This document provides an outline for a tutorial on graph signal processing (GSP) for machine learning. It will include a brief introduction to GSP, key GSP tools for machine learning, and how GSP can help address challenges related to exploiting data structure, improving efficiency/robustness, and enhancing model interpretability. Applications of GSP for machine learning will also be discussed. The tutorial will conclude with a summary of open challenges and new perspectives in the field.

1. Graph Signal Processing for Machine
Learnin
g
A Review and New Perspectives
ICASSP Tutorial, June 2021
Xiaowen Dong, Dorina Thanou, Laura Toni
,
Michael Bronstein, Pascal Frossard

2. ICASSP Tutorial, June 2021
Part I
I
GSP for ML: Additional challenges

3. Outline
3
• Brief introduction to graph signal processing (GSP)
• Key GSP tools for machine learning
• Challenge I: GSP for exploiting data structure
• Challenge II: GSP for improving ef
fi
ciency and robustness
• Challenge III: GSP for enhancing model interpretability
• Applications
• Summary, open challenges, and new perspectives

4. Outline
3
• Brief introduction to graph signal processing (GSP)
• Key GSP tools for machine learning
• Challenge I: GSP for exploiting data structure
• Challenge II: GSP for improving ef
fi
ciency and robustness
• Challenge III: GSP for enhancing model interpretability
• Applications
• Summary, open challenges, and new perspectives

5. Outline
3
• Brief introduction to graph signal processing (GSP)
• Key GSP tools for machine learning
• Challenge I: GSP for exploiting data structure
• Challenge II: GSP for improving ef
fi
ciency and robustness
• Challenge III: GSP for enhancing model interpretability
• Applications
• Summary, open challenges, and new perspectives

6. GSP for improving ef
fi
ciency and robustness
B. Robustness against topological noise
C. Improvement on computational ef
fi
ciency
4
A. Improvement on data ef
fi
ciency and robustnes
s

7. GSP for improving ef
fi
ciency and robustness
B. Robustness against topological noise
fi
ciency and robustnes
s

8. A. Improvement on data ef
fi
ciency and robustness
• In many classical ML applications, data is scarce or noisy
5

9. A. Improvement on data ef
fi
ciency and robustness
• In many classical ML applications, data is scarce or noisy
5
• Need for prior knowledge to improve performance:
- Graph-based regularization could help in that direction

10. A.1. Graph signal smoothness as loss function
• Cross entropy loss is widely used in (semi-)supervised learning
frameworks. However:
- Underlying manifold structure is not preserved: Inputs of the same class tend to
be mapped to the same outpu
t
- One-hot-bit encoding is independent of the input distribution and network
initialization: slow training process
6

11. A.1. Graph signal smoothness as loss function
• Cross entropy loss is widely used in (semi-)supervised learning
frameworks. However:
- Underlying manifold structure is not preserved: Inputs of the same class tend to
be mapped to the same outpu
t
- One-hot-bit encoding is independent of the input distribution and network
initialization: slow training process
6
• An alternative GSP inspired loss function
:
- Maximizes the distance between images of distinct classes
- Minimizes a loss based on the smoothness of the label signals on a similarity
graph

12. A.1. Graph signal smoothness as loss function
• Cross entropy loss is widely used in (semi-)supervised learning
frameworks. However:
- Underlying manifold structure is not preserved: Inputs of the same class tend to
be mapped to the same outpu
t
- One-hot-bit encoding is independent of the input distribution and network
initialization: slow training process
6
Smoothness of label
signal for class c
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SG(sc) = sT
c Lsc =
X
u,v2V
Wv,u sc(v) sc(u)
2
• An alternative GSP inspired loss function
:
- Maximizes the distance between images of distinct classes
- Minimizes a loss based on the smoothness of the label signals on a similarity
graph

13. A.1. Graph signal smoothness as loss function
• Cross entropy loss is widely used in (semi-)supervised learning
frameworks. However:
- Underlying manifold structure is not preserved: Inputs of the same class tend to
be mapped to the same outpu
t
- One-hot-bit encoding is independent of the input distribution and network
initialization: slow training process
6
<latexit sha1_base64="jDy8S+Shcq0QB6YPbKAmN3WDYtQ=">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</latexit>
LGk
(f) =
C
X
c=1
SG(sc) =
X
sc(v)sc(u)=0,8c
exp( ↵kf(xv) f(xu)k)
Smoothness of label
signal for class c
<latexit sha1_base64="MHw39RZxbBulPYRL9+7KyYyxido=">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</latexit>
SG(sc) = sT
c Lsc =
X
u,v2V
Wv,u sc(v) sc(u)
2
• An alternative GSP inspired loss function
:
- Maximizes the distance between images of distinct classes
- Minimizes a loss based on the smoothness of the label signals on a similarity
graph

14. Experimental validation
• Clustered like embeddings for CIFAR-1
0
• Robustness to cluster deviations
7
Bontonou et al., “Introducing graph smoothness loss for training deep learning architectures,” IEEE Data Science
Workshop 2019

15. A.2. Latent geometry graphs
• Capture the latent geometry of intermediate layers of deep networks
using graphs
• Quantify the expressive power of each layer by measuring the label
variation
• Impose desirable properties into these representations by applying
GSP constraints
- Robustness: Enforce smooth label variations between consecutive layers
8
Lassance et al., “Laplacian networks: bounding indicator function smoothness for neural networks robustness”,
APSIPA Trans. on Sign. and Infor. Process., 2021
Lassance et al., “Representing Deep Neural Networks Latent Space Geometries with Graphs ”, Algorithms, 2021

16. A.3. Graph regularization for dealing with noisy labels
• Main challenge: Binary classi
fi
cation with noisy labels
• Proposed approach (DynGLR): A two step learning proces
s
- Graph learning: extract deep feature maps and learn a graph that maximizes/
minimizes similarity between two nodes that have same/opposite labels (G-Net
)
- Classi
fi
er learning: alternate between re
fi
ning the graph (H-Net) and restoring
the corrupted classi
fi
er signal (W-Net) through graph Laplacian regularization
(GLR)
9

17. A.3. Graph regularization for dealing with noisy labels
• Main challenge: Binary classi
fi
cation with noisy labels
• Proposed approach (DynGLR): A two step learning proces
s
- Graph learning: extract deep feature maps and learn a graph that maximizes/
minimizes similarity between two nodes that have same/opposite labels (G-Net
)
- Classi
fi
er learning: alternate between re
fi
ning the graph (H-Net) and restoring
the corrupted classi
fi
er signal (W-Net) through graph Laplacian regularization
(GLR)
9
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Y r
= arg min
B
kY r 1
Bk2
2 + µr
BT
Lr
B GLR

18. Experimental validation
• Classi
fi
cation error curves for different levels of noisy label
s
- Phoneme: contains nasal and oral sound
s
- CIFAR-10: subselection of two classes, airplane and ship
• Introducing GLR is bene
fi
cial especially in the high noise regime
10
Ye et al., “Robust Deep Graph Based Learning for Binary Classi
fi
cation” IEEE Tans. on Sign. and Inform.
Process. over Net. 2020

19. A.4. Graph regularization for zero shot learning
• Zero shot learning: Exploit models trained with supervision, and
some mapping to a semantic space, to recognise objects as belonging
to classes with unseen examples during trainin
g
• Key assumption: Regularize the space of the continuous manifolds
and the map between them by minimising the isoperimetric loss (IPL)
11
Visual embeddings
Semantic embeddings
<latexit sha1_base64="tmcolBD1/UXo39MYdOOLS5qBPs0=">AAAB63icbVA9SwNBEJ2LXzF+RS1tFoNgIeFOIlpYBGwsI5oPTI6wt9lLluzuHbt7QjjyF2wsFLH1D9n5b9xLrtDEBwOP92aYmRfEnGnjut9OYWV1bX2juFna2t7Z3SvvH7R0lChCmyTikeoEWFPOJG0aZjjtxIpiEXDaDsY3md9+okqzSD6YSUx9gYeShYxgk0mPZ+i+X664VXcGtEy8nFQgR6Nf/uoNIpIIKg3hWOuu58bGT7EyjHA6LfUSTWNMxnhIu5ZKLKj209mtU3RilQEKI2VLGjRTf0+kWGg9EYHtFNiM9KKXif953cSEV37KZJwYKsl8UZhwZCKUPY4GTFFi+MQSTBSztyIywgoTY+Mp2RC8xZeXSeu86l1U3btapX6dx1GEIziGU/DgEupwCw1oAoERPMMrvDnCeXHenY95a8HJZw7hD5zPHxmXjZk=</latexit>
Z, S

20. Graph-based approximation of IPL
• Isoperimetric loss (IPL): measures the
fl
ow through a closed
neighborhood relative to the area of its boundar
y
12

