In applications throughout science and engineering one is often faced with
the challenge of solving an ill-posed inverse problem, where the number of
available measurements is smaller than the dimension of the model to be
estimated. However in many practical situations of interest, models are
constrained structurally so that they only have a few degrees of freedom
relative to their ambient dimension. This paper provides a general framework to
convert notions of simplicity into convex penalty functions, resulting in
convex optimization solutions to linear, underdetermined inverse problems.

Suppose we have samples of a \emph{subset} of a collection of random
variables. No additional information is provided about the number of latent
variables, nor of the relationship between the latent and observed variables.
Is it possible to discover the number of hidden components, and to learn a
statistical model over the entire collection of variables? We address this
question in the setting in which the latent and observed variables are jointly
Gaussian, with the conditional statistics of the observed variables conditioned
on the latent variables being specified by a graphical model.

In this paper we introduce a novel flow representation for finite games in
strategic form. This representation allows us to develop a canonical direct sum
decomposition of an arbitrary game into three components, which we refer to as
the potential, harmonic and nonstrategic components. We analyze natural classes
of games that are induced by this decomposition, and in particular, focus on
games with no harmonic component and games with no potential component. We show
that the first class corresponds to the well-known potential games.