SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Definition 3. The set of functions approximated by  $\mathbf{PEs}$ $\mathcal{F}_{\mathrm{PE}}$  is formed by all the functions  $f \in L_{\perp}^{2}$  such that: for all  $\varepsilon > 0$ , there is  $\gamma \in \Gamma$  such that
$$
\mathbb {P} \left(\| \mathrm {P E} _ {\gamma} (S, Z) - \iota_ {X} f \| _ {\mathrm {M S E}} \geqslant \varepsilon\right) \xrightarrow [ n \rightarrow \infty ]{} 0. \tag {5}
$$
Note that, as before,  $\gamma$  may depend on  $\varepsilon$ . When passing PEs as input to GNNs,  $\mathcal{F}_{\mathrm{PE}}$  serves as the base space  $\mathcal{B}$ , and the space of interest to characterize the functions well approximated by the whole
architecture  $\Phi_{\theta, \gamma}$  is therefore  $\mathcal{F}_{\mathbf{S}}(\mathcal{F}_{\mathrm{PE}})$ . In fact, by simple Lipschitz property: for any  $f \in \mathcal{F}_{\mathbf{S}}(\mathcal{F}_{\mathrm{PE}})$  and  $\varepsilon > 0$ , there are  $\theta \in \Theta, \gamma \in \Gamma$  such that
$$
\mathbb {P} \left(\| \Phi_ {\theta , \gamma} (S, Z) - \iota_ {X} f \| _ {\mathrm {M S E}} \geqslant \varepsilon\right) \xrightarrow [ n \to \infty ]{} 0
$$
In the rest of the section, we therefore aim to characterize  $\mathcal{F}_{\mathrm{PE}}$  for several representative examples. We first briefly comment on observed node features, then move on to PEs. Proofs are in App. B.
4.1 Node features
A first, simple example, is when observed node features are actually a sampling of some function  $Z = \iota_X f^{(0)}$ . This is a convenient choice that is often adopted in the literature [24, 25, 23, 8, 29]. In this case, by adopting the identity  $\mathrm{PE}_{\gamma}(S,Z) = Z$ , it is immediate that  $\mathcal{F}_{\mathrm{PE}} = \{f^{(0)}\}$ . A more realistic example is the presence of centered noise:
$$
Z = \iota_ {X} f ^ {(0)} + \nu \in \mathbb {R} ^ {n \times d _ {0}} \tag {6}
$$
where  $\nu = [\nu_{1},\dots ,\nu_{n}]$  and the  $\nu_{i}$  are i.i.d. noise vectors with  $\mathbb{E}\nu_{i} = 0$  and  $\operatorname {Cov}(\nu_i) = C_\nu$ . This time,  $\mathcal{F}_{\mathrm{PE}}$  cannot contain directly  $f^{(0)}$ , as the Law of Large Numbers (LLN) gives
$$
\left\| Z - \iota_ {X} f ^ {(0)} \right\| _ {\mathrm {M S E}} ^ {2} = \| \nu \| _ {\mathrm {M S E}} ^ {2} \xrightarrow [ n \to \infty ]{\mathcal {P}} \operatorname {T r} \left(C _ {\nu}\right) > 0
$$
However, when applying the graph shift matrix at least once, one obtains convergent PEs.
Proposition 4. Consider the adjacency matrix (ex. 1) or normalized Laplacian (ex. 2). If the node features are a noisy sampling (6) and the PE are defined  $\mathrm{PE}_{\gamma}(S,Z) = SZ$ , then,  $\mathcal{F}_{\mathrm{PE}} = \{\mathbf{S}f^{(0)}\}$ .
Of course this may not be the only possibility for removing noise from node features, and moreover it is not clear how realistic the node features model (6) actually is. The study of more refined models linking graph structure and node features is a major path for future work.
