In independent component analysis (ICA) the common task is to achieve either spatial or temporal independence by linearly mapping into a feature space. If the data possesses both spatial and temporal structures such as a sequence of images or 3d-scans taken at fixed time intervals, we can require the transformed data to be as independent as possible in both domains. First introduced by Stone using a joint entropy energy function, spatiotemporal ICA is a promising method for real-world data analysis. We propose a novel algorithm for performing spatiotemporal ICA by jointly diagonalizing various source conditions such as higher-order cumulants of the mixtures, both in time and in space. Similar to algebraic ICA algorithms, this provides a robust method for data analysis, which is confirmed by simulations.

The goal of multidimensional independent component analysis (MICA) lies in the linear separation of data into statistically independent groups of signals. In this work, we give an elementary proof for the uniqueness of this problem in the case of equally sized subspaces, showing that the separation matrix is essentially unique except for row permutation and scaling. The proof is based on the reinterpretation of groupwise independence as factorization of the joint characteristic function. We then employ this property to propose a novel algorithm for robustly performing MICA. Simulation results demonstrate the reliability of our method. 1.

this paper, we treat polynomial nonlinearities, especially second-order monomials or quadratic forms. These represent a relatively simple class of nonlinearities, which can be investigated in detail

Abstract. Dimension reduction provides an important tool for preprocessing large scale data sets. A possible model for dimension reduction is realized by projecting onto the non-Gaussian part of a given multivariate recording. We prove that the subspaces of such a projection are unique given that the Gaussian subspace is of maximal dimension. This result therefore guarantees that projection algorithms uniquely recover the underlying lower dimensional data signals. An important open problem in signal processing is the task of efficient dimension reduction, i.e. the search for meaningful signals within a higher dimensional data set. Classical techniques such as principal component analysis hereby define ‘meaningful ’ using second-order statistics (maximal variance), which may often be inadequate for signal detection, e.g. in the presence of strong noise. This contrasts to higher order models including projection pursuit [1, 2] or non-Gaussian subspace analysis (NGSA) [3, 4]. While the former extracts a single non-Gaussian independent component from the data set, the latter tries to detect a whole non-Gaussian subspace within the data, and no assumption of independence within the subspace is made.

Abstract. In this article, we consider high-dimensional data which contains a low-dimensional non-Gaussian structure contaminated with Gaussian noise and propose a new method to identify the non-Gaussian subspace. A linear dimension reduction algorithm based on the fourth-order cumulant tensor was proposed in our previous work [4]. Although it works well for sub-Gaussian structures, the performance is not satisfactory for super-Gaussian data due to outliers. To overcome this problem, we construct an alternative by using Hessian of characteristic functions which was applied to (multidimensional) independent component analysis [10,11]. A numerical study demonstrates the validity of our method. 1

Comon showed using the Darmois-Skitovitch theorem that under mild assumptions a real-valued random vector and its linear image are both independent if and only if the linear mapping is the product of a permutation and a scaling matrix. In this work, a much simpler, direct proof is given for this theorem and generalized to the case of random vectors with complex values. The idea is based on the fact that a random vector is independent if and only if locally the Hessian of its logarithmic density is diagonal.

Abstract. Independent Subspace Analysis (ISA) is a generalization of ICA. It tries to find a basis in which a given random vector can be decomposed into groups of mutually independent random vectors. Since the first introduction of ISA, various algorithms to solve this problem have been introduced, however a general proof of the uniqueness of ISA decompositions remained an open question. In this contribution we address this question and sketch a proof for the separability of ISA. The key condition for separability is to require the subspaces to be not further decomposable (irreducible). Based on a decomposition into irreducible components, we formulate a general model for ISA without restrictions on the group sizes. The validity of the uniqueness result is illustrated on a toy example. Moreover, an extension of ISA to subspace extraction is introduced and its indeterminacies are discussed. With the increasing popularity of Independent Component Analysis, people started to get interested in extensions. Cardoso [2] was the first to formulate

The decomposition of surface electromyogram data sets (s-EMG) is studied using blind source separation techniques based on sparseness; namely independent component analysis, sparse nonnegative matrix factorization, and sparse component analysis. When applied to artificial signals we find noticeable di#erences of algorithm performance depending on the source assumptions. In particular, sparse nonnegative matrix factorization outperforms the other methods with regards to increasing additive noise. However, in the case of real s-EMG signals we show that despite the fundamental di#erences in the various models, the methods yield rather similar results and can successfully separate the source signal. This can be explained by the fact that the di#erent sparseness assumptions (super-Gaussianity, positivity together with minimal 1-norm and fixed number of zeros respectively) are all only approximately fulfilled thus apparently forcing the algorithms to reach similar results, but from di#erent initial conditions.

An important aspect of successfully analyzing data with blind source separation is to know the indeterminacies of the problem, that is how the separating model is related to the original mixing model. If linear independent component analysis (ICA) is used, it is well known that the mixing matrix can be found in principle, but for more general settings not many results exist. In this work, only considering random variables with bounded densities, we prove identifiability of the postnonlinear mixing model with analytic nonlinearities and calculate its indeterminacies. A simulation confirms these theoretical findings.

Ac,V,,u version of the Darmois--Skitovitc theorem is proved using a multivariate extension of the latter by Ghurye and Olkin. This makes it possible tocuO,,#Gu theindeterminacI; of independentcndepend analysis (ICA) withcthu#C variables and cu,EGEuACC Furthermore, the multivariate Darmois--Skitovitc theorem is used to show uniqueness of multidimensional ICA, where only groups ofsourc# are mutually independent.