13,309 research outputs found
Regularization with Approximated Maximum Entropy Method
We tackle the inverse problem of reconstructing an unknown finite measure
from a noisy observation of a generalized moment of defined as the
integral of a continuous and bounded operator with respect to .
When only a quadratic approximation of the operator is known, we
introduce the approximate maximum entropy solution as a minimizer of a
convex functional subject to a sequence of convex constraints. Under several
assumptions on the convex functional, the convergence of the approximate
solution is established and rates of convergence are provided.Comment: 16 page
Takeuchi's Information Criteria as a form of Regularization
Takeuchi's Information Criteria (TIC) is a linearization of maximum
likelihood estimator bias which shrinks the model parameters towards the
maximum entropy distribution, even when the model is mis-specified. In
statistical machine learning, regularization (a.k.a. ridge regression)
also introduces a parameterized bias term with the goal of minimizing
out-of-sample entropy, but generally requires a numerical solver to find the
regularization parameter. This paper presents a novel regularization approach
based on TIC; the approach does not assume a data generation process and
results in a higher entropy distribution through more efficient sample noise
suppression. The resulting objective function can be directly minimized to
estimate and select the best model, without the need to select a regularization
parameter, as in ridge regression. Numerical results applied to a synthetic
high dimensional dataset generated from a logistic regression model demonstrate
superior model performance when using the TIC based regularization over a
and a penalty term
Fast optimization of Multithreshold Entropy Linear Classifier
Multithreshold Entropy Linear Classifier (MELC) is a density based model
which searches for a linear projection maximizing the Cauchy-Schwarz Divergence
of dataset kernel density estimation. Despite its good empirical results, one
of its drawbacks is the optimization speed. In this paper we analyze how one
can speed it up through solving an approximate problem. We analyze two methods,
both similar to the approximate solutions of the Kernel Density Estimation
querying and provide adaptive schemes for selecting a crucial parameters based
on user-specified acceptable error. Furthermore we show how one can exploit
well known conjugate gradients and L-BFGS optimizers despite the fact that the
original optimization problem should be solved on the sphere. All above methods
and modifications are tested on 10 real life datasets from UCI repository to
confirm their practical usability.Comment: Presented at Theoretical Foundations of Machine Learning 2015
(http://tfml.gmum.net), final version published in Schedae Informaticae
Journa
A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
We develop a high-order kinetic scheme for entropy-based moment models of a
one-dimensional linear kinetic equation in slab geometry. High-order spatial
reconstructions are achieved using the weighted essentially non-oscillatory
(WENO) method, and for time integration we use multi-step Runge-Kutta methods
which are strong stability preserving and whose stages and steps can be written
as convex combinations of forward Euler steps. We show that the moment vectors
stay in the realizable set using these time integrators along with a maximum
principle-based kinetic-level limiter, which simultaneously dampens spurious
oscillations in the numerical solutions. We present numerical results both on a
manufactured solution, where we perform convergence tests showing our scheme
converges of the expected order up to the numerical noise from the numerical
optimization, as well as on two standard benchmark problems, where we show some
of the advantages of high-order solutions and the role of the key parameter in
the limiter
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