Learning Adaptive Multiresolution Transforms via Meta-Framelet-based Graph Convolutional Network
Authors: Tianze Luo
Zhanfeng Mo
Sinno Jialin Pan
Venue: ICLR 2024 poster (2024)
Keywords: Graph neural networks
graph multiresolution analysis
Primary area: learning on graphs and other geometries & topologies
TL;DR: We propose the MM-FGCN, a novel framework designed to learn adaptive graph multiresolution transforms, resulting in the attainment of state-of-the-art performance in various graph representation learning tasks.

Abstract: Graph Neural Networks are popular tools in graph representation learning that capture the graph structural properties. However, most GNNs employ single-resolution graph feature extraction, thereby failing to capture micro-level local patterns (high resolution) and macro-level graph cluster and community patterns (low resolution) simultaneously. Many multiresolution methods have been developed to capture graph patterns at multiple scales, but most of them depend on predefined and handcrafted multiresolution transforms that remain fixed throughout the training process once formulated. Due to variations in graph instances and distributions, fixed handcrafted transforms can not effectively tailor multiresolution representations to each graph instance. To acquire multiresolution representation suited to different graph instances and distributions, we introduce the Multiresolution Meta-Framelet-based Graph Convolutional Network (MM-FGCN), facilitating comprehensive and adaptive multiresolution analysis across diverse graphs. Extensive experiments demonstrate that our MM-FGCN achieves SOTA performance on various graph learning tasks.

Published as a conference paper at ICLR 2024
LEARNING ADAPTIVE MULTIRESOLUTION TRANS- FORMS VIA MAFTA-PRAMELET-BASED GRAPH CONVO- LUTIONAL NETWORK
Tianze Luo"! Zhanfeng Mo Sinno Jialin Pan
Nanyang Technological University, Singapore; The Chinese University of Hong Kong it tianze001, zhanfengCO1 contu.edu... sinnopan@cubk. edu. hk
ABSTRACT
Graph Neural Networks are popular tools in graph representation learning that cap- ture the graph structural properties. However, most GNNs employ single-resolution graph feature extraction, thereby failing to capture micro-level local patters (high resolution) and macro-level graph cluster and community patterns (low resolution) simultaneously. Many multiresolution methods have been developed to capture graph patterns at multiple scales, but most them depend on predefined and hand- crafted multiresolution transforms that remain fixed throughout the training process once formulated. Due to variations in graph instances and distributions, fixed handcrafted transforms can not effectively tailor multiresolution representations each graph instance. To acquire multiresolution representation suited to different graph instances and distributions, we introduce the Multiresolution -rea--Fraeeee- based Graph Convolutional Network (MM-FGCN), facilitating comprehensive and adaptive multiresolution analysis across diverse graphs. Extensive experiments demonstrate that our MM-FGCN achieves SOTA performance various graph learning tasks. The code available on GitHut
INTRODUCTION
The ubiquity of graph-structurekt data (Zhou et al. 2020 Wu al. 2020 Sanchez-Gonzaler et al. 2018 Fout et al. 2017]] Hamaguchi et al.) 2017) in today's 's interconnected society has sparked immense interest in the machine learning community for processing and analysis of such data, which leverages mathematical representations like graphs to capture interdependencies between data entities. Graph neural network (GNN) has found widespread adoption due to case of implementation and quality of prediction. Recent research (Balcilar al. 2021 Geng et al. (023)) underscores that most GNN models, including GCN (Kipf & Welling 2017). GAT (Thekumparampil et al.] (018), and GraphSage (Hamilton et al.) 2017b), fundamentally operate low-pass filters in the context of graph signal processing (Chang et al. (02T). They generate smooth node embeddings using low-resolution features, where neighboring graph nodes share similar graph features, and local feature aggregation leads to informative representations.
However, capturing solely low-resolution information is insufficient for achieving comprehensive graph representation. Low-resolution information represents graph signals that vary smoothly over the graph and are associated with low-frequency graph signals, whereas high-resolution information encompasses local disruption and detailed patterns that are associated with high-frequency graph signals. Thus, it is also crucial to capture the fine- grained graph details at high rish-tesolution levels. For instance, GNNs may fail disassortative graphs (Liu et al. 2022a; Pei et al. 2020 Suresh et al.) 202T), where locally connected nodes often exhibit different features and labels. This heterogeneity emphasizes the necessity of using the high-pass graph filters to capture the disruptive local patterns (Liu et al., 2022a Pei et al. 2020). another example, for social network data, high- and low- frequency components represent micro and macro-level dynamics respectively. While the micro- level highlights individual interactions, revealing personal influences, the macro-level captures
Both authors contributed equally to this research. https///ituub..mm/lzz/ http://///mmttttgggaaaeameeeeeee Magprppmmmmlirreooooonneteteeeee framell

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Published as a conference paper at ICLR 2024
Fitee veries
(High resolution)
(Mid resolution)
(Low resolution)
AG
AG
M
Fiar bank nodeet seaiio
(a) Filter Banks of Convantional Graph Wavelet Transforms
(b) Our Proposed MM-FGCN with Multiresolution
Figure Comparison of the filter banks of the conventional graph wavelet transforms with our proposed MM-FGCN with learnable multiresolution filter banks, We plot three levels of resolutions and each resolution level contains one low- pass filter and two high-pas filters.
communities, clusters, and motif's, shedding light on broader social relations and group behaviors. Therefore, GNN that relies solely on features from one or a few resolution levels fails to capture comprehensive graph representation. necessitating the use of multiresolution graph analysis.
Recent advancements in multiresolution graph representation learning fall into two main categories. i.e. the 1) graph-structure based approach (González & Ortega 2019 Bacciu et al. 2023 Geng et al.
2023; Xu et al., 2019c), which usually adopts down-sampling methods to partition the graph into multiple resolutions, or adopt specially designed GNN such as Graph-U-Net (Gao & Ji 2019) to capture the graph features at different resolutions. 2) Graph-spectral- the methods under this category adopt low-pass and high-pass spectral filters (Zhu et al. 2021 Chien al.] (020). Other methods adopt wavelet transforms (Zheng al. 2021 a to project graph signals to graph signal subspaces of different resolution levels. The wavelet frame transform provides an efficient way to obtain representations based on features of various scales.
approach, where some of
Most current multiresolution methods rely on either heuristic, inflexible spatial down- and up- sampling strategies, fixed, manually crafted spectral filters. For instance, the MR-GNN model (Xul cet al. (019cc employs multi-hop convolution layers with receptive fields of fixed size. UFGCony (Zheng et al. 202 and WFTG (Dong) 2017b) leverage deliberately designed graph framelet transform to discern graph signals across various resolutions. Furthermore, PyGNN (Geng et al. 2023) utilizes manually devised downsampling technique to categorize graph signals into dif- ferent frequency levels. However, the reliance of these methods on fixed multiresolution analysis strategies imposes significant limitations on obtaining eigh-performing representations. practice, various graph instances and distributions may manifest distinct scales and resolution factors, with their discriminative information residing at different resolution levels. Additionally, designing an appropriate multiresolution transform demands deep understanding of the dataset-specific inductive bias, making it hard to generalize to other domains. Thus, employing fixed multiresolution analysis strategies fails to customize appropriate multiresolution transform for individual graph instances.
To address this limitation, it is crucial to learn an adaptive multiresolution representation that can be automatically tailored to diverse graph instances and distributions. This motivates us establish meta-learner to generate the customized feature transform and multiresolution analysis strategy for each individual graph instance. In this paper, we introduce the Multiresolution Graph Convolution Network (MM-FGCN), a spectral-based method designed to learn adaptive multiresolution representations for different graph instances. For each input graph instance, the MM- FGCN first generates the meta-frai framelet generator, which consists of set of customized band-pan filters in the frequency domain. The meta- petalframlet generator in turn induces set of framelet- based multiresolution bases. Then, the input graph feature is decomposed into multiresolution components through projections onto each multiresolution basis. Finally, these multiresolution components are manipulated and passed to successive layers for downstream feature processing. As visualized in Figure [ our MM-FGCN creates an adaptive multiresolution transform for each graph instance by learning customized stratified multiresolution frequency partition in the frequency domain. In contrast, traditional filter-based and wavelet-bass based methods are confined to employing a fixed multiresolution analysis strategy across the entire graph dataset.
Contributions. In this paper, we propose a novel MM-FGCN for adaptive multiresolution represen- tation learning. The contribution of this paper is three-fold.
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We introduce Multiresolution Meta-Framelet System (MMFS) (Section 4. able multiresolution bases that can be simply constructed based on a set of meta-band band-pass filters (Section
set of learn-
We show that MMFS induces a series of progressive resolution graph signal spaces that inherently possess denseness, tightness, and dilation and translation properties (Section 4.1) Thus, the multiresolution decomposition and reconstruction for any graph signal can be achieved by projections onto each basis in MMFS.
Based on the MMFS-based multiresolution transform, we propose the Multiresolution Meta--Fraeeet-bhee Graph Convolutional Network (MM-FGCN) (Section 4.3) for adap- tive multiresolution graph signal processing. Extensive experiments show that our model achieves state-of-te of-t-of-theeart performance compared other baseline methods.
