SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The bias term depends on the anti-symmetric part of  $\Theta$  with respect to  $\mathcal{G}$ , whereas the variance term captures the difference of the dimension between the space of linear maps  $\mathbb{R}^{Nd} \to \mathbb{R}^{Nk}$  (measured by  $N^2 dk$ ), and the space of  $\mathcal{G}$ -equivariant linear maps (measured by  $\left(\chi_{\psi|_{\mathcal{G}}} \mid \chi_{\phi|_{\mathcal{G}}}\right)$ ; a proof can be found in [72, Section 2.2]). The bias term reduces to zero for  $\mathcal{G} \leq \mathcal{A}_G$  and we recover Theorem 1 in [8] when  $\mathcal{G} = \mathcal{A}_G$ . Notably, by using a larger group  $\mathcal{G}$ , the dimension of the equivariant space measured by  $\left(\chi_{\psi|_{\mathcal{G}}} \mid \chi_{\phi|_{\mathcal{G}}}\right)$  is smaller, and thus the variance term increases; meanwhile, the bias term decreases due to extra (symmetry) constraints on the estimator. We remark that  $\left(\chi_{\psi|_{\mathcal{G}}} \mid \chi_{\phi|_{\mathcal{G}}}\right)$  depends on  $d, k$  as well: For  $d = k = 1$ ,  $\phi(g) = \psi(g) \in \mathbb{R}^{N \times N}$  are standard permutation matrices; For general  $d > 1, k > 1$ ,  $\phi(g) \in \mathbb{R}^{Nd \times Nd}$ ,  $\psi(g) \in \mathbb{R}^{Nk \times Nk}$  are block-diagonal matrices, where each block is a permutation matrix.
To understand the significance of the risk gap, we note that (see Appendix B.1) when  $n > Nd + 1$ , the risk of the least square estimator is
$$
\mathbb {E} \left[ \| Y - \hat {\Theta} ^ {\top} X \| _ {F} ^ {2} \right] = \sigma_ {\xi} ^ {2} \frac {N d}{n - N d - 1} + \sigma_ {\xi} ^ {2}. \tag {7}
$$
Thus, when  $n$  is small enough so that the risk gap in Theorem 2 is dominated by the variance term, enforcing equivariance gives substantial generalization gain of order  $\frac{N^2dk - (\chi_{\psi|\mathcal{G}}|\chi_{\phi|\mathcal{G}})}{n - Nd - 1}$ .
Example 3.1. In Appendix C.1 we construct an example with  $\mathcal{X} = \mathbb{R}^3$ ,  $\mathcal{Y} = \mathbb{R}^3$ , and  $x \sim \mathcal{N}(0, \sigma_X^2 I_d)$ . We consider a target linear function  $f^*$  that is  $S_2$ -equivariant and approximately  $S_3$ -equivariant, and compare the estimators  $f_{\psi_{S_2}(\hat{\Theta})}$  versus  $f_{\psi_{S_3}(\hat{\Theta})}$ . When the number of training samples  $n$  is small,
Figure 1: Overview of Definitions for Approximate Symmetries: (a) Graph  $G$  and its coarsened graph  $G'$ ; (b) Induced graphons  $\mathcal{W}_G, \mathcal{W}_{G'}$  with small cut distance  $\delta_{\square}$  and their symmetries  $\mathcal{A}_G = S_2, \mathcal{A}_{G'} = \{e\}$ . The induced symmetry group  $\mathcal{G}_{G \to G'}$  yields more symmetries for  $G$ ; (c) Approximate equivariant mapping  $f^*$  that is close to  $f_{\mathcal{G}_{G \to G'}}^*$ .
using a  $S_{3}$ -equivariant least square estimator yields better test error than the  $S_{2}$ -equivariant one, as shown in figure inset. The dashed vertical line denotes the theoretical threshold  $n^{*} \approx 35$ , before which using  $S_{3}$  yields better generalization than  $S_{2}$ . This is an example where enforcing more symmetries than the target (symmetry) results on better generalization properties.
