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A Problem Properties
A.1 FGW Distance Metric
Fused Gromov-Wasserstein Distance [26] are defined in Eq.1. It can also be rewritten as follows:
$$
\operatorname {F G W} \left(G _ {1}, G _ {2}\right) = \min  _ {\boldsymbol {\pi} \in \Pi \left(\boldsymbol {\mu} _ {1}, \boldsymbol {\mu} _ {2}\right)} \left\langle (1 - \alpha) \boldsymbol {D} + \alpha L \left(\boldsymbol {A} _ {1}, \boldsymbol {A} _ {2}\right) \otimes \boldsymbol {\pi}, \boldsymbol {\pi} \right\rangle , \tag {8}
$$
where  $\pmb{D} = \{d(\pmb{x}_1^{(i)},\pmb{x}_2^{(j)})\}_{ij}$  is the distance matrix of node features, and  $L(A_{1},A_{2}) = \{|A_{1}(i,k) - A_{2}(j,\ell)|\}_{i,j,k,\ell}$  is a 4-dimensional tensor that depicts the structural distances. We denote  $\langle U,V\rangle = \mathrm{tr}(U^{\top}V)$  as the matrix scalar product. The  $\otimes$  operator conducts tensor-matrix multiplication as:  $\mathcal{L}\otimes T\stackrel {\mathrm{def.}}{=}\left(\sum_{k,\ell}\mathcal{L}_{i,j,k,\ell}T_{k,\ell}\right)_{i,j}$ . Then according to Proposition 1 in [28], when we take the  $\ell_2$ -norm to calculate the structural distance, we have:
$$
L \left(\boldsymbol {A} _ {1}, \boldsymbol {A} _ {2}\right) \otimes \boldsymbol {\pi} = c _ {\boldsymbol {A} _ {1}, \boldsymbol {A} _ {2}} - 2 \boldsymbol {A} _ {1} \boldsymbol {\pi} \boldsymbol {A} _ {2}, \tag {9}
$$
where  $c_{\pmb{A}_1,\pmb{A}_2} = \pmb{A}_1^{\odot 2}\pmb{\mu}_1\pmb{1}_{n_2}^\top +\pmb{1}_{n_1}\pmb{\mu}_2^\top \pmb{A}_2^{\odot 2}$ , and  $\odot 2$  denotes the Hadamard square (i.e.,  $\pmb{U}^{\odot 2} = \pmb{U} \odot \pmb{U}$ ). Taking Eq.9 into Eq.8, we can rewrite the minimizer as:
$$
\begin{array}{l} \mathrm {R H S} = (1 - \alpha) \langle \boldsymbol {D}, \boldsymbol {\pi} \rangle + \overbrace {\alpha \langle \boldsymbol {A} _ {1} ^ {\odot 2} \boldsymbol {\mu} _ {1} \boldsymbol {1} _ {n _ {2}} ^ {\top} , \boldsymbol {\pi} \rangle + \alpha \langle \boldsymbol {1} _ {n _ {1}} \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {A} _ {2} ^ {\odot 2} , \boldsymbol {\pi} \rangle} ^ {- 1} - 2 \alpha \langle \boldsymbol {A} _ {1} \boldsymbol {\pi} \boldsymbol {A} _ {2}, \boldsymbol {\pi} \rangle \\ = \underbrace {\alpha \left(\boldsymbol {A} _ {1} ^ {\odot 2} \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {\mu} _ {1} + \boldsymbol {A} _ {2} ^ {\odot 2} \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {\mu} _ {2}\right)} _ {\text {c o n s t a n t}} + \langle (1 - \alpha) D - 2 \alpha \boldsymbol {A} _ {1} \pi \boldsymbol {A} _ {2}, \pi \rangle \tag {10} \\ \end{array}
$$