2 RELATED WORK
Multi-resolution Graph Representation Learning. Graph representation learning with multi- resolution techniques aims to provide multi-scale views of the graph data to better understand the local/detailed and global/overall information. Conventional methods adopt the techniques from computer visions, constructing different views based on the graph structure. González & Ortega (2019) adopt the downsampling method to retrieve the graph at different resolutions and perform graph matching. MR-GNN (Xu et al.] (019c) adopt several weighted graph convolution layers to learn graph representations at different granularities. Geng et al. (2023) propose the Pyramid Graph Neural Network which converts graphs with multi-scale downsampling. Another stream of graph multiresolution analysis focuses designing graph spectral filters Dong (2017b); Mallat 1989) to decompose graph signals into multiple resolutions. Zhhnng et al. (2021a) introduce tight graph framelet system with multiresolution, deliberately process graph signals different scales. FAGCN (Bo et al. 202T) enhances graph convolution with frequency adaptation that integrates different frequencies via an attention mechanism, and GPR-GNN (Chien et al., (020)) iteratively combines multiple frequencies with generalized pagerank. However, the capability of the aforementioned model is limited by the fixed spectral filter.
3 PRELIMINARY
We focus on undirected graphs represented as (X, A) with n nodes. Here, X Raadd represents node features, and A E R is the adjacency matrix, where A [ >> 0 if an edge exists between node i and j. and A ji = otherwise. The Laplacian matrix of the graph is defined as L = D A, where D = diag( (A1,) is the diagonal degree matrix with D i Σ === A[i and 1, is an all-one vector of size n. Without specification, (: denotes the inner product, in] denotes {1, n3.
Graph Representation Learning. For any graph data G sampled from the graph domain G, graph representation learning aims to learn a graph representation fo(-) : R with which we can embed each node of G into h-dimensional compact vector, facilitating downstream tasks such as node classification and graph classification. desirable graph representation should be able to capture the essential graph structural properties.
Spectral Graph Signal Processing. A graph signal xx.) generally refers to a G + R mapping. As ax()) assigns value each of the nodes, it is represented by a vector x R", where x(i] corresponds to the graph signal value assigned to the i-th node of G. In spectral graph signal processing (Kipf & Welling 2017 Shumann al. 2013), the graph Laplacian L plays a crucial role in graph modeling and analy vii, and it is tightly related to graph structural properties, including clusterability (Chiplunkar et al. 2018). connectivity (Fiedler) 1989), node distance, etc. fact, serves as graph shift operator which enables to transform graph signal into the frequency domain and manipulate its frequency components. Suppose the eigendecomposition of the graph Laplacian is L UAU the graph spectrum refers to the diagonal eigenvalue matrix A = diag(A) ..., An), and the spectral bases is the collection of eigenvectors U = (μj.... u,)). Thus, a graph signal x can be transformed into the frequency domain via graph Fourier transform & ((u,, x)... (u,, x)) U x. and it can be reconstructed from its frequency components x via the inverse graph Fourier transform x E, x) u, UX. Furthermore, one can apply smooth filter go manipulate
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frequency components of x by the spectral convolution (Kipf & Welling 2017
g0 (L) x _ Ugo X.
In machine learning practice, applying spectral convolution to the graph feature X (which can be viewed as a d-dimensional graph signal) provides us with informative graph representation. Different implementations of filter go lead to desirable graph representations for different purposes, such as Gasteiger et al.
classification, denoising, smoothing, and abnormally detection (Xu et al., 2019a
2019
Li ct al.
102 Tang et al. 2022).
Spectral Graph Multiresolution Analysis. Classic multiresolution analysis (Mallat 1989 Cohen et al. 1993) aims to decompose signal into multiple components of varying resolutions, which can then be processed individually to provide a comprehensive representation of the signal. Let L (R) be the measurable, qquare-integrable
one-dimensional functions, where the inner product of x, E (R) is (I, 2) = (f)d). Given a resolution factor > 1, the multiresolution decomposition for signals in (R) is determined by a series of progressive resolution function spaces (v.)r. where each V,, is subspace of (R), and V, C V, if < r'. The {V,), is expected to satisfy the denseness, dilation property, and translation property (Mallat) 2006). ensuring that V, collects the rrrreoouttion signals, and the multiresolution decomposition of any given signal achieved by projecting it into each subspace V,.
Denseness: } contains sufficient information to represent and reconstruct any signal, that is, the union of {vr)r is dense in L (R),, and the intersection of {V+hr is (0). Dilation property: signals in V, can be derived from signals in V,++1 by scaling them using a resolution factor of y, that is, (t) E V/ - (tv(()) = (yy) E V++1, where D, is the dilation operator.
Translation property: when a signal x is translated for s in the spatial domain, its -"- resolution component translates for the same amount in the frequency domain, that is, (T,x) = (rrr) where Pr (IR) + V,, is the projection to V,, and T,x(-) = 2(s .) is the translation operator.
The goal of multiresolution analysis is to determine a set of bases (Wrr): that spans the desirable V,, satisfying the denseness, dilation, and translation properties. Morcover, the 7 resoouuttii component
of a signal x should be derivable from its projection onto each basis, i.c. (x(x)= -x (Wrer x) Dri- Thus, the multiresolution decomposition of can be achieved by = Σ, (x(x) Er. (4/ 2) Vr--
For instance, proper choice of V, the collection of piecewise constant functions over [->", and gir can be set as the associated Haar-like wavelets (Dong] 2017b).
For multiresolution graph analysis, one needs to extend the dilation and translation properties the graph signal domain (where each graph signal is represented by a vector in IR") and determine the multiresolution graph bases {4pri)ra C 1 To this end, one needs define the spatial dilation and translation operators for graph signals by generalizing the scalar multiplication and node subtraction to the graph domain. According to the harmonic analysis theory (Stein) 1993] Gavish et al., (0100). the graph dilation and translation operators can be defined based on the graph Fourier transform. Consider graph signal &p E Rn generated by a one-dimensional filter g, i.e. 4p = -xx (Ax.) ug, the --dilaiion and o-translation of sp are defined as
D,40 Eg(1As) Ub, Vy > 0, and Topp Σ Eg(Ax)uple uk. Ye E G,
& respectively. Therefore, finding the desirable multiresolution bases is equivalent to identifying a set of filters (r.))r such that the bases {spriv bri, generated by Prie Ee Iri (Ax Jue()) Un satisfies the aforementioned conditions.
Finally, desirable set (sPrin must exhibit tighiness. The set of bases is called tight if and only if ||x|2 =xx|==Er,,> ((priv; holds for arbitrary x. Intuitively, tightness ensures that the projection operator onto these bases preserves the overall energy (norm) of the original graph signal. It's worth noting that this property, while essential, is less restrictive than orthogonality. As guaranteed by the polarization identity, it enables multiresolution decomposition via x Er..v (Fpriv x) Priv-
This decomposition be equivalently expressed = Emi.o Pr
$Φ x, where + is an n-by-- multiresolutton transform matrix, with each column representing basis Priv. and N is the total number of bases. As the multiresolution transform matrix is defined by concatenating the multiresolution bases, will use these two terms interchangeably throughout the rest of the paper.
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4 METHODOLOGY
We propose the Multiresolution sita--Framelctbbasee Graph Convolution Network (MM-FGCN), designed for adaptive multiresolution representation learning for varying graph instances. In Section 4.1 and Section 4.2 we construct set of learnable multiresolution bases, termed Multiresolution Petaa-rammeeet System (MMFS). Our MMFS inherently possesses tightness and spans progressive multiresolution graph signal subspaces that satisfy denseness, dilation, and translation properties. For each graph, MM-FGCN first calculates the adaptive MMFS and the associated multiresolution trans- form matrix. This matrix enables to decompose and manipulate the multiresolution components of the graph feature, yielding comprchensive graph representations (Section 4.3
4.1 MUL.TIRESOLUTION META-FRAMELET SYSTEM
As mentioned in Section | learning the adaptive multiresolution bases is essential for obtaining a comprchensive graph representation. Suppose N is the total number of multiresolution bases, straightforward approach to learn the multiresolution transform matrix via a neural network Mg MM, A). However, without additional constraints,
+ R parameterized by &, such that this directly learned & may fail to meet the tightness property " I, making the multiresolution decomposition infcasible. Even if we impose constraints on o to ensure tightness, denseness, translation, and dilation properties, the constrained optimization process becomes challenging to solve due to numerical instability. Additionally, learning dense n x N matrix requires an excessive amount of parameters, leading to significant computational overhead.
To address these limitations, we construct a set of learnable multiresolution bases with much fewer parameters, called the Multiresolution Meta-Framelet System (MMFS). MMFS consists of a set of learnable graph framelets, each generated by spectral meta-filter. Individually, these meta-filter filters are distinguished by their trainable bandwidth parameters and specific resolution levels, all while sharing common trainable resolution factor. The following arguments show that MMFS is born to be tight, and it spans progressive multiresolution spaces that possess denseness, dilation, and translation properties. Hence, multiresolution decomposition be achieved by using the MMFS-based multiresolution transform.
