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Rep the Set: Neural Networks for Learning Setrepresentations
K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis
Data Science and Mining group (DaSciM),Laboratoire d’Informatique (LIX), Ecole Polytechnique, France
http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, FrancePreprint available at: https://arxiv.org/abs/1904.01962
April 26, 2019
1 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Data Science and Mining @ Ecole Polytechnique, France
2 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Data Science and Mining @ Ecole Polytechnique, France
Research Topics
Machine Learning and AI
AI and Data Science methods (degeneracy, similarity, deep learning,multi-label classification)
Applications to: Text Mining/NLP, Social nets, Web marketing/advertising,Time Series
J. Read, M. Vazirgiannis
Operations Research and Mathematical programming
Optimization for Energy apps
Distance Geometry, protein conformation
C. d’Ambrosio, L. Liberti
3 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Data Science and Mining @ Ecole Polytechnique, France
Graph of Words: graph based text/NLP
4 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Data Science and Mining @ Ecole Polytechnique, France
Graph of Words: graph based text/NLP
5 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Data Science and Mining @ Ecole Polytechnique, France
Machine/Deep Learning methods for Graphs
6 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Data Science and Mining @ Ecole Polytechnique, France
Machine/Deep Learning methods for Graphs
7 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Data Science and Mining @ Ecole Polytechnique, France
Industrial Collaborations and Projects
8 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Machine Learing on Sets
Typical ML algorithms (i.e. regression or classification) designed for fixeddimensionality objects.
s imalrity learning between sets should be invariant to permutation:challenging task
supervised tasks: set output label invariant or equivariant to thepermutationi its elements.
population statistics estimation, giga-scale cosmology, nano-scale quantumchemistry.
unsupervised tasks, “set” representation needs to be learned.
set expansion - assume a set of similar objects - find similar to the setextensions, i.e. extend the set {lion, tiger, leopard} with cheetah
web marketing extend a set high-value customers with similar people.
astrophysics: assuming set of interesting celestial objects, find similar ones insky surveys.
9 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Background and State of the art
NNs for sets became very popular isnpired by computer vision problems suchas the automated classification of point clouds. Proposed architectures haveachieved state-of-the-art results on many different tasks.
Base approaches: PointNet [Qi et.al., CVPR2017] and DeepSets [Zaheeret.al., NIPS2017]
transform sets’ elements vectors using several NN layers into newrepresentations
apply some permutation-invariant function to the emerging vectors togenerate representations for the sets.
Pointnet: max pooling, DeepSets: vector sum
representation of the set is then passed on to a standard architecture (e.g.,fully connected layers,nonlinearities, etc).
Other efforts: PointNet++, SO-Net
10 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Motivation and Contribution
Data objects decomposed into sets of simpler objects: natural to representeach object as the set of its components or parts.
Conventional ML algorithms operate on vectors / sequences. Thus unable toprocess sets as
sets may vary in cardinality
set elements lack a meaningful ordering
Challenge: Sets as input to Neural Network Architectures
Contribution: RepSet: a new neural network architecture, handlingexamples as sets of vectors.
computes the correspondences between an input set and some hidden sets bysolving a series of network flow problems.
resulting representation fed to a NN architecture to produce the output.
allows end-to-end gradient-based learning.
Experimental evaluation: favorable on classification (text, graph) tasksoutperforming satet of the art
11 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset architecture - Permutation Invariance
Assume an example X represented as a set X = {v1, v2, . . . , vn} ofd-dimensional vectors, vi ∈ Rd . (i.e the embeddings of X ′s elements)Objective: design architecture whose output is invariant for all n!permutations of X elements => permutation invariant function.
propose a novel permutation invariant layercontains m “hidden sets” Y1,Y2, . . . ,Ym of d-dimensional vectors (samedim as X elements)based on bipartite graph matchingits components are trainable,elements of a hidden set Yi correspond to the columns of a trainable matrixW(i).each column of matrix W(i) is a vector u ∈ Yi .
12 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset architecture - Similarity via graph matching
to measure the similarity between X and each one of the hidden sets Yi :comparing their components.
capitalize on network flow algorithms - specifically bipartite matching:compute optimal mapping between the elements of X and the elements ofeach hidden set Yi .
Each edge e connects a vertex in X to one in Yi .
Matching M: subset of edges - each node in X connects to one in Yi .
optimal solution is interpreted as similarity between node sets X and Yi .
The bipartite graph is a matrix |X | × |Y |, cell values from {0,1}13 / 27
K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset architecture - bipartite matching Optimization
Assume a set of vectors, X = {v1, v2, . . . , v|X |} and a hidden setY = {u1,u2, . . . ,u|Y |}, the bipartite matching between the elements of thetwo sets is solving the optimization problem:
|X |∑i=1
|Y |∑j=1
xij f (vi ,uj)
subject to:
|X |∑i=1
xij ≤ 1 ∀j ∈ {1, . . . , |Y |}
|Y |∑j=1
xij ≤ 1 ∀i ∈ {1, . . . , |X |}
xij ≥ 0 ∀i ∈ {1, . . . , |X |},∀j ∈ {1, . . . , |Y |}
(1)
f (vi ,uj) differentiable function, and xij = 1 if component i of X assigned tocomponent j of Yi , 0 otherwise.we defined f (vi ,uj) = ReLU(v>i uj).