21. • A graph-based approximation of IPL:
- Treat visual embeddings as vertices in a graph, with af
fi
nities as edges,
and visual-to-embedding map as a linear function on the grap
h
- Seek a non-parametric deformation represented by changes in the connectivity
matrix, that minimises the IP
L
- Approximate IPL loss by using spectral reduction: Spectral graph wavelets of
the semantic embedding space
- Construct a new graph based on the spectral embedding
s
- Perform clustering to map clusters to labels
Graph-based approximation of IPL
• Isoperimetric loss (IPL): measures the
fl
ow through a closed
neighborhood relative to the area of its boundar
y
12
<latexit sha1_base64="7xt4b3CmVDnRxDG63sGidf+g/oA=">AAAB/nicbVDLSgNBEJyNrxhfq+LJy2CQRJCwK4q5CAEPeoxoHpCsYXYymwyZfTDTK4RlwV/x4kERr36HN//GSbIHjRY0FFXddHe5keAKLOvLyC0sLi2v5FcLa+sbm1vm9k5ThbGkrEFDEcq2SxQTPGAN4CBYO5KM+K5gLXd0OfFbD0wqHgZ3MI6Y45NBwD1OCWipZ+5d3SelFF/gcndIILlNj3GrdNQzi1bFmgL/JXZGiihDvWd+dvshjX0WABVEqY5tReAkRAKngqWFbqxYROiIDFhH04D4TDnJ9PwUH2qlj71Q6goAT9WfEwnxlRr7ru70CQzVvDcR//M6MXhVJ+FBFAML6GyRFwsMIZ5kgftcMgpirAmhkutbMR0SSSjoxAo6BHv+5b+keVKxzyrWzWmxVs3iyKN9dIDKyEbnqIauUR01EEUJekIv6NV4NJ6NN+N91pozspld9AvGxzdFIpO3</latexit>
G
0
= (Ŝ, W0
)
<latexit sha1_base64="8PArUU3gijpamSXDc22p2p6gwf0=">AAAB+HicbVBNS8NAEJ3Ur1o/GvXoZbEI9VISUexFKHjx4KFS+wFtKJvtpl262YTdjVBDf4kXD4p49ad489+4bXPQ1gcDj/dmmJnnx5wp7TjfVm5tfWNzK79d2Nnd2y/aB4ctFSWS0CaJeCQ7PlaUM0GbmmlOO7GkOPQ5bfvjm5nffqRSsUg86ElMvRAPBQsYwdpIfbvYG2GdNqboGg3Ld2eNvl1yKs4caJW4GSlBhnrf/uoNIpKEVGjCsVJd14m1l2KpGeF0WuglisaYjPGQdg0VOKTKS+eHT9GpUQYoiKQpodFc/T2R4lCpSeibzhDrkVr2ZuJ/XjfRQdVLmYgTTQVZLAoSjnSEZimgAZOUaD4xBBPJzK2IjLDERJusCiYEd/nlVdI6r7iXFef+olSrZnHk4RhOoAwuXEENbqEOTSCQwDO8wpv1ZL1Y79bHojVnZTNH8AfW5w/4L5Hz</latexit>
Ŝ = g(L)S
<latexit sha1_base64="fMsTCSc6iNqmBR5A9AKX8HFwNOs=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBU9kVxeKp4EGPFe0HdJeSTbNtaDZZk2yhLP0dXjwo4tUf481/Y9ruQasPBh7vzTAzL0w408Z1v5zCyura+kZxs7S1vbO7V94/aGmZKkKbRHKpOiHWlDNBm4YZTjuJojgOOW2Ho+uZ3x5TpZkUD2aS0CDGA8EiRrCxUuAnml2hG+Qbie575YpbdedAf4mXkwrkaPTKn35fkjSmwhCOte56bmKCDCvDCKfTkp9qmmAywgPatVTgmOogmx89RSdW6aNIKlvCoLn6cyLDsdaTOLSdMTZDvezNxP+8bmqiWpAxkaSGCrJYFKUc2RdnCaA+U5QYPrEEE8XsrYgMscLE2JxKNgRv+eW/pHVW9S6q7t15pV7L4yjCERzDKXhwCXW4hQY0gcAjPMELvDpj59l5c94XrQUnnzmEX3A+vgE9VJER</latexit>
: G ! S
<latexit sha1_base64="0xGFLkHF8pbRJjg2mpWGmf2Zm8M=">AAAB8XicbVBNSwMxEJ31s9avqkcvwSJUkLIrir0IBQ96rGA/aLuUbJptQ7PJkmSFsvRfePGgiFf/jTf/jWm7B219MPB4b4aZeUHMmTau++2srK6tb2zmtvLbO7t7+4WDw4aWiSK0TiSXqhVgTTkTtG6Y4bQVK4qjgNNmMLqd+s0nqjST4tGMY+pHeCBYyAg2VmrfoRtUap+j5lmvUHTL7gxomXgZKUKGWq/w1e1LkkRUGMKx1h3PjY2fYmUY4XSS7yaaxpiM8IB2LBU4otpPZxdP0KlV+iiUypYwaKb+nkhxpPU4CmxnhM1QL3pT8T+vk5iw4qdMxImhgswXhQlHRqLp+6jPFCWGjy3BRDF7KyJDrDAxNqS8DcFbfHmZNC7K3lXZfbgsVitZHDk4hhMogQfXUIV7qEEdCAh4hld4c7Tz4rw7H/PWFSebOYI/cD5/AKeujuo=</latexit>
G = (Z, W)

22. Experimental validation
• Embeddings in AWA
1
• IPL regularization generates more compact embedding
s
• Error decrease of 9.8% and 44% for AWA1 and CUB dataset
13
2D t-SNE, no IPL 2D t-SNE, with IPL
Deutch et al., “Zero shot learning with the isoperimetric loss”, AAAI, 2020

23. A.5. GSP for multi-task learning
• Multi-task learning: Learn simultaneously several related tasks
- Helps improve generalisation performance
• Often data are collected in a distributed fashio
n
- Each node can communicate only with local neighbors to solve a task
14
a) single task learning b) multi-task learning
Nassif et al., “Multi-task learning over graphs”, IEEE Signal Process. Mag., 2020

24. Spectral regularization for multi-task learning
• The graph captures the correlation between the tasks, i.e., nodes of
the graph
• The goal of each node is to compute the parameters that minimise
an objective function
• The relationship between the tasks can be exploited by imposing a
regularization of the cost function on the task graph
• Regularization examples
:
- Smoothness of tasks in the graph
- Graph spectral regularization
15
<latexit sha1_base64="r6f93I8FXWPyNIcB/xxSF92PiKU=">AAACCHicbZDLSgMxFIYzXmu9jbp0YbAIdVNmRFEXQsGNuKpgL9AZh0yatmGSzJBklDJ06cZXceNCEbc+gjvfxrSdhbb+EPj4zzmcnD9MGFXacb6tufmFxaXlwkpxdW19Y9Pe2m6oOJWY1HHMYtkKkSKMClLXVDPSSiRBPGSkGUaXo3rznkhFY3GrBwnxOeoJ2qUYaWMF9t5DEN058AJ6SPY8TkWQGWcIr4OobOAwsEtOxRkLzoKbQwnkqgX2l9eJccqJ0Jghpdquk2g/Q1JTzMiw6KWKJAhHqEfaBgXiRPnZ+JAhPDBOB3ZjaZ7QcOz+nsgQV2rAQ9PJke6r6drI/K/WTnX3zM+oSFJNBJ4s6qYM6hiOUoEdKgnWbGAAYUnNXyHuI4mwNtkVTQju9Mmz0DiquCcV5+a4VD3P4yiAXbAPysAFp6AKrkAN1AEGj+AZvII368l6sd6tj0nrnJXP7IA/sj5/AB3BmLw=</latexit>
w0
k = arg min
wk
Jk(wk)
<latexit sha1_base64="SYHQnvASBWBTJ7Joczb8IMqE06k=">AAAB6HicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0cIiYGOZgPmA5Ah7m7lkzd7esbsnhCO/wMZCEVt/kp3/xk1yhSY+GHi8N8PMvCARXBvX/XYKa+sbm1vF7dLO7t7+QfnwqKXjVDFssljEqhNQjYJLbBpuBHYShTQKBLaD8d3Mbz+h0jyWD2aSoB/RoeQhZ9RYqTHulytu1Z2DrBIvJxXIUe+Xv3qDmKURSsME1brruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfzQ+dkjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCGz/jMkkNSrZYFKaCmJjMviYDrpAZMbGEMsXtrYSNqKLM2GxKNgRv+eVV0rqoeldVt3FZqd3mcRThBE7hHDy4hhrcQx2awADhGV7hzXl0Xpx352PRWnDymWP4A+fzB9IJjO0=</latexit>
k
<latexit sha1_base64="p9SpMXVuM3Kf+Nud1SVYSc4hny4=">AAACNnicbVDLSiNBFK32NU58RV3OpjAIiUroFgd1IQizGBEUFWML6aSprlTHoquqm6pqJRT9VW7mO2bnxoUibv0Eq2MWvg5cOJxzL/feE2WMKu26d87Y+MTk1I/pn5WZ2bn5heri0oVKc4lJC6cslZcRUoRRQVqaakYuM0kQjxjxo+RP6fvXRCqainM9yEiHo76gMcVIWymsHvndNbgHAyT7AaciNH4BD7umz9KoqPuN0lI5D02y5xXd48Mwqd+ESQOuB7FE2AREo8JsFmd1fwP+bVTCas1tukPAr8QbkRoY4SSs/g96Kc45ERozpFTbczPdMUhqihkpKkGuSIZwgvqkbalAnKiOGb5dwFWr9GCcSltCw6H6fsIgrtSAR7aTI32lPnul+J3XznW80zFUZLkmAr8tinMGdQrLDGGPSoI1G1iCsKT2VoivkA1E26TLELzPL38lF5tN73fTPd2q7e+O4pgGv8AKqAMPbIN9cABOQAtgcAvuwAN4dP45986T8/zWOuaMZpbBBzgvr1VKqTY=</latexit>
W⇤
= arg min
W
Jglob
(W) =
N
X
k=1
Jk(wk) +
⌘
2
R(W, G)
<latexit sha1_base64="ud2SGMnpgd1lLqFufKkIH9ZDiH4=">AAAB+nicbVBNSwJRFL1jX2ZfYy3bPJLAIGQmitwEQotatLBQR9BJ3jyf+vDNB++9KWTyp7RpUUTbfkm7/k1PnUVpBy4czrmXe+/xIs6ksqxvI7O0vLK6ll3PbWxube+Y+d2GDGNBaJ2EPBRND0vKWUDriilOm5Gg2Pc4dbzh5cR3HqiQLAxqahRR18f9gPUYwUpLHTN/V3SO0dURukDOfQ3dIKdjFqySNQVaJHZKCpCi2jG/2t2QxD4NFOFYypZtRcpNsFCMcDrOtWNJI0yGuE9bmgbYp9JNpqeP0aFWuqgXCl2BQlP190SCfSlHvqc7fawGct6biP95rVj1ym7CgihWNCCzRb2YIxWiSQ6oywQlio80wUQwfSsiAywwUTqtnA7Bnn95kTROSvZZybo9LVTKaRxZ2IcDKIIN51CBa6hCHQg8wjO8wpvxZLwY78bHrDVjpDN78AfG5w8V+JFF</latexit>
R(W, G) = WT
LW
<latexit sha1_base64="d7vS7+Ai8B3yKZQg0Dknb5AgP/I=">AAAB/XicbVDLSgNBEOz1GeNrfdy8DAYhAQm7opiLEPCgBw9RkmwgWcPsZDYZMvtgZlaIIfgrXjwo4tX/8ObfOEn2oIkFDUVVN91dXsyZVJb1bSwsLi2vrGbWsusbm1vb5s5uXUaJILRGIh6Jhocl5SykNcUUp41YUBx4nDpe/3LsOw9USBaFVTWIqRvgbsh8RrDSUtvcv8s7x+iqgC6Qc19F3fxNATltM2cVrQnQPLFTkoMUlbb51epEJAloqAjHUjZtK1buEAvFCKejbCuRNMakj7u0qWmIAyrd4eT6ETrSSgf5kdAVKjRRf08McSDlIPB0Z4BVT856Y/E/r5kov+QOWRgnioZkushPOFIRGkeBOkxQovhAE0wE07ci0sMCE6UDy+oQ7NmX50n9pGifFa3b01y5lMaRgQM4hDzYcA5luIYK1IDAIzzDK7wZT8aL8W58TFsXjHRmD/7A+PwBqTCSGw==</latexit>
R(W, G) = WT
g(L)W

