$ containing a vertex from $T$,
we can construct a closed walk in $G$ by replacing a vertex $q_i^j$ by a path in $G[Q_i]$.
This closed walk induces a simple cycle $C$ in $G$ containing a vertex in $T$.
Hence, $C'$ also contains a vertex from $X$.

Let $X$ be a solution for $I'$.
We assume without loss of generality that $X \cap \{ q_i^j \mid i \in [c],j \in [k+1] \} = \emptyset$.
Suppose towards a contradiction that $X$ is not a solution for $I$.
Let $C$ be an cycle in $G$ satisfying $V(C) \cap X = \emptyset \not = V(C) \cap T$.

We construct a closed walk $C'$ in $G'$ from $C$ by replacing each maximal consecutive subpath in $C$
containing only vertices from $Q_i$ with $q_i^1$, for all $i \in [c]$.
Note that $C$ contains a simple cycle which intersects $T'$ but not $X$—a contradiction.

Now we are finally ready to prove <ref>.

First, we apply <ref> on $I$ and
hence assume that $V^\infty \setminus T = \emptyset$.
Furthermore, by <ref> we assume that <ref> are not applicable.
Thus, $G[V^\infty \cap T]$ is an independent set.
We now create $k+1$ copies of each vertex in $V^\infty \cap T$ such that
we cannot remove all of them, even if there are deletable.
However, we have to be careful what kind of new cycles this creates.
We do the following.
Let $V^\infty \cap T = \{v_1,v_2,\dots,v_p\}$ and take a fresh set of
vertices $Q_i :=\{ q_i^j \mid j \in [k+1]\}$ for each $i \in [p]$.
We construct $G' := (V',E')$, where

\begin{align*}
V' := &(V \setminus (V^\infty \cap T)) \cup \bigcup_{i=1}^p Q_i  \cup \{ w_i',w_i \mid \{v_i,w\} \in E \}, \text{ and}\\
E' := &\{ \{v,w\} \in E \mid v,w \in V'\} \cup \\
&\{ \{w_i,w\},\{w_i,w_i'\}, \{w_i',q_i^j\}  \mid \{v_i,w\} \in E, i \in [p]\} , j\in[k+1] \}.
\end{align*}

Output $I'=(G',T',k)$,
where $T' := (T \setminus V^\infty) \cup \{ w_i \mid \{v_i,w\} \in E \}$.
Since $|\{ q_i^j \mid i \in [c],j \in [k+1] \}| \leq |V|(k+1)$ and
we create for each edge at most $k+2$ new edges, $G'$ is constructed after $O(k(|V|+|E|))$ time.
Together with the preprocessing of <ref> this gives an overall running time of $O(k^2(|V|+|E|))$.

We now claim that $I$ is a -instance
if and only if $I'$ is a -instance of Subset Feedback Vertex Set.

Let $X$ be a solution for $I$.
Observe that $X \subseteq V'$, because $V \setminus V' \subseteq V^\infty$.
Assume towards a contradiction that there is a cycle $C'$ in $G'$ such that $V(C') \cap X = \emptyset$.
Assume without loss of generality that $C'$ is a shortest of the set of cycles satisfying $V(C') \cap X = \emptyset$.
Hence, for each $i \in [p]$ we have that  $|V(C') \cap Q_i| \leq 1$,
and since  $G[V^\infty \cap T]$ is an independent set,
a vertex $q_i^j \in  V(C') \cap Q_i$ has two neighbors $w'_i,u'_i$, where $w,u \in V$.
Thus, we can construct a cycle $C$ in $G$ by replacing each subpath $w'_i,q_i^j,u'_i$
with the vertex $v_i \in T \cap V^\infty$, where $q_i^j \in  V(C') \cap Q_i$ and $i\in[p],j\in[k+1]$.
This contradicts $X$ being a solution for $I$.

Let $X'$ be a solution for $I'$.
For all $i \in [p]$, we assume without loss of generality that

$X' \cap Q_i = \emptyset$:
This can be assumed, because $|Q_i| > k \geq |X'|$ and all vertices in $Q_i$ have the same neighborhood.

$X' \cap \{ w_i \mid \{v_i,w\} \in E \} = \emptyset$:
This can be assumed, because these vertices are of degree two and thus a vertex $w_i$ can be replaced by its origin $w$.

$X' \cap \{ w_i' \mid \{v_i,w\} \in E \} = \emptyset$:
Such a vertex $w_i'$ can be replaced by its origin $w$ as well,
because for each cycle $C'$ in $G'$ passing through $T'$ which does not include $w$,
we know that $v_i'q_i^jw_i'q_i^{j'}u_i'$ is a subpath of $C'$ for some $v,u\in V$, $q_i^j,q_i^{j'} \in Q_i$, and $j,j' \in [k+1]$.
Hence $C'-\{q_i^j,w_i\}$ is also a cycle in $G'$ that contains a vertex from $T'$.
Thus, there is $V(C')\cap (X\setminus \{w_i'\}) \not= \emptyset$.

