10,603 research outputs found
Gradient Hard Thresholding Pursuit for Sparsity-Constrained Optimization
Hard Thresholding Pursuit (HTP) is an iterative greedy selection procedure
for finding sparse solutions of underdetermined linear systems. This method has
been shown to have strong theoretical guarantee and impressive numerical
performance. In this paper, we generalize HTP from compressive sensing to a
generic problem setup of sparsity-constrained convex optimization. The proposed
algorithm iterates between a standard gradient descent step and a hard
thresholding step with or without debiasing. We prove that our method enjoys
the strong guarantees analogous to HTP in terms of rate of convergence and
parameter estimation accuracy. Numerical evidences show that our method is
superior to the state-of-the-art greedy selection methods in sparse logistic
regression and sparse precision matrix estimation tasks
A Proximal-Gradient Homotopy Method for the Sparse Least-Squares Problem
We consider solving the -regularized least-squares (-LS)
problem in the context of sparse recovery, for applications such as compressed
sensing. The standard proximal gradient method, also known as iterative
soft-thresholding when applied to this problem, has low computational cost per
iteration but a rather slow convergence rate. Nevertheless, when the solution
is sparse, it often exhibits fast linear convergence in the final stage. We
exploit the local linear convergence using a homotopy continuation strategy,
i.e., we solve the -LS problem for a sequence of decreasing values of
the regularization parameter, and use an approximate solution at the end of
each stage to warm start the next stage. Although similar strategies have been
studied in the literature, there have been no theoretical analysis of their
global iteration complexity. This paper shows that under suitable assumptions
for sparse recovery, the proposed homotopy strategy ensures that all iterates
along the homotopy solution path are sparse. Therefore the objective function
is effectively strongly convex along the solution path, and geometric
convergence at each stage can be established. As a result, the overall
iteration complexity of our method is for finding an
-optimal solution, which can be interpreted as global geometric rate
of convergence. We also present empirical results to support our theoretical
analysis
A Proximal Stochastic Gradient Method with Progressive Variance Reduction
We consider the problem of minimizing the sum of two convex functions: one is
the average of a large number of smooth component functions, and the other is a
general convex function that admits a simple proximal mapping. We assume the
whole objective function is strongly convex. Such problems often arise in
machine learning, known as regularized empirical risk minimization. We propose
and analyze a new proximal stochastic gradient method, which uses a multi-stage
scheme to progressively reduce the variance of the stochastic gradient. While
each iteration of this algorithm has similar cost as the classical stochastic
gradient method (or incremental gradient method), we show that the expected
objective value converges to the optimum at a geometric rate. The overall
complexity of this method is much lower than both the proximal full gradient
method and the standard proximal stochastic gradient method
Neutrino Masses and Heavy Triplet Leptons at the LHC: Testability of Type III Seesaw
We study LHC signatures of Type III seesaw in which SU(2)_L triplet leptons
are introduced to supply the heavy seesaw masses. To detect the signals of
these heavy triplet leptons, one needs to understand their decays to standard
model particles which depend on how light and heavy leptons mix with each
other. We concentrate on the usual solutions with small light and heavy lepton
mixing of order the square root of the ratio of light and heavy masses,
(m_\nu/M_{\nu_R})^{1/2}. This class of solutions can lead to a visible
displaced vertex detectable at the LHC which can be used to distinguish small
mixing and large mixing between light and heavy leptons. We show that, in this
case, the couplings of light and heavy triplet leptons to gauge and Higgs
bosons, which determine the decay widths and branching ratios, can be expressed
in terms of light neutrino masses and their mixing. Using these relations, we
study heavy triplet lepton decay patterns and production cross section at the
LHC. If these heavy triplet leptons are below a TeV or so, they can be easily
produced at the LHC due to their gauge interactions from being non-trivial
representations of SU(2)_L. We consider two ideal production channels, 1)
E^+E^- \to \ell^+\ell^+ \ell^-\ell^- jj (\ell=e,\mu,\tau) and 2) E^\pm N \to
\ell^\pm \ell^\pm jjjj in detail. For case 1), we find that with one or two of
the light leptons being \tau it can also be effectively studied. With judicious
cuts at the LHC, the discovery of the heavy triplet leptons as high as a TeV
can be achieved with 100 fb^{-1} integrated luminosity.Comment: 39 pages, 36 figures, accepted version by PR
Modelling the Dynamic Relationship between Systematic Default and Recovery Risk
Default correlation modelling is becoming the most popular problem in the
field of credit derivatives pricing. An increase in default risk would cause the
recovery rate to change correspondingly. Correlation between default and
recovery rates has a noticeable effect on risk measures and credit derivatives
pricing.
After an introduction, we review the most recent literature covering default
correlation and the relationship between default and recovery rates. We
adopt the copula methodology to focus on estimating the default correlations
rather than focus on modelling probabilities of default, we then use stress
testing to compare the distributions of the probability of default under different
copula functions. We develop a Gamma-Beta model to link the recovery
rate directly with the individual probability of default, this is instead of an
extended one factor model to relate them by a systematic common factor.
One factor models are re-examined to explore correlated recovery rates under
three distributions: the Logit-normal, the Normal and the Log-normal. By
analyzing the results respectively obtained from these two classes of modelling
scheme, we argue that the direct dependence (Gamma-Beta) model
behaves better, in estimating the recovery rate given individual probability
of default and in suggesting a better indication of their relationship. Finally,
we apply default correlation and the correlated recovery rate to portfolio risk
modelling. We conclude that if the recovery rates are independent stochastic
variables, the expected losses in a large portfolio might be underestimated
because the uncorrelated recovery risks can be diversified, so the correlation
between default rate and recovery risk can not be neglected in the applications.
Here, we believe the first time, the recovery rate depends on individual
default probability by means of a closed formula
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