4 Generalization with Approximate Symmetries
Large graphs are typically asymmetric. Therefore, the assumption of  $f^{*}$  being  $\mathcal{A}_G$ -equivariant in Section 3 becomes trivial. Yet, graphon theory asserts that graphs can be seen as living in a "continuous" metric space, with the cut distance (Definition 1). Since graphs are continuous entities, the combinatorial notion of exact symmetry of graphs is not appropriate. We hence relax the notion of exact symmetry to a property that is "continuous" in the cut distance. The regularity lemma [73, Theorem 5.1] asserts that any large graph can be approximated by a coarser graph in the cut distance. Since smaller graphs tend to exhibit more symmetries than larger ones, we are motivated to consider the symmetries of coarse-grained versions of graphs as their approximate symmetries (Definition 2, 3). We then present the notion of approximately equivariant mappings (Definition 4), which allows us to precisely characterize the bias-variance tradeoff (Corollary 3) in the approximate symmetry setting based on Lemma 1. Effectively, we reduce selecting the symmetry group to choosing the coarsened graph  $G'$ . Moreover, our symmetry model selection perspective allows for exploring different coarsening procedures and choosing the one that works best for the problem.
Definition 1 (Induced graphon). Let  $G$  be a (possibly edge weighted) graph with node set  $[N]$  and adjacency matrix  $A = \{a_{i,j} \in [0,1]\}_{i,j=1}^{N}$ . Let  $\mathcal{P}_N = \{I_N^n = \left(\frac{n-1}{N}, \frac{n}{N}\right)\}_{n=1}^N$  be the equipartition of  $[0,1]$  to  $N$  intervals (where formally the last interval is the closed interval  $[\frac{N-1}{N}, 1]$ ). We define the graphon  $W_G$  induced by  $G$  as
$$
W _ {G} (x, y) = \sum_ {i, j = 1} ^ {N} a _ {i, j} \mathbb {1} _ {I _ {N} ^ {i}} (x) \mathbb {1} _ {I _ {N} ^ {j}} (y),
$$
where  $\mathbb{1}_{I_N^i}$  is the indicator function of the set  $I_N^i\subset [0,1]$ ,  $i = 1,\dots ,N$
We induce a graphon from an edge-node weighted graph  $G' = ([M], A, b)$ , with rational node weights  $b_i$ , as follows. Let  $N \in \mathbb{N}$  be a value such that the node weights can be written in the form  $b = \{b_m = \frac{q_m}{N}\}_{m=1}^M$ , for some  $q_m \in \mathbb{N}_{\geq 0}$ ,  $m = 1, \ldots, M$ . We blow-up  $G'$  to an edge weighted graph  $\overline{G}_N$  of  $\sum q_m$  nodes by splitting each node  $m$  of  $G'$  into  $q_m$  nodes  $\{n_m^j\}_{j=1}^{q_m}$  of weight 1, and defining the adjacency matrix  $\overline{A}_N$  with entries  $\overline{a}_{n_m^j, n_{m'}^{j'}} = a_{m, m'}$ . Note that for any two choices  $N_1, N_2 \in \mathbb{N}$  in the above construction,  $\delta_\square(W_{\overline{G}_N}, W_{\overline{G}_N}) = 0$ , where the infimum in the definition of  $\delta_\square$  is realized on some permutation of intervals in  $[0, 1]$ , as explained below. We hence define the induced graphon of the edge-node weighted graph  $G'$  by  $W_{G'} := W_{\overline{G}_N}$  for some  $N$ , where, in fact, each choice of  $N$  gives a representative of an equivalence class of piecewise constant graphons with zero  $\delta_\square$  distance.