System). Given the number of resolution levels R > O. for the each resolution level [R], define I spectral meta-filters { "....11....991111 These meta-filters are mappings from the interval [0, 1] itself, and they are paramelerized by vector w E S. The collection of the R x meta-filters is called the meta-framelet generator: We define the meta-framelet learner as Mg(+) : G + x Rt,, a neural network that maps any graph instance G = G A) to a specific meta-framelet generator w and resolution factor The Multiresolution System (MMFS) is defined as set of graph signals (spriv), ,, where
Definition (Multiresolution
Priv Σ gh. (y-. .Ax) use [v] uk:
Mc((X A and un the k-thh eigenvalue and eigenvector of the graph Laplacian L. and J is the smallest value such that 7 -J+R >maxe (L) 1. The MMFS-based multiresolution transform matrix is defined as the concatenation of { {Priivh }. that is
where (w
PMM A (U J+1 (+1A) UT...., U 19 ((A((-1+RA)A
Definition iillustrates the construction of MMFS based on the meta-framelee generator. Here, Priv represents the basis comprising a r-resolution dilation and translation w.r.t the e-th node. At the r-resolution level, the meta-filter gh, filtrates the information localized around the e-th node. Notably, the efficient computation of 4 This can be achieved by circumventing the need for cigen-decompositionn of L through the application of Chebyshev approximation (Defferrard et al.) 2016b) to 9+% ( LL)).. The subsequent proposition offers a construction for the meta-framel generator, ensuring that the MMFS meets the criteria of tightness, denseness, translation, and dilation. The proof is available in Appendix D
Proposition (MMFS- MMMFS-based Multiresolution Decomposition), Following the notations in Definition
7 suppose the meta-framelet generator satisfies
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(2) +... g4,, (f, =1 VA [0,
gf,((" = gi, X+ 9.. (7- (1-
(r Ar > 1, i [1]
then following the construction in Definition the MMFS induced by (pria) forms tight bases
system. Here, the indices (r, (r,i,0 v) are iterated over v E [n], with (r,i) drawn from the set (R) x 1); 1Ir R. For any graph signal x RR, the multiresolution transform matrix is MM E [ the multiresolution decomposition achieved by
Σ (4Priv x) Priv 2MM RMM MM X, V,k,0
X
where X + 2 PMM is the multiresolution transform. Moreover, let V = rpeie the resulting subspaces {Vr}r turn out be a series of progressive resolution space that possess denseness, dilation, and translation properties.
Proposition shows that, once the meta-filters of the 1-resolution level are determined, a desirable MMFS can be constructed in a stratified and iterative manner. As visualized in Figure [ the r- resolution level meta-filters (97.1...., ",,,,, induce unitary partition within the support of 1r+1.1 which is the low-pass filter of the (r + 1)-reoolution level.
4.2 META- FRAMEEER GENERATOR
To implement AMFS-based transform, the remaining step is to design the formulation of the feta-framelet generator {9T.1 g such that Et 1. This inspires to simply set (9.. {h,} as 7 band-pass filters to partition the [0, 1] interval into / regions. In this paper, we instantiate each gf,, as meta-band-pass filter based on polynomial splines in Han al. (2016), i.c.
E 00, c u +,,
sin( m(A-e it., 11
E (c-1 - 0--1 ---
+
(A) &
46,-1
(3)
1,
A E E
cos( 7(1 - tes)
A E (e - -i,
where {c1, (1,, C1-1, are parameters encoded in Specifically, for any Ω (2(1-1),
define
Σ wbil é a min(c. C-11 c++1 c.),
(4)
where a E (0, 1/2) is a predefined hyperparameter and it holds that = cp & C1 < ... < c-- Ci = Notably, the parameterization of the meta-framelet generator uses only 2(1 1) parameters, significantly reducing the budget compared to the dense n-by- x (I- x n) multiresolution transform matrices. Intuitively, the meta-filters adaptively decompose graph features into spectral channels and process frequency components at various resolution levels, leading to flexible and comprehensive graph representation.
4.3 MULTIRESOLUTION META-FRAMMEN FRAMELET-BSSED GRAPH CONVOLUTION NETWORK
Leveraging the efficient construction and computation of MMFS-base multiresolution transform matrix 4MM in Proposition [
based Graph Convolution (MM-FGConV) and its associated graph pooling counterpart (MM-FGPool). These operators serve meta-analogs to the conventional graph convolution and graph pooling methods (Defferrard al. 2016b). The MM-FGPool operator simply defined as MMFGPool (H; X, A) A
now establish the Multiresolution
1 4 MM H. where the meta-framelet coefficients are aggregated and concatenated the output of the readout of the final classifier. The computation of MM-FGCony is illustrated in Figure 2 and its details presented in Algorithm
An L-layrr MMFS- based Graph Convolutional Network (MM-FGCN) is defined by MMFGCN (A, & h MMFGPoole (H, Wr: A)),, H (MMFGConve... (H, A, X)W,,,), V E [Z]]
(5)
(6)
0 vec ((01)E)12), 2 vec ((Wi)eiu))
(7)

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Input Graph
MM a
IEAP
Portitons
Mr
-
(A
+nit
Figure 2: the computation of MM-FGConv operator with a mnet-fframelet learner Me and learnable filter O.
Algorithm 1 MM-FGGony
A Algorithm 2 Meta-training MM-FGCN
1: Input: graph data G (X, A). graph Laplacian
1: Input: graph datasct S, MM-FGCN parameters
L. meta-framelet generators {97.). ). a meta-
(0, w; E). the empirical loss Z(: Learning rates
neural network MMg. learnable diagonal matrix
31, B2 O.
O. the T-order Chebyshev polynomial approxima-
Output: optimized MM-FGCN (0", w")
tion Chebyshev,
Split dataset Smetay Smain +
2: Output: X, the output processed graph signal.
4: for t in [T') do
3: J + [log, (Prmax (L)) R]
B MiniBatch(SSmooe
+- B1 VeLB (O, w;
4: (w.7) + MMe(X, A)
B' +- MiniBatchh (Smoin)
5: Pr.i Chebyshev, (g.,)) ) for E
(0, w: {) - (0, w: {) Vo
6: 4MM + (y J++ - L) 200
(7 17 (+
9: end for
7: X <- MMM 2 MM X
10: (0", :=: --*) (0, w:
8: return X
11: return (0" ...." :=")
Figure 3: Left: the computation of MMFS-based multiresolution graph convolution operator. Right: implemen-
tation of MM-FGCN meta-training algorithm.
is the meta- fett-frrameet-baseed graph convolutional operator as defined in Algorithm h is fixed classifier (e.g. softmax), W, R xd, are learnable weight matrices, and (-) is the activation function. We define (0, w) as the base-parameters, and define & as the partmeaareesess. By design, the MM-FGCN permutational invariant (Maron et al.) (2019), and is equipped with learnable multiresolution transform that adapts to each graph instance.
where MetaFGConve..
Following the optimization paradigm introduced in MAMI. (Finn et al.) 2017 Hospedales al., 2022), we employ meta-learning to train the MM-FGCN model. We aim to acquire multiresolu- tion transformation that enables the MM-FGCN backbone to adaptively and effectively represent individual graph instances. Specifically, our objective is
min Ls(O, (* (0)), s.t. (0( arg min C(O, E),
(8)
(s(
Σ
I(MMFGCNO, w:e (G),y);
S
(G.y)ES
where I.(:.) a loss function, egg. the cross entropy. As outlined in Algorithm we partition the training data into two sets: meta-training set and a standard training set. In each iteration, initiate an update to the meta-parameter &, denoted as g', through gradient descent computed on batch of meta-train vreaa-trainin data. Subsequently, proceed update all parameters (0, w, {) using the full gradient evaluation at (0, w, g') based on the standard training data.
5 EXPERIMENTS
5.1 NODE CLASSIFICATION
Datasets. We conduct experiments on both assortative and disassortative graph datasets. dataset is called assortative if its neighboring nodes usually have similar labels and features (Ma al.
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Table 1: Test accuracy (in percentage) for citation networks with standard deviation after +. The results with the best performance are highlighted with
Method
Assettative
Disnseetative
Citeseer
Pobmed
Comell
Wiscomsin
Chsmeleon
Squireel
MLP
56.1 1: 1.1
46.5 1.3
0.7.7
61.6 :: 6.3
81.3 7.1
$1.9 1: 5.3
48.5 1: 3.0
31.5 =: 1.4
SPECTRAL (Bruna 2014
71.3 4: 1.4
68.9 2 0.8
T3.9 0.6
32.1
2 7.2
57.5 7.5
56.7 2
62.4 2
58.8 :: 2.2
CHERYSHEV Darierane
2016a
81.2 + 1.2
00.8 0.9
74.4 +.6
63.1 + 7.8
50.1 + 7.2
16.3
60.2 000+-22
56.4 + 2.5
GWNN ct 12019
82.8 0.9
71.7
79.1 1.0.
56.8
1081110 7.6
65.11669
61.2 1 4.9
63.7 128
55.4 2.3
MPNN
16.
02.3
68
5.4
60.8
68.1
GRAPPNADE
0.6
54.2
7.
5-.
62.1
LANCZOSNEIT 110
1.5
60.11.2
68.2
54.1
GCN Kipt Wolmi
70.3+
7200004
61.1
67.6
GAT Wil.eos
M0T3
72.5 0.7
7..0+:0.3
36.3
57.0 6.1
57.8
66.0
$1.3
81.7 1,3
71A U.S
79.2 0.3
37.4 2 7.8
614
58.3 7.2
62.7
28,0 = 188
GIN-+ AArREE 13133
81.6
71.5 0.8
79.1 +044
59.2
+ 6.5
60.5 + 6.2
61.1 +668
61 1144-22
37.2
2022
83.1 111==00 0.7
72.3 + 0.8
79.5 044
67.2 = 7.7
79.5 + 5.6
76.9 :44
68.2 2.5
58.
