Two directional Laplacian pyramids with application to data imputation

Modeling and analyzing high-dimensional data has become a common task in various fields and applications. Often, it is of interest to learn a function that is defined on the data and then to extend its values to newly arrived data points. The Laplacian pyramids approach invokes kernels of decreasing widths to learns a given dataset and a function defined over it in a multi-scale manner. Extension of the function to new values may then be easily performed. In this work, we extend the Laplacian pyramids technique to model the data by considering two-directional connections. In practice, kernels of decreasing widths are constructed on the row-space and on the column space of the given dataset and in each step of the algorithm the data is approximated by considering the connections in both directions. Moreover, the method does not require solving a minimization problem as other common imputation techniques do, thus avoids the risk of a non-converging process. The method presented in this paper is general and may be adapted to imputation tasks. The numerical results demonstrate the ability of the algorithm to deal with a large number of missing data values. In addition, in most cases, the proposed method generates lower errors compared to existing imputation methods applied to benchmark dataset.