Relational Factor Graphs

1
Relational Factor Graphs

• Lin Liao
• Joint work with Dieter Fox

2
A Running Example
Collective classification of a persons
significant places
3
Features to Consider

• Local features
• Temporal time of day, day of week, duration
• Geographic near restaurants, near stores
• Pair-wise features
• Transitions which place follows which place
• Global features
• Aggregates number of homes or workplaces

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Which Graphical Model?

• Option 1 Bayesian networks and Probabilistic
  Relational Models
• But the pair-wise relations may introduce cycles

Place 2
Place 1
Place 3
Place 4
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Which Graphical Model?

• Option 2 Markov networks and Relational Markov
  Networks
• But aggregations can introduce huge cliques and
  lose independence relations.

Number of homes
Place 2
Place 1
Place 3
Place 4
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Motivation

• We want a relational probabilistic model that is
• Suitable to represent both undirected relations
  (e.g., pair-wise features) and directed relations
  (e.g., deterministic aggregation)
• Able to address some of the computational issues
  at the template level

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Outline

• Representation
• Factor graphs Kschischang et al. 2001, Frey
  2003
• Relational factor graphs
• Inference
• Belief propagation
• Inference templates
• Summation template based on FFT
• Experiments

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Factor Graph

• Undirected factor graph Kschischang et al. 2001
• Bipartite graph that includes both variable nodes
  (x1,,xN) and factor nodes (f1,,fM)
• Joint distribution of variables is proportional
  to the product of factor functions

x1
x3
f2
f1
f3
x4
x2
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Factor Graph

• Directed factor graph Frey 2003
• Allow some edges to be directed so as to unify
  Bayesian networks and Markov networks
• A valid graph should have no directed cycles

x1
x3
f2
f1
f3
x4
x2
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Markov Network to Factor Graph
Markov network
Factor graph
Factors represent the potential functions
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Bayesian Network to Factor Graph
Bayesian network
Factor graph
Factors represent the conditional probability
table
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Unify MN and BN
Aggregate features
Number of homes
Aggregation factor

Place labels
Local features
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Relational Factor Graph

• A set of factor templates that can be used to
  instantiate (directed) factor graphs given data
• Representation template
• Use SQL (similar to RMN)
• Guarantee no directed cycles
• Inference template
• Optimization within a factor (discussed later)

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Place Labeling Schema
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Place Labeling Transition Features
Pair-wise factor
Label1
Label2
Label3
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Place Labeling Aggregate Features
Aggregate feature
Num of homes

Bool variables
Home?
Home?
Home?
Label1
Label2
Label3
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Outline

• Representation
• Factor graphs Kschischang et al. 2001, Frey
  2003
• Relational factor graphs
• Inference
• Belief propagation
• Inference templates
• Summation template based on FFT
• Experiments

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Inference in Factor Graph

• Belief propagation two types of messages
• Message from variable x to factor f
• Message from factor f to variable x

nx factors adjacent to x nf variables adjacent
to f
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Inference Templates

• Simplest case specify the function f(nf) and use
  the above formula to compute message f -gt x
• Problem complexity is exponential in the number
  of factor arguments. This can be very expensive
  for aggregation factors
• Inference templates allow users to specify
  optimized algorithms at the template level
• Be in general form and easy to be shared
• Support template level complexity analysis

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Summation Templates
xout

..
xin1
xin2
xin7
xin8
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Summation Forward Message

• Compute the distribution of the sum of
  independent variables xin1, . , xin8

xout

..
xin1
xin2
xin7
xin8
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Summation Forward Message

• Convolution tree each node can be computed using
  FFT total complexity O(nlog2n)

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Summation Backward Message

• Message from xout defines a prior distribution of
  the sum. For each value of xin2, compute the
  distribution of sum and weighted by the prior

xout

..
xin1
xin2
xin7
xin8
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Summation Backward Message

• If we reuse the results cached for the forward
  message, complexity becomes O(nlogn)

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Summation Templates

• By using convolution tree, FFT, and caching, the
  average complexity of passing a message through
  summation factor is O(nlogn), instead of
  exponential.

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Learning

• Estimate the weights for probabilistic factors
  (local features, pair-wise features, and
  aggregate features)
• Optimize the weights to maximize the conditional
  likelihood of the labeled training data
• The same algorithm as RMN

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Experiments

• Two data sets
• Single data set one persons GPS data for 4
  months
• Multiple data set one-week GPS data from 5
  subjects
• Six candidate labels Home, Work, Shopping,
  Dining, Friend, Others
• Get the geographic knowledge from Microsoft
  MapPoint Web Service

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How Much Aggregates Help

• Test on multiple data set leave-one-subject-cro
  ssvalidation
• Test on single data set crossvalidation (train
  on 1 month, test on 3 months)

Error rate Multiple Single
No aggregate 28 9
With aggregate 18 6
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How Efficient the Optimized BP
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Summary

• Relational factor graph is
• SQL (directed) factor graph
• It is
• Suitable to represent both undirected relations
  and directed relations
• Convenient to use no directed cycles
• Able to address computation issues at the
  template level