MLMLP-Clrrrs 20h11
68.5 1.0
61.23 1.6
71.8 0.0
84.9 1 5.7
5..4
5731-43
SO.T 2.2
33.7 1.5
NLGCN Liu oi 20220
1..4 1.5
1.4
0.7
57.6
5.5
65.5
902203
70.129
112
NL.GAT Tineeral 20110
80.1 00.1+1.. 1.3
71.2+11 1.5
7.77
G4.7 + 7.6
02.++
7..9+733
66.7+114
56.8 2.5
Geom CLIke
10:0
80.0 1.2
0.
79.8 1..
8.6
57.5
58.2
603117
33.3
61.32
7924
4.
66.4
63.0
724+44
UEGCONV R42az0ggg
20240
12.7
79.9
17.2
60.1
MM-FGGN (Ours)
84.4" 1
73.9" 100
80.7"
88.9" 1 8.3
86.1"
88.5"
73.97 2.1
67.5
1:2
Table 2: Performance comparison for graph property prediction. QM7 is a regression task in MSE; others are for classification in test accuracy in percentage. The results with the best performance are highlighted with *
Pooling Operators
PROTEINS
Mutagenicity (t)
D&D (t)
NC11 (t)
Ogbg (t)
QM7 (J)
TOPKPool
73.48
3.57
79.84
2.46
74.87
4.12
75.11 1 3.45
78.14
0.62
175.41
3.16
AttentionPool
73.93 1 5.37
80.25 + 2:22
77.48
1: 2.65
74.94
= 1.27
74.41
2.12
177.99 = 2.22
SAGPool
75.80
2.91
79.86
2.36
74.96
3.60
76.30
1.53
75.26
2.29
41.93 1. 1.14
SUMPool
74.91 4.08
80.69
3.26
78.91
+ 3.37
76.96
1.70
77.41 1.16
42.09
MAX
73.57
3.94
78.83 1 1.70
75.80
4.11
75.96
1.82
78.16
1.33
177.48 4.70
MEAN
73.13 3.11
80.37
2.44
76.89
2.23
73.70
2.55
78.21
0.90
177.40 4.69
UFGPool-SUM
77.77 - 2.60
81.59 = 1.40
$0.92 + 1.68
77.58
1.21
78.80 = 0.56
41.71
UPGPoOl-PPETRRM
2.400
1.28
79.83
78.36
U.77
41.67 0.95
MM-FGPool (Ours)
78.07' 2.36
83.91' + 1.32
81.51" # 1.55
78.57'
+ 0.82
79.12" + 0.85
41.19" 0.88
2022), as observed in citation networks and community networks. In contrast, disassortative datasets, such as co-occurrence networks and webpage linking networks, consist of numerous nodes with identical labels that are distant from one another. In this paper, we evaluate the performance of our MM-FGCN on assortative datasets, including Cora, Citeseer, and Pubmed (Sen et al.] 2008), as well as disassortative datasets, including Cornell (Craven et al. 1998), Texas (Craven et al. 1998). Wisconsin (Craven et al. 1998), Chameleon (Rozemberczki et al. (02T)) and Squirrel (Rozemberczki cct al. 2021). For assortative datasets, following the configuration in (Kipf & Welling 2016). we allocate 20 nodes per class for training, 1,000 nodes for testing. and 500 for validation. As for disassortative datasets, we divide each dataset into training, validation, and test sets using a split ratio of 60%:20%:20%% All experimental results are averaged over 10 independent repetitions.
Baselines. We benchmark MM-FGCN against various competitive basclines on node classifica- tion tasks, including MLP, CHERYSHE (Defferrard et al.) 2016a), GCN (Kipf & Welling 2016).
SPECTRAL AL. CNN (Bruna et al. 2014), GWNN (Xu et al. 2019b). MPNN (Gilmer et al. 2017).
GRAPHSAGE (Hamilton al. 2017a), LANCZOSNET (Liao al. 2019), GAT (Veličković et al.
2018), Non-Local GNN (Liu el al. 102 Geom-GCN (Pei et al. 22220), two variants of UFGCony
(Zheng al. 2021a), i.e. UFGConvShrinkage and RFl,,n and PyGNN (Geng et al.) 2023).
We adhere to the original implementations of the baseline models as described in their respective papers. As for MM-FGCN, the implementation details are elaborated in the Appendix[A]
Results. As presented in Table [ our proposed MM-FGCN model demonstrates state-of-the-art performance compared to all baseline models on both assortative and disassortative datasets. For disassortative datasets, compared to GCN, MM-FGCN achieves a significant performance gain of 34.7%, 25%, and 28.9% on the Cornel, Texas, and Wisconsin datasets, respectively. This evidence highlights that in disassortative datasets, where the node homophily is diminished and conventional models based on low-pass filters such as GCN struggle to capture effective graph representations. In contrast, MM-FGCN demonstrates its capability of learning multiresolution framelet transform that dynamically adapts to the characteristics of each graph dataset. More experiments and results on node classifications are elaborated in the Appendix B
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Published as a conference paper at ICLR 2024
Table 3: Ablation study on the meta framelet learner and the meta-learning algorithm. Test accuracy (in
percentage) with standard deviation after I. are reported.
Methaal
Graph Classfication
Noda Classifcation
Munagenickt
D&D
NCII
Cius
Cliesers
Coracil
Touas
Chanclcon
Squireel
type inceripp
81.6tt 1.5
Qundneissyppe
718-14
69.7+00
76.7
67.5+277
Trimull MM-FGCN (Ours)
11.+1.7
75.-00
0.
72.2+0077
7.2+
626-+2
50--
+.5
=00
+3.9"
4..2
5.2
GRAPH CLASSIFICATION
We assess the efficacy of MM-FGCN on 6 benchmark graph classification and regression datasets, including D&D (Dobson Doig, 2003), PROTEINS (Dobson & Doig 2003). NC11 (Wale et 2008), Mutagenicity (Kazius al, 2005). Ogbg-molhiv (Hu et al. 2020). and QM7 (Blum & Reymond 2009). Following the configuration of Zheng et al. (202 la). each dataset split into training, validation, and test set by ratio of 80%, 10%, and 10%. The results averaged over 10 independent repetitions. We also compare MM-FGPool with graph classification methods based on the conventional GCN backbone together with various state-of-the-art pooling strategies, including SUM, MEAN, MAX pooling. TOPK (Gao & Ji 2019), AttentionPool (Li et al. (2016), SAGPool (Lee et al. 2019), UFGPool-SUM, and JLFGPool-SPECTRRR
(Zheng et al.) 2021a). la). The
implementation details can be found in Appendix [ The results are shown in Table 2 and our model achieves the highest performance among all the baselines on the five datasets, demonstrating the effectiveness of MM-FGPool in aggregating graph information on various datasets.
5.3 ABLATION STUDIES
To validate the benefits of using a meta- meta-framelet learner, in Table 3, we show the MM-FGCN variants with handerafted filters (Dong) 2017b) (e.g. (a) Haar-type, (b) linear-type, (c) quadratic-type framelet filters). To assess the performance improvement achieved by the meta-learning algorithm elaborated in Algorithm|
2. we compare it against direct training scheme where both and & are updated simultaneously, as shown in row (d) trainable framelet transforms of Table 3 According to the results, models with trainable meta-framesr generators outperform those with fixed and handcrafted graph transforms, highlighting the necessity of using trainable graph transforms for enhanced performance. Furthermore, using a mnta--frmmeee learner indeed brings performance gain compared to using directly trainable filters, showing that the meta-framelet learner enhances the capacity of MM-FGCN. We also show that meta-learning contributes improvement in the generalization performance of MM-FGCN, leading to more discriminative graph representations. Extra ablation studies on the hyperparameters of MM-FGCN and visualizations are detailed in Appendix B
6 CONCLUSION
In this paper, we present MM-FGCN, spectral-based model for adaptive multiresolution repre- sentation learning for varying graph instances. Our MM-FGCN model equipped with set of trainable multiresolution bases, which can be simply and efficiently constructed based on set of sata-band--ass filters. By optimizing the meta-filters, MM-FGCN learns an adaptive frequency parti- tion of the graph spectrum domain, enabling us to perform a customized multiresolution transform on each graph instance. Comprehensive experiments show that our proposed method exhibits high performance and adaptivity to various types of graphs, including graph and node classification for dissortative and assortative graphs from various domains.
ACKNOWLEDGEMENTS
This research is supported, in part, by Alibaba Group through the Alibaba Innovative Research (AIR) Program and Alibaba-NTU Singapore Joint Research Institute (JRI), Nanyang Technological University, Singapore. Tianze Luo wishes to extend his gratitude to the Alibaba Innovative Research (AIR) Program, NTU, Singapore, for their support. Sinno J. Pan thanks the support of the Hong Kong Jockey Club Charities Trust to the JC STEM Lab of Integration of Machine Learning and Symbolic Reasoning and the Microsoft Research Asia collaborative research grant.

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Published as a conference paper at ICLR 2024
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A EXPERIMENT DETAILS
A. DETAILS ON GRAPH DATASETS
We show the statistics of the datasets for node classification in Table For each dataset, we list its graph statistics, data split, and homophily score, which is computed as follows.
Homophily. We adopt the homophily indicator H()) of the graph from Liu et al. (021b), which can be calculated as:
1 V
fu: u E ((() and y(u) = y(v)}
H()) =
[V(e)
PEV
where | {u: u and v(u) y(t)} denotes the number of o' s directly connected nodes who have the same label as and [W(') the number of neighbouring nodes of v. Intuitively, high (G) indicates an assortative graph and vice versa.
