## Penalised regression

On regression estimation with penalties on the model.
Practically this means choosing appropriate smoothing to do good model selection, and possibly using some clever optimisation method.
Related to compressed sensing but here we consider sampling complexity,
the effect of measurement noise, and more general penalties than just $$\ell_1$$.

To discuss:

LARS, LASSO, Group LASSO, de-biassed LASSO, Elastic net, etc.

In nonparametric statistics we might estimate simultaneously what look like
many, many parameters, which we constrain in some clever fashion,
which usually boils down to something we can interpret as a “smoothing”
parameters, controlling how many parameters we still have to model
from a subset of the original.

The “regularisation” nomenclature claims descent from Tikhonov, (eg TiGl65 etc) who wanted to solve ill-conditioned integral and differential equations, so it’s slightly more general.
“Smoothing” seems to be common in the
spline and
kernel estimate communities of
Wahba (Wahb90) and Silverman (Silv82) et al,
who usually actually want to smooth curves.

Penalization” has a geneology unknown to me, but is probably the least abstruse for common usage.

These are, AFAICT, more or less the same thing.
“smoothing” is more common in my communities which is fine,
but we have to remember that “smoothing” an estimator might not always infer smooth dynamics in the estimand;
it could be something else being smoothed, such as variance in the estimate of parameters of a rough function.

In every case, you wish to solve an ill-conditioned inverse problem, so you tame it by adding a penalty to solutions you feel one should be reluctant to accept.

TODO: make comprehensible

TODO: examples

TODO: discuss connection with model selection

TODO: discuss connection with compressed sensing.

The real classic approach here is spline smoothing of functional data.
More recent approaches are things like sparse regression.

## Debiassed LASSO

See GBRD14 and Geer14c.

## Implementations

I’m not going to mention LASSO in (generalised) linear regression,
since everything does that these days (Oh alright,
Jerome Friedman’s glmnet for R is the fastest,
and has a MATLAB version.

But SPAMS (C++, MATLAB, R, python) by Mairal himself, looks interesting.
It’s an optimisation library for many various in sparse problems.

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See original: Penalised regression