25. Summary so far
• Graphs can be used to capture hidden dependencies of any type of
data
• The domain of the graph is application speci
fi
c
- Latent space of feature
s
- Manifold approximation
- Distributed agents
• GSP tools, and in particular, graph regularization help towards
imposing desirable propertie
s
- Better node embedding
s
- Cleaner signals
- More robust models
16

26. GSP for improving ef
fi
ciency and robustness
A. Improvement on data ef
fi
ciency and robustness
C. Improvement on computational ef
fi
ciency
17
B. Robustness against topological noise

27. GSP for improving ef
fi
ciency and robustness
A. Improvement on data ef

28. B. Robustness against topological noise
• In the context of graph-structured data, stability can be de
fi
ned with
respect to perturbation to the underlying topology
18

29. B. Robustness against topological noise
• In the context of graph-structured data, stability can be de
fi
ned with
respect to perturbation to the underlying topology
18
Why is it important
?
• noisy/unreliable
graph
• adversarie
s
• transferabilit
y
• changes through
time

30. 19
B.1. Stability of graph
fi
lters
Levie et al., “On the transferability of spectral graph
fi
lters,” SampTA, 2019.

31. Stability of graph
fi
lters
20
Kenlay et al., “On the stability of polynomial spectral graph
fi
lters,” ICASSP, 2020.

32. B.2. Stability of graph neural networks
21
Gama et al., “Stability properties of graph neural networks,” IEEE TSP, 2020
.
Ruiz et al., “Graph neural networks: Architectures, stability, and transferability,” Proceedings of the IEEE, 2021.

33. Stability of graph neural networks
22
Gama et al., “Stability properties of graph neural networks,” IEEE TSP, 2020
.
Ruiz et al., “Graph neural networks: Architectures, stability, and transferability,” Proceedings of the IEEE, 2021.

34. Stability of graph neural networks
• Perturbations that do not modify the degree distribution of the graph
23
Kenlay et al., “On the stability of graph convolutional neural networks under edge rewiring,” ICASSP, 2021.
• stability under double-edge rewiring

35. B.3. Interpretable stability bounds
24
Kenlay et al., “Interpretable stability bounds for spectral graph
fi
lters,” ICML, 2021.
PGD
Edge addition
Edge deletion
Robust PGD
Edge addition
Edge deletion
Robust
• how does stability depend on topological properties of
perturbation?
BA graph 3-regular graph

36. B.3. Interpretable stability bounds
24
Kenlay et al., “Interpretable stability bounds for spectral graph
fi
lters,” ICML, 2021.
PGD
Edge addition
Edge deletion
Robust PGD
Edge addition
Edge deletion
Robust
• how does stability depend on topological properties of
perturbation?
• spectral graph
fi
lters are most stable i
f
- adding or deleting edges between high degree node
s
- not perturbing too much around any one node
BA graph 3-regular graph

37. B.4. Robustness beyond stability
• GSP in the presence of topological uncertainty (more in part III) [2, 4
]
• Filters on stochastic time-evolving graphs [1
]
• Stochastic graph
fi
lters built on sequence of randomly perturbed
graphs [3]
25
[1] Isu
fi
et al., “Filtering random graph processes over random time-varying graphs,” IEEE TSP, 2017
.
[2] Ceci and Barbarossa, “Graph signal processing in the presence of topology uncertainties,” IEEE TSP, 2020
.
[3] Gao et al., “Stochastic graph neural networks,” ICASSP, 2020
.
[4] Miettinen et al., “Modelling graph errors: Towards robust graph signal processing,” arXiv, 2020.

38. GSP for improving ef
fi
ciency and robustness
C. Improvement on computational ef
fi
ciency
A. Improvement on data ef
fi
ciency and robustness
B. Robustness against topological noise
26

39. GSP for improving ef
fi
ciency and robustness
C. Improvement on computational ef
fi
ciency
A. Improvement on data ef

40. - Eigendecomposition of a large matrix is expensiv
e
- Training and deploying GNNs remains challenging due to high memory
consumption and inference latency
27
https://en.wikipedia.org/wiki/Social_network_analysis

41. C. Improvement on computational complexity
• Many ML algorithms require a large amount of resources (i.e., space,
runtime, communication) for their computation
- Eigendecomposition of a large matrix is expensiv
e
- Training and deploying GNNs remains challenging due to high memory
consumption and inference latency
27
Can we use the graph structure and classical GSP operators to
alleviate some of these issues?
https://en.wikipedia.org/wiki/Social_network_analysis