The items in the overbrace become a constant because the marginals of  $\pi$  are fixed as  $\mu_{1}$  and  $\mu_{2}$ , respectively. Therefore, we can remove the constant term and formulate an equivalent optimization objective denoted as  $\overline{\mathrm{FGW}}(G_1, G_2)$ , which can be rewritten as:
$$
\begin{array}{l} \overline {{\mathrm {F G W}}} (G _ {1}, G _ {2}) = \min  _ {\boldsymbol {\pi} \in \Pi (\boldsymbol {\mu} _ {1}, \boldsymbol {\mu} _ {2})} \langle (1 - \alpha) \boldsymbol {D} - 2 \alpha \boldsymbol {A} _ {1} \boldsymbol {\pi} \boldsymbol {A} _ {2}, \boldsymbol {\pi} \rangle \\ = \min  _ {\boldsymbol {\pi} \in \Pi (\boldsymbol {\mu} _ {1}, \boldsymbol {\mu} _ {2})} (1 - \alpha) \operatorname {t r} \left(\boldsymbol {\pi} ^ {\top} \boldsymbol {D}\right) - 2 \alpha \operatorname {t r} \left(\boldsymbol {\pi} ^ {\top} \boldsymbol {A} _ {1} \boldsymbol {\pi} \boldsymbol {A} _ {2}\right). \tag {11} \\ \end{array}
$$
We denote the above optimization objective as a function  $f(\pi)$  w.r.t.  $\pi$ , and we call it the FGW function in the main text.
A.2 Mirror Descent and the Relaxed FGW Solver in FGwMixup*
Definition 1 (Normal Cone). Given any set  $\mathcal{C}$  and point  $x\in \mathcal{C}$ , we can define normal cone as:
$$
\mathcal {N} _ {\mathcal {C}} (x) = \{g: g ^ {\top} x \geq g ^ {\top} y, \forall y \in \mathcal {C} \}
$$
There are some properties of normal cone. The normal cone is always convex, and  $\operatorname{Proj}_{\mathcal{C}}(x + z) = x, \forall x \in \mathcal{C}, z \in \mathcal{N}_{\mathcal{C}}(x)$  always holds.
Proposition 3 (The Update of Mirror Descent). The update of Mirror Descent takes the form of  $x^{(k + 1)} = \arg \min_{x\in \mathcal{X}}D_{\phi}(x,y^{(k + 1)})$ , where  $\phi (\cdot)$  is a Bregman Projection, and  $\nabla \phi (y^{(k + 1)}) = \nabla \phi (\omega^{(k)}) - \nabla f(\pi) / \rho$ .
Proof. For a constrained optimization problem  $\min_{x\in \mathcal{X}}f(x),\mathcal{X}\subset \mathcal{D}$ , Mirror Descent iteratively updates  $x$  with  $x^{(k + 1)} = \arg \min_{x\in \mathcal{X}}\nabla f(x^{(k)})^{\top}x + \rho D_{\phi}(x,x^{(k)})$  with Bregman divergence  $D_{\phi}(x,y)\coloneqq \phi (x) - \phi (y) - \langle \nabla \phi (y),x - y\rangle$ . We further have:
$$
\begin{array}{l} x ^ {(k + 1)} = \arg \min  _ {x \in \mathcal {X}} \nabla f (x ^ {(k)}) ^ {\top} x + \rho D _ {\phi} (x, x ^ {(k)}) \\ \Rightarrow 0 \in \nabla f \left(x ^ {(k)}\right) ^ {\top} / \rho + \nabla \phi \left(x ^ {(k + 1)}\right) - \nabla \phi \left(x ^ {(k)}\right) + \mathcal {N} _ {\chi} \left(x ^ {(k + 1)}\right) \tag {12} \\ \Rightarrow x ^ {(k + 1)} = \arg \min  _ {x \in \mathcal {X}} D _ {\phi} (x, y ^ {(k + 1)}), \text {w h e r e} \nabla \phi (y ^ {(k + 1)}) = \nabla \phi (x ^ {(k)}) - \nabla f (x ^ {(k)}) / \rho \\ \end{array}
$$
Figure 3: The illustration of the MD procedure.