Hence $X' \subseteq V \setminus V^\infty$.
Now assume towards a contradiction that there is a cycle $C$ in $G$ which does not contain a vertex in $X'$.
Hence there $v_i \in V(C) \cap (T \cap V^\infty)$, otherwise $C$ is also a cycle in $G'$.
We construct a cycle $C'$ in $G'$ from $C$ by replacing each subpath $u,v_i,w$ in $C$ with
$u,u_i,u'_i,v_i,w'_i,w_i,w$, for all $v_i \in V(C)\cap (T \cap V^\infty)$.
This contradicts $X'$ being a solution because $u_i \in T'$.

By using <ref> we now can compute a minimum by any known Subset Feedback Vertex Set algorithms.
Thus, <ref> together with the
algorithm of Iwata et al. [42] imply <ref>.

§ CONCLUSION

We have analyzed the (parameterized) computational complexity of
, a canonical variant of the problem of finding temporal
paths, where the waiting time at every vertex is restricted.

Unlike its non-restless counterpart or the “walk-version”, this problem turns out to be computationally hard,
even in quite restricted cases.
On the positive side, we give an efficient algorithm to find short restless temporal paths
and we could identify structural parameters of the underlying graph and of the temporal graph itself
that allow for fixed-parameter algorithms.

§.§ Acknowledgements.

We thank the referees for their careful reading and constructive comments which
significantly improved the presentation of these results.

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# Trash to Treasure:
Using text-to-image models to inform the design of physical artefacts

Amy Smith1, Hope Schroeder2, Ziv Epstein2, Michael Cook3, Simon Colton1,4,
Andrew Lippman2

###### Abstract

Text-to-image generative models have recently exploded in popularity and
accessibility. Yet so far, use of these models in creative tasks that bridge
the 2D digital world and the creation of physical artefacts has been
understudied. We conduct a pilot study to investigate if and how text-to-image
models can be used to assist in upstream tasks within the creative process,
such as ideation and visualization, prior to a sculpture-making activity.
Thirty participants selected sculpture-making materials and generated three
images using the Stable Diffusion text-to-image generator, each with text
prompts of their choice, with the aim of informing and then creating a
physical sculpture. The majority of participants (23/30) reported that the
generated images informed their sculptures, and 28/30 reported interest in
using text-to-image models to help them in a creative task in the future. We
identify several prompt engineering strategies and find that a participant’s
prompting strategy relates to their stage in the creative process. We discuss
how our findings can inform support for users at different stages of the
design process and for using text-to-image models for physical artefact
design.

## Background and Motivations

Text-to-image deep learning models are generative AI techniques that
synthesize images from text inputs. Implementations of this technology such as
Midjourney, DALLE-2, and Stable Diffusion (Rombach et al. 2022) have exploded
in popularity in the last year. The company Stability AI has released the
models, weights, and code for Stable Diffusion, allowing it to be used
publicly, and Midjourney’s Discord interface has reached over 1 million users.
Increased access to this technology has accelerated the development of open-
source tools and resources for creativity and design. As a result, we have
seen artists using AI-generated imagery as part of their visual design
process. Two examples include a Cosmopolitan magazine cover designed in tandem
with DALLE-2, (Liu 2022), and an artist using Midjourney to win an art
competition (Harwell 2022). This democratization has raised questions
regarding how such models can be used in a wide variety of tasks, such as idea
visualization (Epstein et al. 2020; Epstein, Schroeder, and Newman 2022;
Rafner et al. 2021; Smith and Colton 2022).

These tools offer new possibilities for navigating the creative process.
Design research suggests a need for flexible pathways for creative
computational assistance _early_ on in the design process (Mothersill and Bove
2017). Many AI-empowered co-creative tools focus on support for the later
stages of design, when participants exploit, refine, or implement ideas, but
are less involved in the earliest stages of co-creation, when participants are
in the “explore” phase (Hwang 2022).

In contrast to using the generative model for directly creating digital media
content, a growing body of work explores the use of generative AI for upstream
tasks in the creative process, such as ideation and visualization, that have a
range of downstream tangible outcomes. These can include tattoos (TattoosAI
2022), fashion (Levine 2022), community values (Epstein, Schroeder, and Newman
2022), and visualizations of the future (Rafner et al. 2021). A key challenge
for using generative AI in upstream creative tasks, such as ideation and idea
visualization, is the fact that the affordances of physical materials could
differ from those of generated imagery, and users may be frustrated bridging
that distance to bring their idea into reality. Work by Dang et al. (2022)
shows that lack of support in the trial and error process of prompt
engineering (tactics for refining prompts to synthesize desired outputs) can
be frustrating for users, motivating further investigation into users’
prompting strategies. Connecting a user’s needs, based on their prompting
strategy, to their design stage could help provide individualized support for
a user’s design goals.

