Systems and methods for generating predictive matrix-variate T models

1. A computer-implemented method to predict one or more missing elements from a partially observed matrix, comprising:

a. receiving one or more user item ratings;

b. generating a model parameterized by matrices U, S, V;

c. applying the model to display an item based on one or more predicted missing elements;

d. applying the model at run-time and determining the matrix value U T i , S, V j , which is output as a rating.

2. The method of claim 1 , wherein the generating the model comprises:

a. defining an object function J(U, S, V) using a matrix variate T model;

b. initializing U, S, V;

c. determining a quadratic convex upper bound of J as Q(U, S, V);

d. fixing two of U, S, V and updating the remaining variable to minimize Q; and

e. outputting U, S, V.

3. The method of claim 1 , comprising applying the model at run-time and requesting a user to rate one or more items.

4. The method of claim 1 , comprising generating a rating.

5. The method of claim 1 , wherein the model comprises a predictive matrix-variate t model.

6. The method of claim 1 , comprising predicting missing elements from the generated model.

7. The method of claim 6 , wherein the predicting elements uses an EM-style optimization method.

8. The method of claim 6 , wherein the predicting elements uses an optimization method that sequentially minimizes a convex upper-bound of the log-likelihood.

9. The method of claim 6 , comprising predicting by modes.

10. The method of claim 6 , comprising predicting by means.

11. A computer-implemented method to predict missing elements from a partially-observed matrix, comprising:

treating the matrix as a single sample drawn from a matrix-variate t distribution;

using a matrix variate t model (MVTM) to predict the missing elements;

displaying an item based on one or more predicted missing elements; and

applying the matrix variate t model at run-time and determining the matrix value U T i , S, V i , which is output as a rating.

12. The method of claim 11 , wherein the MVTM generalizes a range of known probabilistic models.

13. The method of claim 11 , comprising automatically performing model selection to encourage sparse predictive models.

14. The method of claim 11 , comprising predicting missing elements from the matrix variate t model.

15. The method of claim 14 , wherein the predicting elements uses an optimization method that sequentially minimizes a convex upper-bound of the log-likelihood.

16. The method of claim 1 , comprising applying Maximum Margin Matrix Factorization (MMMF) to perform mode estimation.

17. The method of claim 1 , comprising applying Stochastic Relational Models (SRMs) to estimate a covariance matrix.

18. The method of claim 1 , comprising applying collaborative filtering to predict missing elements in the matrix.