This updating procedure is vividly depicted by Fig.3. For each update of  $x^{(k)}$ , MD conducts the following steps:
① Apply mirror mapping  $(\nabla \phi(\cdot))$  on  $x^{(k)}$  from the primal space  $\mathcal{D}$  to the dual space  $\nabla \phi(\mathcal{D})$ .
② Take a gradient step in the dual space, which is  $\delta^{(k + 1)} = \nabla \phi (x^{(k)}) - \nabla f(x^{(k)}) / \rho$
(3) Inversely mapping  $(\nabla \phi^{-1}(\cdot))$  the dual space back to the primal space, which is  $y^{(k + 1)} = \nabla \phi^{-1}(\delta^{(k + 1)})$ , also written as  $\nabla \phi (y^{(k + 1)}) = \nabla \phi (x^{(k)}) - \nabla f(x^{(k)}) / \rho$ .
(4) The updated point may fall out of the constraints of  $\mathcal{X}$ . Thus, Bregman projection is conducted to project  $y^{(k + 1)}$  to  $x^{(k + 1)}\in \mathcal{X}$ , where  $x^{(k + 1)} = \arg \min_{x\in \mathcal{X}}D_{\phi}(x,y^{(k + 1)})$ .
The relaxed FGW conducts projections to constraints of rows and columns alternately, which is written as  $\min_{\pi} f(\pi) + \mathbb{I}_{\{\pi \in \Pi_1\}} + \mathbb{I}_{\{\pi \in \Pi_2\}}$ , where  $\mathbb{I}$  is the indicator function. We adopt the operator splitting strategy, and the optimization can be formulated as:
$$
F _ {\rho} (\pi , \omega) = f (\pi) + f (\omega) + \mathbb {I} _ {\{\pi \in \Pi_ {1} \}} + \mathbb {I} _ {\{\omega \in \Pi_ {2} \}} \tag {13}
$$
We apply Mirror Descent algorithm with Bregman divergence  $D_{\phi}(x,y)$  to solve this problem, whose update takes the form of:
$$
\pi^ {(k + 1)} = \arg \min  _ {\pi \in \Pi_ {1}} \nabla f (\pi) ^ {\top} \pi + \rho D _ {\phi} (\pi , \omega^ {(k)}) \tag {14}
$$
$$
\omega^ {(k + 1)} = \arg \min _ {\omega \in \Pi_ {2}} \nabla f (\omega) ^ {\top} \omega + \rho D _ {\phi} \left(\omega , \pi^ {(k + 1)}\right)
$$
Let's take the MD update of  $\pi$  as an example. Taking relative entropy as  $\phi(x) = \sum_{i} x_i \log x_i$ , we can solve  $y^{(k+1)}$  (conducting steps ① to ③) with exact analytical solutions. Specifically,  $\nabla \phi(x) = \sum_{i} (1 + \log x_i)$ , and the update w.r.t.  $\pi$  writes:
$$
\begin{array}{l} \pi^ {(k + 1)} = \arg \min  _ {\pi \in \Pi_ {1}} D _ {\phi} (\pi , y ^ {(k + 1)}), \text {w h e r e} \log (y ^ {(k + 1)}) = \log (\omega^ {(k)}) - \nabla f (\omega^ {(k)}) / \rho \\ \Rightarrow \pi^ {(k + 1)} = \arg \min  _ {\pi \in \Pi_ {1}} D _ {\phi} (\pi , y ^ {(k + 1)}), \text {w h e r e} y ^ {(k + 1)} = \omega^ {(k)} \odot \exp (- \nabla f (\omega^ {(k)}) / \rho). \tag {15} \\ \end{array}
$$
Furthermore, under this setting, we also have an exact solution of the projection step (4)  $\arg \min_{\pi \in \Pi_1}D_\phi (\pi ,y^{(k + 1)}) = \mathrm{diag}(\mu_1.\big{/}y^{(k + 1)}1_{n_2})y^{(k + 1)}$ . Then the MD update procedure can be formulated according to Prop. 3 as follows:
$$
\pi^ {(k + 1)} \leftarrow \omega^ {(k)} \odot \exp (- \nabla f \left(\omega^ {(k)}\right) / \rho), \pi^ {(k + 1)} \leftarrow \operatorname {d i a g} \left(\mu_ {1}. / \pi^ {(k + 1)} 1 _ {n _ {2}}\right) \pi^ {(k + 1)} \tag {16}
$$
Furthermore, the sub-gradient of our FGW optimizing objective w.r.t.  $\pi$  is calculated as follows:
$$
\nabla f (\pi) = \nabla_ {\pi} \operatorname {F G W} \left(G _ {1}, G _ {2}\right) = (1 - \alpha) D - 4 \alpha A _ {1} \pi A _ {2}. \tag {17}
$$
Taking Eq.17 into Eq.16, we have the update of  $\pi$ , and a similar procedure is conducted for  $\omega$ . Setting  $\rho$  to  $1 / \gamma$ , the final update procedure is shown in Line 8-11 in Alg.2.