4.2 Positional Encodings
In this section, we consider classical PEs computed solely from the graph structure and show how they articulate with our framework. We consider two examples that are the most-often used in the literature: PEs as eigenvectors of the graph shift matrix [12, 13] (actually a recent variant that account for sign indeterminancy [31]), and PE based on distance-encoding [30] (again a variant that, as we will see, generalize other architectures [49]). For most of the results below, we will focus on two representative cases of kernels, that include many practical examples. We remark that these yield sufficient conditions to establish our results, but by no mean necessary. Other cases could be examined in future work.
Example a (Stochastic Block Models). In this case, the space of latent variables  $\mathcal{X} = \{1,\dots ,K\}$  is finite, each element corresponds to a community label. The kernel  $w$  is represented by a matrix  $C\stackrel {\mathrm{def.}}{=}[w(\ell ,k)]\in \mathbb{R}_{+}^{K\times K}$  that gives the probability of connection between communities  $\ell$  and  $k$ , and  $P\in \mathbb{R}_+^K$  is a probability vector of size  $K$  that sum to 1.
Example b (P.s.d. kernel). Here we assume that  $w$  is positive semi-definite (p.s.d.). This includes for instance the Gaussian kernel.
For any symmetric matrix (resp. self-adjoint compact operator)  $M$ , we denote by  $\lambda_i^M$  its eigenvalues and  $u_i^M$  its eigenvectors (resp. eigenfunctions), with any arbitrary choice of sign or basis here. Since in all our examples operators are either p.s.d. or finite-rank, the eigenvalues are ordered as such: first the non-zero eigenvalues by decreasing order (from positive to negative), then all zero eigenvalues.
4.2.1 Eigenvectors and SignNet
It has been proposed [12, 13] to feed the first  $q$  eigenvectors of the graph into the GNN, for a fixed  $q$ . A potential problem with this approach is the sign ambiguity of the eigenvectors, or even the basis ambiguity in case of eigenvalues with multiplicities. Here we consider only the sign ambiguity for simplicity: we will assume that the first eigenvalue of  $\mathbf{S}$  are distinct. The sign ambiguity was alleviated in [31] by taking a sign-invariant function: considering an eigenvector  $u_{i}^{S}$  of  $S$ ,
$$
\left(\mathbf {Q} f\right) \left(u _ {i} ^ {S}\right) \stackrel {\text {d e f .}} {=} f \left(u _ {i} ^ {S}\right) + f \left(- u _ {i} ^ {S}\right) \in \mathbb {R} ^ {n \times p} \tag {7}
$$
Figure 1: Illustration of the role of the SignNet architecture and of the renormalization by  $\sqrt{n}$  of the eigenvectors on synthetic data, with a latent space  $\mathcal{X} = [-1,1]$  ( $x$ -axis), a Gaussian kernel  $w$ , and uniform distribution  $P$ . Blue dots represent a graph from the training set, orange dot a test graph that is twice bigger. From left to right: eigenvectors with renormalization (with a different sign for the two graphs), eigenvectors without, PEs with, and PEs without, with the regression test errors of a GNN trained using these PE with or without renormalization. We observe that SignNet indeed fixed the sign ambiguity. The absence of renormalization yields inconsistent PEs across graphs of different sizes, which results in a high test error on test graphs than training graphs.
where  $f: \mathbb{R} \to \mathbb{R}^p$  is a function applied to each coordinate of  $u_i^S$  to preserve permutation-equivariance. The resulting function is sign-invariant, and one can parameterized  $f$ . Given the first  $q$  eigenvectors  $u_i^S$  and a collection of MLPs  $f_{\gamma_i}^{\mathrm{MLP}}: \mathbb{R} \to \mathbb{R}^{p_i}$  for some output dimensions  $p_i$ , the PE considered in this subsection concatenates the outputs:
$$
\mathrm {P E} _ {\gamma} (S) = \left[ \left(\mathbf {Q} f _ {\gamma_ {i}} ^ {\mathrm {M L P}}\right) \left(\sqrt {n} u _ {i} ^ {S}\right) \right] _ {i = 1} ^ {q} \in \mathbb {R} ^ {n \times p} \tag {8}
$$
where  $p = \sum_{i=1}^{q} p_i$  and the MLP are applied element-wise. The parameter  $\gamma$  gathers the  $\gamma_i$ . The equation (8) involves a renormalization of the eigenvectors  $u^S$  by the square root of the size of the graph  $\sqrt{n}$ : indeed, as  $u_i^S$  is normalized in  $\mathbb{R}^n$ , this is necessary for consistency across different graph sizes. See Sec. 4.2.3 for a discussion and some numerical illustrations.