Table 4: Statistics of the oldecclassiiication datasets used in our experiments. The homophily level of the dataset can be used to distinguish assortative and disassortative graph datasets.
Cor
Citeneer
Puhmed
Chumeleon
Squirel
Yours
Wircomal
Homophily
0:25
0,06
Splits
140,
10001/ 000
130/5500/
60/5500/1, 000
6055, (P95//006//003
(000/237//10 /209%
0096/299
*Nodes
2. 70s
3, 3,227
19, 717
277
3,
180
163
251
#bdges
5,4229
4, 7.722
838
36, 101
217. 077
206
309
499
#Restures
1,453
3, 7/705
500
2,325
2, 060
1, 7.703
1, 1,303
1. 7.003
@Classes
7
3
5
5
For the graph classification task, we adopt Mutagenicity, D&D, NCII, geog-molhiv, and QM7 datasets. The D&D and PROTEINS datasets are used for protein structure classification, which aims to categorize proteins into enzyme and non-enzyme structures. The NCI1 dataset is used for identifying chemical compounds that inhibit lung cancer cells. The Mutagenicity dataset is used for recognizing mutagenic molecular compounds that have the potential for drug development. The QM7 dataset used for predicting the atomization energy value of molecules. The Ogbg-Molhiv is molecular property prediction dataset for predicting whether a molecule inhibits HIV virus replication not. All the datasets contain more than 1,000 graphs with varying graph structures (in terms of the average number of nodes and edges, the average degree of nodes) and node features. The statistics of each dataset are displayed in Table
Table 5: Summary of the datasets for the graph property prediction tasks.
Datasets
PROTEINS
Mutagemicity
D&D
NCII
oghg-molhiv
QM7
Graphs
1,113
4,337
1,178
4,110
41,127
7,165
Min Nodes
4
30
2
4
Max Nodes
620
417
5,748
111
222
23
Avg Nodes
39
284
26
15
Avg Edges
73
716
123
Features
3
14
89
37
0
Classes
2
2
2
1 (R)
A.2 IMPLEMENTATION DETAILS
Eyper-parameters
We implement our model using PyTorch. We set the default number of filters as four, which is suitable for most of the datasets. The default Chebyshev approximation order is set to 6. The dimension of hidden variables is searched from {16, 32, 64). and the level of filters are selected from {2, 3, 4, 53. Other hyperparameters are set at: 0.001 for the learning rate, 0.001 for weight decay, 0.5 for dropout, and 2 for the number of MM-FGConv layers. These hyper-parameters are used for both node and graph classification tasks. For the graph classification task, we further apply our proposed MM-FGPool operation as elaborated in Section 4.3, followed by linear classifier. For baseline methods in node classification, we adopt the code from the author's 's released implementation with the default settings. For graph classification, adopt the experiment setting form /Zhng et al.
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(2021a), where two-layer GCN networks followed by the pooling methods listed in Table 2 Specifically, this experiment setting is comparable to our model's design, where we also adopt two convolutional layers and one pooling layer.
Design of the neural network. MMg
In practice, the neural network MMg is set to be 2-layer standard GCN network that maps graph data (X, A) Raxd IR=x into 2(7 1)-dimnnsional vector ( i.e. the parameters of the meta-filters (gr,)
Specifically, we set (x,X.)) Rd/2xd/2 and W3 R4/2x((1-1) are trainable weights, L is the normalized graph Laplacian matrix computed from the adjacency matrix A, and (-)) is the ReLU activation function. We denote the collection of all the trainable weights { (W1, W2, Wa).
= 1/n. 1 oL (LX W3, where W1 E Raxd/2 W2
After obtaining the 2(7 dimmennsioaa output E R2(1-1) from MM. and Equation (4) to determine the exact formulation of the meta-filters (or, To determine the formulation of (r, i)-th meta-filter, we first decode its lower- and the upper-cutoff positions and margins, i.e. G, Cs+1, Ets and E++1, from w using Equation (3). Then, the formulation of gi.; (the bandpass starting from + c to G++1 + E(+1) can be directly derived from Equation (4). Finally, starting from g..: we can determine g7 in an iterative manner, using the second condition in Proposition 2. The visualization example of the meta-filters is shown in Figure and Figure 2 of the manuscript.
Equation (3)
Split of Smela and Smain
The Smeta and Smain are the data randomly split from the training data. our experiment, we take 80% of the training data the Smain and 20% as Smenn In practice, our MM-FGCN shares the same train and test dataset splitting scheme as other baseline methods. We first train the MM-FGCN by vottatraining (Algorithm 2) on the training data. Then, fix the trained parameters 0, and evaluate the accuracy of the MM-FGCN on the testing dataset. The dataset splitting in line 3 of Algorithm 2 is done randomly within the training data. Hence, in our experiments, the MM-FGCN is evaluated under the same data resource as other baselines.
B EXTRA EXPERIMENTS
B. EXPERIMENTS ON RANDOM 60%/20%/22% SPLITS
The main results of the full sets of node classification experiments with statistics of datasets are summarized in Table and Table For fair comparison with the state-of-the-art methods, we list the additional experiments for node classification on homophily graph datasets with 60%/20%//00% split. Corresponding results are shown in Table
Cora
Citeseer
Pubmed
MIP
76.44 + 0.30
76.25 + 0.28
86.43 + 0.13
GCN
87.78 + 0.96
81.39 + 1.23
88.9 0..2
GAT
76.70 1 0.42
67.20
1 0.46
83.28 1 0.12
GraphSAGE
86.58 0.26
7..34
86.85 0.11
Geom-GCN
85.27 1.28
77.99
0.92
90.05 0.17
ACM-GCN
88.62 1.22
81.68 0.99
90.66
GCNII
88.98 1.33
81.58 + 1.3
89.8 0.30
MM-FGCN (Ours)
89,89* + 1.12
82.97' + 0.85
91.52' + 0.21
Table 6: Test accuracy for classifications on homophily graphs under 60%/20%//0% random split.
B.2 EXPERIMENTS ON FIXED 48%/32%%/20% SPLITS
We further conduct the node classification experiments on data split with fixed 48%//2%%/20% according to Pei al. 2020). The corresponding results are shown in Table
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Cornell
Wisconsin
Toxas
Chameleon
Squirel
Cora
Cleeseer
Pubmed
Gomm-GCN
60.54 + 3.67
64.51
3.66
66.76 2.72
60.00 - 2.81
38
0.92
86.35 + 1.57
78.02 + 1.15
80.35 + 0.47
GGCN
S1.08 + 6.63
86.86
3.29
84.86
4.55
71.14 1.84
56.17
1.58
87.56
1.05
77.14
1.45
89.15 = 0.37
H2GCN
82.70 5.22
S1.65
84.86 7:23
00.11221 2.15
36.48
1.80
87.87
77.11
1.7
0.38
77.81
1.
87.61
76.26
Geom-GCC GCN
60.54
64.51 3.66
06.76 2.72
00:00
38.15 0.92
86.35
78.02 1.15
80.5 0.47
ACM-GCN
85.14 + 6.07
88.43 3.22
87.84 4.4
09:14 1.91
$6.19
1.49
87.91
+ 0.96
17.32 1.7
90.00
= 0.52
GCNO
77.96 + 3.79
80.39
=: 3.40
77.57 = 3.83
63,86 = 3.01
38.47 2 1.58
56.37 à 1.25
90.15 0.443
NLMLP
84.9 1 5.7
57.3 4.3
85.4 3,8
50.7 2.22
33.7
78
19
8 1.55
NLGON
57.6
18.11:10
56.7
7.3
63.7
56.8
88.2
MM--PGN (Ours)
87.35
6.18
89.02' 5.41
89,31' 1. 1.36
72.61" 1.84
61.4" 1. 1.22
89.35" 1 1.15
79.86 1. 1.42
91.42 1. 0.01
Table 7: Test accuracy for node classífication results under fixed 48%/32%6/20% split.
B.3 PERTURBATION RESILIENCE OF MM-FGCN.
In this study, we add extra experiments to assess the perturbation resilience of our MM-FGCN against noise perturbation present in input graph data, which is ubiquitous in real-world datasets. Particularly, we train the MM-FGCN with corrupted data that are contaminated by random noise of various magnitudes. The noise magnitude is controlled by the noise ratios, which are defined as the amount of randomly deleted edges (or randomly flipped binary-valued features) divided by the number of untainted edges (or features). We then investigate how the performance of the resultant models varies when the noise level change from O to 1. As illustrated in Figure 4 our MM-FGCN consistently outperforms the baselines with remarkable margin even under the presence of considerable noise. Thus, the MM-FGCN demonstrates a strong noise resilience making it a highly promising solution for real-world applications.
Cora Structure Perturbation
Cora Feature Perturbation
0.85
Ours
Ours
0.80
GCN
0.80
GCN
Test Accuracy
GAT
Test Accuracy
GAT
0.75
0.75
0.70
0.70
0.65
0.65
0.60
0.2
0.4
0.6
0.8
0.000
0.025
0.050
0.075
0.100
0.125 0.150
Noise Ratio
Noise Ratio
Figure 4: Noise resilience experiments with the edge (left) and feature (right) noise perturbations on Cora.