Definition 2. Let  $G'$  be an edge-node weighted graph with  $M$  nodes and node weights  $b = \{b_m = \frac{q_m}{N'}\}_{m=1}^M$ , satisfying  $\{q_m\}_m \subset \mathbb{N}_{\geq 0}$  and  $q_m = N$ , and let  $G$  be an edge weighted graph with  $N$
nodes. We say that  $G'$  coarsens  $G$  up to error  $\epsilon$  if
$$
\delta_ {\square} \left(W _ {G}, W _ {G ^ {\prime}}\right) <   \epsilon .
$$
Suppose that  $G'$  coarsens  $G$ . This implies the existence of an assignment of nodes of  $G$  to nodes of  $G'$ . Namely, the supremum underlying the definition of  $\delta_{\square}$  over the measure preserving bijections  $\phi : [0,1] \to [0,1]$  is realized by a permutation  $\Pi : [N] \to [N]$  applied on the intervals  $\{I_N^i\}_{i=1}^N$ . With this permutation, the assignment  $C_{G \to G'}$  defined by  $[N] \ni \Pi(n_m^j) \mapsto m \in [M]$ , for every  $m = 1, \ldots, M$  and  $j = 1, \ldots, q_m$ , is called a cluster assignment of  $G$  to  $G'$ .
Let  $G'$ , with  $M$  nodes, be a edge-node weighted graph that coarsens the simple graph  $G$  of  $N$  nodes. Let  $C_{G \to G'}$  be a cluster assignment of  $G$  to  $G'$ . Let  $\mathcal{A}_{G'}$  be the automorphism group of  $G'$ . Note that two nodes can belong to the same orbit of  $\mathcal{A}_{G'}$  only if they have the same node weight. Hence, for every two nodes  $m, m'$  in the same orbit of  $\mathcal{A}_{G'}$  (i.e., equivalent clusters), there exists a bijection  $\psi_{m,m'}: C_{G \to G'}^{-1}\{m\} \to C_{G \to G'}^{-1}\{m'\}$ . We choose a set of such bijections, such that for every  $m, m'$  on the same orbit of  $\mathcal{A}_{G'}$
$$
\psi_ {m, m ^ {\prime}} = \psi_ {m ^ {\prime}, m} ^ {- 1}.
$$
We moreover choose  $\psi_{m,m}$  as the identity mapping for each  $m = 1,\dots ,M$ . We identify each element  $g$  of  $\mathcal{A}_{G^{\prime}}$  with an element  $\overline{g}$  in a permutation group  $\overline{\mathcal{A}}_{G^{\prime}}$  of  $[N]$ , called the blown-up symmetry group, as follows. Given  $g\in \mathcal{A}_{G^{\prime}}$ , for every  $m\in [M]$  and every  $n\in C_{G\to G'}^{-1}\{m\}$ , define
$$
\bar {g} n := \psi_ {m, g m} n.
$$
Definition 3. Let  $G'$ , with  $M$  nodes, be a edge-node weighted graph that coarsen the (simple or weighted) graph  $G$  of  $N$  nodes. Let  $C_{G \to G'}$  be a cluster assignment of  $G$  to  $G'$  with cluster sizes  $c_1, \ldots, c_M$ . Let  $\mathcal{A}_{G'}$  be the automorphism group of  $G'$ , and  $\overline{\mathcal{A}}_{G'}$  be the blown-up symmetry group of  $[N]$ . For every  $m = 1, \ldots, M$ , let  $S_{c_m}$  be the symmetry group of  $C_{G \to G'}^{-1}\{m\}$ . We call the group of permutations
$$
\mathcal {G} _ {G \rightarrow G ^ {\prime}} := \left(\mathcal {S} _ {c _ {1}} \times \mathcal {S} _ {c _ {2}} \dots \times \mathcal {S} _ {c _ {M}}\right) \rtimes \overline {{\mathcal {A}}} _ {G ^ {\prime}} \subseteq \mathcal {S} _ {N}
$$
the symmetry group of  $G$  induced by the coarsening  $G'$ . We call any element of  $\mathcal{G}_{G \to G'}$  an approximate symmetry of  $G$ .
Specifically, every element  $s \in G_{G \to G'}$  can be written in coordinates by
$$
s = s _ {1} s _ {2} \dots s _ {M} a \sim (s _ {1}, \dots , s _ {M}, a),
$$
for a unique choice of  $s_j \in S_{c_j}$ ,  $j = 1, \dots, M$ , and  $a \in \overline{\mathcal{A}}_{G'}$  (see Appendix B.2 for details).