## Study Design

We conduct a pilot study to examine the impact of introducing AI-generated
images into the early stages of a design task in a physical medium. The
experiment was advertised as a community activity at a research university to
“turn trash into treasure” by making artistic sculptures out of discarded
materials. Once participants had joined the activity, they were were
instructed by the facilitator to choose 3-5 pieces of material from a box with
a range of objects like test tubes, pipe cleaners, foam pieces, and wires.

Once participants had chosen their sculpture materials, we explained that they
would give the facilitator three text prompts which would generate three
images. We asked them to consider how these images could inform their
sculpture design. We then asked participants for their first prompting phrase,
which the facilitator used as input for Stable Diffusion. The facilitator then
asked the participant the following whilst the image was being generated: 1)
“Why did you choose that prompt?” and 2) “What are you expecting to see?” Once
the facilitator had written down the responses, the generated image was
revealed to the participant. The facilitator repeated this process of
prompting, reflection, and image reveal up to another two times. After the
visualization stage, the participant was given 3 minutes to build a sculpture
using their existing materials, as well as adhesives like tape and hot glue.
Once the sculpture had been completed, the facilitator asked 1) “Was your
sculpture informed by your generated images?” and 2) “Would you use a text-to-
image model like Stable Diffusion for a creative task again?”

After the activity, we set out to measure each participant’s level of
conceptual exploration through their prompting journey. The concept of
semantic distance is a popular one in the evaluation of the creative processes
(Kenett 2019) and is one we believe extends to exploration through semantic
space in the early creative process. To operationalize this, we transformed
user prompts into sentence embeddings,111We used the sentence transformer:
https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2‘all-
MiniLM-L6-v2’ from HuggingFace due to its performance capturing semantic
meaning. then measured the cosine distance between a user’s first, second, and
third prompts, taking the average distance over these to characterize a user’s
level of conceptual exploration from prompt to prompt. We also qualitatively
analyzed the notes from each participant interview as well as the sculptures
each produced.

## Results

### Generated images informed final designs

Of the 30 participants, 27 produced at least two prompt and image pairs. Of
those 27, 24 produced all three images (the remaining three produced just one
image). 23/30 participants self-reported that their sculptures were informed
by the images they saw, and 28/30 participants reported that they would use
text-to-image models again for a creative task. Figures 1, 2, 3, and 4 show
examples of participants’ three prompt and images pairs and the created
sculptures that have strong visual links to the final design highlighted.

Figure 1 shows an example of strong image influence on sculpture design for
someone who did not have a sculpture idea going into the process. The
rightmost image is a photo of the sculpture they built, which is striking in
its visual similarity to the third generated image in particular. Some
congruences, including color of materials which were not changeable, are
highlighted in colored boxes on the image.

Figure 1: Visual elements inform sculpture design of a building.

Figure 2 shows the journey of someone who knew from the onset they wanted to
make a sculpture of a crab. The three images are outputs from the prompts:
“crab, ocean, spider, seaweed, plush toy of a crab” →“crab, ocean, spider,
seaweed, plush toy of a crab, colour red” →“crab, ocean, spider, seaweed,
plush toy of a crab, colour red, insect, lobster.” Despite already knowing
what they wanted to make, the participant said the images gave reminders of a
crab’s features so they could create it in real life. The participant tweaked
their second prompt to include “colour red” because their physical materials
were also red. Common entities like crabs are easy to generate thanks to their
high visual determinacy, but some users with more abstract ideas than the
example discussed above reported frustration visualizing what they had in
mind, even if they knew what they wanted to see in the images.

Figure 2: Visual elements inform sculpture design of a crab.

Of the participants who reported that the images were not informative to their
sculpture (n = 7), three explained that this was because images did not
sufficiently relate to the materials given and they could not build what they
saw. Some of these participants expressed frustration in formulating a text
prompt that would produce what they wanted to see in an image. Only 36.2% of
images contained elements that the participants expected to see, indicating
many visual elements were unexpected. 40% of participants (12/30) directly
referenced their chosen materials, like “box” or “sponge.” 222We do not find
statistically significant differences in the distribution of average cosine
distances of user prompts and whether or not the prompt contained mentions of
physical materials (t = 1.01, p = 0.32), material qualities (t = -1.25, p =
0.22), or colors (t = 0.12 , p = 0.91). The frustration some users had
translating between materials and expected images suggests a need to better
support users in translating between physical materials and prompt wording
when ideating for physical artefacts.

