Nonparametric Latent Feature Models for Link Prediction

1
Nonparametric Latent Feature Modelsfor Link
Prediction

• Kurt T. Miller, Thomas L. Griffiths, Michael I.
  Jordan
• NIPS 2009
• Presented by Minhua Chen, 06.04.2010.

2
Problem Formulation

• Link prediction in Social Network (Binary Matrix
  completion)

1 0 ? ?
0 1 ? 0
1 ? 1 ?
0 1 0 1
Y
Yij 1 person i is linked to person j. Yij
0 person i is not linked to person j. Yij ?
unobserved entry to be filled in.

• Linkage can stand for different relations,
  e.g., friends or not, colleagues or not.
• If the network is a directed graph, then Y can
  be asymmetric.
• Observed entries auxiliary information
  (optional) ? unobserved entries

3
Methods

• Class-based model
• Entities are clustered into classes.
• Linkage is determined by which classes they
  belong to.
• Models Infinite Relational Model (IRM)
• Mixed Membership Stochastic
  Blockmodel (MMSB)
• Disadvantage clustering description is too
  coarse, not expressive.
• Latent-feature model
• Interactions between latent-features
  determine the linkage.
• This paper extends it to a nonparametric
  model using IBP.
• Number of latent features can be inferred as
  well as their interactions.

4
Model

• Define Z to be a binary NK matrix with N people
  and K latent features.
• Define W to be a KK weighting matrix for the K
  latent features.
• The model is
• Or expressed in more details

5
Results on Synthetic Data
(c) Ground truth of Z (d) Generated
Y (e) Inferred Z Although the
missing values are imputed correctly, the
inferred Z is different from ground truth. This
indicates that the model is unidentifiable.
6
Results on Multi-Task Data

• The Countries data contains 54 relation matrices
  among 14 countries, along with 90 given
  covariates.
• The Alyawarra data contains 26 kinship
  relationship matrices of 104 people in the
  Alyawarra tribe in Central Australia.
• For each dataset, 80 of the data is used for
  training and the rest 20 is used for testing.

• LFRM outperforms IRM and MMSB with proper
  initialization.

7
Results on Single-Task Data
AUC performance
LFRM w/IRM 0.9509
LFRM rand 0.9466
IRM 0.8906
MMSB 0.8705

• 234 authors who published with the most other
• people in NIPS 1-17 are used, and their
• coauthorship matrix is constructed.