14 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset architecture - Learning and output
Given input set X and the m hidden sets Y1,Y2, . . . ,Ym, formulate mbipartite matching problems,
solving we end up with an m-dimensional vector vX : hidden representationof set X .
This m-dimensional vector can be used as features for different machinelearning tasks such as set regression or set classification. For instance, in thecase of a set classification problem with |C| classes, the output is computedas follows:
pX = softmax(W(c) vX + b(c)) (2)
where W(c) ∈ Rm×|C| is a matrix of trainable parameters and b(c) ∈ R|C| isthe bias term. We use the negative log likelihood of the correct labels astraining loss:
L = −∑X
log pXi (3)
where i is the class label of set X . Note that we can create a deeperarchitecture by adding more fully-connected layers.
15 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset architecture - Learning and output
Given input set X and the m hidden sets Y1,Y2, . . . ,Ym, formulate mbipartite matching problems,end up with an m-dimensional vector vX : hidden representation of set X .Can be used as features for different machine learning tasks such as setregression or set classification. For set classification with |C| classes, theoutput is computed as:
pX = softmax(W(c) vX + b(c)) (4)
We use the negative log likelihood of thecorrect labels as training loss:
L = −∑X
log pXi (5)
where i is the class label of set X .* The architecture supports permutationinvariance (proof in the paper)
16 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset architecture - Tackling the complexity of the bipartite matching
major weakness the computational complexity: maximum cardinalitymatching in a weighted bipartite graph with n vertices and m edges takestime O(mn + n2 log n), with the classical Hungarian algorithm.Prohibitive for very large datasets.ApproxRepSet: approximation of bipartite matching problem involvingoperations that can be performed on a GPUAssuming an input set of vectors, X = {v1, v2, . . . , v|X |} and a hidden setY = {u1,u2, . . . ,u|Y |}. Assume |X | ≥ |Y |, optimization becomes:
max
|X |∑i=1
|Y |∑j=1
xij f (vi ,uj)
subject to:
|X |∑i=1
xij ≤ 1 ∀j ∈ {1, . . . , |Y |}
xij ≥ 0 ∀i ∈ {1, . . . , |X |},∀j ∈ {1, . . . , |Y |}
(6)
relaxed formulation of the problem - constraint has been removed.17 / 27
K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Experimental Evaluation - Synthetic Data
18 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Experimental Evaluation - Text Categorization
19 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Experimental Evaluation - Text Categorization
Classification test error of the proposed architecture and the baselines on the 8text categorization datasets.
20 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Experimental Evaluation - Text Set Extension
Terms of the employed pre-trained model that are most similar to the elementsand centroids of elements of5 hidden sets
21 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Experimental Evaluation - Graph classification
22 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Experimental Evaluation - Graph Classification
Classification accuracy (± standard deviation) of proposed architecture(s) and thebaselines. For MU-TAG, PROTEINS ( bioinformatics datasets ) the nodeembeddings that we generated do not incorporateinformation about them.
23 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Experimental Evaluation - Runtimes
Runtimes with respect to the number of hidden setsm, the size of the hiddensets—Yi—(left)and embeddings with different dimensions (right).
24 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Experimental Evaluation - Runtimes
Runtimes with respect to the number of input setsN(left) and the size of theinput sets—Xi—(right).
25 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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Repset - Conclusion
Machine learning with sets is increasingly important
Sets may vary in cardinality and their elements lack a meaningful ordering:
standard machine learning algorithms fail to learn high-quality representations.
We proposed RepSet, a neural network approach for learning set representations.
exhibits powerful permutation invarianceproperties.
computes mappings between input sets andsome hidden sets by solving a graphmatching/network flow problems.
Since matching/network flow algorithms aredifferentiable, we can use standardbackpropagation for learning the parameters ofthe hidden sets.
for large sets we introduced a relaxedversion(ApproxRepSet) - fast matrix operations andscales to very large datasets.
Repsets performs favorably on text/ graph
classification.
Future Work
: apply Repset on Group
Recommentation (i.e.
gaming)
26 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations
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THANK YOU !
AcknowledgementsDr. I. Nikolentzos, Dr. K. Skianis, Dr. P. Meladianos
http://www.lix.polytechnique.fr/dascim/
Software and data sets:http://www.lix.polytechnique.fr/dascim/software datasets/
Repset preprint available at: https://arxiv.org/abs/1904.01962
27 / 27K. Skianis, G. Nikolentzos, S. Limnios, M. Vazirgiannis Data Science and Mining group (DaSciM), Laboratoire d’Informatique (LIX), Ecole Polytechnique, France http://www.lix.polytechnique.fr/dascim/ Ecole Polytechnique, France Preprint available at: https://arxiv.org/abs/1904.01962Rep the Set: Neural Networks for Learning Set representations