Kronecker Product Model Research Articles

ESTIMATION OF THE KRONECKER COVARIANCE MODEL BY QUADRATIC FORM

We propose a new estimator, the quadratic form estimator, of the Kronecker product model for covariance matrices. We show that this estimator has good properties in the large dimensional case (i.e., the cross-sectional dimension n is large relative to the sample size T). In particular, the quadratic form estimator is consistent in a relative Frobenius norm sense provided ${\log }^3n/T\to 0$ . We obtain the limiting distributions of the Lagrange multiplier and Wald tests under both the null and local alternatives concerning the mean vector $\mu $ . Testing linear restrictions of $\mu $ is also investigated. Finally, our methodology is shown to perform well in finite sample situations both when the Kronecker product model is true and when it is not true.

Estimation of a multiplicative correlation structure in the large dimensional case

We propose a Kronecker product model for correlation or covariance matrices in the large dimensional case. The number of parameters of the model increases logarithmically with the dimension of the matrix. We propose a minimum distance (MD) estimator based on a log-linear property of the model, as well as a one-step estimator, which is a one-step approximation to the quasi-maximum likelihood estimator (QMLE). We establish rates of convergence and central limit theorems (CLT) for our estimators in the large dimensional case. A specification test and tools for Kronecker product model selection and inference are provided. In a Monte Carlo study where a Kronecker product model is correctly specified, our estimators exhibit superior performance. In an empirical application to portfolio choice for S&P500 daily returns, we demonstrate that certain Kronecker product models are good approximations to the general covariance matrix.

A Kronecker Product Model for Repeated Pattern Detection on 2D Urban Images.

Repeated patterns (such as windows, balconies, and doors) are prominent and significant features in urban scenes. Therefore, detection of these repeated patterns becomes very important for city scene analysis. This paper attacks the problem of repeated pattern detection in a precise, efficient and automatic way, by combining traditional feature extraction with a Kronecker product based low-rank model. We introduced novel algorithms that extract repeated patterns from rectified images with solid theoretical support. Our method is tailored for 2D images of building façades and tested on a large set of façade images.

R-optimal designs for multi-factor models with heteroscedastic errors

In this paper, we consider the R-optimal design problem for multi-factor regression models with heteroscedastic errors. It is shown that a R-optimal design for the heteroscedastic Kronecker product model is given by the product of the R-optimal designs for the marginal one-factor models. However, R-optimal designs for the additive models can be constructed from R-optimal designs for the one-factor models only if sufficient conditions are satisfied. Several examples are presented to illustrate and check optimal designs based on R-optimality criterion.

Dimension of Marginals of Kronecker Product Models

A Kronecker product model is an exponential family whose sufficient statistics matrix factorizes as a Kronecker product of two matrices, one assigned to a visible set of variables and the other to a hidden set of variables. We estimate the dimension of the set of visible marginal probability distributions by the maximum rank of the Jacobian in the limit of large parameters. The limit is described by the tropical morphism: a piecewise linear map with pieces corresponding to slicings of the visible matrix by the normal fan of the hidden matrix. We obtain combinatorial conditions under which the model has the expected dimension, equal to the minimum of the number of natural parameters and the dimension of the ambient probability simplex. Furthermore, we prove that the binary restricted Boltzmann machine always has the expected dimension.

On the multicell processing capacity of the cellular MIMO uplink channel in correlated rayleigh fading environment

In the context of cellular systems, it has been shown that multicell processing can eliminate inter-cell interference and provide high spectral efficiencies with respect to traditional interference-limited implementations. Moreover, it has been proved that the multiplexing sum-rate capacity gain of multicell processing systems is proportional to the number of base station (BS) antennas. These results have been also established for cellular systems, where BSs and user terminals (UTs) are equipped with multiple antennas. Nevertheless, a common simplifying assumption in the literature is the uncorrelated nature of the Rayleigh fading coefficients within the BSUT MIMO links. In this direction, this paper investigates the ergodic multicell-processing sum-rate capacity of the Gaussian MIMO cellular multiple-access channel in a correlated fading environment. More specifically, the multiple antennas of both BSs and UTs are assumed to be correlated according to the Kronecker product model. Furthermore, the current system model considers Rayleigh fading, uniformly distributed UTs over a planar coverage area and power-law path loss. Based on free probabilistic arguments, the empirical eigenvalue distribution of the channel covariance matrix is derived and it is used to calculate both optimal joint decoding and minimum mean square error (MMSE) filtering capacity. In addition, numerical results are presented, where the per-cell sum-rate capacity is evaluated while varying the cell density of the system, as well as the level of fading correlation. In this context, it is shown that the capacity performance is greatly compromised by BS-side correlation, whereas UT-side correlation has a negligible effect on the system's performance. Furthermore, MMSE performance is shown to be greatly suboptimal but more resilient to fading correlation in comparison to optimal decoding.

The spatiotemporal MEG covariance matrix modeled as a sum of Kronecker products

The single Kronecker product (KP) model for the spatiotemporal covariance of MEG residuals is extended to a sum of Kronecker products. This sum of KP is estimated such that it approximates the spatiotemporal sample covariance best in matrix norm. Contrary to the single KP, this extension allows for describing multiple, independent phenomena in the ongoing background activity. Whereas the single KP model can be interpreted by assuming that background activity is generated by randomly distributed dipoles with certain spatial and temporal characteristics, the sum model can be physiologically interpreted by assuming a composite of such processes. Taking enough terms into account, the spatiotemporal sample covariance matrix can be described exactly by this extended model. In the estimation of the sum of KP model, it appears that the sum of the first 2 KP describes between 67% and 93%. Moreover, these first two terms describe two physiological processes in the background activity: focal, frequency-specific alpha activity, and more widespread non-frequency-specific activity. Furthermore, temporal nonstationarities due to trial-to-trial variations are not clearly visible in the first two terms, and, hence, play only a minor role in the sample covariance matrix in terms of matrix power. Considering the dipole localization, the single KP model appears to describe around 80% of the noise and seems therefore adequate. The emphasis of further improvement of localization accuracy should be on improving the source model rather than the covariance model.

G-optimal designs for multi-factor experiments with heteroscedastic errors

We consider constructing G-optimal designs for multi-factor experiments when the errors are heteroscedastic. The method is illustrated with examples assuming the Kronecker product models and the additive models. Sufficient conditions are given so that G-optimal designs for the multi-factor experiments can be constructed from the optimal designs for the single-factor experiments.