In post-interviews, the reasons participants reported that images were
informative were diverse. Some found that the lateral concepts introduced by
the images gave them new conceptual ideas, like the addition of a river in
Figure 1 perhaps indicating they were in an early design stage where
exploration was particularly useful. Others honed existing ideas by getting
implementation inspiration from the images, like in Figure 2 perhaps
indicating a later design stage focused on execution. These findings confirm
those in Epstein, Schroeder, and Newman (2022), which showed that AI-generated
images are helpful in visualizing ideas for two main reasons: they give
lateral insight as well as implementation ideas.

Figure 3: Visual elements relate to sculpture design of a garden. Figure 4:
Visual elements inform sculpture design of a robot.

Two participants who did not find the images helpful built their sculpture
immediately after their first image, and explained that the strength of their
pre-visualization ideas meant that they believed no images would further
inform the sculptures they would make. This might suggest that seeing images
is less helpful for users with an already concrete vision, but we do not find
strong evidence to support that claim. Of the individuals with a design idea
before starting the activity, 3/8 found seeing the images helpful and 5 did
not. Of the individuals with no design idea before, 15/23 found seeing the
images helpful and 8 did not, a 44% difference. This provides suggestive
evidence (t = -1.365, p = 0.183) that participants who did not have an idea
may find the generated images more helpful.

### Distinct prompting styles emerge

Participants’ wording choices for prompts ranged in their degree of conceptual
exploration. We observed and named some patterns we saw in users’ prompting
choices. The most minimal amount of conceptual exploration that we observed
was a pattern we call the “refiner” style. In these instances, the participant
started with a prompt and made minor edits to it. It was the most common
pattern we saw (n = 15). For example: Prompt 1: ‘geometric ominous dystopian
creature metallic’. Prompt 2: ‘geometric ominous dystopian creature _neon
heart beating_ ’. Prompt 3: ‘geometric ominous creature neon heart beating
_rainbow_ ’. The participant described here kept changing the final terms to
try to see the image they wanted.

The next pattern observed was the “rephraser” prompting style (n = 5). Here,
conceptual subject matter remained the same, but changes to the wording or
order of the prompt changed substantially. Example: Prompt 1: ‘idyllic cyber
vista with a bottle robot’. Prompt 2: ‘dasani water bottle robot in front of
the Windows XP wallpaper’. Prompt 3: ‘robot that looks like a water bottle in
front of a grassy hill’. Each prompt contains a reference to a robot, but the
way they refer to the same background of a grassy hill changes.

A few participants did even more conceptual exploration. The “explorer”
prompting style (n = 2) we describe consisted of three conceptually unrelated
prompts. Example: Prompt 1: ‘my printed map required too much plastic’. Prompt
2: ‘I’m exhausted but I’m still having fun’. Prompt 3: ‘my hovercraft is full
of eels’. The small size of this category shows that most people had some idea
of what they were exploring.

These described styles form an exploration gradient where “explorers” have the
most semantic distance traveled, “rephrasers” have a main idea but still
explore around it, and “refiners” focus on exploitation of a single string of
words.

We observed that a remaining 7 participants did not fit neatly in to these
qualitatively defined prompting categories, often using a mix of styles we
have described. For example, the user who built the sculpture in Figure 1
prompted as follows: Prompt 1: “Hamburg is the most beautiful city in the
world”, Prompt 2: “sun and shiny water, colourful buildings made of trash and
ropes”, Prompt 3: “green red and white buildings in front of a shiny river
with a white fence on the side”. The first prompt appears to be exploratory,
whereas the second two describe a more concrete scene in two similar but
distinct ways, reminiscent of behavior we see in those using a “rephraser”
style. The final three participants to account for only gave a single prompt,
so their prompting style does not form a style that can be attributed over
time.

Because not all participants fit neatly into the prompting styles we
qualitatively described, we use a computational measure of semantic distance
to objectively characterize all prompting journeys in the rest of our
analysis. We calculate the average conceptual distance a participant traveled
through their prompting journey by taking the cosine distance between the text
embeddings of first and second prompts, then the second and third prompts,
then averaging the two distances. For participants with two prompts, we treat
the single distance between the two prompts as the ”average” distance. For
participants with a single prompt, where no distance between prompts was
traveled, we defined the distance as 0.

Figure 5: Histogram of participants’ conceptual exploration across their
prompting session, measured by average cosine distance. Shown above histogram:
mean cosine distance within a prompting style and 95% confidence intervals.
The prompting styles appear to occur at different average cosine distances.

