1,263 research outputs found

    Covariance Estimation in High Dimensions via Kronecker Product Expansions

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    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 rr 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

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    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

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    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 L1×L2×L3L_1\times L_2\times L_3 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 L1×L2×L3L_1\times L_2\times L_3 lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic L×1×1L\times 1\times 1 lattice chain in a 3D rectangular "tube" with LL up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large LL.Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1408.383

    A constructive arbitrary-degree Kronecker product decomposition of tensors

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    We propose the tensor Kronecker product singular value decomposition~(TKPSVD) that decomposes a real kk-way tensor A\mathcal{A} into a linear combination of tensor Kronecker products with an arbitrary number of dd factors A=∑j=1Rσj Aj(d)⊗⋯⊗Aj(1)\mathcal{A} = \sum_{j=1}^R \sigma_j\, \mathcal{A}^{(d)}_j \otimes \cdots \otimes \mathcal{A}^{(1)}_j. We generalize the matrix Kronecker product to tensors such that each factor Aj(i)\mathcal{A}^{(i)}_j in the TKPSVD is a kk-way tensor. The algorithm relies on reshaping and permuting the original tensor into a dd-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 Aj(1),…,Aj(d)\mathcal{A}^{(1)}_j,\ldots,\mathcal{A}^{(d)}_j 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

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    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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