B.4 ABLATION ON THE NUMBER OF THE META-FRAMELES GENERATORS
We analyze how the size of the meta-frameles generator (i.e. a set of spectral filters), 1, affects the performance of MM-FGCN, With an insufficient amount of meta- feltefii the model may fail to learn the optimal frequency partition and cannot disentangle graph signals into desirable frequency components. Intuitively, an overly small I may hinder the learning of discriminative graph represen- tations. Conversely, a large improves the precision of frequency partition learning but also increases computational expenses. This requires us to strike balance between the by selectively choosing the value L.
We evaluate the performance of MM-FGCN with different choices of I over Cora, Citeseer, Chameleon, Squirrel, and D&D datasets. As shown in Figure (a), the Meta-FCGN is able to achieve of-te--f-theaan performance by constructing the meta-framelet generator with only 3 filters. Overall, the model performance is stable and robust across different choices of I. In general, we recommend setting I = for effective and efficient implementation.
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B.5 ABLATION ON THE ORDER OF CHEBYSHEV APPROXIMATION
Recall that the Chebyshev approximation trick Defferrard et al. (2016b) is applied in Algorithm 1 for efficient computation of each 9e,, (L.) Broadly speaking, using higher-order Chebyshev approximation leads to smaller approximation error w.r.t the meta-generator but creates a greater computational overhead. As illustrated in Figure (b), the performance of MM-FGCN is robust to Chebyshev approximation of different orders. Empirically, the MM-FGCN achieves optimal performance with a Chebyshev approximation of an order greater than 4. In contrast, a low-order Chebyshev approximation order incurs undesirable approximation error, which hinders graph representation learning and impairs model generalization. We recommend using a higher than 4-order Chebyshev approximation for good model performance.
Ablation on Different Filter Numbers
Ablation on Chebyshev Approx. Order
85
85
80
80
Test Test Accuracy
Test Accuracy
75
75
Cora
Cora
Citeseer
70
Citeseer
Chameleon
Chameleon
Squirrll
65
Squirrel
D&D
D&D
6
7
8
2
No. Filters
Chebyshev Approx. Order
(a) Ablation on number filters used in the meta- framelet generator.
(b) Ablation on the order of Chebyshev approxima- tion.
Figure 5: Ablation studies on MM-FGCN's hyperparameters.
B.6 VISUALIZATION OF MM-FGCN REPRESENTATION
In this section, we empirically show that the MM-FGCN is able to produce more discriminative graph representations than conventional GCN, both assortative (e.g. Cora) and disassortative (e.g. Coreell) datasets. In order to assess the quality of the learned graph representation, we visualize the hidden features generated by the penultimate layer of both MM-FGCN and GCN via t-SNE van der Maaten & Hinton (2008). As shown in Figure 6 the graph representation of our MM-FGCN is more spatially clustering than GCN, especially for the disassortative dataset that is more challenging for classification. In fact, the learned feature of MM-FGCN is strongly correlated to the label, which significantly facilitates node and graph classification tasks. In summary, the visualization of the learned hidden features validates that the adaptiveness and expressiveness of our MM-FGCN are beneficial to learning discriminative graph representations.
We also demonstrate the effectiveness of the MM-FGCN by visualising the filters learned by MM- FGCN Figure [ We can observe that the filters learned from Cora dataset (assortative) more concentrate on the low-frequency signals than the filters learned from the Cornell dataset (disassortative). Due to the assortative properties, aggregating local information with low- low-frequency signals benefit the model's performance on the Cora dataset. In contrast, elaborated in Bo et al. 2021), high-frequency signals are useful for disassortative networks, which corresponds to our learned multi-resolution filters since more filters concentrate the high-frequency part, providing a comprehensive feature extraction on the high-frequency signal. This phenomenon shows the adaptivity of our MM-FGCN different types of graphs
B.7 ROBUSTNESS ANALYSIS
We show that our MM-FGCN inherently possesses greater robustness compared to conventional GCN models, without relying on any specific robust data augmentation techniques in Appendix B.4 Furthermore, we observe that the robustness of our MM-FGCN can further enhanced when combined with data augmentation methods. We assess the robustness of our MM-FGCN in the face
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MM-FGGN filters Cora
MM-FGGN filters Cornell
1.0
1.0
0.8
0.8
0.6
(o MevI
9/X)
heen
Cot Cet
Dretn
0.4
(9ft hemm
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
A
MM-FGCN (Cora)
MM-FGCN (Cornell)
00000
GCN (Cora)
GCN (Corelll)
Figure 6: Meta-framelet generator (row 1) and feature visualization (row 2-3) on the test and validation sets Cora (left, assortative) and Cornell (right, disassortative) datasets using MM-FGCN and GCN.
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Global Evasion
Global Poisoning
Adversarial accuracy (%)
80
No defense
80
W/ Jaccard defense
78
W/ GRAND deferse
76
76
Non-adaptive
74
ZZZ
Adaptive attack
72
72
GCN
Ours
GCV
Oura
GCW
Oure
GCN
Oura
GON
aura
GDN
Oura
GCN OURS
4
2
0.2
0.4
0.6
0.8
0.2
0.8
0.8
Noise ratio
Nolse ratio
Figure 7: Top: the adversarial accuracy of MM-FCGN under graph attack. Bottom: representation distortion of each perturbation level.
of both adaptive and non-adaptive graph attacks Mujkanovic et al. (2022). These attacks encompass graph poisoning, which involves perturbing the adjacency matrix before training, and graph evasion, which perturbs the adjacency matrix after training. We only compare our method with GCN but also evaluate two prevalent graph defense techniques, namely GRAND defense and Jaccard defense. We present the results on Cora in the top row of Figure 7 demonstrating that our MM-FGCN showcases substantially higher adversarial accuracy than GCN when subjected to graph attacks. Moreover, by incorporating GRAND and Jaccard defense techniques, the adversarial accuracy of MM-FGCN can be further enhanced. This highlights that the inherent robustness gain of MM-FGCN independent of the benefits offered by graph defense methods, demonstrating our orthogonal contributions.
We also validate our MM-FGCN's immunity against realist graph topology perturbations. To evaluate the robustness of well-trained model, introduce noise of varying magnitudes into the adjacency matrix. Subsequently, we assess the deviation between the perturbed hidden representation and the original using the & distance metric. As depicted in the bottom demonstrates a distinct 'potential well', where the potential refers to the distance from the perturbed feature to the original, unperturbed feature. The results in Figure 7 show that MM-FGCN has the capability to learn robust representations that remain intact by minor perturbations in the graph adjacency matrix. In contrast, conventional GCN experiences a progressively linear increase in representation distortion as the magnitude of perturbation grows.
of Figure 7 MM-FGCN
B.8 ABLATION STUDIES ON META-LEARNING
We compare the performance of MM-FGCN, both with and without meta-training, on the node classification datasets including Cora, Citeseer, PubMed, Cornell, Texas, Wisconsin, Chameleon, and Squirrel; and graph classification datasets including PROTEINS, Mutagenicity, D&D, NCHI,
Ogbg-molhiv, and QM7. We show the ablation study results in Table g and Table 9
Texas 7.9
Squiree
MBM FGCN N wio meta-learanee MMC
$2.9 2 0.3
12.2 1 0.7
80.1202
78.
83
823 = 4.8
62.6 = 2.6
59.2 - 2.2
0.5
73.9
0.6
80.7
2
0.2
88.9
86.1
43
88.5
7397
211
67.5
Table Comparison of with and without meta-learning framework on the node classification tasks.
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PROTEINS (1)
Mutagenicity (1)
(&D
NC
Oghg-molhine (1)
QM7 (2)
MM-FGPeel meta-lcarning
77.86
82.
144
81,0
75.9
78.57
0.73
MGM-FGPoo
78.07 1: 2.36
83.91
1.32
81.31 1.5
78.37 0.82
79.12 0.88
41.19 0.88
Table 9: Comparison of with and without meta-learning framework on the graph classification tasks.
As shown in Table and Table 9 in practice, using meta-training (Algorithm 2) leads a consistent performance gain on varying datasets in both graph and node classification settings. Moreover, as shown in Figure the meta-training process accelerates the convergence of MM-FGCN and improves the generalization performance of MM-FGCN.
In intuition, using different data batches in the meta-step and main-step of Algorithm 2 helps MM-FGCN to avoid overfitting In the meta-step, the meta-framelet learner & is first optimized to improve the meta-framelet for current backbone model A. Based on this optimized meta-framelet representation, the whole model is then evaluated and optimized with another newly sampled data batch. Using two identically distributed but distinct data batches, the -emeta-framlie transform will be cautiously updated until achieving good performance on both the meta- and main-data batches. Consequently, we anticipate that the meta-training process encourages MM-FGCN to find meta- framelet representations that can be generalized to different batches drawn from the underlying distribution, rather than merely following the gradient calculated on the current data batch, thus reducing the risk of overfitting.
How does meta-learning work?
Following Hospedales et al. (0022), in this paper, Meta-learning is understood 'learning-to taroing-tt--laaan to-learn', which refers to the process of improving the base learning algorithm over multiple learning episodes, i.e. the base- and meta-learning episodes. During base-learning, base-learning algorithm solves the base task, such as graph/node classification. During meta-learning, a meta-algorithm updates the base learning algorithm such that the model learns improve the meta objective.
