1,131 research outputs found
Structured sparsity-inducing norms through submodular functions
Sparse methods for supervised learning aim at finding good linear predictors
from as few variables as possible, i.e., with small cardinality of their
supports. This combinatorial selection problem is often turned into a convex
optimization problem by replacing the cardinality function by its convex
envelope (tightest convex lower bound), in this case the L1-norm. In this
paper, we investigate more general set-functions than the cardinality, that may
incorporate prior knowledge or structural constraints which are common in many
applications: namely, we show that for nondecreasing submodular set-functions,
the corresponding convex envelope can be obtained from its \lova extension, a
common tool in submodular analysis. This defines a family of polyhedral norms,
for which we provide generic algorithmic tools (subgradients and proximal
operators) and theoretical results (conditions for support recovery or
high-dimensional inference). By selecting specific submodular functions, we can
give a new interpretation to known norms, such as those based on
rank-statistics or grouped norms with potentially overlapping groups; we also
define new norms, in particular ones that can be used as non-factorial priors
for supervised learning
Quasi-concave density estimation
Maximum likelihood estimation of a log-concave probability density is
formulated as a convex optimization problem and shown to have an equivalent
dual formulation as a constrained maximum Shannon entropy problem. Closely
related maximum Renyi entropy estimators that impose weaker concavity
restrictions on the fitted density are also considered, notably a minimum
Hellinger discrepancy estimator that constrains the reciprocal of the
square-root of the density to be concave. A limiting form of these estimators
constrains solutions to the class of quasi-concave densities.Comment: Published in at http://dx.doi.org/10.1214/10-AOS814 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Data Filtering for Cluster Analysis by -Norm Regularization
A data filtering method for cluster analysis is proposed, based on minimizing
a least squares function with a weighted -norm penalty. To overcome the
discontinuity of the objective function, smooth non-convex functions are
employed to approximate the -norm. The convergence of the global
minimum points of the approximating problems towards global minimum points of
the original problem is stated. The proposed method also exploits a suitable
technique to choose the penalty parameter. Numerical results on synthetic and
real data sets are finally provided, showing how some existing clustering
methods can take advantages from the proposed filtering strategy.Comment: Optimization Letters (2017
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