Free Component Analysis (FCA)
We describe a method for unmixing mixtures of freely independent random variables in a manner analogous to the independent component analysis (ICA) based method for unmixing independent random variables from their additive mixtures. Random matrices play the role of free random variables in this context so the method we develop, which we call Free component analysis (FCA), unmixes matrices from additive mixtures of matrices. We describe the theory, the various algorithms, and compare FCA to ICA. We show that FCA performs comparably to, and often better than, ICA in every application, such as image and speech unmixing, where ICA has been known to succeed. Our computational experiments suggest that not-so-random matrices, such as images and spectrograms of waveforms are (closer to being) freer ‘in the wild’ than we might have theoretically expected. …

Approximate Entropy
In statistics, an approximate entropy (ApEn) is a technique used to quantify the amount of regularity and the unpredictability of fluctuations over time-series data. …

Functional Linear Array Model (FLAM)
The functional linear array model (FLAM) is a unified model class for functional regression models including function-on-scalar, scalar-on-function and function-on-function regression. Mean, median, quantile as well as generalized additive regression models for functional or scalar responses are contained as special cases in this general framework. Our implementation features a broad variety of covariate effects, such as, linear, smooth and interaction effects of grouping variables, scalar and functional covariates. Computational efficiency is achieved by representing the model as a generalized linear array model. While the array structure requires a common grid for functional responses, missing values are allowed. Estimation is conducted using a boosting algorithm, which allows for numerous covariates and automatic, data-driven model selection. To illustrate the flexibility of the model class we use three applications on curing of resin for car production, heat values of fossil fuels and Canadian climate data (the last one in the electronic supplement). These require function-on-scalar, scalar-on-function and function-on-function regression models, respectively, as well as additional capabilities such as robust regression, spatial functional regression, model selection and accommodation of missings. An implementation of our methods is provided in the R add-on package FDboost. …

Stochastic Multidimensional Scaling
Multidimensional scaling (MDS) is a popular dimensionality reduction techniques that has been widely used for network visualization and cooperative localization. However, the traditional stress minimization formulation of MDS necessitates the use of batch optimization algorithms that are not scalable to large-sized problems. This paper considers an alternative stochastic stress minimization framework that is amenable to incremental and distributed solutions. A novel linear-complexity stochastic optimization algorithm is proposed that is provably convergent and simple to implement. The applicability of the proposed algorithm to localization and visualization tasks is also expounded. Extensive tests on synthetic and real datasets demonstrate the efficacy of the proposed algorithm. …