In this paper, the base objective refers to 'learning a graph/node classifier using features based on a framelet transform parameterized by w', as shown Equation (8) and Equation (9). We aim to design a meta-objective that refers to 'finding good w such that the w- w-base- base-objective can be achieved well'. To this end, as shown the second term of Equation (8), we set the meje-obeective to be 'finding the with the lowest classification loss, for any given network backbone parameter 0', The intuition behind this is that for any model backbone, we always want to choose the most compatible w-framelet transform, such that the model backhone 0 can fully release its power, achieving classification loss. By combining the base- and meta-objectives, we establish the meta-lcarning training paradigm of MM-FGCN as bi-level optimization problem in Equation (8).
Unlike meta-learning models, where the base-learning algorithm is continuously improved during the meta-learning episode, standard machine learning model is trained on specific task using fixed, handcrafted base-learning algorithms. For instance, standard GCNs are trained to predict graph labels by minimizing the classification loss, using human-designed features like the graph Fourier transform. On the contrary, our MM-FGCN is able to learn suitable meta- frameeft transform that is adaptive to varying graph instances and distributions via the meta-learning paradigm.
In nutshell, using meta-learning improves the performance MM-FGCN on various graph learning scenarios. In intuition, meta-learning encourages MM-FGCN to learn meta-framelet representations that generalize well.
B.9 EXPERIMENTS ON LARGE-SCALL GRAPH DATASETS
Our MM-FGCN can be applied to large graphs, e.g. OGBN datasets, and achieves a high perfor- mance. We conduct experiments on the OGBN-arxiv and OGBN-produe datasets, and the results listed in the following table.
To train our MM-FGCN, for OGBN-arxiv, we use the 128-dimensional feature vector as the node feature of each node; for OGBN-produch we use the (00-dimensional feature vector as each node's 's feature. It is worth noting that some state-of-ther the-the-att methods (such SinTeeGTTPE--RvvG TAPE+RevGAT, and Sim SimTeG+TAPE TAOG++AAGGraa AGE) adopt the raw title and abstract form the OGBN- arxiv and detailed product descriptions from OGBN- as the node features, and achieve better
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Training on Cora
Training on Core
10-
0.8
0.7
WiC mida-sarrng
Wie metalserrnn wath muta leaming
Opoch
(a) Accuracy on the test set
(b) Accuracy on the train set
Figure 8: Comparison of MM-FGCN with and w/o meta-training on the Cora dataset. We report the
train and test accuracy of each training step.
results due using a good NLP module in extracting node features. Since we mainly compare the performance of the GNN module, we neglect the gain from using NLP modules such as GIANT-XRT or TAPE, etc. Instead, we directly use the preprocessed feature vectors provided both datasets on model and our baseline models.
Training details
To train the OGBN-Arxiv, we use the full-batch training. In the OGBN-products dataset, we use neighbor sampling with size 10 (similar UniMP [10]) to sample the subgraph during training. Other hyperparameter settings follow experiments other datasets elaborated in the Appendix. show the corresponding results in the following tables.
Method
Test Ace. 55.50 + 0.23
Val. Acc.
MLP
57.65 = 0.12
NODE2VEC
70.07 = 0.13
71.29 = 0.13
GRAPHZOOM
0.118
72.20 = 0.07
P&L. C&S
71.26
73.00 + 0..0
GRAPHSAGE
72.77 = 0.11
GCN
0.224
73.00 = 0..17
DEEPERGCN
0.17
72.62 = 0.14
SIGN
0.11
73.23 0.06
GAAN
0.11
UFGCONV-R
0.12
73,21 = 0.05
UniMP
7
74.50 + 0.05
Exphormer
7
DRGAT
74.16 0.07
75.34 - 0.02
Ours
74.25 + 0.15
75.46 0.08
Table 10: Experiment on the OGBN-Arxiv dataset.
Method
Test Acc.
Val. Acc.
MLP
55.50 0.23
57.65 0.12
GCN-Cluster
78.97 + 0.36
92.12 + 0.09
GAT-Cluster
79.23 0.78
89.85 0.22
GAT-NeighborSamlling
79.45 + 0.59
GraphSAINT
80.27 0.26
DeeperGCN
80.90 0.20
92.38 + 0.09
UniMP
82.56 0.31
93.08 0.17
AGDN
83.34 0.27
92.29 + 0.10
Ours
84.03 0.23
93.57 0.12
Table 11: Experiment on the
dataset.
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100lie 120
02-
02
08
08
0.2
0.2
30
0.2
(a) step O
(b) step 20
(c) step 50
(d) step 100
WWW.GON Elber on Camuel
MWNTCON Filam Cameel
WWw.GOO thars Commel
wocricnnlBer Cormel
LNR Back
2.4
0.6
2.4
0.5
0.4
0.4
0.5
(e) step O
(1) step 20
(g) step 50
(h) step 100
Figure 9: From (a) to (d): Variation of filters during the training process on the Cora dataset. From (c) to (h): Variation of filters during the training process on the Cornell dataset.
B.10 VARIATION OF FILTERS DURING THE TRAINING PROCESS
We visualize the variation of filters during the training process on the Cora and Cornell datasets. We train our model on both datasets for 100 epochs and visualize the filters at training steps 0, 20, 50, and 100.
Recall that the Cora dataset contains graph of a citation network, where each node represents scientific publication and the node labels are the research domain. Thus, the node label (i.e. the graph signal of interest) varies smoothly across the nodes, emphasizing the importance of low-frequency information in learning good graph representations.
As shown in Figure (a. b, c, d), we observe that MM-FGCN automatically learns to focus more on establishing the low-frequency representations: at the initial stage of the training process, the meta-filters are evenly separated and do not show special preference the graph signals specific frequencies. During the training process, the filters tend to concentrate on the low-frequency part of the graph signal to retrieve more refined low-frequency information of the graph.
The Cornell dataset contains a website network, where each node represents a web page that is manually classified into five categories: student, project, course, staff, and faculty. Edges represent hyperlinks between them. In this case, nodes with the same labels are not likely to link to each other. Thus, the node label (i.e. the graph signal of interest) varies drastically across neighbouring nodes, showing that high-frequency information is essential in learning good graph representations.
As shown in Figure (e, f, g. h), observe that MM-FGCN automatically learns focus more on establishing the high-frequency representations: at the initial stage of the training process, the meta-filters are evenly separated and do not show special preference the graph signals specific frequencies. During the training process, the filters tend to concentrate on the high-frequency part of the graph signal to refine the high-frequency representations of the graph.
B.11 TIME COMPLEXITY ANALYSIS
The complexity of a single forward pass of the meta-framelet convolution is approximately times as the cost of standard graph convolution, where is the number of meta- framelets. In practice, our MM-FGCN can be computed efficiently, without significant overhead.
Suppose we instantiate the meta-learner Me as L-layer GCN, we use a k-order Chebyshev approximation for the n x n graph Laplacian matrix L, and we set the number of filters as r. Then, for each d- d-dimensional input graph feature H R x the computational complexity of meta-framed convolution is O(rn x n x d) + O(n x n x d) + O(r x k2 x n²) = O(rn (d + 12)). In this setting, the standard graph convolutional layer costs 2d)
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Published as a conference paper at ICLR 2024
Experiments training and inference time cost
In the following Table and Table I3, we compare both the training and inference time cost of our MM-FGCN against training other baselines, including GCN, GAT, GraphSage and SVD-GCN, We report the time cost of one training epoch in Table and we evaluate the time cost of on single forward pass in Table T3 Specifically, each training epoch of MM-FGCN includes both the base- and the meta-update step. And we set the Chebyshev approximation order & and the number of meta-framelets & 4 and 4. We conduct the experiments on varying GNN architectures, with varying parameter budgets (e.g. model size). The experiments are conducted on single 40G
A100 GPU.
Model
Paranieess
Ass. IT)
Hime/pochh
$ Faauniiiss
Asc: (11
Tinelspoch... (1)
Parunciess
Asc: (1)
GCN
11,246
99.4
[1.012
$1.23%
K1.2
11.15
100,314
12.0
[101
GAT
H0,134
79.5
0.00]
50.422
81.7
01002
101,684
$2.5
0.024
GraphSaac [4]
HU076
73.9
01244
51,682
80.9
0.281
137,216
$1.9
0.315
SVD-GGNN [15)
10.37
79.5
0.235
53.412
81.4
0.254
195,452
62.5
0.261
MM--OOCN (our) w/s meto-nining
10,674
62.2
0.045
52.546
K3.7
0,050
001,046
66.0
0.833
MM-FOCN (o0m)
10674
82.6
0.064
52.546
64.4
0.088
101,046
$4.5
0.61
Table 12: Comparison of accuracy and training runtime (in seconds) on Cora datasets. Although our model's 's runtime is slightly larger than GCN and GAT, our method remains faster than GraphSage and SVD-GCN.
Pinmeters
Aoc. (1)
Inference [.)
Acc: (1)
Inference Time (J)
Parameters
Ace: (7)
Inference Time (
GCN GAT
R0244
90.4
B0011
31,238
81.2
0.015
10002004
82.0
0.017
10.124
79.6
8.019
50,422
81.7
0.020
101.844
82.5
0.025
GraphSage [4]
100076
78.9
0.231
51,682
80.9
0.275
107.216
81.9
0.302
SVD SVD.GEN (15)
10.578
79.6
0.227
53,412
81.4
0.247
106.452
82.5
0.258
MM-RGON (ouns)
106674
82.6
0.47
52,546
84,4
0.050
101,046
84.5
0.052
Table 13: Comparison of accuracy and inference time (in seconds) under different parameter sizes the Cora dataset.
