Abstract-A mean-square error lower bound for the discretetime nonlinear filtering problem is derived based on the Van Trees (posterior) version of the Cramér-Rao inequality. This lower bound is applicable to multidimensional nonlinear, possibly non-Gaussian, dynamical systems and is more general than the previous bounds in the literature. The case of singular conditional distribution of the one-step-ahead state vector given the present state is considered. The bound is evaluated for three important examples: the recursive estimation of slowly varying parameters of an autoregressive process, tracking a slowly varying frequency of a single cisoid in noise, and tracking parameters of a sinusoidal frequency with sinusoidal phase modulation.

We propose a new low-complexity approximate joint diagonalization (AJD) algorithm, which incorporates nontrivial block-diagonal weight matrices into a weighted least-squares (WLS) AJD criterion. Often in blind source separation (BSS), when the sources are nearly separated, the optimal weight matrix for WLS-based AJD takes a (nearly) block-diagonal form. Based on this observation, we show how the new algorithm can be utilized in an iteratively reweighted separation scheme, thereby giving rise to fast implementation of asymptotically optimal BSS algorithms in various scenarios. In particular, we consider three specific (yet common) scenarios, involving stationary or block-stationary Gaussian sources, for which the optimal weight matrices can be readily estimated from the sample covariance matrices (which are also the target-matrices for the AJD). Comparative simulation results demonstrate the advantages in both speed and accuracy, as well as compliance with the theoretically predicted asymptotic optimality of the resulting BSS algorithms based on the weighted AJD, both on large scale problems with matrices of the size 100 2 100.Index Terms-Approximate joint diagonalization (AJD), autoregressive processes, blind source separation (BSS), nonstationary random processes.

CANDECOMP/PARAFAC (CP) has found numerous applications in wide variety of areas such as in chemometrics, telecommunication, data mining, neuroscience, separated representations. For an order-tensor, most CP algorithms can be computationally demanding due to computation of gradients which are related to products between tensor unfoldings and Khatri-Rao products of all factor matrices except one. These products have the largest workload in most CP algorithms. In this paper, we propose a fast method to deal with this issue. The method also reduces the extra memory requirements of CP algorithms. As a result, we can accelerate the standard alternating CP algorithms 20-30 times for order-5 and order-6 tensors, and even higher ratios can be obtained for higher order tensors (e.g.,). The proposed method is more efficient than the state-of-the-art ALS algorithm which operates two modes at a time (ALSo2) in the Eigenvector PLS toolbox, especially for tensors with order and high rank.

We propose an improved version of algorithm FastICA which is asymptotically efficient, i.e., its accuracy attains the Cramér-Rao lower bound provided that the probability distribution of the signal components belongs to the class of generalized Gaussian distribution. Its computational complexity is only slightly (about three times) higher than that of ordinary symmetric FastICA. Simulation section shows superior performance of the algorithm compared with JADE, and of non-parametric ICA.

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