Mulivariate Density Estimation from Lower Dimensional Slices

We introduce a new method of statistical density estimation based on projections of higher dimensional data on lower dimensional subspaces. The problem of estimating densities is expected to be more well-posed in lower dimensional subspace due to the fact that in lower dimensions data points can be viewed as relatively less sparse (thus, alleviating the problems arising from curse of dimensionality). While the problem of reconstruction of the higher dimensional density function from low dimensional densities is reminiscent of tomographic reconstruction problem, the reconstruction turns out to be nonunique unless additional constraints are imposed. One such constraint that we consider in the present paper is the maximum entropy criterion. Alternatively maximum likelihood estimated could also be used. Different data models for the projected data can be assumed within this framework. Among other models, we consider the gaussian mixture model for this purpose, and show that an Expectation Maximization strategy for parameter update can be used to solve the problem. While updates of means and covariances can be obtained more or less in a manner similar to the standard EM algorithm ala e.g., Dempster-Schafer[8], computation of best directions of projection, when built into the EM algorithm, essentially involves solving an additional nonlinear optimization problem that is interesting in its own right, and has recently appeared in other statistical problems [5, 171. Interesting connections of special cakes of this optimization problem with known results on the theory of stochastic matrices are pointed out in the context of our discussion. Simple numerical examples are worked out with both real and synthetic data to validate the efficacy of the proposed method.