Figure 5 shows a histogram of average cosine distances across the prompting
journeys of our 27 participants who gave at least 2 prompts, along with the
average and standard deviation of the subgroup we qualitatively grouped into
“refiner,” “rephraser,” “inconsistent,” and “explorer.” The participants we
describe as “refiners” have the lowest average cosine distance of 0.178, while
the participants we described as “explorers” had the highest average cosine
distance of 0.793. The participants we describe as “rephraser” and
“inconsistent” fall in the middle range.

We visualize three participant journeys, one from each of the main styles we
have described, through semantic space in Figure 7. We use TSNE to reduce
dimensionality of the shared embedding space to two dimensions, and draw
arrows from points representing a participant’s first to second to third
prompts. The path between green dots in Figure 7 (representing a single user’s
prompting journey) shows considerable conceptual exploration using an
“explorer” prompting style. The participant journey represented by blue dots
shows a less exploratory “rephraser” style, and the user represented by purple
dots shows the “refiner” prompting style. The prompts and corresponding images
generated for these three participants can be seen in Figure 6.

Figure 6: Prompts and image pairs for the different prompting styles
illustrated in Figure 3. From top to bottom: ‘refiner’, ‘rephraser’ and
‘explorer’

### Prompting Style differs by Design Stage

We next investigate the relationship between the design stage a user
indicated, based on whether they described in our post-interview having
started the task with a sculpture idea or not, and the average semantic
distance they traveled. Because we gave participants the design task of
creating a sculpture, participants skipped what Hwang (2022) calls the “Q&A”
stage of defining the creative task. By constraining the category of materials
and type of creative output (sculpture), we further define the design
problem’s “artefact type” and “media type,” per the Mothersill and Bove (2017)
conceptualization of the design process. Those who came into the visualization
stage of the exercise with no idea what to make started in the “wandering
stage,” wherein creatives explore possible strategies and incubate ideas. Some
who started visualization with a sculpture idea skipped this stage and were
already in the “hands-on stage” of developing solutions, or were already
moving to the “camera-ready stage,” which focuses on selecting and
implementing ideas, to use Hwang’s taxonomy. Using these stages of the
creative process, we compare the average semantic distance traveled for the
group of participants who had sculpture ideas before visualizing with
generative AI to those who did not.

We found that the amount of conceptual exploration a participant did was lower
for participants who said they had a sculpture idea at the visualization stage
than those who did not (t = -2.94, p = 0.006). We also found that participants
who stopped early, generating fewer than 3 images, employed less conceptual
exploration in their images than those who did not (t = 4.31, p <0.001). All
participants who stopped after one (n = 3) or two images (n = 4) had an idea
of what to make before starting their visualization.

Figure 7: TSNE plot of embedded representations of three user prompting
sessions where users range in degree of conceptual exploration while prompting

## Future Work and Limitations

Interesting preliminary findings emerged from this pilot study, but these
findings could be investigated more precisely with changes to the study design
that are informed by this first experiment. A similar experiment that allows
participants to choose their materials after visualization, instead of before,
could investigate whether there is a relationship between prompting styles and
the objects people select to build with. Similarly, one could imagine
comparing two groups, one that used Stable diffusion for ideation, and one
without access to the tool.

We could also ask participants upfront whether they have a sculpture idea or
not before their prompting journey starts, as in this pilot study the design
stages recorded by facilitators were determined by participants’ post-
interviews and this data was therefore emergent. Participants’ comments
reflecting on their design stages in their post-interview may have been
affected by seeing the generated images already. We also gave participants a
maximum of three tries with Stable Diffusion in this experiment for the sake
of time, but more complex prompting behavior may emerge for users across more
interactions with the technology. Anecdotally, we also saw that some users
were distracted by the technology itself, so future work could directly
observe the effect of user familiarity with the tool on the user’s prompting
journey, and we are unsure how user familiarity with the tool may have
affected the prompting journeys we describe.

The embedding representations we chose for prompts could also use the CLIP or
BERT tokenizers that were used in Stable Diffusion for greater conceptual
similarity to the tokenization process of text to image models, and a
sensitivity analysis of individual tokens on these embeddings could be done to
deepen conclusions.

## Conclusion

Most participants found seeing the images generated by a text-to-image model
informative to their final design of a sculpture. The average semantic
distance a participant traveled in their prompting journey during the
visualization activity was lower if they had a sculpture idea to start the
activity than if they did not. This shows that prompting decisions relate to a
participant’s design stage. Participants who started visualization with
sculpture ideas already in mind used image generation as an opportunity to
“exploit” or refine ideas, traveling less average semantic distance than those
who were unsure what to build and used the images to explore. To better
support creators, text-to-image tools could identify a user’s semantic
distance traveled over a prompting session to suggest hints that are useful to
their current design stage.