Proof. From Corollary 4.6 of [59] we have: for any Sinkhorn iteration optimizing  $\pi \in \Pi(\mu, \nu)$  on two different marginals alternately  $\mu, \nu$ , let  $t_1 = \inf \{t \geq 0 : H(\mu_{2t}|\mu) \leq 1\} - 1$ , when iterations  $t \geq 2t_1$ :
$$
H \left(\mu_ {2 t} \mid \mu\right) + H \left(\mu \mid \mu_ {2 t}\right) \leq 1 0 \frac {C _ {1} ^ {2} \vee H \left(\pi_ {*} \mid R\right)}{\left(\lfloor t / 2 \rfloor - t _ {1}\right) t} = \mathcal {O} \left(t ^ {- 2}\right),
$$
$$
H \left(\pi_ {2 t} \mid \pi_ {*}\right) + H \left(\pi_ {*} \mid \pi_ {2 t}\right) \leq 5 \frac {C _ {1} ^ {2} \vee \left(H \left(\pi_ {*} \mid R\right) ^ {1 / 2} C _ {1}\right)}{\sqrt {\left(\lfloor t / 2 \rfloor - t _ {1}\right) t}} = \mathcal {O} \left(t ^ {- 1}\right), \tag {18}
$$
where  $C_1$  is a constant, and  $R$  is a fixed reference measure.
Definition 2 (Bounded Linear Regularity, Definition 5.6 in [60]). Let  $C_1, C_2, \dots, C_N$  be closed convex subsets of  $\mathbb{R}^n$  with a non-empty intersection  $C$ . We call the set  $\{C_1, C_2, \dots, C_N\}$  is bounded linearly regular if for every bounded subset  $\mathbb{B}$  of  $\mathbb{R}^n$ , there exists a constant  $\kappa > 0$  such that
$$
d (x, C) \leq \kappa \max  _ {i \in \{1, \dots , N \}} d (x, C _ {i}), \forall x \in \mathbb {B}, \text {w h e r e} d (x, \mathcal {C}) := \min  _ {x ^ {\prime} \in \mathcal {C}} \| x - x ^ {\prime} \|.
$$
The BLR condition naturally holds if all the  $C_i$  are polyhedral sets.
Definition 3 (Strong Convexity). A differentiable function  $f(\cdot)$  is strongly convex with the constant  $m$ , if the following inequality holds for all points  $x, y$  in its domain:
$$
\left(\nabla f (x) - \nabla f (y)\right) ^ {\top} (x - y) \geq m \| x - y \| ^ {2},
$$
where  $\| \cdot \|$  is the corresponding norm.
In Mirror Descent algorithm, the Bregman divergence  $D_h(x,y)$  requires  $h$  to be strongly convex.