As can be expected, the eigenvectors of  $S$  generally converge to the eigenfunctions of  $\mathbf{S}$ , under a spectral gap assumption. We provide the theorem below which handles all of our running examples. We suppose that the relevant eigenvalues have single multiplicities, to only have sign ambiguity.
Theorem 2. Consider either SBM (ex. a) or p.s.d. kernel (ex. b), and either adjacency matrix (ex. 1) or normalized Laplacian (ex. 2). Fix  $q$ , assume the first  $q + 1$  eigenvalues  $\lambda_1^{\mathbf{S}}$ , ...,  $\lambda_{q+1}^{\mathbf{S}}$  of  $\mathbf{S}$  are two-by-two distinct. We define
$$
\mathcal {F} _ {\mathrm {E i g}} \stackrel {\text {d e f .}} {=} \left\{\left[ \left(\mathbf {Q} f _ {i}\right) \circ u _ {i} ^ {\mathbf {S}} \right] _ {i = 1} ^ {q} \mid f _ {i} \in \mathcal {C} _ {\mathrm {L i p}} \left(\mathbb {R}, \mathbb {R} ^ {p _ {i}}\right), p _ {i} \in \mathbb {N} ^ {*} \right\} \tag {9}
$$
Then  $\mathcal{F}_{\mathrm{PE}} = \overline{\mathcal{F}_{\mathrm{Eig}}}$
Hence  $\mathcal{F}_{\mathrm{PE}}$  contains the eigenfunctions of  $\mathbf{S}$ , modified by the SignNet architecture to account for the sign indeterminancy. We further discuss this space in Sec. 4.2.3. An illustration is provided in Fig. 1.
4.2.2 Distance-encoding PEs
In [30], the authors propose to define PEs through the aggregation of a set of "distances"  $\xi (i,j)$  from each node  $i$  to a set  $j\in V_{T}$  of target nodes (typically, labelled nodes in semi-supervised learning, or anchor nodes selected randomly [56]):
$$
\left(\mathrm {P E} _ {\gamma}\right) _ {i,:} = \operatorname {A G G} \left(\left\{\xi (i, j) \mid j \in V _ {T} \right\}\right)
$$
where AGG is an aggregation function that acts on (multi-)sets, and  $\xi(i,j)$  is selected in [30] as random-walk based distances  $\xi(i,j) = [(AD_A^{-1})_{ij},\ldots,((AD_A^{-1})^q)_{ij}]\in \mathbb{R}^q$ . For simplicity, since here we do not consider any particular set of target nodes, we just consider  $V_{T} = V$  the set of all nodes. Moreover, to use our convergence results, we replace the random walk matrix with our graph shift matrix  $S$ . As aggregation, we opt for the deep-set architecture [58], which applies an MLP on each  $\xi(i,j)$  then a sum. Deep sets can approximate any permutation-invariant function. As we will see below, with the proper normalization to ensure convergence, we obtain:
$$
\mathrm {P E} _ {\gamma} = \frac {1}{n} \sum_ {j} f _ {\gamma} ^ {\mathrm {M L P}} \left(n \cdot \left[ S e _ {j}, \dots , S ^ {q} e _ {j} \right]\right) \in \mathbb {R} ^ {n \times q}
$$
where  $f_{\gamma}^{\mathrm{MLP}}: \mathbb{R}^q \to \mathbb{R}^p$  is applied row-wise and  $e_j \in \mathbb{R}^n$  are one-hot basis vectors. We note that a similar architecture was proposed in a different line of work: it was called Structured Message Passing by [49], or Structured GNN by [25]. In these works, the inspiration is to give nodes unique identifiers,
e.g., one-hot encodings  $e_i$ . However, this process is not equivariant. To restore equivariance, [49] propose a deep-set pooling in the "node-id" dimension  $\mathrm{PE}_{\gamma}(S) = \sum_{j} \Phi_{\gamma}(S, e_j)$ , where  $\Phi_{\gamma}$  is itself a permutation-equivariant GNN, and the equivariance of  $\mathrm{PE}_{\gamma}$  is restored. By choosing  $\Phi_{\gamma}(\mathbf{S}, e_j) = n^{-1} f_{\gamma}^{\mathrm{MLP}}(n \cdot [Se_j, \dots, S^q e_j])$  (which is a valid choice for a message-passing GNN), we obtain exactly distance-encoding PEs above.
