Joni Virta

The squared symmetric FastICA estimator

In this paper we study the theoretical properties of the deflation-based FastICA method, the original symmetric FastICA method, and a modified symmetric FastICA method, here called the squared symmetric FastICA. This modification is obtained by replacing the absolute values in the FastICA objective function by their squares. In the deflation-based case this replacement has no effect on the estimate since the maximization problem stays the same. However, in the symmetric case we obtain a different estimate which has been mentioned in the literature, but its theoretical properties have not been studied at all. In the paper we review the classic deflation-based and symmetric FastICA approaches and contrast these with the squared symmetric version of FastICA in a unified way. We find the estimating equations and derive the asymptotical properties of the squared symmetric FastICA estimator with an arbitrary choice of nonlinearity. This allows the main contribution of the paper, i.e., efficiency comparison of the estimates in a wide variety of situations using asymptotic variances of the unmixing matrix estimates. HighlightsThe formal definition of the squared symmetric FastICA estimator is given.The asymptotical properties of the squared symmetric FastICA are derived.Asymptotic variances of different FastICA estimators are compared.

Asymptotic and Bootstrap Tests for the Dimension of the Non-Gaussian Subspace

Dimension reduction is often a preliminary step in the analysis of large data sets. The so-called non-Gaussian component analysis searches for a projection onto the non-Gaussian part of the data, and it is then important to know the correct dimension of the non-Gaussian signal subspace. In this letter, we develop asymptotic as well as bootstrap tests for the dimension based on the popular fourth-order blind identification method.

Independent component analysis for tensor-valued data

Abstract In preprocessing tensor-valued data, e.g., images and videos, a common procedure is to vectorize the observations and subject the resulting vectors to one of the many methods used for independent component analysis (ICA). However, the tensor structure of the original data is lost in the vectorization and, as a more suitable alternative, we propose the matrix- and tensor fourth order blind identification (MFOBI and TFOBI). In these tensorial extensions of the classic fourth order blind identification (FOBI) we assume a Kronecker structure for the mixing and perform FOBI simultaneously on each direction of the observed tensors. We discuss the theory and assumptions behind MFOBI and TFOBI and provide two different algorithms and related estimates of the unmixing matrices along with their asymptotic properties. Finally, simulations are used to compare the method’s performance with that of classical FOBI for vectorized data and we end with a real data clustering example.

JADE for Tensor-Valued Observations

ABSTRACT Independent component analysis is a standard tool in modern data analysis and numerous different techniques for applying it exist. The standard methods however quickly lose their effectiveness when the data are made up of structures of higher order than vectors, namely, matrices or tensors (e.g., images or videos), being unable to handle the high amounts of noise. Recently, an extension of the classic fourth-order blind identification (FOBI) specially suited for tensor-valued observations was proposed and showed to outperform its vector version for tensor data. In this article, we extend another popular independent component analysis method, the joint approximate diagonalization of eigen-matrices (JADE), for tensor observations. In addition to the theoretical background, we also provide the asymptotic properties of the proposed estimator and use both simulations and real data to show its usefulness and superiority over its competitors. Supplementary material including the proofs of the theorems and the codes for running the simulations and the real data example are available online.

Blind source separation of tensor-valued time series☆

Abstract The blind source separation model for multivariate time series generally assumes that the observed series is a linear transformation of an unobserved series with temporally uncorrelated or independent components. Given the observations, the objective is to find a linear transformation that recovers the latent series. Several methods for accomplishing this exist and three particular ones are the classic SOBI and the recently proposed generalized FOBI (gFOBI) and generalized JADE (gJADE), each based on the use of joint lagged moments. In this paper we generalize the methodologies behind these algorithms for tensor-valued time series. We assume that our data consists of a tensor observed at each time point and that the observations are linear transformations of latent tensors we wish to estimate. The tensorial generalizations are shown to have particularly elegant forms and we show that each of them is Fisher consistent and orthogonal equivariant. Comparing the new methods with the original ones in various settings shows that the tensorial extensions are superior to both their vector-valued counterparts and to two existing tensorial dimension reduction methods for i.i.d. data. Finally, applications to fMRI-data and video processing show that the methods are capable of extracting relevant information from noisy high-dimensional data.

Applying fully tensorial ICA to fMRI data

There are two aspects in functional magnetic resonance imaging (fMRI) data that make them awkward to analyse with traditional multivariate methods — high order and high dimension. The first of these refers to the tensorial nature of observations as array-valued elements instead of vectors. Although this can be circumvented by vectorizing the array, doing so simultaneously loses all the structural information in the original observations. The second aspect refers to the high dimensionality along each dimension making the concept of dimension reduction a valuable tool in the processing of fMRI data. Different methods of tensor dimension reduction are currently gaining popularity in literature, and in this paper we apply two recently proposed methods of tensorial independent component analysis to simulated task-based fMRI data. Additionally, as a preprocessing step we introduce a novel extension of PCA for tensors. The simulations show that when extracting a sufficiently large number of principal components, the tensor methods find the task signals very reliably, something the standard temporal independent component analysis (tICA) fails in.

Tensorial blind source separation for improved analysis of multi-omic data

There is an increased need for integrative analyses of multi-omic data. We present and benchmark a novel tensorial independent component analysis (tICA) algorithm against current state-of-the-art methods. We find that tICA outperforms competing methods in identifying biological sources of data variation at a reduced computational cost. On epigenetic data, tICA can identify methylation quantitative trait loci at high sensitivity. In the cancer context, tICA identifies gene modules whose expression variation across tumours is driven by copy-number or DNA methylation changes, but whose deregulation relative to normal tissue is independent of such alterations, a result we validate by direct analysis of individual data types.

On the Optimal Non-linearities for Gaussian Mixtures in FastICA

In independent component analysis we assume that the observed vector is a linear transformation of a latent vector of independent components, our objective being the estimation of the latter. Deflation-based FastICA estimates the components one-by-one by repeatedly maximizing the expected value of some function measuring non-Gaussianity, the derivative of which is called the non-linearity. Under some weak assumptions, the asymptotically optimal non-linearity for extracting sources with a specific density is given by the location score function of the density. In this paper we look into the consequences of this result from the viewpoint of estimating Gaussian location and scale mixtures. As one of our results we justify the common use of hyperbolic tangent, tanh, as a non-linearity in blind clustering by showing that it is optimal for estimating certain Gaussian mixtures. Finally, simulations are used to show that the asymptotic optimality results hold in various settings also for finite samples.