Proof. Recall that the fixed point set of relaxed FGW is  $\mathcal{X}_{rel} = \{\pi^{*}\in C_{1},\omega^{*}\in C_{2}:0\in\nabla f(\pi^{*}) + \rho (\nabla h(\omega^{*}) - \nabla h(\pi^{*})) + \mathcal{N}_{C_{1}}(\pi^{*}),$  and  $0\in \nabla f(\omega^{*}) + \rho (\nabla h(\pi^{*}) - \nabla h(\omega^{*})) + \mathcal{N}_{C_2}(\omega^*)\}$  Let  $p\in \mathcal{N}_{C_1}(\pi^*)$  and  $q\in \mathcal{N}_{C_2}(\omega^*)$ $\mathcal{X}_{rel}$  is denoted as:
$$
\mathcal {X} _ {r e l} = \left\{\pi^ {*} \in C _ {1}, \omega^ {*} \in C _ {2}: \nabla f \left(\pi^ {*}\right) + \rho (\nabla h \left(\omega^ {*}\right) - \nabla h \left(\pi^ {*}\right)) + p = 0, p \in \mathcal {N} _ {C _ {1}} \left(\pi^ {*}\right) \right. \tag {19}
$$
$$
\nabla f \left(\omega^ {*}\right) + \rho \left(\nabla h \left(\pi^ {*}\right) - \nabla h \left(\omega^ {*}\right)\right) + q = 0, q \in \mathcal {N} _ {C _ {2}} \left(\omega^ {*}\right) \}
$$
We denote  $\hat{\pi} = \mathrm{Proj}_{C_1\cap C_2}(\pi^*)$ . According to Definition 2, as  $C_1$  and  $C_2$  are both polyhedral constraints and satisfy the BLR condition, we have:
$$
\begin{array}{l} \left\| \hat {\pi} - \pi^ {*} \right\| + \left\| \hat {\pi} - \omega^ {*} \right\| \leq 2 \left\| \hat {\pi} - \pi^ {*} \right\| + \left\| \pi^ {*} - \omega^ {*} \right\| \\ = 2 d \left(\pi^ {*}, C _ {1} \cap C _ {2}\right) + \left\| \pi^ {*} - \omega^ {*} \right\| \tag {20} \\ \leq (2 \kappa + 1) \| \pi^ {*} - \omega^ {*} \|. \\ \end{array}
$$
The first inequality holds based on the triangle inequality of the norm distances, and the second inequality holds based on the definition of point-to-space distance in Def.2. Moreover, according to the definition of  $\mathcal{X}_{rel}$ , for any  $(\pi^{*},\omega^{*})\in \mathcal{X}_{rel}$ , we have:
$$
\nabla f \left(\omega^ {*}\right) ^ {\top} (\hat {\pi} - \pi^ {*}) + \rho (\nabla h (\pi^ {*}) - \nabla h (\omega^ {*})) ^ {\top} (\hat {\pi} - \pi^ {*}) + p ^ {\top} (\hat {\pi} - \pi^ {*}) = 0, p \in \mathcal {N} _ {C _ {1}} (\pi^ {*})
$$
$$
\nabla f \left(\pi^ {*}\right) ^ {\top} (\hat {\pi} - \omega^ {*}) + \rho (\nabla h (\omega^ {*}) - \nabla h (\pi^ {*})) ^ {\top} (\hat {\pi} - \omega^ {*}) + q ^ {\top} (\hat {\pi} - \omega^ {*}) = 0, q \in \mathcal {N} _ {C _ {2}} (\omega^ {*}) \tag {21}
$$
Summing up the equations above, we have:
$$
