# Probabilistic Latent Semantic Analysis

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1 Probabilistic Latent Semantic Analysis Thomas Hofmann Presentation by Ioannis Pavlopoulos & Andreas Damianou for the course of Data Mining & Exploration 1

2 Outline Latent Semantic Analysis o Need o Overview o Drawbacks Probabilistic Latent Semantic Analysis o Solution to drawbacks of LSA o Comparison with LSA and document clustering o Model Construction Evaluation of PLSA 2

3 Need for Latent Semantic Analysis Applications o Compare documents in the semantic (concept) space o Relations between terms o Compare documents across languages o Given: Bag of words Find: matching documents in the semantic space Problems addressing o Synonymy ex: buy - purchase o Polysemy ex: book (verb) - book (noun) 3

4 LSA Overview Capturing the meaning among words Addressing polysemy and synonymy Key Idea o Dimensionality reduction of word-document co-occurence matrix o Construction of Latent Semantic space From: Documents Words To: Documents Concepts Words LSA may classify documents together even if they don t have common words! 4

5 LSA Concept Singular Value Decomposition (SVD) Given N which is the word-document co-occurence matrix, compute: N = UΣVt where: Σ is the diagonal matrix with the singular values of N U, V two orthogonal matrices 5

6 LSA SVD 6

7 LSA Concept Dimensionality Reduction Keep the K largest singular values which show the dimensions with the greatest variance between words and documents Discarding the lowest dimensions is supposed to be equivalent to reducing the "noise" Terms and documents are converted to points in a K- Dimensional latent space Results do not introduce well defined probabilities and thus, are difficult to interpret 7

8 Probabilistic LSA Overview Implemented to address: Automated Document Indexing Same concept to LSA o Dimensionality Reduction o Construction of a latent space BUT.. Sound Statistical foundations o Well defined probabilities o Explicable results 8

9 Probabilistic LSA Aspect Model Generative model based on the Aspect model o Latent variables z are introduced and relate to documents d. o z << d, as the same z i may be associated with more than one documents o z performs as a bottleneck and results in dimensionality reduction 9

10 Probabilistic LSA Model Multinomial Mixtures Multinomials Mixing weights Joint probability shows the probability of a word w to be inside a document d Word distributions are combinations of the factors P(w z) and the mixing weights P(z d) 10

11 Probabilistic LSA Model Conditional Independence assumption o Documents and Words are independent given z Thus, equivalently: 11

12 Probabilistic LSA Model fitting Expectation Maximization Standard procedure for latent variable models E-step: Compute the posteriors for the latent variables z M-step: Update the parameters 12

13 Probabilistic LSA Space Sub-simplex dimensionality K-1 << D-1 13

14 Tempered EM Avoid overfitting training data Introduce a regularization term β 14

15 Tempered EM - Concept Add a term β < 1 in the E step. Used to dampen probabilities in M step. Accelerate model fitting procedure compared to other methods (ex. annealing) Perform EM iterations and then decrease β until performance on held-out data deteriorates. 15

16 PLSA vs LSA Great PLSA advantages on the modeling side o Well defined probabilities o Interpretable directions in the Probabilistic Latent Semantic space as multinomial word distributions o Better model selection and complexity control (TEM) Important LSA drawbacks in the same side o Not defined properly normalized probabilities o No obvious interpretations of LS space directions o Selection of dimensions based on ad-hoc heuristics Potential computational advantage of LSA over PLSA (SVD vs EM which is an iterative method) 16

17 Aspect Model vs Clusters Document Clustering Aspect Model Documents Cluster aspect PLSA: Documents are not related to a single cluster flexibility, effective modeling 17

18 Evaluation perplexity Perplexity: Measures how well a prob. distribution can make predictions. Low perplexity more certain predictions, better model PLSA evaluation method: Extract probabilities from LSA Unigram model as baseline PLSA evaluation results PLSA better than LSA TEM better than EM PLSA allows Z > rank(n) (N is the co-oc. Matrix) 18

19 Evaluation Automatic Indexing Given a short document (query q) find the most relevant documents Baseline term matching s(d,q): cosine scoring method combined with term frequencies LSA: Linear combination of s(d,q) and the one derived from the latent space PLSA: Evaluation of similarities of P(z d) & P(z q) 19

20 Evaluation Precision & Recall Precision & Recall: Popular measures in Information Retrieval. 20

21 Evaluation Precision & Recall For intermediate values of recall, the precision of PLSA is almost 100% better than the baseline method!!! 21

22 Evaluation Polysemy Results show advantage of PLSA over polysemy 22

23 Conclusion Documents are represented as vectors of word frequencies There is no syntactic relation or word ordering but co occurences still provide useful semantic insights about the document topics PLSA is a generative model based on this idea. It can be used to extract topics from a collection of documents PLSA significantly outperforms LSA thanks to its probabilistic basis. 23

24 References D.M. Blei, A.Y. Ng, and M.I. Jordan, Latent dirichlet allocation, J. Mach. Learn. Res., vol. 3, 2003, pp T. Hofmann, Unsupervised Learning by Probabilistic Latent Semantic Analysis, Machine Learning, vol. 42, Jan. 2001, pp T. Hofmann, Probabilistic latent semantic analysis, In Proc. of Uncertainty in Artificial Intelligence, UAI 99, 1999, pp DEERWESTER, S., DUMAIS, S., LANDAUER, T., FURNAS, G., AND HARSHMAN, R Indexing by latent semantic analysis. J. Amer. Soc. Info. Sci. 41,

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