9,263 research outputs found
Adaptive Reduced Rank Regression
We study the low rank regression problem , where and are and dimensional
vectors respectively. We consider the extreme high-dimensional setting where
the number of observations is less than . Existing algorithms
are designed for settings where is typically as large as
. This work provides an efficient algorithm which
only involves two SVD, and establishes statistical guarantees on its
performance. The algorithm decouples the problem by first estimating the
precision matrix of the features, and then solving the matrix denoising
problem. To complement the upper bound, we introduce new techniques for
establishing lower bounds on the performance of any algorithm for this problem.
Our preliminary experiments confirm that our algorithm often out-performs
existing baselines, and is always at least competitive.Comment: 40 page
Subspace Tracking and Least Squares Approaches to Channel Estimation in Millimeter Wave Multiuser MIMO
The problem of MIMO channel estimation at millimeter wave frequencies, both
in a single-user and in a multi-user setting, is tackled in this paper. Using a
subspace approach, we develop a protocol enabling the estimation of the right
(resp. left) singular vectors at the transmitter (resp. receiver) side; then,
we adapt the projection approximation subspace tracking with deflation and the
orthogonal Oja algorithms to our framework and obtain two channel estimation
algorithms. We also present an alternative algorithm based on the least squares
approach. The hybrid analog/digital nature of the beamformer is also explicitly
taken into account at the algorithm design stage. In order to limit the system
complexity, a fixed analog beamformer is used at both sides of the
communication links. The obtained numerical results, showing the accuracy in
the estimation of the channel matrix dominant singular vectors, the system
achievable spectral efficiency, and the system bit-error-rate, prove that the
proposed algorithms are effective, and that they compare favorably, in terms of
the performance-complexity trade-off, with respect to several competing
alternatives.Comment: To appear on the IEEE Transactions on Communication
High Dimensional Semiparametric Scale-Invariant Principal Component Analysis
We propose a new high dimensional semiparametric principal component analysis
(PCA) method, named Copula Component Analysis (COCA). The semiparametric model
assumes that, after unspecified marginally monotone transformations, the
distributions are multivariate Gaussian. COCA improves upon PCA and sparse PCA
in three aspects: (i) It is robust to modeling assumptions; (ii) It is robust
to outliers and data contamination; (iii) It is scale-invariant and yields more
interpretable results. We prove that the COCA estimators obtain fast estimation
rates and are feature selection consistent when the dimension is nearly
exponentially large relative to the sample size. Careful experiments confirm
that COCA outperforms sparse PCA on both synthetic and real-world datasets.Comment: Accepted in IEEE Transactions on Pattern Analysis and Machine
Intelligence (TPMAI
Randomized Dimension Reduction on Massive Data
Scalability of statistical estimators is of increasing importance in modern
applications and dimension reduction is often used to extract relevant
information from data. A variety of popular dimension reduction approaches can
be framed as symmetric generalized eigendecomposition problems. In this paper
we outline how taking into account the low rank structure assumption implicit
in these dimension reduction approaches provides both computational and
statistical advantages. We adapt recent randomized low-rank approximation
algorithms to provide efficient solutions to three dimension reduction methods:
Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and
Localized Sliced Inverse Regression (LSIR). A key observation in this paper is
that randomization serves a dual role, improving both computational and
statistical performance. This point is highlighted in our experiments on real
and simulated data.Comment: 31 pages, 6 figures, Key Words:dimension reduction, generalized
eigendecompositon, low-rank, supervised, inverse regression, random
projections, randomized algorithms, Krylov subspace method
Diffusion Maps Kalman Filter for a Class of Systems with Gradient Flows
In this paper, we propose a non-parametric method for state estimation of
high-dimensional nonlinear stochastic dynamical systems, which evolve according
to gradient flows with isotropic diffusion. We combine diffusion maps, a
manifold learning technique, with a linear Kalman filter and with concepts from
Koopman operator theory. More concretely, using diffusion maps, we construct
data-driven virtual state coordinates, which linearize the system model. Based
on these coordinates, we devise a data-driven framework for state estimation
using the Kalman filter. We demonstrate the strengths of our method with
respect to both parametric and non-parametric algorithms in three tracking
problems. In particular, applying the approach to actual recordings of
hippocampal neural activity in rodents directly yields a representation of the
position of the animals. We show that the proposed method outperforms competing
non-parametric algorithms in the examined stochastic problem formulations.
Additionally, we obtain results comparable to classical parametric algorithms,
which, in contrast to our method, are equipped with model knowledge.Comment: 15 pages, 12 figures, submitted to IEEE TS
A nonparametric empirical Bayes approach to covariance matrix estimation
We propose an empirical Bayes method to estimate high-dimensional covariance
matrices. Our procedure centers on vectorizing the covariance matrix and
treating matrix estimation as a vector estimation problem. Drawing from the
compound decision theory literature, we introduce a new class of decision rules
that generalizes several existing procedures. We then use a nonparametric
empirical Bayes g-modeling approach to estimate the oracle optimal rule in that
class. This allows us to let the data itself determine how best to shrink the
estimator, rather than shrinking in a pre-determined direction such as toward a
diagonal matrix. Simulation results and a gene expression network analysis
shows that our approach can outperform a number of state-of-the-art proposals
in a wide range of settings, sometimes substantially.Comment: 20 pages, 4 figure
Signal processing methods for EEG data classification
Imperial Users onl
Subspace Leakage Analysis and Improved DOA Estimation with Small Sample Size
Classical methods of DOA estimation such as the MUSIC algorithm are based on
estimating the signal and noise subspaces from the sample covariance matrix.
For a small number of samples, such methods are exposed to performance
breakdown, as the sample covariance matrix can largely deviate from the true
covariance matrix. In this paper, the problem of DOA estimation performance
breakdown is investigated. We consider the structure of the sample covariance
matrix and the dynamics of the root-MUSIC algorithm. The performance breakdown
in the threshold region is associated with the subspace leakage where some
portion of the true signal subspace resides in the estimated noise subspace. In
this paper, the subspace leakage is theoretically derived. We also propose a
two-step method which improves the performance by modifying the sample
covariance matrix such that the amount of the subspace leakage is reduced.
Furthermore, we introduce a phenomenon named as root-swap which occurs in the
root-MUSIC algorithm in the low sample size region and degrades the performance
of the DOA estimation. A new method is then proposed to alleviate this problem.
Numerical examples and simulation results are given for uncorrelated and
correlated sources to illustrate the improvement achieved by the proposed
methods. Moreover, the proposed algorithms are combined with the pseudo-noise
resampling method to further improve the performance.Comment: 37 pages, 10 figures, Submitted to the IEEE Transactions on Signal
Processing in July 201
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