\begin{array}{l} \nabla f \left(\omega^ {*}\right) ^ {\top} (\hat {\pi} - \pi^ {*}) + \nabla f \left(\pi^ {*}\right) ^ {\top} (\hat {\pi} - \omega^ {*}) + \rho (\nabla h (\pi^ {*}) - \nabla h (\omega^ {*})) ^ {\top} (\omega^ {*} - \pi^ {*}) + p ^ {\top} (\hat {\pi} - \pi^ {*}) + q ^ {\top} (\hat {\pi} - \omega^ {*}) \\ \stackrel {(a)} {=} \nabla f (\omega^ {*}) ^ {\top} (\hat {\pi} - \pi^ {*}) + \nabla f (\pi^ {*}) ^ {\top} (\hat {\pi} - \omega^ {*}) - \rho (D _ {h} (\pi^ {*}, \omega^ {*}) + D _ {h} (\omega^ {*}, \pi^ {*})) + p ^ {\top} (\hat {\pi} - \pi^ {*}) + q ^ {\top} (\hat {\pi} - \omega^ {*}) = 0 \tag {22} \\ \end{array}
$$
The  $(a)$  equation holds based on the definition of Bregman Divergence  $D_{h}(x,y)\coloneqq h(x) - h(y) - \langle \nabla h(y),x - y\rangle$ . Furthermore, due to the property of normal cones  $\mathcal{N}_{C_1}(\pi^*),\mathcal{N}_{C_2}(\omega^*)$ , we have  $p^\top (\pi -\pi^*)\leq 0,\forall \pi \in C_1$  and  $q^{\top}(\omega -\omega^{*})\leq 0,\forall \omega \in C_{2}$ . Taking  $\pi$  and  $\omega$  as  $\hat{\pi}\in C_1\cap C_2$ , we have  $p^\top (\hat{\pi} -\pi^*) + q^\top (\hat{\pi} -\omega^*)\leq 0$ . Therefore, from 22 we have the following inequality:
$$
\begin{array}{l} \nabla f \left(\omega^ {*}\right) ^ {\top} (\hat {\pi} - \pi^ {*}) + \nabla f \left(\pi^ {*}\right) ^ {\top} (\hat {\pi} - \omega^ {*}) - \rho \left(D _ {h} \left(\pi^ {*}, \omega^ {*}\right) + D _ {h} \left(\omega^ {*}, \pi^ {*}\right)\right) \geq 0 \\ \Rightarrow \rho \left(D _ {h} \left(\pi^ {*}, \omega^ {*}\right) + D _ {h} \left(\omega^ {*}, \pi^ {*}\right)\right) \leq \nabla f \left(\omega^ {*}\right) ^ {\top} (\hat {\pi} - \pi^ {*}) + \nabla f \left(\pi^ {*}\right) ^ {\top} (\hat {\pi} - \omega^ {*}) \\ \leq \| f \left(\omega^ {*}\right) \| \left\| \hat {\pi} - \pi^ {*} \right\| + \| \nabla f \left(\pi^ {*}\right) \| \left\| \hat {\pi} - \omega^ {*} \right\| \tag {23} \\ \stackrel {(b)} {\leq} C _ {f} \left(\| \hat {\pi} - \pi^ {*} \| + \| \hat {\pi} - \omega^ {*} \|\right) \\ \stackrel {(2 0)} {\leq} C _ {f} (2 \kappa + 1) \| \pi^ {*} - \omega^ {*} \|. \\ \end{array}
$$
The  $(b)$  inequality holds as the optimization objective is a quadratic function with bounded constraints, which means the gradient norm has a natural constant upper bound  $C_f$ . For the left hand side of the inequality above, the sum of Bregman divergences is also lower bounded by
$$
D _ {h} \left(\pi^ {*}, \omega^ {*}\right) + D _ {h} \left(\omega^ {*}, \pi^ {*}\right) = \left(\nabla h \left(\pi^ {*}\right) - \nabla h \left(\omega^ {*}\right)\right) ^ {\top} \left(\pi^ {*} - \omega^ {*}\right) \geq \sigma \| \pi^ {*} - \omega^ {*} \| ^ {2}, \tag {24}
$$
as  $h(\cdot)$  is  $\sigma$ -strongly convex in the definition of Bregman Divergence. Therefore, combining (24) and (23), we have:
$$
\left\| \pi^ {*} - \omega^ {*} \right\| \leq \frac {C _ {f} (2 \kappa + 1)}{\sigma \rho} \tag {25}
$$
