Some new phenomena in high-dimensional statistics and optimization
Statistical
models
in
which
the
ambient
dimension
is
of
the
same
order
or
larger
than
the
sample
size
arise
frequently
in
different
areas
of
science
and
engineering.
Examples
include
sparse
regression
in
genomics;
graph
selection
in
social
network
analysis;
and
low-rank
matrix
estimation
in
video
segmentation.
Although
high-dimensional
models
of
this
type
date
back
to
seminal
work
of
Kolmogorov
and
others,
they
have
been
the
subject
of
especially
intensive
study
over
the
past
decade,
and
have
interesting
connections
to
many
branches
of
applied
mathematics
and
computer
science,
including
random
matrices
and
algorithms,
concentration
of
measure,
convex
geometry,
and
information
theory.
In
this
talk,
we
discuss
various
issues
in
high-dimensional
statistics,
including
vignettes
on
phase
transitions
in
high-dimensional
graph
recovery,
and
randomized
sketching
for
large-scale
optimization.