In words, we consider the symmetry of the graph  $G$  of  $N$  nodes not by its own automorphism group, but via the symmetry of its coarsened graph  $G'$ . The nodes in the same orbit in  $G$  are either in the same cluster of  $G'$  (i.e., from the same coarsened node), or in the equivalent clusters of  $G'$  (i.e., they belong to the same orbit of the coarsened graph and share the same cluster size). See Figure 1 (and Figure 9 in Appendix A.2) for examples.
Definition 4. Let  $G$  be a graph with  $N$  nodes, and  $\mathcal{X} = \mathbb{R}^{N\times d}$  be the space of graph signals. Let  $X\sim \mu$  where  $\mu$  is a  $S_N$ -invariant measure on  $\mathbb{R}^{N\times d}$ . We call  $f^{*}:\mathbb{R}^{N\times d}\to \mathbb{R}^{N\times k}$  an approximately equivariant mapping if there exists a function  $\kappa :\mathbb{R}_{+}\to \mathbb{R}_{+}$  satisfying  $\lim_{\epsilon \to 0}\kappa (\epsilon) = 0$  (called the equivariance rate), such that for every  $\epsilon >0$  and every edge-node weighted graph  $G^{\prime}$  that coarsen  $G$  up to error  $\epsilon$ ,
$$
\left\| f ^ {*} - f _ {\mathcal {G} _ {G \to G ^ {\prime}}} ^ {*} \right\| _ {\mu} \leq \kappa (\epsilon),
$$
where  $f_{\mathcal{G}_{G \to G'}}^* = \mathcal{Q}_{\mathcal{G}_{G \to G'}}(f)$  is the intertwining projection of  $f^*$  with respect to  $\mathcal{G}_{G \to G'}$ .
Example 4.1. Here we give an example of an approximately equivariant mapping in a natural setting. Suppose that  $G$  is a random geometric graph, namely, a graph that was sampled from a metric-probability space  $\mathcal{M}$  by randomly and independently sampling nodes  $\{x_{n}\}_{n = 1}^{N} \subset \mathcal{M}$ . Hence,  $G$  can be viewed as a discretization of  $\mathcal{M}$ , and the graphs that coarsen  $G$  can be seen as coarser discretizations of  $\mathcal{M}$ . Therefore, the approximate symmetries are local deformations of  $G$ , namely, permutations that swap nodes if they are close in the metric of  $\mathcal{M}$ . This now gives an interpretation for approximately equivariance mappings of geometric graphs: these are mappings that are stable to local deformations. In Appendix C.2 we develop this example in detail.
We are ready to present the bias-variance tradeoff in the setting with approximate symmetries. The proofs are deferred to Appendix B.2.
Corollary 3 (Risk Gap via Graph Coarsening). Let  $\mathcal{X} = \mathbb{R}^{N\times d}$ ,  $\mathcal{Y} = \mathbb{R}^{N\times k}$  be the input and output graph signal spaces on a fixed graph  $G$ . Let  $X\sim \mu$  where  $\mu$  is a  $\mathcal{S}_N$ -invariant distribution on  $\mathcal{X}$ . Let  $Y = f^{*}(X) + \xi$ , where  $\xi \in \mathbb{R}^{N\times k}$  is random, independent of  $X$  with zero mean and finite variance, and  $f^{*}:\mathbb{R}^{N\times d}\to \mathbb{R}^{N\times k}$  be an approximately equivariant mapping with equivariance rate  $\kappa$ . Then, for any  $G^{\prime}$  that coarsens  $G$  up to error  $\epsilon$ , for any  $f\in V$ , we have
$$
\Delta (f, \bar {f} _ {\mathcal {G} _ {G \to G ^ {\prime}}}) = \underbrace {- 2 \langle f ^ {*} , f _ {\mathcal {G} _ {G \to G ^ {\prime}}} ^ {\perp} \rangle_ {\mu}} _ {m i s m a t c h} + \underbrace {\left\| f _ {\mathcal {G} _ {G \to G ^ {\prime}}} ^ {\perp} \right\| _ {\mu} ^ {2}} _ {c o n s t r a i n t} \geq - 2 \kappa (\epsilon) \left\| f _ {\mathcal {G} _ {G \to G ^ {\prime}}} ^ {\perp} \right\| _ {\mu} + \left\| f _ {\mathcal {G} _ {G \to G ^ {\prime}}} ^ {\perp} \right\| _ {\mu} ^ {2}
$$
Notably, Corollary 3 illustrates the tradeoff explicitly in the form of choosing the coarsened graph  $G'$ : If we choose  $G'$  close to  $G$  such that the coarsening error  $\epsilon$  and  $\| f_{\mathcal{G}_{G \to G'}}^{\perp} \|_{\mu}$  are small, then the mismatch term is close to zero; meanwhile the constraint term is also small since  $\mathcal{A}_{\mathcal{G}}$  is typically trivial. On the other hand, if we choose  $G'$  far from  $G$  that yields large coarsening error  $\epsilon$ , then the constraint term  $\| f_{\mathcal{G}_{G \to G'}}^{\perp} \|_{\mu}^{2}$  also increases.
