1,263 research outputs found
Covariance Estimation in High Dimensions via Kronecker Product Expansions
This paper presents a new method for estimating high dimensional covariance
matrices. The method, permuted rank-penalized least-squares (PRLS), is based on
a Kronecker product series expansion of the true covariance matrix. Assuming an
i.i.d. Gaussian random sample, we establish high dimensional rates of
convergence to the true covariance as both the number of samples and the number
of variables go to infinity. For covariance matrices of low separation rank,
our results establish that PRLS has significantly faster convergence than the
standard sample covariance matrix (SCM) estimator. The convergence rate
captures a fundamental tradeoff between estimation error and approximation
error, thus providing a scalable covariance estimation framework in terms of
separation rank, similar to low rank approximation of covariance matrices. The
MSE convergence rates generalize the high dimensional rates recently obtained
for the ML Flip-flop algorithm for Kronecker product covariance estimation. We
show that a class of block Toeplitz covariance matrices is approximatable by
low separation rank and give bounds on the minimal separation rank that
ensures a given level of bias. Simulations are presented to validate the
theoretical bounds. As a real world application, we illustrate the utility of
the proposed Kronecker covariance estimator for spatio-temporal linear least
squares prediction of multivariate wind speed measurements.Comment: 47 pages, accepted to IEEE Transactions on Signal Processin
Regularized Block Toeplitz Covariance Matrix Estimation via Kronecker Product Expansions
In this work we consider the estimation of spatio-temporal covariance
matrices in the low sample non-Gaussian regime. We impose covariance structure
in the form of a sum of Kronecker products decomposition (Tsiligkaridis et al.
2013, Greenewald et al. 2013) with diagonal correction (Greenewald et al.),
which we refer to as DC-KronPCA, in the estimation of multiframe covariance
matrices. This paper extends the approaches of (Tsiligkaridis et al.) in two
directions. First, we modify the diagonally corrected method of (Greenewald et
al.) to include a block Toeplitz constraint imposing temporal stationarity
structure. Second, we improve the conditioning of the estimate in the very low
sample regime by using Ledoit-Wolf type shrinkage regularization similar to
(Chen, Hero et al. 2010). For improved robustness to heavy tailed
distributions, we modify the KronPCA to incorporate robust shrinkage estimation
(Chen, Hero et al. 2011). Results of numerical simulations establish benefits
in terms of estimation MSE when compared to previous methods. Finally, we apply
our methods to a real-world network spatio-temporal anomaly detection problem
and achieve superior results.Comment: To appear at IEEE SSP 2014 4 page
Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach
This paper introduces and analyses the new grid-based tensor approach to
approximate solution of the elliptic eigenvalue problem for the 3D
lattice-structured systems. We consider the linearized Hartree-Fock equation
over a spatial lattice for both periodic and
non-periodic problem setting, discretized in the localized Gaussian-type
orbitals basis. In the periodic case, the Galerkin system matrix obeys a
three-level block-circulant structure that allows the FFT-based
diagonalization, while for the finite extended systems in a box (Dirichlet
boundary conditions) we arrive at the perturbed block-Toeplitz representation
providing fast matrix-vector multiplication and low storage size. The proposed
grid-based tensor techniques manifest the twofold benefits: (a) the entries of
the Fock matrix are computed by 1D operations using low-rank tensors
represented on a 3D grid, (b) in the periodic case the low-rank tensor
structure in the diagonal blocks of the Fock matrix in the Fourier space
reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems
in a box with Dirichlet boundary conditions are treated numerically by our
previous tensor solver for single molecules, which makes possible calculations
on rather large lattices due to reduced numerical
cost for 3D problems. The numerical simulations for both box-type and periodic
lattice chain in a 3D rectangular "tube" with up to
several hundred confirm the theoretical complexity bounds for the
block-structured eigenvalue solvers in the limit of large .Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with
arXiv:1408.383
A constructive arbitrary-degree Kronecker product decomposition of tensors
We propose the tensor Kronecker product singular value decomposition~(TKPSVD)
that decomposes a real -way tensor into a linear combination
of tensor Kronecker products with an arbitrary number of factors
. We generalize the matrix Kronecker product to
tensors such that each factor in the TKPSVD is a -way
tensor. The algorithm relies on reshaping and permuting the original tensor
into a -way tensor, after which a polyadic decomposition with orthogonal
rank-1 terms is computed. We prove that for many different structured tensors,
the Kronecker product factors
are guaranteed to inherit this structure. In addition, we introduce the new
notion of general symmetric tensors, which includes many different structures
such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors.Comment: Rewrote the paper completely and generalized everything to tensor
Channel Covariance Matrix Estimation via Dimension Reduction for Hybrid MIMO MmWave Communication Systems
Hybrid massive MIMO structures with lower hardware complexity and power
consumption have been considered as a potential candidate for millimeter wave
(mmWave) communications. Channel covariance information can be used for
designing transmitter precoders, receiver combiners, channel estimators, etc.
However, hybrid structures allow only a lower-dimensional signal to be
observed, which adds difficulties for channel covariance matrix estimation. In
this paper, we formulate the channel covariance estimation as a structured
low-rank matrix sensing problem via Kronecker product expansion and use a
low-complexity algorithm to solve this problem. Numerical results with uniform
linear arrays (ULA) and uniform squared planar arrays (USPA) are provided to
demonstrate the effectiveness of our proposed method
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