Probabilistic LSA

This is a probabilistic extension to Latent Semantic Analysis


Problem

Notation and problem:

  • $D_1, \ ... \ , D_n$ are documents
  • $T_1, \ ... \ , T_k$ are topics (sort of "clusters")
  • each document may belong to several topics - so these "clusters" are Fuzzy
  • probability of $D_i$ belonging to $T_j$ is $P(T_j \mid D_i)$
  • but cluster membership is secondary in this problem
  • the main problem is to find latent topics that generated documents - which is why it's called Topic Modeling
  • let $t_1, \ ... \ , t_d$ be $d$ terms from the lexicon
  • then the probability that $t_l$ occurs in $T_j$ is $P(t_l \mid T_j)$


Learning

Thus, we need to estimate the following probabilities:

  • $P(T_j \mid D_i)$ and $P(t_l \mid T_j)$
  • usually parameters are learned via maximum likelihood methods like Expectation Maximization


We need to learn $P(T_j \mid D_i)$ and $P(t_l \mid T_j)$

  • $P(t_l \mid D_i)$ can be expressed via them:
  • $P(t_l \mid D_i) = \sum\limits_{j=1}^k p(t_l \mid T_i) \, P(T_j \mid D_i)$
  • thus, for each $t_l$ and $D_i$ we can generate $n \times d$ matrix of probabilities
  • these probabilities are learned from term-document matrix $X$: $X_{il}$ is # of times $t_l$ occurred in $D_i$
  • so we can use Maximum Likelihood Estimator to maximize the product of probabilities of terms we observed


Optimization:

  • we will optimize the log likelihood $\sum_{i,l} X_{il} \cdot \log P(t_l, D_i)$
  • s.t. $\sum_l P(t_l \mid T_j) = 1$ for all $T_j$ and $\sum_j P(T_j \mid D_i) = 1$ for all $D_i$
  • can use Lagrange Multipliers for this


Latent Dirichlet Allocation

is an extension of Probabilistic LSA

  • model term-topic probabilities and topic-document probabilities with Dirichlet Distribution
  • so LDA is a Bayesian version of PLSA
  • but LSA overfits less than PLSA because it has less parameters to fit


References

  • Hofmann, Thomas. "Probabilistic latent semantic analysis." 1999. [1]

Sources

  • Aggarwal, Charu C., and ChengXiang Zhai. "A survey of text clustering algorithms." Mining Text Data. Springer US, 2012. [2]
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