42. C.1. Complexity of spectral clustering
• Spectral clustering : Given a graph capturing the structure of
data points,
fi
nd the partition of the nodes into cluster
s
• Algorithm: Given the graph Laplacian matrix
- Compute the
fi
rst eigenvector
s
- Compute the embedding of each node in
:
- Run K-Means with Euclidean distance on the embeddings
to compute the cluster
s
• Bottlenecks: When are larg
e
- Computing the eigendecomposition is expensiv
e
- The complexity of K-means is high
28
<latexit sha1_base64="lG8/m/tuw4KYjUZGrvnE+q/C4y4=">AAAB6HicbVA9SwNBEJ3zM8avqKXNYhCswp0oWlgELLRMwHxAcoS9zVyyZm/v2N0TwpFfYGOhiK0/yc5/4ya5QhMfDDzem2FmXpAIro3rfjsrq2vrG5uFreL2zu7efungsKnjVDFssFjEqh1QjYJLbBhuBLYThTQKBLaC0e3Ubz2h0jyWD2acoB/RgeQhZ9RYqX7XK5XdijsDWSZeTsqQo9YrfXX7MUsjlIYJqnXHcxPjZ1QZzgROit1UY0LZiA6wY6mkEWo/mx06IadW6ZMwVrakITP190RGI63HUWA7I2qGetGbiv95ndSE137GZZIalGy+KEwFMTGZfk36XCEzYmwJZYrbWwkbUkWZsdkUbQje4svLpHle8S4rbv2iXL3J4yjAMZzAGXhwBVW4hxo0gAHCM7zCm/PovDjvzse8dcXJZ47gD5zPH5t5jMk=</latexit>
G
<latexit sha1_base64="9G4Yhmc71Jj9i88hAaoQMRNdfH0=">AAAB6HicbVA9SwNBEJ3zM8avqKXNYhCswp0oWlgEbKwkAfMByRH2NnPJmr29Y3dPCEd+gY2FIrb+JDv/jZvkCk18MPB4b4aZeUEiuDau++2srK6tb2wWtorbO7t7+6WDw6aOU8WwwWIRq3ZANQousWG4EdhOFNIoENgKRrdTv/WESvNYPphxgn5EB5KHnFFjpfp9r1R2K+4MZJl4OSlDjlqv9NXtxyyNUBomqNYdz02Mn1FlOBM4KXZTjQllIzrAjqWSRqj9bHbohJxapU/CWNmShszU3xMZjbQeR4HtjKgZ6kVvKv7ndVITXvsZl0lqULL5ojAVxMRk+jXpc4XMiLEllClubyVsSBVlxmZTtCF4iy8vk+Z5xbusuPWLcvUmj6MAx3ACZ+DBFVThDmrQAAYIz/AKb86j8+K8Ox/z1hUnnzmCP3A+fwCmFYzQ</latexit>
N
<latexit sha1_base64="YMl63/TQai6qmJHmFj6L7Gqyfns=">AAAB6HicbVA9SwNBEJ3zM8avqKXNYhCswp0oWlgEbASbBMwHJEfY28wla/b2jt09IRz5BTYWitj6k+z8N26SKzTxwcDjvRlm5gWJ4Nq47rezsrq2vrFZ2Cpu7+zu7ZcODps6ThXDBotFrNoB1Si4xIbhRmA7UUijQGArGN1O/dYTKs1j+WDGCfoRHUgeckaNler3vVLZrbgzkGXi5aQMOWq90le3H7M0QmmYoFp3PDcxfkaV4UzgpNhNNSaUjegAO5ZKGqH2s9mhE3JqlT4JY2VLGjJTf09kNNJ6HAW2M6JmqBe9qfif10lNeO1nXCapQcnmi8JUEBOT6dekzxUyI8aWUKa4vZWwIVWUGZtN0YbgLb68TJrnFe+y4tYvytWbPI4CHMMJnIEHV1CFO6hBAxggPMMrvDmPzovz7nzMW1ecfOYI/sD5/AGhiYzN</latexit>
K
Data points Graph structure Detected clusters
<latexit sha1_base64="4WCNov6V0zGqZSoHlnZclCgpAcI=">AAAB6HicbVA9SwNBEJ3zM8avqKXNYhCswp0oWlgEbCwsEjAfkBxhbzOXrNnbO3b3hHDkF9hYKGLrT7Lz37hJrtDEBwOP92aYmRckgmvjut/Oyura+sZmYau4vbO7t186OGzqOFUMGywWsWoHVKPgEhuGG4HtRCGNAoGtYHQ79VtPqDSP5YMZJ+hHdCB5yBk1Vqrf90plt+LOQJaJl5My5Kj1Sl/dfszSCKVhgmrd8dzE+BlVhjOBk2I31ZhQNqID7FgqaYTaz2aHTsipVfokjJUtachM/T2R0UjrcRTYzoiaoV70puJ/Xic14bWfcZmkBiWbLwpTQUxMpl+TPlfIjBhbQpni9lbChlRRZmw2RRuCt/jyMmmeV7zLilu/KFdv8jgKcAwncAYeXEEV7qAGDWCA8Ayv8OY8Oi/Ou/Mxb11x8pkj+APn8wejDYzO</latexit>
L
<latexit sha1_base64="aVetB5EXBB9JZ00ye6Q1GtusioY=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Koko9ljwInhpwX5AG8pmO2nXbjZhdyOU0F/gxYMiXv1J3vw3btsctPXBwOO9GWbmBYng2rjut7O2vrG5tV3YKe7u7R8clo6OWzpOFcMmi0WsOgHVKLjEpuFGYCdRSKNAYDsY38789hMqzWP5YCYJ+hEdSh5yRo2VGvf9UtmtuHOQVeLlpAw56v3SV28QszRCaZigWnc9NzF+RpXhTOC02Es1JpSN6RC7lkoaofaz+aFTcm6VAQljZUsaMld/T2Q00noSBbYzomakl72Z+J/XTU1Y9TMuk9SgZItFYSqIicnsazLgCpkRE0soU9zeStiIKsqMzaZoQ/CWX14lrcuKd11xG1flWjWPowCncAYX4MEN1OAO6tAEBgjP8ApvzqPz4rw7H4vWNSefOYE/cD5/AKBVjMk=</latexit>
K
<latexit sha1_base64="npjqqTLsIfJNa+vNNy/pUjoUBC0=">AAACA3icbZDLSsNAFIYnXtt6i7oR3QwWwUUJSUHsRii6EbqpYNNCG8JkOmmHTi7MTIQaCt34Km5cKMWtL+FO8GGctF1o6w8DH/85hzPn92JGhTTNL21ldW19YzOXL2xt7+zu6fsHtogSjkkDRyziLQ8JwmhIGpJKRloxJyjwGGl6g5us3nwgXNAovJfDmDgB6oXUpxhJZbn6se3W4BVs265VgrZbLkHDMDKqOa5eNA1zKrgM1hyK1aPH7/x4cl139c9ON8JJQEKJGRKibZmxdFLEJcWMjAqdRJAY4QHqkbbCEAVEOOn0hhE8U04X+hFXL5Rw6v6eSFEgxDDwVGeAZF8s1jLzv1o7kX7FSWkYJ5KEeLbITxiUEcwCgV3KCZZsqABhTtVfIe4jjrBUsRVUCNbiyctglw3rwjDvrGK1AmbKgRNwCs6BBS5BFdyCOmgADMbgGbyCN+1Je9Em2vusdUWbzxyCP9I+fgCaLJdB</latexit>
VK = [V1, V2, ..., VK]
<latexit sha1_base64="rQIWe/ZeKLEVmRNgWjZ+GiHG+l4=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7gpjOgI1lAuYCyRJmJ2eTMbOzy8ysEJaUVjYWitj6FKl8CDufwZdwcik08YeBj/8/hznn+DFnSjvOl5VZWV1b38hu5ra2d3b38vsHdRUlkmKNRjySTZ8o5ExgTTPNsRlLJKHPseEPrid54x6lYpG41cMYvZD0BAsYJdpYVdHJF5yiM5W9DO4cClcf4+r3w/G40sl/trsRTUIUmnKiVMt1Yu2lRGpGOY5y7URhTOiA9LBlUJAQlZdOBx3Zp8bp2kEkzRPanrq/O1ISKjUMfVMZEt1Xi9nE/C9rJTooeSkTcaJR0NlHQcJtHdmTre0uk0g1HxogVDIzq037RBKqzW1y5gju4srLUD8vuhdFp+oWyiWYKQtHcAJn4MIllOEGKlADCgiP8Awv1p31ZL1ab7PSjDXvOYQ/st5/AJz5kTU=</latexit>
n
<latexit sha1_base64="CgJ6ISMxcuZ4V82taY/EjRVO6ZA=">AAAB6nicbZDLSsNAFIZP6q1Gq1WXbgZLwVVJBLHLgiCCm4r2Am0ok+mkHTqZhLkIJfQR3LhQxK34ID6CO9/G6WWhrT8MfPz/Ocw5J0w5U9rzvp3c2vrG5lZ+293ZLeztFw8OmyoxktAGSXgi2yFWlDNBG5ppTtuppDgOOW2Fo8tp3nqgUrFE3OtxSoMYDwSLGMHaWnfN3k2vWPIq3kxoFfwFlGqFT1O+cj/qveJXt58QE1OhCcdKdXwv1UGGpWaE04nbNYqmmIzwgHYsChxTFWSzUSeobJ0+ihJpn9Bo5v7uyHCs1DgObWWM9VAtZ1Pzv6xjdFQNMiZSo6kg848iw5FO0HRv1GeSEs3HFjCRzM6KyBBLTLS9jmuP4C+vvArNs4p/XvFu/VKtCnPl4RhO4BR8uIAaXEMdGkBgAI/wDC8Od56cV+dtXppzFj1H8EfO+w+DVZAu</latexit>
VK
<latexit sha1_base64="t3o3FSibv3vdoC4tdqDAt3NQYR8=">AAACAHicbVDLSsNAFJ3UV62vqqCgm8EquCqJIHYjFNwIbio0baGJYTKZtEMnkzAzEUroxu9w58aFIm7Fr3DnHwj+hNPHQlsPXDiccy/33uMnjEplmp9Gbm5+YXEpv1xYWV1b3yhubjVknApMbByzWLR8JAmjnNiKKkZaiSAo8hlp+r2Lod+8JULSmNdVPyFuhDqchhQjpSWvuOskXepxeA5t7+qmDp2AMIU87ZTMsjkCnCXWhJSqe99f7zv3hzWv+OEEMU4jwhVmSMq2ZSbKzZBQFDMyKDipJAnCPdQhbU05ioh0s9EDA3iklQCGsdDFFRypvycyFEnZj3zdGSHVldPeUPzPa6cqrLgZ5UmqCMfjRWHKoIrhMA0YUEGwYn1NEBZU3wpxFwmElc6soEOwpl+eJY2TsnVaNq+tUrUCxsiDfXAAjoEFzkAVXIIasAEGA/AAnsCzcWc8Gi/G67g1Z0xmtsEfGG8/NR+ZMg==</latexit>
n = UT
K n
von Luxburg, “A tutorial on spectral clustering”, Statistics and Computing, 2007
<latexit sha1_base64="yQD8RvOSbjQEP7ARDyGpNHI6280=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4kJKIogcPBS+CIBVMW2hD2Ww37dLdTdjdCCX0N3jxoIhXf5A3/42bNgdtfTDweG+GmXlhwpk2rvvtLC2vrK6tlzbKm1vbO7uVvf2mjlNFqE9iHqt2iDXlTFLfMMNpO1EUi5DTVji6yf3WE1WaxfLRjBMaCDyQLGIEGyv596fortyrVN2aOwVaJF5BqlCg0at8dfsxSQWVhnCsdcdzExNkWBlGOJ2Uu6mmCSYjPKAdSyUWVAfZ9NgJOrZKH0WxsiUNmqq/JzIstB6L0HYKbIZ63svF/7xOaqKrIGMySQ2VZLYoSjkyMco/R32mKDF8bAkmitlbERlihYmx+eQhePMvL5LmWc27qLkP59X6dRFHCQ7hCE7Ag0uowy00wAcCDJ7hFd4c6bw4787HrHXJKWYO4A+czx8wGo2Z</latexit>
N, K <latexit sha1_base64="m1Kucc5AEjVEMHBGxL3IlDwNbe8=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBahXkpSFD14KHgRBK1gP6CNZbPdtEs3m7i7EUron/DiQRGv/h1v/hs3bQ7a+mDg8d4MM/O8iDOlbfvbyi0tr6yu5dcLG5tb2zvF3b2mCmNJaIOEPJRtDyvKmaANzTSn7UhSHHictrzRZeq3nqhULBT3ehxRN8ADwXxGsDZS+7Z8c/1QPS70iiW7Yk+BFomTkRJkqPeKX91+SOKACk04Vqrj2JF2Eyw1I5xOCt1Y0QiTER7QjqECB1S5yfTeCToySh/5oTQlNJqqvycSHCg1DjzTGWA9VPNeKv7ndWLtn7sJE1GsqSCzRX7MkQ5R+jzqM0mJ5mNDMJHM3IrIEEtMtIkoDcGZf3mRNKsV57Ri352UahdZHHk4gEMogwNnUIMrqEMDCHB4hld4sx6tF+vd+pi15qxsZh/+wPr8Af2ujps=</latexit>
O(NK2
)
<latexit sha1_base64="m1Kucc5AEjVEMHBGxL3IlDwNbe8=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBahXkpSFD14KHgRBK1gP6CNZbPdtEs3m7i7EUron/DiQRGv/h1v/hs3bQ7a+mDg8d4MM/O8iDOlbfvbyi0tr6yu5dcLG5tb2zvF3b2mCmNJaIOEPJRtDyvKmaANzTSn7UhSHHictrzRZeq3nqhULBT3ehxRN8ADwXxGsDZS+7Z8c/1QPS70iiW7Yk+BFomTkRJkqPeKX91+SOKACk04Vqrj2JF2Eyw1I5xOCt1Y0QiTER7QjqECB1S5yfTeCToySh/5oTQlNJqqvycSHCg1DjzTGWA9VPNeKv7ndWLtn7sJE1GsqSCzRX7MkQ5R+jzqM0mJ5mNDMJHM3IrIEEtMtIkoDcGZf3mRNKsV57Ri352UahdZHHk4gEMogwNnUIMrqEMDCHB4hld4sx6tF+vd+pi15qxsZh/+wPr8Af2ujps=</latexit>
O(NK2
)
<latexit sha1_base64="p5bpnj8sJcLrP380zrfcvO5JfnM=">AAACCHicbZDLSgMxFIYz9VbrbdSlC4NFcKFlpih2IxR04bKCvUBnGDJp2oYmmSHJCGXapRtfxY0LRdz6CO58G9NpF9r6Q+DjP+dwcv4wZlRpx/m2ckvLK6tr+fXCxubW9o69u9dQUSIxqeOIRbIVIkUYFaSuqWakFUuCeMhIMxxcT+rNByIVjcS9HsbE56gnaJdipI0V2Ic3QSpO+RheQW/kxX0aCHgGM+DeKCgHdtEpOZngIrgzKIKZaoH95XUinHAiNGZIqbbrxNpPkdQUMzIueIkiMcID1CNtgwJxovw0O2QMj43Tgd1Imic0zNzfEyniSg15aDo50n01X5uY/9Xaie5W/JSKONFE4OmibsKgjuAkFdihkmDNhgYQltT8FeI+kghrk13BhODOn7wIjXLJvSg5d+fFamUWRx4cgCNwAlxwCargFtRAHWDwCJ7BK3iznqwX6936mLbmrNnMPvgj6/MHt/GYew==</latexit>
Dn,m = k n mk2