Corollary 4 (Bias-Variance-Tradeoff via Graph Coarsening). Consider the same linear regression setting in Theorem 2, except now  $f^{*}$  is an approximately equivariant mapping with equivariance rate  $\kappa$ , and  $\mathcal{G} = \mathcal{G}_{G\to G^{\prime}}$  is controlled by  $G^{\prime}$  that coarsens  $G$  up to error  $\epsilon$ . Denote the canonical permutation representations of  $\mathcal{G}_{G\to G^{\prime}}$  on  $\mathcal{X}, \mathcal{Y}$  as  $\phi^{\prime}, \psi^{\prime}$ , respectively. Let  $(\chi_{\psi^{\prime}}\mid \chi_{\phi^{\prime}}) = \int_{\mathcal{G}_{G\to G^{\prime}}} \chi_{\psi^{\prime}}(g)\chi_{\phi^{\prime}}(g)\mathrm{d}\lambda (g)$  denote the inner product of the characters. If  $n > Nd + 1$  the risk gap is bounded by
$$
\mathbb {E} \left[ \Delta \left(f _ {\hat {\Theta}}, f _ {\Psi_ {\mathcal {G} _ {G \to G ^ {\prime}}} (\hat {\Theta})}\right) \right] \geq - 2 \kappa (\epsilon) \sqrt {\sigma_ {\xi} ^ {2} \frac {N ^ {2} d k - (\chi_ {\psi^ {\prime}} \mid \chi_ {\psi^ {\prime}})}{n - N d - 1}} + \sigma_ {\xi} ^ {2} \frac {N ^ {2} d k - (\chi_ {\psi^ {\prime}} \mid \chi_ {\psi^ {\prime}})}{n - N d - 1}.
$$
We illustrate our theory in three real-world tasks for learning on a fixed graph: image inpainting, traffic flow prediction, and human pose estimation<sup>1</sup>. For image inpainting, we demonstrate the bias-variance tradeoff via graph coarsening using  $\mathcal{G}$ -Net; For the other two applications, we show that the tradeoff not only emerges from  $\mathcal{G}$ -Net that enforces strict equivariance, but also  $\mathcal{G}$ -Net augmented with symmetry-breaking modules, allowing us to recover standard GNN architectures as a special case. Concretely, we consider the following variants (more details in Appendix D.4.2):
1.  $\mathcal{G}$ -Net with strict equivariance using equivariant linear map  $f_{\mathcal{G}}$ .
2.  $\mathcal{G}$ -Net augmented with graph convolution  $Af_{\mathcal{G}}(x)$ , denoted as  $\mathcal{G}$ -Net(gc).
3.  $\mathcal{G}$ -Net augmented with graph convolution and learnable edge weights:  $\mathcal{G}$ -Net(gc+ew).
4.  $\mathcal{G}$ -Net augmented with graph locality constraints  $(A \odot f_{\mathcal{G}})(x)$  and learnable edge weights, denoted as  $\mathcal{G}$ -Net(pt+ew).