Based on this important result, we analyze the distance between  $\mathcal{X}$  and  $\mathcal{X}_{rel}$ , which is given by  $\mathrm{dist}(\frac{\pi^* + \omega^*}{2},\mathcal{X})\coloneqq \min_{x\in \mathcal{X}}\left\| \frac{\pi^* + \omega^*}{2} -x\right\|$ ,  $\pi^{*}\in C_{1},\omega^{*}\in C_{2}$ . By adding up the two equations in 19 depicting the conditions on  $\pi^{*}$  and  $\omega^{*}$  of  $\mathcal{X}_{rel}$ , we have:
$$
\nabla f \left(\pi^ {*}\right) + \nabla f \left(\omega^ {*}\right) + p + q = 0 \stackrel {(c)} {\Rightarrow} \nabla f \left(\frac {\pi^ {*} + \omega^ {*}}{2}\right) + \frac {p + q}{2} = 0 \tag {26}
$$
The  $(c)$  derivation holds due to the linear property of the subgradient of FGW w.r.t.  $\pi$  (see Eq.17). Invoking this equality, we present the following upper bound analysis on  $\mathrm{dist}(\frac{\pi^* + \omega^*}{2},\mathcal{X})$ :
$\mathrm{dist}\left(\frac{\pi^{*} + w^{*}}{2},\mathcal{X}\right)\stackrel {(d)}{\leq}\epsilon \left\| \frac{\pi^{*} + w^{*}}{2} -\mathrm{proj}_{C_1\cap C_2}\left(\frac{\pi^{*} + w^{*}}{2} -\nabla f\left(\frac{\pi^{*} + w^{*}}{2}\right)\right)\right\|$
$$
\begin{array}{l} \stackrel {(e)} {=} \epsilon \left\| \frac {\pi^ {*} + w ^ {*}}{2} - \operatorname {p r o j} _ {C _ {1} \cap C _ {2}} \left(\frac {\pi^ {*} + p + w ^ {*} + q}{2}\right) \right\| \\ \stackrel {(f)} {=} \frac {\epsilon}{2} \left\| \operatorname {p r o j} _ {C _ {1}} \left(\pi^ {*} + p\right) + \operatorname {p r o j} _ {C _ {1}} \left(w ^ {*} + q\right) - 2 \operatorname {p r o j} _ {C _ {1} \cap C _ {2}} \left(\frac {\pi^ {*} + p + w ^ {*} + q}{2}\right) \right\| \\ \stackrel {(g)} {\leq} \frac {M \epsilon}{2} \| \pi^ {*} - w ^ {*} \|, \tag {27} \\ \end{array}
$$
The  $(d)$  inequality comes from the Luo-Tseng error bound condition [61] and Prop.3.1 in [29]. The  $(e)$  holds based on Eq.26. The  $(f)$  equality holds based on the property of the normal cone, which is  $\mathrm{Proj}_C(x + z) = x, \forall x \in C, z \in \mathcal{N}_C(x)$ . The  $(g)$  inequality holds by invoking Lemma 3.2 in [29], which tells  $\left\| \mathrm{proj}_{C_1}(x) + \mathrm{proj}_{C_2}(y) - 2 \mathrm{proj}_{C_1 \cap C_2}\left(\frac{x + y}{2}\right) \right\| \leqslant M \left\| \mathrm{proj}_{C_1}(x) - \mathrm{proj}_{C_2}(y) \right\|$ . Finally, combining (25) and (27), we can obtain the desired result, i.e.,
$$
\forall \left(\pi^ {*}, \omega^ {*}\right) \in \mathcal {X} _ {r e l}, \operatorname {d i s t} \left(\frac {\pi^ {*} + \omega^ {*}}{2}, \mathcal {X}\right) \leq \frac {(2 \kappa + 1) \epsilon C _ {f} M}{2 \sigma \rho} = \tau / \rho \tag {28}
$$
where  $\tau := \frac{(2\kappa + 1)\epsilon C_f M}{2\sigma}$  is a constant.
