# Pixel Club Seminar: Reading into the MAP: how mean is it ?

- Speaker:
- Remi Gribonval (INRIA)
- Date:
- Tuesday, 19.10.2010, 14:30
- Place:
- Room 337-8 Taub Bld.

Should penalized least squares regression be interpreted as
Maximum A Posteriori estimation?
Penalized least squares regression is often used for signal denoising and
inverse problems, and is commonly interpreted in a Bayesian framework as a
Maximum A Posteriori (MAP) estimator, the penalty function being the
negative logarithm of the prior. For example, the widely used quadratic
program (with an $\ell^1$ penalty) associated to the LASSO / Basis Pursuit
Denoising is very often considered as the MAP under a Laplacian prior in the
context of additive white Gaussian noise (AWGN) reduction. The objective of
this talk is to highlight the fact that, while this is {\em one} possible
Bayesian interpretation, there can be other equally acceptable Bayesian
interpretations. Therefore, solving a penalized least squares regression
problem with penalty $\phi(x)$ need not be interpreted as assuming a prior
$C\cdot \exp(-\phi(x))$ and using the MAP estimator. In particular, I will
show that for {\em any} prior $p_X(x)$, the conditional mean estimator can
be interpreted as a MAP with some prior $C \cdot \exp(-\phi(x))$.
Vice-versa, for {\em certain} penalties $\phi(x)$, the solution of the
penalized least squares problem is indeed the {\em conditional mean}, with a
certain prior $p_X(x)$. In general we have $p_X(x) \neq C \cdot
\exp(-\phi(x))$.

If time allows I will also discuss recent joint work with Volkan Cevher
(EPFL) and Mike Davies (University of Edinburgh) on "compressible priors",
in connection with sparse regularization in linear inverse problems such as
compressed sensing. I will show in particular that Laplace distributed
vectors cannot be considered as typically "compressible": they are not
sufficiently well approximated by sparse vectors to be recovered from a
low-dimensional random projection by, e.g., L1 minimization. This may be
somewhat of a surprise considering that L1 minimization is associated to MAP
estimation under the Laplace prior!