43. Compressive spectral clustering
• A GSP perspectiv
e
- Computation of eigenvectors is bypassed by
fi
ltering random graph signals
- K-means is performed on a subset of randomly selected node
s
- The cluster centers of the whole graph are interpolate
d
• Algorithm:
- Estimate fro
m
- Generate random graph signals in matri
x
- Filter them with a polynomial
fi
lter , i.e.,
- If
- Draw randomly sample nodes, and run K-means to obtain cluster
center
s
- Interpolate the centers by exploiting the sampling theory of bandlimited graph
signals
29
<latexit sha1_base64="oZ9+8Oq+F9d+NMa1aTxqVSN5Wdo=">AAAB+nicbVDLSgMxFM3UV62vqV26CZaCqzIjiF0W3AjdVLAP6Awlk2ba0GQyJBm1jN35G25cKOLWb/AD3OkH+AV+gOljoa0HLhzOuZd77wliRpV2nA8rs7K6tr6R3cxtbe/s7tn5/aYSicSkgQUTsh0gRRiNSENTzUg7lgTxgJFWMDyb+K0rIhUV0aUexcTnqB/RkGKkjdS18xx6KI6luIE1j4k+rHXtolN2poDLxJ2TYrVQuvt++/qsd+13rydwwkmkMUNKdVwn1n6KpKaYkXHOSxSJER6iPukYGiFOlJ9OTx/DklF6MBTSVKThVP09kSKu1IgHppMjPVCL3kT8z+skOqz4KY3iRJMIzxaFCYNawEkOsEclwZqNDEFYUnMrxAMkEdYmrZwJwV18eZk0j8vuSdm5cIvVCpghCw7AITgCLjgFVXAO6qABMLgG9+ARPFm31oP1bL3MWjPWfKYA/sB6/QEs5pe5</latexit>
m ⇡ K log K
<latexit sha1_base64="BeWjLlkDkSsPlDgjnzrFa7oBMek=">AAAB8HicbVDLSgMxFL3js45Wqy7dBEvBVZkRxC4LgrisYB/SDiWTybShSWZIMkIZ+hVuXCjixoWf4Se4829MHwttPXDhcM653EeYcqaN5307a+sbm1vbhR13d6+4f1A6PGrpJFOENknCE9UJsaacSdo0zHDaSRXFIuS0HY6upn77gSrNEnlnxikNBB5IFjOCjZXue9xGI9wf9Utlr+rNgFaJvyDlevEzq1y7741+6asXJSQTVBrCsdZd30tNkGNlGOF04vYyTVNMRnhAu5ZKLKgO8tnCE1SxSoTiRNmSBs3U3x05FlqPRWiTApuhXvam4n9eNzNxLciZTDNDJZkPijOOTIKm16OIKUoMH1uCiWJ2V0SGWGFi7I9c+wR/+eRV0jqv+hdV79Yv12swRwFO4BTOwIdLqMMNNKAJBAQ8wjO8OMp5cl6dt3l0zVn0HMMfOB8/Qs+S8Q==</latexit>
k
<latexit sha1_base64="vOjr189k79o/loNSrTdd5UoYf58=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7gpjOgI2FRQLmAskSZidnkzGzs8vMrBCWlFY2ForY+hSpfAg7n8GXcHIpNPrDwMf/n8Occ/yYM6Ud59PKLC2vrK5l13Mbm1vbO/ndvbqKEkmxRiMeyaZPFHImsKaZ5tiMJZLQ59jwB5eTvHGHUrFI3OhhjF5IeoIFjBJtrOp1J19wis5U9l9w51C4eB9Xv+4Px5VO/qPdjWgSotCUE6VarhNrLyVSM8pxlGsnCmNCB6SHLYOChKi8dDroyD42TtcOImme0PbU/dmRklCpYeibypDovlrMJuZ/WSvRQclLmYgTjYLOPgoSbuvInmxtd5lEqvnQAKGSmVlt2ieSUG1ukzNHcBdX/gv106J7VnSqbqFcgpmycABHcAIunEMZrqACNaCA8ABP8GzdWo/Wi/U6K81Y8559+CXr7RtpcZET</latexit>
L
<latexit sha1_base64="6wRg1FGqPupP4V/JYIbq3pYUego=">AAAB6HicbZC7SgNBFIbPxluMt6ilIItBsAq7gpjOgI1lAuYCSQizs2eTMbOzy8ysEJaUVjYWitj6FKl8CDufwZdwcik08YeBj/8/hznneDFnSjvOl5VZWV1b38hu5ra2d3b38vsHdRUlkmKNRjySTY8o5ExgTTPNsRlLJKHHseENrid54x6lYpG41cMYOyHpCRYwSrSxqn43X3CKzlT2MrhzKFx9jKvfD8fjSjf/2fYjmoQoNOVEqZbrxLqTEqkZ5TjKtROFMaED0sOWQUFCVJ10OujIPjWObweRNE9oe+r+7khJqNQw9ExlSHRfLWYT87+sleig1EmZiBONgs4+ChJu68iebG37TCLVfGiAUMnMrDbtE0moNrfJmSO4iysvQ/286F4UnapbKJdgpiwcwQmcgQuXUIYbqEANKCA8wjO8WHfWk/Vqvc1KM9a85xD+yHr/AY3RkSs=</latexit>
d
<latexit sha1_base64="FEFMRGFUIPT2A1P98ANRLhut344=">AAACBXicbVA9SwNBEN2LXzF+nVpqsRgEq3AniAELAzZWEoP5gFwMe5u9ZMne3rG7J4TjQGz8KzYWitha2tv5H0Rs7N1LUmjig4HHezPMzHNDRqWyrHcjMzM7N7+QXcwtLa+srpnrGzUZRAKTKg5YIBoukoRRTqqKKkYaoSDIdxmpu/2T1K9fESFpwC/UICQtH3U59ShGSkttc7sCHcqh4yPVc724klzGZ46iPpGwk7TNvFWwhoDTxB6T/PHX5/fH9etRuW2+OZ0ARz7hCjMkZdO2QtWKkVAUM5LknEiSEOE+6pKmphzpPa14+EUCd7XSgV4gdHEFh+rviRj5Ug58V3em18pJLxX/85qR8oqtmPIwUoTj0SIvYlAFMI0EdqggWLGBJggLqm+FuIcEwkoHl9Mh2JMvT5PafsE+KFjndr5UBCNkwRbYAXvABoegBE5BGVQBBjfgDjyAR+PWuDeejOdRa8YYz2yCPzBefgBuLJ29</latexit>
R 2 RN⇥d
<latexit sha1_base64="yIDoSqdvoiJSnjW/5xIPnv8D0ww=">AAAB9HicbVDLSsNAFL3xWesr6krcDFbBVUkEscuCmy4r9AVtCJPJpB06mcSZSaGEfkc3LhRx68e482t0+lho64GBwznncu+cIOVMacf5sjY2t7Z3dgt7xf2Dw6Nj++S0pZJMEtokCU9kJ8CKciZoUzPNaSeVFMcBp+1g+DDz2yMqFUtEQ49T6sW4L1jECNZG8mp+3uMmHWJ/OPHtklN25kDrxF2SUvXc/p5e6Ubdtz97YUKymApNOFaq6zqp9nIsNSOcToq9TNEUkyHu066hAsdUefn86Am6NkqIokSaJzSaq78nchwrNY4Dk4yxHqhVbyb+53UzHVW8nIk001SQxaIo40gnaNYACpmkRPOxIZhIZm5FZIAlJtr0VDQluKtfXiet27J7V3Ye3VK1AgsU4AIu4QZcuIcq1KAOTSDwBFN4gVdrZD1bb9b7IrphLWfO4A+sjx/BTJUE</latexit>
H k
<latexit sha1_base64="M7UJxrXa9rQ0ozD8JfmaZtBikVY=">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</latexit>
d ⇠ log N : D̃n,m = k ˜
n
˜
mk2 ⇡ Dn,m
<latexit sha1_base64="uo8ko41WZwk0KTDgGyOy/wdcmIA=">AAACFHicbVBNS8NAEN34bf2KehIvi1VQhJIIoheh4MWjSqtCU8NmM7VLN5uwOxFK6I/woj/FiwdFvHrw5q/RbevBrwcDj/dmdmdelElh0PPenZHRsfGJyanp0szs3PyCu7h0ZtJcc6jzVKb6ImIGpFBQR4ESLjINLIkknEedw75/fg3aiFTVsJtBM2FXSrQEZ2il0N0OUMgYiiBri1D16AHdPAqLQNoXYhZ2eqdbl7UgBoksFKFb9ireAPQv8b9IubriftytY+04dN+COOV5Agq5ZMY0fC/DZsE0Ci6hVwpyAxnjHXYFDUsVS8A0i8FRPbphlZi2Um1LIR2o3ycKlhjTTSLbmTBsm99eX/zPa+TY2m8WQmU5guLDj1q5pJjSfkI0Fho4yq4ljGthd6W8zTTjaHMs2RD83yf/JWc7FX+34p345eo+GWKKrJI1skl8skeq5Igckzrh5Ibck0fy5Nw6D86z8zJsHXG+ZpbJDzivnyyqoTE=</latexit>
˜
n = (H k
R)T
i
<latexit sha1_base64="oZ9+8Oq+F9d+NMa1aTxqVSN5Wdo=">AAAB+nicbVDLSgMxFM3UV62vqV26CZaCqzIjiF0W3AjdVLAP6Awlk2ba0GQyJBm1jN35G25cKOLWb/AD3OkH+AV+gOljoa0HLhzOuZd77wliRpV2nA8rs7K6tr6R3cxtbe/s7tn5/aYSicSkgQUTsh0gRRiNSENTzUg7lgTxgJFWMDyb+K0rIhUV0aUexcTnqB/RkGKkjdS18xx6KI6luIE1j4k+rHXtolN2poDLxJ2TYrVQuvt++/qsd+13rydwwkmkMUNKdVwn1n6KpKaYkXHOSxSJER6iPukYGiFOlJ9OTx/DklF6MBTSVKThVP09kSKu1IgHppMjPVCL3kT8z+skOqz4KY3iRJMIzxaFCYNawEkOsEclwZqNDEFYUnMrxAMkEdYmrZwJwV18eZk0j8vuSdm5cIvVCpghCw7AITgCLjgFVXAO6qABMLgG9+ARPFm31oP1bL3MWjPWfKYA/sB6/QEs5pe5</latexit>
m ⇡ K log K
<latexit sha1_base64="/pkZg139wkTKS2wSw3fmzSav9qo=">AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KKERBS7EQpuBDcV7APSNExuJ+3QyYOZiVBKN/6KGxeKuPUz3Pk3TtsstPXAhcM593LvPUHKmVS2/W0UVlbX1jeKm6Wt7Z3dPXP/oCmTTABtQMIT0Q6IpJzFtKGY4rSdCkqigNNWMLyZ+q1HKiRL4gc1SqkXkX7MQgZEack3j6Ar8DV2wXe6omJZVgX8u67wfLNsW/YMeJk4OSmjHHXf/Or0EsgiGivgRErXsVPljYlQDDidlDqZpCmBIelTV9OYRFR649kDE3yqlR4OE6ErVnim/p4Yk0jKURTozoiogVz0puJ/npupsOqNWZxmisYwXxRmHKsET9PAPSYoKD7ShIBg+lYMAyIIKJ1ZSYfgLL68TJrnlnNp2fcX5Vo1j6OIjtEJOkMOukI1dIvqqIEATdAzekVvxpPxYrwbH/PWgpHPHKI/MD5/AJsklHU=</latexit>
cr
= [cr
1, ..., cr
K]
<latexit sha1_base64="57JLmAcWc6oPu/WM8bWUM5IzQBA=">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</latexit>
c̃j = arg min
cj 2RN
kMcj cr
j k2
2 + cT
j g(L)cj
Trembley et al., “Compressing spectral clustering”, ICML, 2016
Trembley et al., “Approximating spectral clustering by sampling: A review”, Sampling Techniques for Supervised
or Unsupervised Tasks, Springer 2020