C Complete Mixup Procedure
We provide the complete mixup procedure of our method in Algorithm 3. We conduct augmentation only on the training set, and we mix up graphs from every pair of classes. For a specific mixup of two
Datasets	graphs	avg nodes	med nodes	avg edges	med edges	feat dim	classes
PROTEINS	1113	39.05	26	72.82	98	3	2
NCI1	4110	29.76	27	32.30	58	37	2
NCI109	4127	29.57	26	32.13	58	38	2
IMDB-B	1000	19.77	17	193.06	260	N/A	2
IMDB-M	1500	13.00	10	131.87	144	N/A	3
Table 6: Detailed statistics of experimented datasets.
graphs, we apply our proposed algorithm FGWMixup $_{(*)}$  (Alg.1, 2) to obtain the synthetic mixup graph  $\tilde{G} = (\tilde{\mu}, \tilde{X}, \tilde{A})$ . However, the numerical solution does not ensure a discrete adjacency matrix  $\tilde{A}$ . Thus, we adopt a thresholding strategy to discretize  $\tilde{A}$ , setting entries below and above the threshold to 0 and 1, respectively. The threshold is linearly searched between the maximum and minimum entries of  $\tilde{A}$ , aiming to minimize the density difference between the mixup graph and the original graphs. The mixup graphs are added to the training set for augmentation, and the training, validating and testing procedures follow the traditional paradigms.
Algorithm 3 The whole mixup procedure
1: Input: Training set  $\mathcal{G} = \{(G_i, y_i)\}_i$ , mixup ratio  $\beta$   
2: Output: Augmented training set  $\tilde{\mathcal{G}}$   
3:  $N = |\mathcal{G}|, N_y = |\{y_i\}_i|$ ,  $\tilde{\mathcal{G}} = \mathcal{G}$   
4: for  $i \neq j$  in range  $(N_y)$  do  
5: for  $k$  in range  $(2\beta N / N_y(N_y - 1))$  do  
6: Randomly sample  $G_i = (\pmb{\mu}_i, \pmb{X}_i, \pmb{A}_i)$  from class  $y_i$  and  $G_j = (\pmb{\mu}_j, \pmb{X}_j, \pmb{A}_j)$  from  $y_j$   
7: Generate new mixup graph and its label  $\tilde{G}, \tilde{y} = \text{FGWmixup}(_*)(G_i, G_j), \tilde{G} = (\tilde{\pmb{\mu}}, \tilde{\pmb{X}}, \tilde{\pmb{A}})$ .  
8: Search the threshold  $\theta$  to discretize  $\tilde{A}$ :  $\tilde{A} \gets \{1 \text{ if } a_{ij} \geq \theta \text{ else } 0\}_{ij}$ , which minimizes the density differences between  $\tilde{A}$  and  $A_1$ ,  $A_2$ .  
9:  $\tilde{\mathcal{G}} \gets \tilde{\mathcal{G}} \cup \{\tilde{G}\}$   
10: end for  
11: end for  
12: return  $\tilde{\mathcal{G}}$
D.1 Dataset Information
PROTEINS, NCI1, NCI109, IMDB-BINARY (IMDB-B) and IMDB-MULTI (IMDB-M) are the five benchmark graph classification datasets used in our experiments. When preprocessing the datasets, we remove all the isolated nodes (i.e., 0-degree nodes) in order to avoid invalid empty message passing of MPNNs. We present the detailed statistics of the preprocessed datasets in Table 6, including the number of graphs, the average/median node numbers, the average/median edge numbers, the dimension of node features, and the number of classes.
D.2 Backbones
We adopt two categories of GNN frameworks as the backbones. The first category is the virtual-node-enhanced Message Passing Neural Networks (MPNNs), including vGCN and vGIN. The second category is the Graph Transformer-based networks, including Graphormer and GraphormerGD. The details of these GNN layers are listed as follows: