Partially labeled classification with Markov random walks

1
A discussion on
Partially labeled classification with Markov
random walks
By M. Szummer and T. Jaakkola
Xuejun Liao 15 June 2006
2

• Neighborhood graph (undirected)

Wik gives the (on-step) connection strength datum
i and k

• The graph induces a Markov random walk with
  one-step transitions

• t-step Markov random walk

where A is the one-step transition matrix with
Aijp(ki)

• Assuming uniform initial distribution p(i)

3

• Transduction

• Likelihood for labeled data

• Maximizing the likelihood gives

E-step
M-step
4

• Estimation based on margin maximization

where C denotes the number of classes and NC(k)
gives the number of labeled points in the same
class as k, and

• The solution to this linear program can be found
  in closed form

• For each data k, choose tk as the smallest
  number of transitions needed to reach a labeled
  datum from datum k.

5

• An example