  High-dimensional big data appears in many research fields such as image
recognition, biology and collaborative filtering. Often, the exploration of
such data by classic algorithms is encountered with difficulties due to `curse
of dimensionality' phenomenon. Therefore, dimensionality reduction methods are
applied to the data prior to its analysis. Many of these methods are based on
principal components analysis, which is statistically driven, namely they map
the data into a low-dimension subspace that preserves significant statistical
properties of the high-dimensional data. As a consequence, such methods do not
directly address the geometry of the data, reflected by the mutual distances
between multidimensional data point. Thus, operations such as classification,
anomaly detection or other machine learning tasks may be affected.
  This work provides a dictionary-based framework for geometrically driven data
analysis that includes dimensionality reduction, out-of-sample extension and
anomaly detection. It embeds high-dimensional data in a low-dimensional
subspace. This embedding preserves the original high-dimensional geometry of
the data up to a user-defined distortion rate. In addition, it identifies a
subset of landmark data points that constitute a dictionary for the analyzed
dataset. The dictionary enables to have a natural extension of the
low-dimensional embedding to out-of-sample data points, which gives rise to a
distortion-based criterion for anomaly detection. The suggested method is
demonstrated on synthetic and real-world datasets and achieves good results for
classification, anomaly detection and out-of-sample tasks.