44. C.2. Adaptive
fi
lters and per-node weighting in GNN
30
Tailor et al., “Adaptive
fi
lters and aggregator fusion for ef
fi
cient graph convolutions,” arXiv, 2021.

45. Adaptive
fi
lters and per-node weighting in GNN
31
• Operations can be computed with the standard compressed
sparse row (CSR) algorithm (SpMM
)
• Lower memory consumption (no message materialisation)
Graph attention network Ef
fi
cient graph convolution
• Requires only memory
<latexit sha1_base64="oYgfCvOlWOLs4yzAZMee0dSG8Fo=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXsquKPZY8OJJK9gPaJeSTbNtaJJdkqxQlv4FLx4U8eof8ua/MdvuQVsfDDzem2FmXhBzpo3rfjuFtfWNza3idmlnd2//oHx41NZRoghtkYhHqhtgTTmTtGWY4bQbK4pFwGknmNxkfueJKs0i+WimMfUFHkkWMoJNJt1X784H5Ypbc+dAq8TLSQVyNAflr/4wIomg0hCOte55bmz8FCvDCKezUj/RNMZkgke0Z6nEgmo/nd86Q2dWGaIwUrakQXP190SKhdZTEdhOgc1YL3uZ+J/XS0xY91Mm48RQSRaLwoQjE6HscTRkihLDp5Zgopi9FZExVpgYG0/JhuAtv7xK2hc176rmPlxWGvU8jiKcwClUwYNraMAtNKEFBMbwDK/w5gjnxXl3PhatBSefOYY/cD5/AAe8jYo=</latexit>
O(N)

46. C.3. Scalable inception graph network
32
Frasca et al., “SIGN: Scalable inception graph neural networks,” ICML Workshop, 2020.