In [25], powerful universality results were shown for this choice of architecture in the case of non-random edges  $a_{ij} = w(x_i, x_j)$  and  $q = 1$ . With our notations, they implicitly studied PE functions of the following form:  $\int f(w(\cdot, x)) dP(x)$ . This allows to modify the values of the kernel before computing the degree function, and can therefore break potential indeterminancy such as constant degrees. Unfortunately, their proof technique and the concentration inequalities they use are not true anymore for Bernoulli random edges, which are far more realistic than deterministic weighted edges. Here we show that for a large class of kernels, concentration can be restored when we add an MLP filter on the eigenvalues of  $S$  with ReLU. Our definition of distance-encoding PEs is therefore:
$$
\mathrm {P E} _ {\gamma} = \frac {1}{n} \sum_ {j} f _ {\gamma_ {1}} ^ {\mathrm {M L P}} \left(n \cdot \left[ S _ {\gamma_ {2}} e _ {j}, \dots , S _ {\gamma_ {2}} ^ {q} e _ {j} \right]\right) \tag {10}
$$
where  $S_{\gamma_2} \stackrel{\mathrm{def.}}{=} h_{f_{\gamma_2}^{\mathrm{MLP}}}(S)$  is a filter that applies an MLP  $f_{\gamma_2}^{\mathrm{MLP}}$  on the eigenvalues of  $S$ .
Theorem 3. Consider either SBM (ex. a) or p.s.d. kernel (ex. b), and either adjacency matrix (ex. 1) or normalized Laplacian (ex. 2). Consider the PE (10). We define
$$
\mathcal {F} _ {\text {D i s t}} \stackrel {\text {d e f .}} {=} \left\{\int f ([ \mathbf {S} \delta_ {x} (\cdot), \dots , \mathbf {S} ^ {q} \delta_ {x} (\cdot) ]) d P (x) \mid f \in \mathcal {C} _ {\text {L i p}} ([ 0, 1 ] ^ {q}, \mathbb {R} ^ {p}), p \in \mathbb {N} ^ {*} \right\} \tag {11}
$$
where  $\mathbf{S}\delta_{x} \stackrel{\mathrm{def.}}{=} \{z \mapsto w_{\mathbf{S}}(z, x)\}$  by abuse of notation. Then  $\mathcal{F}_{\mathrm{Dist}} \subset \mathcal{F}_{\mathrm{PE}}$ .
Note that here we only have an inclusion  $\mathcal{F}_{\mathrm{Dist}} \subset \mathcal{F}_{\mathrm{PE}}$  instead of an equality as in Thm. 2: indeed, we show that the PE (10) can approximate functions in  $\mathcal{F}_{\mathrm{Dist}}$ , but they may converge to other functions. Nevertheless, as a consequence of our analysis, all the universality results of [25, Sec. 5.3] are valid with the choice of PE (10), see Appendix C for a reminder using our notations. This is a strict, and non-trivial improvement over [25], as their results were only derived for non-random edges. For this, Theorem 3 relies mostly on a new concentration inequality for Bernoulli matrices with ReLU filters in Frobenius norm, that we give below since it is of independent interest.
