We present a method for estimating conditionally Gaussian random vectors with random covariance matrices, which uses techniques from the field of machine learning. Such models are typical in communication systems, where the covariance matrix of the channel vector depends on random parameters, e.g., angles of propagation paths. If the covariance matrices exhibit certain Toeplitz and shift-invariance structures, the complexity of the minimum mean squared error (MMSE) channel estimator can be reduced to @math floating point operations, where @math is the channel dimension. While in the absence of structure the complexity is much higher, we obtain a similarly efficient (but suboptimal) estimator by using the MMSE estimator of the structured model as a blueprint for the architecture of a neural network. This network learns the MMSE estimator for the unstructured model, but only within the given class of estimators that contains the MMSE estimator for the structured model. Numerical simulations with typical spatial channel models demonstrate the generalization properties of the chosen class of estimators to realistic channel models.
If we have a ULA at the base station, we know that the covariance matrix has to be a Toeplitz matrix. Thus, we should choose @math as the set of positive semi-definite Toeplitz matrices. In this case, the ML estimate can no longer be given in closed form and iterative methods have to be used @cite_23 @cite_11 @cite_5 .
@cite_5 @cite_23 @cite_11
1999640431 2029301183 2097428361 2762725970
Covariance matrices from stationary time series are Toeplitz. Multichannel and multidimensional processes have covariance matrices of block Toeplitz form. In these cases and many other situations, one knows that the actual covariance matrix belongs to a particular subclass of covariance matrices. This paper discusses a method for estimating a covariance matrix of specified structure from vector samples of the random process. The theoretical foundation of the method is to assume that the random process is zero-mean multivariate Gaussian, and to find the maximum-likelihood covariance matrix that has the specified structure. An existence proof is given and the solution is interpreted in terms of a minimum-entropy principle. The necessary gradient conditions that must be satisfied by the maximum-likelihood solution are derived and unique and nonunique analytic solutions for some simple problems are presented. A major contribution of this paper is an iterative algorithm that solves the necessary gradient equations for moderate-sized problems with reasonable computational ease. Theoretical convergence properties of the basic algorithm are investigated and robust modifications discussed. In doing maximum-entropy spectral analysis of a sine wave in white noise from a single vector sample, this new estimation procedure causes no splitting of the spectral line in contrast to the Burg technique. This paper explores a seemingly counter-intuitive idea: the possibility of accelerating the solution of certain linear equations by adding even more equations to the problem. The basic insight is to trade-off problem size by problem structure. We test this idea on Toeplitz equations, in which case the expense of a larger set of equations easily leads to circulant structure. The idea leads to a very simple iterative algorithm, which works for a certain class of Toeplitz matrices, each iteration requiring only two circular convolutions. In the symmetric definite case, numerical experiments show that the method can compete with the preconditioned conjugate gradient method (PCG), which achieves O(nlogn) performance. Because the iteration does not converge for all Toeplitz matrices, we give necessary and sufficient conditions to ensure convergence (for not necessarily symmetric matrices), and suggest an efficient convergence test. In the positive definite case we determine the value of the free parameter of the circulant that leads to the fastest convergence, as well as the corresponding value for the spectral radius of the iteration matrix. Although the usefulness of the proposed iteration is limited in the case of ill-conditioned matrices, we believe that the results show that the problem size problem structure trade-off deserves further study. In this paper, we propose a second order optimization method to learn models where both the dimensionality of the parameter space and the number of training samples is high. In our method, we construct on each iteration a Krylov subspace formed by the gradient and an approximation to the Hessian matrix, and then use a subset of the training data samples to optimize over this subspace. As with the Hessian Free (HF) method of Martens (2010), the Hessian matrix is never explicitly constructed, and is computed using a subset of data. In practice, as in HF, we typically use a positive definite substitute for the Hessian matrix such as the Gauss-Newton matrix. We investigate the effectiveness of our proposed method on deep neural networks, and compare its performance to widely used methods such as stochastic gradient descent, conjugate gradient descent and L-BFGS, and also to HF. Our method leads to faster convergence than either L-BFGS or HF, and generally performs better than either of them in cross-validation accuracy. It is also simpler and more general than HF, as it does not require a positive semidefinite approximation of the Hessian matrix to work well nor the setting of a damping parameter. The chief drawback versus HF is the need for memory to store a basis for the Krylov subspace. In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.