Markov Random Fields

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Markov Random Fields Conditional Random Fields

• John WinnMSR Cambridge

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Road map

• Markov Random Fields
• What they are
• Uses in vision/object recognition
• Advantages
• Difficulties
• Conditional Random Fields
• What they are
• Further difficulties

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Markov Random Fields
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Examples of use in vision

• Grid-shaped MRFs for pixel labelling e.g.
  segmentation

• MRFs (e.g. stars) over part positions for
  pictorial structures/constellation models.

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Advantages

• Probabilistic model
• Captures uncertainty
• No irreversible decisions
• Iterative reasoning
• Principled fusing of different cues
• Undirected model
• Allows non-causal relationships (soft
  constraints)
• Efficient algorithmsinference now practical for
  MRFs with millions variables can be applied to
  raw pixels.

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Maximum Likelihood Learning
Sufficient statisticsof data
Expected model sufficient statistics
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Difficulty I Inference

• Exact inference intractable except in a few cases
  e.g. small models
• Must resort to approximate methods
• Loopy belief propagation
• MCMC sampling
• Alpha expansion (MAP solution only)

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Difficulty II Learning

• Gradient descent vulnerable to local minima
• Slow must perform expensive inference at each
  iteration.
• Can stop inference early
• Contrastive divergence
• Piecewise training variants
• Need fast accurate methods

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Difficulty III Large cliques

• For images, we want to look at patches not pairs
  of pixels. Therefore would like to use large
  cliques.
• Cost of inference (memory and CPU) typically
  exponential in clique size.
• Example Field of Experts, Black Roth
• Training contrastive divergenceover a week on a
  cluster of 50 machines
• Test Gibbs samplingvery slow?

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Other MRF issues

• Local minima when performing inference in
  high-dimensional latent spaces
• MRF models often require making inaccurate
  independence assumptions about the observations.

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Conditional Random Fields
Lafferty et al., 2001
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Examples of use in vision

• Grid-shaped CRFs for pixel labelling (e.g.
  segmentation), using boosted classifiers.

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Difficulty IV CRF Learning
Sufficient statisticsof labels given the image
Expected sufficient statistics given the image
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Difficulty V Scarcity of labels

• CRF is a conditional model needs labels.
• Labels are expensive increasingly hard to
  define.
• Labels are also inherently lower dimensional than
  the data and hence support learning fewer
  parameters than generative models.