Signal Processing on High-Dimensional Data Through A Low-Dimensional Embedding

In this work, we explore methods for the inverse process to dimensionality reduction in data signal processing.  That is, the recovery of a high-dimensional observation from its low-dimensional counterpart such that the inherent structure in the data is preserved.  Processing high-dimensional data in the ambient space quickly becomes computationally expensive and often impossible.  As a result, methods have been developed to reduce the dimensionality of the data to a level where the desired operations can be performed.  In cases where the operation of interest produces an out-of-sample point, in the low-dimensional space, one has limited tools to reproduce the out-of-sample point in the high-dimensional ambient space.  We propose a method for recovering a “best-fit” high-dimensional realization from its low-dimensional counterpart.  We have performed experiments on data drawn form statistical distributions, low-dimensional smooth manifolds, and image data sets.  These experiments consist of three major parts.  First, we discover a low-dimensional embedding for the high-dimensional data set.  Second, we construct an out-of-sample interpolation point in the low-dimensional space.  Finally, we recover the “best-fit” realization of the interpolation point in the high-dimensional space where the data resides.  Our work is primarily focused on developing an inverse method for the dimensionality reduction technique of Metric Multidimensional Scaling.  Our results suggest proof of concept and we are currently researching comparable methods.