## Acknowledgements

We thank Pip Mothersill and Janet Rafner for helpful comments and feedback. We
thank the 99F crew, participants of an earlier iteration of this workshop, and
Kim Hew-Low for help shaping the workshop.

The vast majority of participants in our study were interested in using a
generative creative tool like this, and they had many ideas for ways they
would do so—- from designing a new kind of battery to generating artwork for
their bedrooms. With thoughtful prompting guidance, we hope that a wide of
variety of creators can use text-to-image models to inspire visionary ideas
and bring those ideas from our screens into reality.

## References

  * Dang et al. (2022) Dang, H.; Mecke, L.; Lehmann, F.; Goller, S.; and Buschek, D. 2022.  How to Prompt? Opportunities and Challenges of Zero-and Few-Shot Learning for Human-AI Interaction in Creative Applications of Generative Models.  _arXiv preprint arXiv:2209.01390_.
  * Epstein et al. (2020) Epstein, Z.; Levine, S.; Rand, D. G.; and Rahwan, I. 2020.  Who Gets Credit for AI-Generated Art?  _iScience_ , 23(9): 101515.
  * Epstein, Schroeder, and Newman (2022) Epstein, Z.; Schroeder, H.; and Newman, D. 2022.  When happy accidents spark creativity: Bringing collaborative speculation to life with generative AI.  _arXiv preprint arXiv:2206.00533_.
  * Harwell (2022) Harwell, D. 2022.  He used AI to win a fine-arts competition. Was it cheating?  _The Washington Post_.
  * Hwang (2022) Hwang, A. H.-C. 2022.  Too Late to Be Creative? AI-Empowered Tools in Creative Processes.  In _Extended Abstracts of the 2022 CHI Conference on Human Factors in Computing Systems_ , CHI EA ’22. New York, NY, USA: Association for Computing Machinery.  ISBN 9781450391566.
  * Kenett (2019) Kenett, Y. N. 2019.  What can quantitative measures of semantic distance tell us about creativity?  _Current Opinion in Behavioral Sciences_ , 27: 11–16.  Creativity.
  * Levine (2022) Levine, G. 2022.  Futuristic Fashion Designs Generated by DALL-E’s AI.
  * Liu (2022) Liu, G. 2022.  The World’s Smartest Artificial Intelligence Just Made Its First Magazine Cover.  _Cosmopolitan_.
  * Mothersill and Bove (2017) Mothersill, P.; and Bove, V. M. 2017.  Humans, Machines and the Design Process. Exploring the Role of Computation in the Early Phases of Creation.  _The Design Journal_ , 20(sup1): S3899–S3913.
  * Rafner et al. (2021) Rafner, J.; Langsford, S.; Hjorth, A.; Gajdacz, M.; Philipsen, L.; Risi, S.; Simon, J.; and Sherson, J. 2021.  Utopian or Dystopian?: Using a ML-Assisted Image Generation Game to Empower the General Public to Envision the Future.  In _Creativity and Cognition_. New York, NY, USA: Association for Computing Machinery.  ISBN 9781450383769.
  * Rombach et al. (2022) Rombach, R.; Blattmann, A.; Lorenz, D.; Esser, P.; and Ommer, B. 2022.  High-resolution image synthesis with latent diffusion models.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 10684–10695.
  * Smith and Colton (2022) Smith, A.; and Colton, S. 2022.  The@ artbhot text-to-image twitter bot.  In _Proceedings of the International Conference on Computational Creativity_.
  * TattoosAI (2022) TattoosAI. 2022.  AI-powered tattoo artist.  https://www.tattoosai.com/.

.


Most of the current paper describes how to estimate magnetic field within the
framework of the Velocity Gradient Technique (VGT). However, the techniques
that are developed in the current paper does not require the prediction from
the gradient technique to work. In the following, we discuss the complementary
techniques that can help us to construct the 3D distribution of magnetic
field.

###  B.1 Obtaining $M_{s}$, $M_{A}$ and the inclination angle $\gamma$

Alfvénic Mach number $M_{A}$: For instance, the ability of various techniques
to estimate the values of $M_{A}$ and $M_{s}$ from observational data were
studied in literature prior to the time of GT appeared. For instance, the
ability of Tsallis statistics to obtain $M_{A}$ was demonstrated in Esquivel &
Lazarian (2010) and Tofflemire et al. (2011). In addition, Esquivel & Lazarian
(2005) obtained empirical relations between the anisotropy of the structure
function and $M_{A}$. In fact, the first demonstration of the effect of
magnetic field strength on the anisotropy of correlations in the channel maps
was presented in Lazarian et al. (2002). The analytical relations between
$M_{A}$ the ratio of the quadropole to monopole moments of the structure
functions of intensity was obtained in LP12 for synchrotron emission and in
KLP17,Kandel et al. (2017b) and Kandel et al. (2018) for the spectroscopic
data. The ability of obtaining the mean $M_{A}$ was demonstrated with
polarization data in Lazarian et al. (2018), while the statistics of $M_{A}$
was carefully studied in Yuen & Lazarian (2020b).