Theorem 4. Consider either SBM (ex. a) or p.s.d. kernel (ex. b), and either adjacency matrix (ex. 1) or normalized Laplacian (ex. 2). Define the Gram matrix  $W = [w_{\mathbf{S}}(x_i, x_j) / n]_{ij}$ . For all  $\varepsilon > 0$ , there is an MLP filter  $S_{\gamma} = h_{f_{\gamma}^{\mathrm{MLP}}}(\mathcal{S})$  such that
$$
\mathbb {P} \left(\left\| S _ {\gamma} - W \right\| _ {\mathrm {F}} \geqslant \varepsilon\right)\rightarrow 0.
$$
The proof of this theorem, given in appendix B.3, is inspired by the so-called USVT estimator [7]. One notes that the use of an MLP graph filter is quite unconventional. A more classical choice is polynomial filters: this avoids the diagonalization of  $S$  by computing  $\sum_{k} a_{k} S^{k}$ , it is for instance the basis for the ChebNet architecture [10]. For the purpose of Theorems 3 and 4, polynomial filters do not work, and ReLU is of crucial importance: indeed, we need the filter to zero-out  $\mathcal{O}(n)$  eigenvalues uniformly in some interval  $[-\tau, \tau]$ . This cannot be done with polynomials with a fixed number of parameters and growing  $n \to \infty$ . On the other hand, when choosing  $f$  as an MLP with ReLU, due to the shape of this non-linearity,  $f_{\gamma_2}^{\mathrm{MLP}}$  can be uniformly 0 on a whole domain. Of course, polynomial filters offer great computational advantages, and perform well in practice, despite their flaw in our asymptotic analysis. Moreover, ReLU is technically non-differentiable. Designing filters that offer both computational advantages and exact approximation is still an open question. In practice, we observe that the ReLU-filter does learn to approximate its expected shape, when we
minimize the reconstruction error  $\| S_{\gamma} - W\|_{\mathrm{F}}$  on synthetic data where  $W$  is known, see Fig. 2.
Figure 2: Illustration of Theorem 4 on synthetic data where  $W$  is known, with a Gaussian kernel. Unfiltered eigenvalues of  $S$  are represented by blue crosses, filtered ones obtained by minimizing  $\min_{\gamma_2} \| S_{\gamma_2} - W \|_F$  by orange dots, and the ideal ReLU-filter used in the proof of Thms. 3 and 4 is represented by a red line.
Approximation power. As mentioned above, in the absence of node features, one may opt for constant input, but this may lead to degenerate situations. PEs aim to counteract that, by increasing GNNs' approximation power. We quickly verify that this is indeed the case for our two examples.
Proposition 5. There are cases where  $\mathcal{F}_{\mathbf{S}}(1)\subset \mathcal{F}_{\mathbf{S}}(\mathcal{F}_{\mathrm{Eig}})$  or  $\mathcal{F}_{\mathbf{S}}(1)\subset \mathcal{F}_{\mathbf{S}}(\mathcal{F}_{\mathrm{Dist}})$  with strict inclusions.
Moreover, as mentioned in the previous section, existing universality results [25] can be generalized in our case, see App. C. Another interesting question is somewhat the opposite: given the already rich class of functions generated by PEs, are GNNs really more powerful?
Proposition 6. There are cases where  $\mathcal{F}_{\mathrm{Eig}} \subset \mathcal{F}_{\mathrm{S}}(\mathcal{F}_{\mathrm{Eig}})$  or  $\mathcal{F}_{\mathrm{Dist}} \subset \mathcal{F}_{\mathrm{S}}(\mathcal{F}_{\mathrm{Dist}})$ , with strict inclusions.
The proof, which is not so trivial, invokes functions with at least one round of message-passing after the computation of PEs, so the additional approximation power does not come only from MLPs. Intuitively, it seems natural that message-passing rounds are useful for other reasons, e.g. noise reduction or smoothing [23]. We leave these complementary lines of investigation for future work.
**Renormalization.** A striking point in our variants of PEs is the presence of various normalization factors by the graph size  $n$  to ensure convergence: the equation (8) involves a renormalization of the eigenvectors  $u^S$  by the square root of the size of the graph  $\sqrt{n}$ , while (10) involves a multiplicative factor  $n$  inside the MLP  $f_{\gamma_1}^{\mathrm{MLP}}$  (the  $1/n$  outside of the sum is more classical). Our analysis shows that these normalization factors are necessary for convergence when  $n \to \infty$ , and more generally for consistency across different graph sizes.