47. Scalable inception graph network
33
Frasca et al., “SIGN: Scalable inception graph neural networks,” ICML Workshop, 2020.

48. Take home message:
GSP for robustness and ef
fi
ciency
34
… for ML
GSP Tools …
Graph signal regularization
Graph
fi
ltering
Graph interpolation
Graph based transforms
Diffusion operators
Better embeddings
Higher classi
fi
cation accuracy in noisy settings
Less complex GNNs
Faster approximation of eigendecomposition
Stability with respect to topological noise

49. Outline
35
• Brief introduction to graph signal processing (GSP)
• Key GSP tools for machine learning
• Challenge I: GSP for exploiting data structure
• Challenge II: GSP for improving ef
fi
ciency and robustness
• Challenge III: GSP for enhancing model interpretability
• Applications
• Summary, open challenges, and new perspectives

50. Outline
35
• Brief introduction to graph signal processing (GSP)
• Key GSP tools for machine learning
• Challenge I: GSP for exploiting data structure
• Challenge II: GSP for improving ef

51. fi
c: Extract relevant knowledge from dat
a
- A.1. Signal analysis: Use GSP to reveal interpretable feature
s
- A.2. Structure inference: Use GSP to learn interpretable structures
36
B. Model speci
fi
c: Improve our understanding of ML model
s
- B.1. Understanding the expressive power of GNN
s
- B.2. A posteriori interpretation of DNN

52. GSP for enhancing interpretability
A. Domain speci
fi
c: Extract relevant knowledge from dat
a
- A.1. Signal analysis: Use GSP to reveal interpretable feature
s
- A.2. Structure inference: Use GSP to learn interpretable structures
36
B. Model speci

53. fi
c
knowledge discover
y
• In neuroscience, GSP tools have been used to improve our
understanding of the biological mechanisms underlying human
cognition and brain disorders
• Analysis in the spectral domain reveals the variation of signals on the
anatomical network
37
[Fig. from Huang’18]

54. A.1.1. GSP for understanding cognitive
fl
exibility
• Cognitive
fl
exibility describes the human ability to switch between
modes of mental functio
n
• Clarifying the nature of cognitive
fl
exibility is critical to understand the
human min
d
• GSP provides a framework for integrating brain network structure,
function, and cognitive measure
s
• It allows to decompose each BOLD signal into aligned and liberal
components
38
Medaglia et al., “Functional Alignment with Anatomical Networks is Associated with Cognitive Flexibility”, Nat.
Hum. Behav., 2018

55. BOLD signal alignment across the brain
• Functional alignment with anatomical networks facilitates cognitive
fl
exibility (lower switch costs
)
- Liberal signals are concentrated in subcortical regions and cingulate cortice
s
- Aligned signals are concentrated in subcortical, default mode, fronto-patietal,
and cingulo-opercular systems
39
BOLD signals
White matter
network
Decomposition of the signals into aligned
and liberal using GFT

56. A.1.2. Structural decoupling index
• Ratio of liberal versus aligned energy in a speci
fi
c nod
e
• Spatial organization of regions according to decoupling index reveals
behaviourally relevant gradient
40
Preti and Van De Ville., “Decoupling of brain function from structure reveals regional behavioral specialization in
humans”, Nat. Comm., 2019
Structurally coupled regions
:
Sensory regions
Structurally decoupled regions
:
Higher-level cognitive regions

57. A.1.3. Understanding ASD through GSP
• Predict Autism Spectrum Disorder (ASD) by exploiting structural and
functional informatio
n
• Discriminative patterns are extracted in the graph Fourier domai
n
• Frequency signatures are de
fi
ned as the variability over time of graph
Fourier modes
41
Itani and Thanou, “Combining anatomical and functional networks for neuropathology identi
fi
cation: A case study
on autism spectrum disorder”, Medical Image Analysis, 2021

58. Interpretable and discriminative features
• Neurotypical patients express a predominant activity in the parieto-
occipital region
s
• ASD patients express high level of activity in the pronto-temporal
areas
42
Neurotypical patients ASD patients

59. A.2. Structure inference
• Given observations on a number of variables, and some prior
knowledge (distribution, model, constraints), learn a measure of
relations between the
m
43
v1
v2
graph signal
?

60. A.2. Structure inference
• Given observations on a number of variables, and some prior
knowledge (distribution, model, constraints), learn a measure of
relations between the
m
43
v1
v2
graph signal
?
How to infer interpretable structure from data
?

61. GSP for structure inference
44
= X
D(G)
y x
given and , infer and x
G
y D
Dong et al., “Learning graphs from data: A signal representation perspective,” IEEE SPM, 2019
.
Mateos et al., “Connecting the dots: Identifying network structure via graph signal processing,” IEEE SPM, 2019.

62. GSP for structure inference
44
= X
D(G)
y x
given and , infer and x
G
y D
D: smoothness
Dong et al., “Learning graphs from data: A signal representation perspective,” IEEE SPM, 2019
.
Mateos et al., “Connecting the dots: Identifying network structure via graph signal processing,” IEEE SPM, 2019.

63. GSP for structure inference
44
= X
D(G)
y x
given and , infer and x
G
y D
D
D
: smoothness
: diffusion
Dong et al., “Learning graphs from data: A signal representation perspective,” IEEE SPM, 2019
.
Mateos et al., “Connecting the dots: Identifying network structure via graph signal processing,” IEEE SPM, 2019.

64. GSP for structure inference
44
= X
D(G)
y x
given and , infer and x
G
y D
D
D
: smoothness
: diffusion
! Examples of signal graph models:
! Smoothness:
! Diffusion:
D(G) =
D(G) = e ⌧L
Dong et al., “Learning graphs from data: A signal representation perspective,” IEEE SPM, 2019
.
Mateos et al., “Connecting the dots: Identifying network structure via graph signal processing,” IEEE SPM, 2019.

65. A.2.1 Imposing domain speci
fi
c priors
• Leads to more interpretable structures
- Genes are typically clustered into pathways
- Bipartite graph structure is more probable for drug discover
y
• Example of spatial and spectral constraints
:
- K-component grap
h
- Connected sparse grap
h
- K-connected d-regular grap
h
- Cospectral graph
45
Kumar et al., “Structured Graph Learning via Laplacian Spectral Constraints”, NeurIPS, 2019
<latexit sha1_base64="5SQcrObhR1p4QPsS58YZuXxHu+s=">AAACV3icbVFdS8MwFE27qXN+TX30JTgEQRntUPRFEHzxUdHpYJ0lvc1mXJqWJBVG6Z8UX/ZXfNG0TnGbFwLnnnNuPk6ChDOlHWdi2ZXq0vJKbbW+tr6xudXY3nlQcSqBdiDmsewGRFHOBO1opjntJpKSKOD0MRhdFfrjK5WKxeJejxPaj8hQsAEDog3lN4QXEf0MhGd3uZ953EyGJMcX2Mu8n9bPXgrG8Yzj5cLNn0bHGHwXe5ziX8voyM1LAsJYqxktKTvw22aDRtNpOWXhReBOQRNN68ZvvHlhDGlEhQZOlOq5TqL7GZGaAad53UsVTQiMyJD2DBQkoqqflbnk+MAwIR7E0iyhccn+nchIpNQ4CoyzSEHNawX5n9ZL9eC8nzGRpJoK+D5okHKsY1yEjEMmKWg+NoCAZOauGJ6JJKDNV9RNCO78kxfBQ7vlnrac25Pm5fk0jhraQ/voELnoDF2ia3SDOgjQO/qwKlbVmlif9rJd+7ba1nRmF82Uvf0FAXSzMg==</latexit>
S = {{ j = 0}k
j=1, c1  k+1  · · ·  p  c2}
<latexit sha1_base64="qLJ+7C357BodzbGolRvXoNBLHTw=">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</latexit>
S = { 1 = 0, c1  2  · · ·  p  c2}
<latexit sha1_base64="PDA7+mb+hulcywb71ndgU8vf+14=">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</latexit>
S = {{ j = 0}k
j=1, c1  k+1  · · ·  p  c2}, diag(L) = d1
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S = { i = f( ˜i), 8i 2 [1, p]}

66. Illustrative example
• Animals datase
t
• Imposing components leads to more semantically meaningful graphs
46
Graphical LASSO 1-component 5-component

67. A.2.2. Learning product graphs
• Cartesian product graphs are useful to explain complex relationships
in multi-domain graph data
47
Kadambari and Chepuri, “Product graph learning from multi-domain data with sparsity and rank constraints,” arXiv, 2020.
• learning graph factors with rank constraints
exact decomposition approximate decomposition

68. Multi-view object clustering
• COIL-20 dataset
:
- 10 objects Object graph of 10 nodes
- Rotation every 10 degrees: 36 views/image View graph with 36 nodes
48
Object graph and its connected components
View graph and its connected components

69. GSP for enhancing interpretability
A. Domain speci
fi
c: Extract relevant knowledge from dat
a
- A.1. Signal analysis: Use GSP to reveal interpretable feature
s
- A.2. Structure inference: Use GSP to learn interpretable structures
49
B. Model speci
fi
c: Improve our understanding of ML model
s
- B.1. Understanding the expressive power of GNN
s
- B.2. A posteriori interpretation of DNNs

