177,435 research outputs found
Explicit and almost sure conditions for K/2 degrees of freedom
It is well known that in K-user constant single-antenna interference channels
K/2 degrees of freedom (DoF) can be achieved for almost all channel matrices.
Explicit conditions on the channel matrix to admit K/2 DoF are, however, not
available. The purpose of this paper is to identify such explicit conditions,
which are satisfied for almost all channel matrices. We also provide a
construction of corresponding asymptotically DoF-optimal input distributions.
The main technical tool used is a recent breakthrough result by Hochman in
fractal geometry.Comment: To be presented at IEEE Int. Symp. Inf. Theory 2014, Honolulu, H
A comment on Stein's unbiased risk estimate for reduced rank estimators
In the framework of matrix valued observables with low rank means, Stein's
unbiased risk estimate (SURE) can be useful for risk estimation and for tuning
the amount of shrinkage towards low rank matrices. This was demonstrated by
Cand\`es et al. (2013) for singular value soft thresholding, which is a
Lipschitz continuous estimator. SURE provides an unbiased risk estimate for an
estimator whenever the differentiability requirements for Stein's lemma are
satisfied. Lipschitz continuity of the estimator is sufficient, but it is
emphasized that differentiability Lebesgue almost everywhere isn't. The reduced
rank estimator, which gives the best approximation of the observation with a
fixed rank, is an example of a discontinuous estimator for which Stein's lemma
actually applies. This was observed by Mukherjee et al. (2015), but the proof
was incomplete. This brief note gives a sufficient condition for Stein's lemma
to hold for estimators with discontinuities, which is then shown to be
fulfilled for a class of spectral function estimators including the reduced
rank estimator. Singular value hard thresholding does, however, not satisfy the
condition, and Stein's lemma does not apply to this estimator.Comment: 11 pages, 1 figur
Excess Optimism: How Biased is the Apparent Error of an Estimator Tuned by SURE?
Nearly all estimators in statistical prediction come with an associated
tuning parameter, in one way or another. Common practice, given data, is to
choose the tuning parameter value that minimizes a constructed estimate of the
prediction error of the estimator; we focus on Stein's unbiased risk estimator,
or SURE (Stein, 1981; Efron, 1986) which forms an unbiased estimate of the
prediction error by augmenting the observed training error with an estimate of
the degrees of freedom of the estimator. Parameter tuning via SURE minimization
has been advocated by many authors, in a wide variety of problem settings, and
in general, it is natural to ask: what is the prediction error of the
SURE-tuned estimator? An obvious strategy would be simply use the apparent
error estimate as reported by SURE, i.e., the value of the SURE criterion at
its minimum, to estimate the prediction error of the SURE-tuned estimator. But
this is no longer unbiased; in fact, we would expect that the minimum of the
SURE criterion is systematically biased downwards for the true prediction
error. In this paper, we formally describe and study this bias.Comment: 39 pages, 3 figure
On Degrees of Freedom of Projection Estimators with Applications to Multivariate Nonparametric Regression
In this paper, we consider the nonparametric regression problem with
multivariate predictors. We provide a characterization of the degrees of
freedom and divergence for estimators of the unknown regression function, which
are obtained as outputs of linearly constrained quadratic optimization
procedures, namely, minimizers of the least squares criterion with linear
constraints and/or quadratic penalties. As special cases of our results, we
derive explicit expressions for the degrees of freedom in many nonparametric
regression problems, e.g., bounded isotonic regression, multivariate
(penalized) convex regression, and additive total variation regularization. Our
theory also yields, as special cases, known results on the degrees of freedom
of many well-studied estimators in the statistics literature, such as ridge
regression, Lasso and generalized Lasso. Our results can be readily used to
choose the tuning parameter(s) involved in the estimation procedure by
minimizing the Stein's unbiased risk estimate. As a by-product of our analysis
we derive an interesting connection between bounded isotonic regression and
isotonic regression on a general partially ordered set, which is of independent
interest.Comment: 72 pages, 7 figures, Journal of the American Statistical Association
(Theory and Methods), 201
Hamiltonian analysis of Poincar\'e gauge theory scalar modes
The Hamiltonian constraint formalism is used to obtain the first explicit
complete analysis of non-trivial viable dynamic modes for the Poincar\'e gauge
theory of gravity. Two modes with propagating spin-zero torsion are analyzed.
The explicit form of the Hamiltonian is presented. All constraints are obtained
and classified. The Lagrange multipliers are derived. It is shown that a
massive spin- mode has normal dynamical propagation but the associated
massless is pure gauge. The spin- mode investigated here is also
viable in general. Both modes exhibit a simple type of ``constraint
bifurcation'' for certain special field/parameter values.Comment: 28 pages, LaTex, submitted to International Journal of Modern Physics
Hamiltonian Analysis of Poincar\'e Gauge Theory: Higher Spin Modes
We examine several higher spin modes of the Poincar\'e gauge theory (PGT) of
gravity using the Hamiltonian analysis. The appearance of certain undesirable
effects due to non-linear constraints in the Hamiltonian analysis are used as a
test. We find that the phenomena of field activation and constraint bifurcation
both exist in the pure spin 1 and the pure spin 2 modes. The coupled spin-
and spin- modes also fail our test due to the appearance of constraint
bifurcation. The ``promising'' case in the linearized theory of PGT given by
Kuhfuss and Nitsch (KRNJ86) likewise does not pass. From this analysis of these
specific PGT modes we conclude that an examination of such nonlinear constraint
effects shows great promise as a strong test for this and other alternate
theories of gravity.Comment: 30 pages, submitted to Int. J. Mod. Phys.
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