154 research outputs found
Maximal Independent Sets for Pooling in Graph Neural Networks
Convolutional Neural Networks (CNNs) have enabled major advances in image
classification through convolution and pooling. In particular, image pooling
transforms a connected discrete lattice into a reduced lattice with the same
connectivity and allows reduction functions to consider all pixels in an image.
However, there is no pooling that satisfies these properties for graphs. In
fact, traditional graph pooling methods suffer from at least one of the
following drawbacks: Graph disconnection or overconnection, low decimation
ratio, and deletion of large parts of graphs. In this paper, we present three
pooling methods based on the notion of maximal independent sets that avoid
these pitfalls. Our experimental results confirm the relevance of maximal
independent set constraints for graph pooling
Maximal Independent Vertex Set applied to Graph Pooling
Convolutional neural networks (CNN) have enabled major advances in image
classification through convolution and pooling. In particular, image pooling
transforms a connected discrete grid into a reduced grid with the same
connectivity and allows reduction functions to take into account all the pixels
of an image. However, a pooling satisfying such properties does not exist for
graphs. Indeed, some methods are based on a vertex selection step which induces
an important loss of information. Other methods learn a fuzzy clustering of
vertex sets which induces almost complete reduced graphs. We propose to
overcome both problems using a new pooling method, named MIVSPool. This method
is based on a selection of vertices called surviving vertices using a Maximal
Independent Vertex Set (MIVS) and an assignment of the remaining vertices to
the survivors. Consequently, our method does not discard any vertex information
nor artificially increase the density of the graph. Experimental results show
an increase in accuracy for graph classification on various standard datasets
Graph-based Time Series Clustering for End-to-End Hierarchical Forecasting
Existing relationships among time series can be exploited as inductive biases
in learning effective forecasting models. In hierarchical time series,
relationships among subsets of sequences induce hard constraints (hierarchical
inductive biases) on the predicted values. In this paper, we propose a
graph-based methodology to unify relational and hierarchical inductive biases
in the context of deep learning for time series forecasting. In particular, we
model both types of relationships as dependencies in a pyramidal graph
structure, with each pyramidal layer corresponding to a level of the hierarchy.
By exploiting modern - trainable - graph pooling operators we show that the
hierarchical structure, if not available as a prior, can be learned directly
from data, thus obtaining cluster assignments aligned with the forecasting
objective. A differentiable reconciliation stage is incorporated into the
processing architecture, allowing hierarchical constraints to act both as an
architectural bias as well as a regularization element for predictions.
Simulation results on representative datasets show that the proposed method
compares favorably against the state of the art
The expressive power of pooling in Graph Neural Networks
In Graph Neural Networks (GNNs), hierarchical pooling operators generate
local summaries of the data by coarsening the graph structure and the vertex
features. Considerable attention has been devoted to analyzing the expressive
power of message-passing (MP) layers in GNNs, while a study on how graph
pooling affects the expressiveness of a GNN is still lacking. Additionally,
despite the recent advances in the design of pooling operators, there is not a
principled criterion to compare them. In this work, we derive sufficient
conditions for a pooling operator to fully preserve the expressive power of the
MP layers before it. These conditions serve as a universal and
theoretically-grounded criterion for choosing among existing pooling operators
or designing new ones. Based on our theoretical findings, we analyze several
existing pooling operators and identify those that fail to satisfy the
expressiveness conditions. Finally, we introduce an experimental setup to
verify empirically the expressive power of a GNN equipped with pooling layers,
in terms of its capability to perform a graph isomorphism test
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