$\mathcal{G}$ -Net serves as a baseline to validate our generalization analysis for equivariant estimators (see Section 3). Yet in practice, we observe that augmenting  $\mathcal{G}$ -Net with symmetry breaking can further improve performance, thereby justifying the analysis for approximate symmetries (see Section 4). In particular,  $\mathcal{G}\text{-Net}(\mathrm{gc})$  and  $\mathcal{G}\text{-Net}(\mathrm{gc + ew})$  are motivated by graph convolutions used in standard GNNs;  $\mathcal{G}\text{-Net}(\mathrm{pt + ew})$  is inspired by the concept of locality in CNNs, where  $A \odot f_{\mathcal{G}}$  effectively restricts the receptive field to the 1-hop neighborhood induced by the graph  $A$ .
5.1 Application: Image Inpainting
We consider a  $28 \times 28$  grid graph as the fixed domain, with grey-scale images as the graph signals. The learning task is to reconstruct the original images given masked images as inputs (i.e., image inpainting). We use subsets of MNIST [74] and FashionMNIST [75], each comprising 100 training samples and 1000 test samples. The input and output graph signals are  $(m_i \odot x_i, x_i)$ , where  $x_i \in \mathbb{R}^{28 \times 28} \equiv \mathbb{R}^{784}$  denotes the image signals and  $m_i$  denotes a random mask (size  $14 \times 14$  for MNIST and  $20 \times 20$  for FashionMNIST). We investigate the symmetry model selection problem by clustering
The source code is available at https://github.com/nhuang37/Approx_Equivariant_Graph_Nets.
the grid into  $M$  patches with size  $d \times d$ , where  $d \in \{28, 14, 7, 4, 2, 1\}$ . Here  $d = 28$  means one cluster (with  $\mathcal{S}_N$  symmetry);  $d = 1$  is 784 singleton clusters with no symmetry (trivial).
We consider  $\mathcal{G}_{G\to G^{\prime}}$  equivariant networks  $\mathcal{G}$  -Net with ReLU nonlinearity. We parameterize the equivariant linear layer  $f:\mathbb{R}^N\rightarrow \mathbb{R}^N$  with respect to  $\mathcal{G}_{G\to G^{\prime}} = (\mathcal{S}_{c_1}\times \dots \times \mathcal{S}_{c_M})\rtimes \overline{\mathcal{A}}_{G^{\prime}}$  using the following block-matrix form (assuming the nodes are ordered by their cluster assignment), with  $f_{kl}$  denoting block matrices, and  $a_{k},b_{k},e_{kl}$  representing scalars:
$$
f = \left[ \begin{array}{l l l} f _ {1 1} & \dots & f _ {1 M} \\ f _ {M 1} & \dots & f _ {M M} \end{array} \right], f _ {k k} = a _ {k} I + b _ {k} \mathbb {1} \mathbb {1} ^ {\top}, f _ {k l} = e _ {k l} \mathbb {1} \mathbb {1} ^ {\top} \text {f o r} k \neq l. \tag {8}
$$
The coarsened graph symmetry  $\mathcal{A}_{G'}$  induces constraints on  $a_{k}, b_{k}, e_{kl}$ . If  $\mathcal{A}_{G'}$  is trivial, then these scalars are unconstrained. In the experiment, we consider a reflection symmetry on the coarsened grid graph, i.e.,  $\mathcal{A}_{G'} = S_2$  which acts by reflecting the left (coarsened) patches to the right (coarsened) patches. Suppose the reflected patch pairs are ordered consecutively, then  $a_{k} = a_{k+1}, b_{k} = b_{k+1}$  for  $k \in \{1, 3, \ldots, M-1\}$ , and  $e_{kl} = e_{k+1, l-1}$
for  $k \in \{1, 3, \dots, M - 1\}$ ,  $l \in \{2, 4, \dots, M\}$  (see Figure inset for an illustration). In practice, we extend the parameterization to  $f: \mathbb{R}^{N \times d} \to \mathbb{R}^{N \times k}$ . More details can be found in Appendix D.2.
Figure 2 shows the empirical risk first decreases and then increases as the group decreases, illustrating the bias-variance tradeoff from our theory. Figure 2 (left) compares a 2-layer  $\mathcal{G}$ -Net with a 1-layer linear  $\mathcal{G}$ -Net, demonstrating that the tradeoff occurs in both linear and nonlinear models. Figure 2 (right) shows that using reflection symmetry of the coarsened graph outperforms the trivial symmetry baseline, highlighting the utility of modeling coarsened graph symmetries.