70. GSP for enhancing interpretability
A. Domain speci
fi
c: Improve our understanding of ML model
s
- B.1. Understanding the expressive power of GNN
s
- B.2. A posteriori interpretation of DNNs

71. B.1. A GSP perspective on the expressive power of
GNNs
• A spectral analysis of GNNs provides a complementary point of view
to classical Weisfeiler-Lehman (WL) tes
t
• One step further in explaining GNN
s
• A common framework for spectral and spatial GNN
s
• The frequency pro
fi
le is de
fi
ned as:
50
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H(l+1)
= (
X
s
C(s)
H(l)
W(l,s)
)
Convolution support
<latexit sha1_base64="upYE1WIxiMm6FCFG4FVcxnBCmUw=">AAACEnicbVC7SgNBFL3rM8bXqqXNYBCSwrAriloIgTSWEbJJIC9mJ5NkyOzsMjMrhCXfYOOv2FgoYmtl5984m6TQxAMDh3Pu4c49fsSZ0o7zba2srq1vbGa2sts7u3v79sFhTYWxJNQjIQ9lw8eKciaop5nmtBFJigOf07o/Kqd+/YFKxUJR1eOItgM8EKzPCNZG6tqFVmXIuirf4ibTw4Vb1GN40EnO3AnKe51quZPkVWGCvELXzjlFZwq0TNw5ycEcla791eqFJA6o0IRjpZquE+l2gqVmhNNJthUrGmEywgPaNFTggKp2Mj1pgk6N0kP9UJonNJqqvxMJDpQaB76ZDLAeqkUvFf/zmrHuX7cTJqJYU0Fmi/oxRzpEaT/mfEmJ5uO0ByKZ+SsiQywx0abFrCnBXTx5mdTOi+5l0bm/yJVu5nVk4BhOIA8uXEEJ7qACHhB4hGd4hTfryXqx3q2P2eiKNc8cwR9Ynz+1SZuN</latexit>
s( ) = diag 1
(UT
C(s)
U)
Eigenvectors of the Laplacian

72. Frequency support of well known architectures
51
Balcilar et al., Analyzing the expressive power of graph neural networks in a spectral perspective, ICLR 2021
<latexit sha1_base64="upYE1WIxiMm6FCFG4FVcxnBCmUw=">AAACEnicbVC7SgNBFL3rM8bXqqXNYBCSwrAriloIgTSWEbJJIC9mJ5NkyOzsMjMrhCXfYOOv2FgoYmtl5984m6TQxAMDh3Pu4c49fsSZ0o7zba2srq1vbGa2sts7u3v79sFhTYWxJNQjIQ9lw8eKciaop5nmtBFJigOf07o/Kqd+/YFKxUJR1eOItgM8EKzPCNZG6tqFVmXIuirf4ibTw4Vb1GN40EnO3AnKe51quZPkVWGCvELXzjlFZwq0TNw5ycEcla791eqFJA6o0IRjpZquE+l2gqVmhNNJthUrGmEywgPaNFTggKp2Mj1pgk6N0kP9UJonNJqqvxMJDpQaB76ZDLAeqkUvFf/zmrHuX7cTJqJYU0Fmi/oxRzpEaT/mfEmJ5uO0ByKZ+SsiQywx0abFrCnBXTx5mdTOi+5l0bm/yJVu5nVk4BhOIA8uXEEJ7qACHhB4hGd4hTfryXqx3q2P2eiKNc8cwR9Ynz+1SZuN</latexit>
s( ) = diag 1
(UT
C(s)
U)

73. Frequency pro
fi
les of known GNNs
52
(i)
fi
rst 5 ChebNet supports (ii)
fi
rst 7 CayleyNet supports
(iii) GCN frequency pro
fi
les (iv) GIN on 1D

74. B.2. A posteriori interpretation of learning architectures
• Hypothesis: Identify relative change in model’s predictio
n
• Model Analysis and Reasoning using Graph-based Interpretability:
- Construct a domain (for interpretability) grap
h
- De
fi
ne an explanation function at the nodes of the grap
h
- Choose the in
fl
uential nodes by applying a high pass graph
fi
lterin
g
- Generate explanations by determining in
fl
uential nodes on the graph
53
Anirudh et al., MARGIN: Uncovering Deep Neural Networks Using Graph Signal Analysis, Frontiers in Big Data, 2021
<latexit sha1_base64="3yisCBUcwqPU4HOow+UOEhzHeNc=">AAAB63icbVBNSwMxEJ2tX7V+VT16CRahXsqutOix4EGPFewHtEvJptk2NMkuSVYoS/+CFw+KePUPefPfmG33oK0PBh7vzTAzL4g508Z1v53CxubW9k5xt7S3f3B4VD4+6egoUYS2ScQj1QuwppxJ2jbMcNqLFcUi4LQbTG8zv/tElWaRfDSzmPoCjyULGcEmk8Lq3eWwXHFr7gJonXg5qUCO1rD8NRhFJBFUGsKx1n3PjY2fYmUY4XReGiSaxphM8Zj2LZVYUO2ni1vn6MIqIxRGypY0aKH+nkix0HomAtspsJnoVS8T//P6iQlv/JTJODFUkuWiMOHIRCh7HI2YosTwmSWYKGZvRWSCFSbGxlOyIXirL6+TzlXNa9Tch3qlWc/jKMIZnEMVPLiGJtxDC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHHwaNlg==</latexit>
f(G)
<latexit sha1_base64="wCXBF8Xi0qMQSbLRPCiYupVJY+g=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBahQimJVPQiVET0WMF+QBvKZrtpl242cXdTKKG/w4sHRbz6Y7z5b9y2OWjrg4HHezPMzPMizpS27W8rs7K6tr6R3cxtbe/s7uX3DxoqjCWhdRLyULY8rChngtY105y2Iklx4HHa9IY3U785olKxUDzqcUTdAPcF8xnB2kjuHbpCxUYJ3ZbQ9Wk3X7DL9gxomTgpKUCKWjf/1emFJA6o0IRjpdqOHWk3wVIzwukk14kVjTAZ4j5tGypwQJWbzI6eoBOj9JAfSlNCo5n6eyLBgVLjwDOdAdYDtehNxf+8dqz9SzdhIoo1FWS+yI850iGaJoB6TFKi+dgQTCQztyIywBITbXLKmRCcxZeXSeOs7JyX7YdKoVpJ48jCERxDERy4gCrcQw3qQOAJnuEV3qyR9WK9Wx/z1oyVzhzCH1ifP9Apj3s=</latexit>
G = (V, E, A)
<latexit sha1_base64="GDAsVCwNxsS2AI09VNnjJbDmqpg=">AAACDnicbVDLSsNAFJ3UV62vqEs3g6VQQUtSKroRKm50V8E+oIlhMp20QyeTMDMRStsvcOOvuHGhiFvX7vwbp20W2nrgwuGce7n3Hj9mVCrL+jYyS8srq2vZ9dzG5tb2jrm715BRIjCp44hFouUjSRjlpK6oYqQVC4JCn5Gm37+a+M0HIiSN+J0axMQNUZfTgGKktOSZhZsiPYIX0BkFJ/AycEZe+b58DJ0gEogxSKFDOWx4Zt4qWVPARWKnJA9S1Dzzy+lEOAkJV5ghKdu2FSt3iISimJFxzkkkiRHuoy5pa8pRSKQ7nL4zhgWtdKC+QBdXcKr+nhiiUMpB6OvOEKmenPcm4n9eO1HBuTukPE4U4Xi2KEgYVBGcZAM7VBCs2EAThAXVt0LcQwJhpRPM6RDs+ZcXSaNcsk9L1m0lX62kcWTBATgERWCDM1AF16AG6gCDR/AMXsGb8WS8GO/Gx6w1Y6Qz++APjM8fgLCZOA==</latexit>
I(i) = kf Afk2
2, 8i 2 V

75. Explanations for image classi
fi
cation (I)
• Nodes of the graph: superpixels from image
s
• Graph edges: relative importance of each superpixel
• Explanation: ratio between size of superpixel corresponding to the
node and the size of the largest superpixel
54

76. Explanations for image classi
fi
cation (II)
55
Original image
MARGIN
Grad-CAM

77. Interpreting decision boundaries
• Use MARGIN to identify samples that are likely to be misclassi
fi
e
d
- Nodes: embeddings of each sample
- Edges: similarity between embedding
s
- Explanation function: local label agreement
56

78. Take home message:
GSP for interpretability
• Interpreting the dat
a
- Graph-based transforms reveal new and interpretable features
- Integrating them into a machine learning framework leads to more accurate and
interpretable model
s
- Imposing application-related constraints in topology inference algorithms
generates interpretable structure
s
• Interpreting the models
- Analysing the spectral behaviour of GNNs provides insights on their expressive
powe
r
- GSP operators contribute towards the a posteriori interpretation of learning
architectures
57

79. Summary
• GSP has shown promising results towards improving different aspects
of classical ML algorithm
s
- Robustness to noisy and limited dat
a
- Robustness to topological nois
e
- Data and model interpretabilit
y
• Presented works are indicative only and non-exhaustiv
e
• Plenty of room for exciting research!
58