C RELATED WORKS ON META GRAPH REPRESENTATION LEARNING
Graph representation learning refers to the process of converting the raw graph data into high dimensional vectors while preserving intrinsic graph properties (Chen al. 2020). Effective graph representations can provide significant insights into graph data and benefit downstream tasks, such as
social analysis (Min et al. 2021 molecular property prediction and generation (Zhang al. 202
Zang & Wang 2020), graph generation (oo et al.
2022 Luo et al. 2023b a). time series analysis (Li
al.
2021b Cui al. 202I.. neural signal processing (Ding al. 2023 Ding & Guan 2023 etc.
Meta-learning (Hospedales et al., 2022 Liu et al. 2022b Finn et al.
2017). also referred to as
learning learn, aims to enhance base learning algorithm through knowledge gained from meta-algorithm. It has been applied to graph representation learning in (Huang & Zitnik] 2020
Xiao et al. 2021 Zügner Günnemann] 2019). Huang & Zitnik (2020) use local subgraphs to transfer subgraph-specinn information and learn transferable knowledge faster via meta gradients. Xiao al. (202T) use a meta-learner relate tasks on graphs describing the relations of their own dimensions to improve few-shot learning. Zügner & Günnemann (2019) optimize the graphs as hyperparameter using meta- gratiengss to solve bi-level optimization problem underlying training- time attacks. While these methods are designed for either transfer learning or multi-task learning scenarios, our proposed meta-learning algorithm for MM-FGCN model aims to improve the base graph representation algorithm, the graph transform of MM-FGCN, by optimizing meta-algorithm, ie. the meta- fratmele learner. Since a perturbation on graph framelets may cause high variation in graph representations, simultaneously updating the meta- meta-framan framelet learner and the base model may be inefficient. Therefore our meta- fratmele algorithm is learned in a bi-level meta-learning framework to enhance the overall learning performance.
D PROOF DETAILS
Proof of Proposition According to the discrete tight framelet transform theory (Theorem 2.1 and Theorem 3.1 in (Dong 2017a)). the series of progressive resolution subspaces (V,), with inherentny V, = span ((yprio )a inherently satisfies the denseness, translation, and dilation properties, making
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Published as a conference paper at ICLR 2024
it set of desirable multiresolution bases for graph domain data. To complete the proof, only need to verify the tightness of the MMFS, ie. PMM 2 XM = x holds for any graph signal x. Let
I ([R) x [I),) : < R2, we have
PMM $
T MM
Σ
Prin
(10)
E
(y (7
(11)
(ni)e
=U
Σ
gort (T- U
(12)
=U
9i,1
(13)
I)) ... If. 20-201010 (o-1t1I)) ) D
Σ IR. P7
Lig 207 (
iE[1
9h, (r (7-
(14)
0 ( (1-12(1-)+)- gi.1 2 ((- U
-U
Σ
Fi, 17 (1-
- U
(15)
I<iETTEERR
Σ
gi. 17 (122)))1))) + 2, (1- U
(16)
1E
(17)
=U
Σ
gi. (7-
2
( (1- UJ II,
(18)
IEIEI
which completes the proof.
26


Reviews:
[{"review_id": "stCa35d4sd", "reviewer": "Reviewer_nDyN", "summary": "This paper use meta-learning to build for adaptive framelet for GNN.\n\nThey evaluate the model's performance on various graph learning tasks.", "questions": "1. Could you provide details on the time cost for meta-learning as well as the inference time for the model?\n2. Would it be possible to evaluate the model on larger datasets like ogbn? I did observe ogbg-molhiv, but I'm referring to graphs with a larger number of nodes rather than the total number of graphs.\n3. An detailed introduction to meta-learning would help readers in gaining a clearer understanding of the experimental setup.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. MM-FGCN is can learn adaptive multiresolution representation.\n\n2. Achieve STOA performance on graph learning tasks.\n\n3. parameterization of the meta-framelet generator uses use fewer parameters than previous method.", "weaknesses": "1. Require additional meta training.\n2. The framelet is defined using a meta-band-pass filter based on polynomial splines, the model use Chebyshev approximation to circumventing the need for eigen-decomposition, but compared with graph-structure-based approach, Chebyshev approximation seems a more expensive.\n3. Following the previous point, it seems it's hard to use the model on large graphs. (ogbn tasks)\n4. The performance is not close to STOA, recent studies show better performance, e.g. [1],[2]\n\n[1] https://arxiv.org/abs/2305.10498\n[2] https://arxiv.org/abs/2105.07634", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sob3Fwnxsa", "reviewer": "Reviewer_oTNL", "summary": "The Multiresolution Meta-Framelet-based Graph Convolutional Network (MM-FGCN) is a novel approach to graph representation learning that allows for adaptive multiresolution analysis across diverse graphs. It achieves state-of-the-art performance on various graph learning tasks. \n\nIn my opinion, adaptive multi-resolution is a promising yet difficult approach for the representation of graphs, lying in the intersection of graph signal processing and graph machine learning. In particular, I appreciate the construction of the framelet filters to satisfy the three properties (where denseness is different from the sparse and redundant representation in wavelet and framelet). The paper extends the framelet theory and presents the solution with relatively few parameters (only $\\Theta, \\omega$) to avoid a large number of parameters for the framelet system, yet achieves significant improvement in the experiments. The theories and proofs are solid in general, and though some of the writing could be improved, I think it is an excellent piece of work.", "questions": "1. How is the neural network $M_\\xi$ formulated? What does the output $\\omega$ look like?     \n2. How is the graph data split to obtain $S_{meta}, S_{main}$?    \n3. Why is the meta training needed for the training procedure?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper proposes a novel approach to design an adaptive learnable set of multi-resolution representations on graphs, with solid theoretical motivation and proof.    \n2. The proposed method significantly improves node classification (especially on disassortative tasks) and graph classification tasks.", "weaknesses": "1.The paper lacks some necessary model  and data descriptions for the implementation, like the specification of neural network $M_\\xi$.    \n2. minor issues:  missing \"translation\" before property near \"Mallat, 2006\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RIyQI236pb", "reviewer": "Reviewer_sJoC", "summary": "This paper proposes a Multiresolution Meta-Framelet-based Graph Convolutional Networks (MM-FGCN), which employs a diverse set of framelets for constructing graph convolution and learns meta-framelet generator networks via the meta-learning scheme. Since most GNNs depend on single-resolution graph feature extraction, they often fail to capture local patterns and community patterns simultaneously. To resolve it, some papers have proposed multi-resolution graph feature extraction-based GNNs. But, they use predefined and hand-crafted multiresolution transforms. To address these issues, this paper designs meta-learning based adaptive multiresolution graph convolution and they have shown the effectiveness with their experiments.", "questions": "- I think MM-FGCN can also be applied to the graph classification tasks. Could you provide the performance of MM-FGCN with MM-FGPool and MMFGCN with the standard graph pooling?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed paper deals with really important research problem. Simultaneously capturing high and low resolution with graph convolution is interesting and important.\n- The proposed method seems novel to me.\n- From their experiments, the proposed meta-framelet-based graph convolutional networks show good performance on various tasks. \n- The paper is well-written and easy to follow. In particular, the preliminary section provides the necessary and detailed information to understand the proposed method.", "weaknesses": "- One of the important details about how to split the meta dataset from the main dataset is missing. It should have been provided to fairly compare the proposed method with other graph neural networks.\n- In the same context, it would be better if you explain why meta learning framelet transforms is better than directly training them. From Table 3, it is easy to know that framelet transforms with meta-learning is more effective compared to the direct training scheme. But, why meta-learning scheme is better than the direct training scheme is mysterious. I think this is related to how to construct the meta dataset.\n- It would be better if the author provided the change of filters according to the training step since the author claimed that the limitation of existing multiresolution transforms works is that they remain fixed throughout the training process. So, I'd like to see the variations of filters during the training and the analysis about it.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PvZ1zAAQYJ", "reviewer": "Reviewer_PtSQ", "summary": "The paper proposes a new method for graph representation learning leverage spectral graph representation learning and meta learning. Experiments show the proposed method shows superior performance on a variety of graph datasets.", "questions": "1. Why Geom-GCN in Table 6 lacks std?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The method is relatively novel, combining several ideas from graph spectral filtering and meta learning.\n2. Adequate ablation studies are performed to show the importance of proposed components.", "weaknesses": "1. The datasets used in the paper are relatively small, given the fact that there are a numerous large-scale graph benchmark datasets nowadays. It may not be a large concern in the early days, e.g. 2016-2017 when GNN was originally proposed, but in today’s standard more larger graphs are expected to evaluate the proposed approach rigorously and reliably.\n2. Lack of some well-known baseline methods such as GIN (Xu, Keyulu, et al. \"How powerful are graph neural networks?.\" arXiv preprint arXiv:1810.00826 (2018).) and graph transformers (e.g. Rampášek, Ladislav, et al. \"Recipe for a general, powerful, scalable graph transformer.\" Advances in Neural Information Processing Systems 35 (2022): 14501-14515.) Given the abundance of such existing methods, I encourage the authors to admit this fact and discuss about the pros and cons of using the proposed MM-FGCN approach in practice. \n3. There is no mentioning on the releasement of the code making reproducibility hard. \n4. Typo, e.g. “denseness, dilation property, and property (Mallat, 2006)” in Section 3.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)