Sonic Mach number $M_{s}$: It its turn, $M_{s}$ was obtained via the analysis
of kurtosis and skewness of the emission and polarization observational data
(see Burkhart et al. 2009; Burkhart & Lazarian 2012; Burkhart et al. 2013,
2014). Combining thus found $M_{s}$ with the $M_{A}$ obtained with the
approaches above ,one can successfully obtain the strength of the POS
component of the magnetic field. The results of $M_{s}$ measurements are very
robust, and they are marginally affected either by shear or by large-scale
magnetic field curvature. Similarly, the calculation of $M_{A}$ is not much
influenced by the galactic shear or any other large-scale shear induced by
non-turbulent motions.

Inclination Angle $\gamma$: To obtain the full magnetic field strength, an
independent way of obtaining $\gamma$ is necessary. Different measures depend
differently on the aforementioned turbulence parameters and $\gamma$. This
provides another avenue for obtaining the 3D vector of the magnetic field.
Several approaches for obtaining $\gamma$ are suggested and explored in Yuen
et al. (2023); Malik et al. (2023, 2024) and Hu et al. (2024); Hu & Lazarian
(2024). If combined with the value of the magnetic field obtained with the DMA
approach, the output provides the full 3D magnetic field vector.

###  B.2 Removing density contributions when computing gradients

As we discussed earlier, MHD turbulence theory is defined in terms of velocity
and magnetic field fluctuations (see Beresnyak & Lazarian 2019). The density
can act either as a passive scalar or show its own non-universal scaling. The
former is mostly the case of subsonic turbulence and this, as pointed out in
§3.5 results in the density gradient technique. However, in general, one can
get more reliable predictions if the density contributions are removed.

Tt is worth emphasizing that the most of the previous predictions on Alfvénic
Mach number in both the LP-series or the KLP series assume a constant density
and negligible thermal broadening effects. While the authors have discussed
qualitatively how these effects change the analytical predictions, it is
unavoidable to deal with both the density and thermal effects in regimes such
as multiphase HI media. To remove the effects of density and thermal
broadening from observed channels, Yuen et al. (2021) proposes the Velocity
Decomposition Algorithm (VDA), an algorithm that is built upon the foundation
of LP00, that applies to every channel map. The velocity caustics map $p_{v}$,
which is the channel map without any density fluctuations, is given by:

$p_{v}=p-\left(\langle pI\rangle-\langle p\rangle\langle
I\rangle\right)\frac{I-\langle I\rangle}{\sigma_{I}^{2}}$ (B1)

where $p$ is the channel map, $I$ is the column density map and
$\langle..\rangle$ is the averaging operator. Yuen et al. (2021) shows that
the VDA method is extremely accurate reflecting the statistics of velocity
fluctuations in sub-sonic media while being fairly accurate in supersonic
media. For our current paper’s purpose, we need the velocity centroid map with
no density effects. The appendix of Yuen et al. (2021) gives a formula for the
constant density velocity centroid:

$V_{n}({\bf X})=\int dvvp_{v}({\bf X},v)$ (B2)

which is exactly the quantity that is investigated in both KLP16 and KLP17.
Note, that in this paper we use capital symbols to denote the 2D vectors, i.e.
${\bf X}$ is the 2D Plane of Sky (POS) vector. In below we are going to
investigate the gradients of centroid or channels assuming density being
constant. The exact formula in obtaining these observables are given by either
Eq.B1 (channels) or Eq.B2 (centroids).

###  B.3 Obtaining 3D magnetic field structures from divergence free condition

It was discussed in Lazarian & Yuen (2018b) that the 3D distribution of POS
magnetic field can be obtained applying velocity gradients to channel maps or
reduced centroids and using the galactic rotation curve. This approach was
applied in González-Casanova & Lazarian (2019) to map the POS magnetic field
using gradients of the velocity centroids. The corresponding map of 3D
distribution of the POS magnetic field was then successfully tested by
comparing the starlight polarization predicted for a number of stars with the
known distances with the actual measured polarization.

