Probabilistic Latent Semantic Indexing

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Table of contents

1 Latent Linear Models 
1.1 Latent Linear Models for Single-View Data
1.1.1 Gaussian Mixture Models
1.1.2 Factor Analysis
1.1.3 Probabilistic Principal Component Analysis
1.1.4 Independent Component Analysis
1.1.5 Dictionary Learning
1.2 Latent Linear Models for Count Data
1.2.1 Admixture and Topic Models
1.2.2 Topic Models Terminology
1.2.3 Probabilistic Latent Semantic Indexing
1.2.4 Latent Dirichlet Allocation
1.2.5 Other Topic Models
1.3 Latent Linear Models for Multi-View Data
1.3.1 Probabilistic Canonical Correlation Analysis
1.4 Overcomplete Latent Linear Models
2 Tensors and Estimation in Latent Linear Models 
2.1 Tensors, Higher Order Statistics, and CPD
2.1.1 Tensors
2.1.2 The Canonical Polyadic Decomposition
2.1.3 Tensor Rank and Low-Rank Approximation
2.1.4 CP Uniqueness and Identifiability
2.2 Higher Order Statistics
2.2.1 Moments, Cumulants, and Generating Functions
2.2.2 CPD of ICA Cumulants
2.2.3 CPD of LDA Moments
2.3 Algorithms for the CP Decomposition
2.3.1 Algorithms for Orthogonal Symmetric CPD
2.3.2 Algorithms for Non-Orthogonal Non-Symmetric CPD
2.4 Latent Linear Models : Estimation and Inference
2.4.1 The Expectation Maximization Algorithm
2.4.2 Moment Matching Techniques
3 Moment Matching-Based Estimation in Topic Models 
3.1 Contributions
3.2 Related Work
3.3 Discrete ICA
3.3.1 Topic Models are PCA for Count Data
3.3.2 GP and Discrete ICA Cumulants
3.3.3 Sample Complexity
3.4 Estimation in the GP and DICA Models
3.4.1 Analysis of the Whitening and Recovery Error
3.5 Experiments
3.5.1 Datasets
3.5.2 Code and Complexity
3.5.3 Comparison of the Diagonalization Algorithms
3.5.4 The GP/DICA Cumulants vs. the LDA Moments
3.5.5 Real Data Experiments
3.6 Conclusion
4 Moment Matching-Based Estimation in Multi-View Models 
4.1 Contributions
4.2 Related Work
4.3 Non-Gaussian CCA
4.3.1 Non-Gaussian, Discrete, and Mixed CCA
4.3.2 Identifiability of Non-Gaussian CCA
4.3.3 The Proof of Theorem 4.3.1
4.4 Cumulants and Generalized Covariance Matrices
4.4.1 Discrete CCA Cumulants
4.4.2 Generalized Covariance Matrices
4.5 Estimation in Non-Gaussian, Discrete, and Mixed CCA
4.6 Experiments
4.6.1 Synthetic Count Data
4.6.2 Synthetic Continuous Data
4.6.3 Real Data Experiment – Translation Topics
4.7 Conclusion
5 Conclusion and Future Work 
5.1 Algorithms for the CP Decomposition
5.2 Inference for Semiparametric Models
A Notation
A.1 The List of Probability Distributions
B Discrete ICA
B.1 The Order-Three DICA Cumulant
B.2 The Sketch of the Proof for Proposition 3.3.1
B.2.1 Expected Squared Error for the Sample Expectation
B.2.2 Expected Squared Error for the Sample Covariance
B.2.3 Expected Squared Error of the Estimator ̂︀ S for the GP/DICA Cumulants
B.2.4 Auxiliary Expressions
C Implementation
C.1 Implementation of Finite Sample Estimators
C.1.1 Expressions for Fast Implementation of the LDA Moments Finite Sample Estimators
C.1.2 Expressions for Fast Implementation of the DICA Cumulants Finite Sample Estimators
C.2 Multi-View Models
C.2.1 Finite Sample Estimators of the DCCA Cumulants