In practice, this is generally not used. Indeed, if the training and testing graphs have roughly the same "range" of sizes  $n \in [n_{\min}, n_{\max}]$ , then a GNN model can learn the proper normalization to perform, which is not the point of view of our analysis  $n \to \infty$ . While in-depth benchmarking of PEs has been done in the literature [13] and is out-of-scope of this paper, we give a small numerical illustration of the effect of normalization in Table 1. We consider a synthetic dataset with a classic classification problem on (unobserved) latent variables  $x_i$  and a Gaussian kernel  $W$ . To emphasize the effect of the normalization, we also examine a situation where the test graphs are much larger than the training graphs while following the same model, which we denote by out-of-dist. Concerning real data, since there are practically no datasets for node-classification with several graphs of sufficiently different size to test the renormalization, we artificially extract many subgraphs from a single large graph (CiteSeer) with labelled nodes to create such a dataset, denoted by CiteSeer-subgraphs. We also directly look at two graph-classification datasets with many graphs of different sizes. Note that, to emphasize the effect of PEs, we discard eventual node features and use only the graph structure.
On synthetic data exactly formed of random graphs of vastly different sizes, the renormalization is of course necessary to obtain good performance, as predicted by our theory: without it, the PEs do not converge when  $n$  grows. On real data, we see that renormalization generally improves performance, and this is more true for IMDB-BINARY, which contains a larger range of graph sizes, and distance-based PEs. Note that here we use relatively small GNNs that are not state-of-the-art (in particular since we do not use node features), as well as a different train/test split than most papers ( $K = 5$  CV-folds instead of  $K = 10$ ): indeed, we do not want our models to learn the proper normalization on the limited range of sizes  $n$  in the dataset, so we limit their number of parameters and use a smaller training set. We do not expect our simple renormalization process to make a significant difference on large-scale benchmarks with state-of-the-art models [13], but this is a pointer in an interesting direction to be explored in the future. In particular, this type of normalization may be useful in real-world scenarios where the test graphs are far larger than the labelled training graphs.
On large random graphs, the manner in which GNNs label nodes can be modelled by functions. The analysis of the resulting function spaces is still in its infancy, and of a very different nature to the studies of graph-tasks, both discrete [54] or in the limit [36]. In this paper, we clarified significantly the nature of the space of functions well-approximated by GNNs on large-graphs, showing that it can be defined by a few extension rules within the space of square-integrable functions. We then showed the usefulness of Positional Encodings by analyzing two popular examples, established new universality results, as
Dataset	Eigenvectors	Distance-encoding
w/ norm.	w/o norm.	w/ norm.	w/o norm.
Synthetic	68.61	65.59	67.31	62.49
Synthetic (out-of-dist)	67.87	62.51	66.80	63.33
CiteSeer-subgraphs	49.45	49.43	48.99	37.09
IMDB-BINARY [55] (graph-classif.)	67.80	66.10	71.10	63.95
COLLAB [55] (graph-classif.)	73.74	74.77	75.65	75.02
Table 1: Test accuracy for GNNs with different PEs, with or without renormalization by the graph size  $n$  . Results for 5-fold cross-validation averaged over 3 experiments.
well as some concentration inequalities of independent interest. Our theory hinted at some process for consistency across graphs of different sizes that can help generalization in practice.
This paper, which in large part consisted in properly defining the objects of interest, is without doubt only a first step in their analysis. Future studies might look at specific settings and derive more useful properties of the space  $\mathcal{F}_{\mathrm{S}}$ , more powerful PEs, a better understanding of their limitations, or more realistic models for node features. In particular, a better connection with the existing WL-based theory for finite small graphs, and associated "powerful" architectures, is a major path for future work.
The authors acknowledge the fundings of ANR GraVa ANR-18-CE40-0005 and ANR GrandMa ANR21-CE23-0006.
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