5,963 research outputs found

### TIGER: A Tuning-Insensitive Approach for Optimally Estimating Gaussian Graphical Models

We propose a new procedure for estimating high dimensional Gaussian graphical
models. Our approach is asymptotically tuning-free and non-asymptotically
tuning-insensitive: it requires very few efforts to choose the tuning parameter
in finite sample settings. Computationally, our procedure is significantly
faster than existing methods due to its tuning-insensitive property.
Theoretically, the obtained estimator is simultaneously minimax optimal for
precision matrix estimation under different norms. Empirically, we illustrate
the advantages of our method using thorough simulated and real examples. The R
package bigmatrix implementing the proposed methods is available on the
Comprehensive R Archive Network: http://cran.r-project.org/

### Adaptive variance function estimation in heteroscedastic nonparametric regression

We consider a wavelet thresholding approach to adaptive variance function
estimation in heteroscedastic nonparametric regression. A data-driven estimator
is constructed by applying wavelet thresholding to the squared first-order
differences of the observations. We show that the variance function estimator
is nearly optimally adaptive to the smoothness of both the mean and variance
functions. The estimator is shown to achieve the optimal adaptive rate of
convergence under the pointwise squared error simultaneously over a range of
smoothness classes. The estimator is also adaptively within a logarithmic
factor of the minimax risk under the global mean integrated squared error over
a collection of spatially inhomogeneous function classes. Numerical
implementation and simulation results are also discussed.Comment: Published in at http://dx.doi.org/10.1214/07-AOS509 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Empirical information on nuclear matter fourth-order symmetry energy from an extended nuclear mass formula

We establish a relation between the equation of state (EOS) of nuclear matter
and the fourth-order symmetry energy $a_{\rm{sym,4}}(A)$ of finite nuclei in a
semi-empirical nuclear mass formula by self-consistently considering the bulk,
surface and Coulomb contributions to the nuclear mass. Such a relation allows
us to extract information on nuclear matter fourth-order symmetry energy
$E_{\rm{sym,4}}(\rho_0)$ at normal nuclear density $\rho_0$ from analyzing
nuclear mass data. Based on the recent precise extraction of
$a_{\rm{sym,4}}(A)$ via the double difference of the "experimental" symmetry
energy extracted from nuclear masses, for the first time, we estimate a value
of $E_{\rm{sym,4}}(\rho_0) = 20.0\pm4.6$ MeV. Such a value of
$E_{\rm{sym,4}}(\rho_0)$ is significantly larger than the predictions from
mean-field models and thus suggests the importance of considering the effects
of beyond the mean-field approximation in nuclear matter calculations.Comment: 7 pages, 1 figure. Presentation improved and discussions added.
Accepted version to appear in PL

### Calibrated Multivariate Regression with Application to Neural Semantic Basis Discovery

We propose a calibrated multivariate regression method named CMR for fitting
high dimensional multivariate regression models. Compared with existing
methods, CMR calibrates regularization for each regression task with respect to
its noise level so that it simultaneously attains improved finite-sample
performance and tuning insensitiveness. Theoretically, we provide sufficient
conditions under which CMR achieves the optimal rate of convergence in
parameter estimation. Computationally, we propose an efficient smoothed
proximal gradient algorithm with a worst-case numerical rate of convergence
\cO(1/\epsilon), where $\epsilon$ is a pre-specified accuracy of the
objective function value. We conduct thorough numerical simulations to
illustrate that CMR consistently outperforms other high dimensional
multivariate regression methods. We also apply CMR to solve a brain activity
prediction problem and find that it is as competitive as a handcrafted model
created by human experts. The R package \texttt{camel} implementing the
proposed method is available on the Comprehensive R Archive Network
\url{http://cran.r-project.org/web/packages/camel/}.Comment: Journal of Machine Learning Research, 201

### Pivotal estimation via square-root Lasso in nonparametric regression

We propose a self-tuning $\sqrt{\mathrm {Lasso}}$ method that simultaneously
resolves three important practical problems in high-dimensional regression
analysis, namely it handles the unknown scale, heteroscedasticity and (drastic)
non-Gaussianity of the noise. In addition, our analysis allows for badly
behaved designs, for example, perfectly collinear regressors, and generates
sharp bounds even in extreme cases, such as the infinite variance case and the
noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds
for $\sqrt{\mathrm {Lasso}}$ including prediction norm rate and sparsity. Our
analysis is based on new impact factors that are tailored for bounding
prediction norm. In order to cover heteroscedastic non-Gaussian noise, we rely
on moderate deviation theory for self-normalized sums to achieve Gaussian-like
results under weak conditions. Moreover, we derive bounds on the performance of
ordinary least square (ols) applied to the model selected by $\sqrt{\mathrm
{Lasso}}$ accounting for possible misspecification of the selected model. Under
mild conditions, the rate of convergence of ols post $\sqrt{\mathrm {Lasso}}$
is as good as $\sqrt{\mathrm {Lasso}}$'s rate. As an application, we consider
the use of $\sqrt{\mathrm {Lasso}}$ and ols post $\sqrt{\mathrm {Lasso}}$ as
estimators of nuisance parameters in a generic semiparametric problem
(nonlinear moment condition or $Z$-problem), resulting in a construction of
$\sqrt{n}$-consistent and asymptotically normal estimators of the main
parameters.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1204 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming

We propose a pivotal method for estimating high-dimensional sparse linear
regression models, where the overall number of regressors $p$ is large,
possibly much larger than $n$, but only $s$ regressors are significant. The
method is a modification of the lasso, called the square-root lasso. The method
is pivotal in that it neither relies on the knowledge of the standard deviation
$\sigma$ or nor does it need to pre-estimate $\sigma$. Moreover, the method
does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle
performance, attaining the convergence rate $\sigma \{(s/n)\log p\}^{1/2}$ in
the prediction norm, and thus matching the performance of the lasso with known
$\sigma$. These performance results are valid for both Gaussian and
non-Gaussian errors, under some mild moment restrictions. We formulate the
square-root lasso as a solution to a convex conic programming problem, which
allows us to implement the estimator using efficient algorithmic methods, such
as interior-point and first-order methods

### New Bounds for Restricted Isometry Constants

In this paper we show that if the restricted isometry constant $\delta_k$ of
the compressed sensing matrix satisfies $\delta_k < 0.307,$ then $k$-sparse
signals are guaranteed to be recovered exactly via $\ell_1$ minimization when
no noise is present and $k$-sparse signals can be estimated stably in the noisy
case. It is also shown that the bound cannot be substantively improved. An
explicitly example is constructed in which $\delta_{k}=\frac{k-1}{2k-1} < 0.5$,
but it is impossible to recover certain $k$-sparse signals

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