[Colloquium] REMINDER: 7/15 TTIC Colloquium: Yuan Yao, Peking University (NOTE UNUSUAL DAY)

[Mary Marre]

When:     Wednesday, July 15th at 11:00 a.m. (note unusual day)

Where:    TTIC, 6045 S Kenwood Avenue, 5th Floor, Room 526

Who:       Yuan Yao, Peking University

Title:        Sparse Recovery via Dynamics

Abstract: In this talk, we recover sparse signals from their noisy linear
measurements by solving nonlinear differential equations (inclusions),
which is based on the notion of inverse scale space (ISS) developed in
applied mathematics. Our goal here is to bring this idea to address a
challenging problem in statistics, \emph{i.e.} finding unbiased and
sign-consistent estimators using dynamics. We call our dynamics
\emph{Bregman ISS} and \emph{Linearized Bregman ISS}. A well-known
shortcoming of LASSO and any convex regularization approaches lies in the
bias of estimators. However, we show that under proper conditions, there
exists a bias-free and sign-consistent point on the solution paths of such
dynamics, which corresponds to a signal that is the unbiased estimate of
the true signal and whose entries have the same signs as those of the true
signs, \emph{i.e.} the oracle estimator. Therefore, their solution paths
are regularization paths better than the LASSO regularization path, since
the points on the latter path are biased when sign-consistency is reached.
We also show how to efficiently compute their solution paths in both
continuous and discretized settings: the full solution paths can be exactly
computed piece by piece, and a discretization leads to \emph{Linearized
Bregman iteration}, which is a simple iterative thresholding rule and easy
to parallelize. Theoretical guarantees such as sign-consistency and minimax
optimal $l_2$-error bounds are established in both continuous and discrete
settings for specific points on the paths. Early-stopping rules for
identifying these points are given. The key treatment relies on the
development of differential inequalities for differential inclusions and
their discretizations, which extends the previous results and leads to
exponentially fast recovering of sparse signals before selecting wrong ones.


This is a joint work with Feng Ruan (Stanford), Jiechao Xiong (PKU),
Stanley Osher and Wotao Yin (UCLA).


Host:  *Jinbo Xu*





Mary C. Marre
Administrative Assistant
*Toyota Technological Institute*
*6045 S. Kenwood Avenue*
*Room 504*
*Chicago, IL  60637*
*p:(773) 834-1757 <%28773%29%20834-1757>*
*f: (773) 357-6970 <%28773%29%20357-6970>*
*mmarre at ttic.edu <mmarre at ttic.edu>*
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