Title: Geometric Models for Dimensionality Reduction in Signal and Data
Processing

                 	Michael Wakin
   Center for the Mathematics of Information
       California Institute of Technology

ABSTRACT:  The burden of processing high-dimensional signals and data can
often be alleviated by exploiting low-dimensional models for the behavior of
the data. Examples of such models include bandlimited signals, signals
having a sparse representation in terms of some basis or dictionary, or
signals clustering along low-dimensional manifolds within the ambient signal
space. These type of models are the basis and justification for a variety of
techniques that aim to alleviate the burden of dimensionality in signal and
data processing (examples include data compression, parameter estimation,
manifold learning, and so on); these can loosely be termed methods for
"dimensionality reduction".

In these talks I will overview several recent developments in dimensionality
reduction, including new insight and analysis into the geometry of
low-dimensional signal models, and radically new methods for dimensionality
reduction inspired by these models.

For the first talk I will focus primarily on Compressive Sensing (CS), an
emerging field based on the revelation that a signal having a sparse
representation in some basis can be recovered from a small number of
projections onto a second basis that is incoherent with the first. (A random
measurement basis typically suffices.) This fact has many promising
implications in signal and data processing and has inspired the development
of dramatically more efficient devices for data collection. Taking a largely
geometric perspective, I will overview several of the basic fundamental
results in CS concerning random matrices, and based on parallels with the
Johnson-Lindenstrauss Lemma and Whitney's Embedding Theorem, I will also
discuss possible extensions of CS from sparsity-based signal models to
manifold-based signal models.