The ability of obtaining 3D POS B-field maps provide another way of obtaining
the actual distribution of 3D magnetic field vectors. Indeed, the magnetic
field is a solenoidal vector three components of which are constrained through
the divergence free condition: $\nabla\cdot{\bf B}=0,$ , meaning that that
only 2 components in Fourier space are independent. Therefore if we are given
2 components, we can restore the 3d one. The corresponding procedure was
suggested in Chepurnov (2021) for restoring the z-component of magnetic field
when two POS component are obtained using the Lazarian & Yuen (2018b)
synchrotron gradient tomography that can restore the 3D POS magnetic field.

We would like to stress that the same procedure is also applicable to studying
magnetic fields using both the POS magnetic fields obtained using the channel
maps or reduced centroids in the presence of galactic rotation curve. The
caveat for the application of the technique is that the magnetic field is
solenoidal in actual 3D space and can lose this property in other coordinates.
For instance, the distortions of the 2D slice for which the POS magnetic field
is determined can result in errors in determining the LOS component of
magnetic field. This means that the POS for the procedure should be
sufficiently coarse grained to avoid problems of the mapping.

##  Appendix C Numerical simulations employed for testing

In Table 1 we describe the set of numerical simulations that we employ to test
our analytical results.

Model | $M_{S}$ | $M_{A}$ | $\beta=2M_{A}^{2}/M_{S}^{2}$ | Resolution | Model | $M_{S}$ | $M_{A}$ | $\beta=2M_{A}^{2}/M_{S}^{2}$ | Resolution
---|---|---|---|---|---|---|---|---|---
b11 | 0.41 | 0.04 | 0.02 | $480^{3}$ | huge-0 | 6.17 | 0.22 | 0.0025 | $792^{3}$
b12 | 0.92 | 0.09 | 0.02 | $480^{3}$ | huge-1 | 5.65 | 0.42 | 0.011 | $792^{3}$
b13 | 1.95 | 0.18 | 0.02 | $480^{3}$ | huge-2 | 5.81 | 0.61 | 0.022 | $792^{3}$
b14 | 3.88 | 0.35 | 0.02 | $480^{3}$ | huge-3 | 5.66 | 0.82 | 0.042 | $792^{3}$
b15 | 7.14 | 0.66 | 0.02 | $480^{3}$ | huge-4 | 5.62 | 1.01 | 0.065 | $792^{3}$
b21 | 0.47 | 0.15 | 0.22 | $480^{3}$ | huge-5 | 5.63 | 1.19 | 0.089 | $792^{3}$
b22 | 0.98 | 0.32 | 0.22 | $480^{3}$ | huge-6 | 5.70 | 1.38 | 0.12 | $792^{3}$
b23 | 1.92 | 0.59 | 0.22 | $480^{3}$ | huge-7 | 5.56 | 1.55 | 0.16 | $792^{3}$
b31 | 0.48 | 0.48 | 2.0 | $480^{3}$ | huge-8 | 5.50 | 1.67 | 0.18 | $792^{3}$
b32 | 0.93 | 0.94 | 2.0 | $480^{3}$ | huge-9 | 5.39 | 1.71 | 0.20 | $792^{3}$
b41 | 0.16 | 0.49 | 18 | $480^{3}$ | h0-1200 | 6.36 | 0.22 | 0.0025 | $1200^{3}$
b42 | 0.34 | 1.11 | 18 | $480^{3}$ | h9-1200 | 10.79 | 2.29 | 0.098 | $1200^{3}$
b51 | 0.05 | 0.52 | 200 | $480^{3}$ | e5r2 | 0.13 | 5.99 | 4363 | $1200^{3}$
b52 | 0.10 | 1.08 | 200 | $480^{3}$ | e5r3 | 0.61 | 0.63 | 2.09 | $1200^{3}$
Ms0.2Ma0.2 | 0.2 | 0.2 | 2 | $480^{3}$ | e6r3 | 5.45 | 0.50 | 0.017 | $1200^{3}$
Ms0.4Ma0.2 | 0.57 | 0.28 | 0.48 | $480^{3}$ | e7r3 | 0.53 | 3.24 | 73.64 | $1200^{3}$
Ms4.0Ma0.2 | 3.81 | 0.18 | 0.00446 | $480^{3}$ | incompressible | 0 | 0.61 | $\infty$ | $512^{3}$
Ms20.0Ma0.2 | 20.59 | 0.18 | 0.00015 | $480^{3}$ |  |  |  |  |
Table 1:  Description of MHD simulation cubes which some of them have been
used in the series of papers about VGT (Yuen & Lazarian, 2017a, b; Lazarian &
Yuen, 2018b, a). $M_{s}$ and $M_{A}$ are the R.M.S values at each the
snapshots are taken. The incompressible simulation is performed by Ho (2022).

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