Figure 2: Bias-variance tradeoff via graph coarsening. Left: 2-layer  $\mathcal{G}$ -Net(blue) and 1-layer linear  $\mathcal{G}$ -equivariant functions (orange), assuming the coarsened graph is asymmetric; Right: 2-layer  $\mathcal{G}$ -Net with both trivial and non-trivial coarsened graph symmetry. See Table 7 for more numerical details.
5.2 Application: Traffic Flow Prediction
The traffic flow prediction problem can be formulated as follows: Let  $X^{(t)} \in \mathbb{R}^{n \times d}$  represent the traffic graph signal (e.g., speed or volume) observed at time  $t$ . The goal is to learn a function  $h(\cdot)$  that maps  $T'$  historical traffic signals to  $T$  future graph signals, assuming the fixed graph domain  $G$ :
$$
\left[ X ^ {(t - T ^ {\prime} + 1)}, \dots , X ^ {(t)}; G \right] \xrightarrow {h (\cdot)} \left[ X ^ {(t + 1)}, \dots , X ^ {(t + T)} \right].
$$
The traffic graph is typically large and asymmetric. Therefore we leverage our approximate symmetries to study the symmetry model selection problem (Section 4). We use the METR-LA dataset which represents traffic volume of highways in Los Angeles (see figure inset). We use both the original graph  $G$  from [76], and its sparsified version  $G_{s}$  which is more faithful to the road geometry. In  $G_{s}$ , the  $(i,j)$ -edge exists if and only if nodes  $i,j$  lie on the
same highway. We first validate our Definition 4 in such dataset (see Appendix D.3.1). In terms of the models, we consider a simplified version of DCRNN model proposed in [76], where the spatial module is modelled by a standard GNN, and the temporal module is modelled via an encoder-decoder architecture for sequence-to-sequence modelling. We follow the same temporal module and extending its GNN module using  $\mathcal{G}$ -Net(gc), which recovers DCRNN when choosing  $S_N$ . We then consider different choices of the coarsened graph  $G'$  (to induce approximate symmetries). As shown in Table 3, using 2 clusters as approximate symmetries yields better generalization error than using 9 clusters, or the full permutation group  $S_N$ .
G-Net(gc)	SN	SC1×SC2	SC1×...×SC9
Graph G*	3.173 ± 0.013	3.150 ± 0.008	3.204 ± 0.006
Graph G	3.106 ± 0.013	3.092 ± 0.008	3.174 ± 0.013
Table 3: Traffic forecasting with different choices of graph clustering. Table shows Mean Absolute Error (MAE) across 3 runs.
G-Net(gc+ew)	S16	Relax-S16	AG=(S2)2	Trivial
MPJPE ↓	42.55 ± 0.88	39.87 ± 0.46	42.18 ± 0.49	41.60 ± 0.32
P-MPJPE ↓	34.48 ± 0.44	31.38 ± 0.14	32.08 ± 0.20	31.69 ± 0.17
Table 4: Human pose estimation with different choices of symmetries. Table shows mean per-joint position error (MPJPE) and MPJPE after alignment (P-MPJPE) across 3 runs.
5.3 Application: Human Pose Estimation
We consider the simple (loopy) graph  $G$  with adjacency matrix  $A \in \{0,1\}^{16 \times 16}$  representing 16 human joints, illustrated on the right. The input graph signals  $X \in \mathbb{R}^{16 \times 2}$  describe the joint 2D spatial location. The goal is to learn a map to reconstruct the 3D pose  $Y \in \mathbb{R}^{16 \times 3}$ . We use the standard benchmark dataset Human3.6M [77] and follow the evaluation protocol in [78]. The generalization performance is measured by mean per joint position error (MPJPE) and MPJPE after alignment (P-MPJPE). We implement a
4-Layer  $\mathcal{G}$ -Net(gc+ew), which recovers the same model architecture in SemGCN [78] when choosing  $\mathcal{G} = S_{16}$  (further details are provided in Appendix D.4.1).