On the Power of Standard Information for Tractability for L2- Approximation in the Randomized Setting

We study the approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability in the randomized setting for Λstd and Λall for the normalized or absolute error criterion. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and some notions of algebraic tractability without any condition. For the absolute error criterion, we show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 98, 101, 102 and almost solve Open Problem 100 as posed by E.Novak and H.Wo´zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.

Easy representation of multivariate functions with low-dimensional terms via Gaussian process regression kernel design: applications to machine learning of potential energy surfaces and kinetic energy densities from sparse data

We show that Gaussian process regression (GPR) allows representing multivariate functions with low-dimensional terms via kernel design. When using a kernel built with HDMR (High-dimensional model representation), one obtains a similar type of representation as the previously proposed HDMR-GPR scheme while being faster and simpler to use. We tested the approach on cases where highly accurate machine learning is required from sparse data by fitting potential energy surfaces and kinetic energy densities.

Approximation properties of multivariate exponential sampling series

In this paper, we generalize the family of exponential sampling series for functions of $n$ variables and study their pointwise and uniform convergence as well as the rate of convergence for the functions belonging to space of $\log$-uniformly continuous functions. Furthermore, we state and prove the generalized Mellin-Taylor's expansion of multivariate functions. Using this expansion we establish pointwise asymptotic behaviour of the series by means of Voronovskaja type theorem.

Gaussian mixture model decomposition of multivariate signals

We propose a greedy variational method for decomposing a non-negative multivariate signal as a weighted sum of Gaussians, which, borrowing the terminology from statistics, we refer to as a Gaussian mixture model. Notably, our method has the following features: (1) It accepts multivariate signals, i.e., sampled multivariate functions, histograms, time series, images, etc., as input. (2) The method can handle general (i.e., ellipsoidal) Gaussians. (3) No prior assumption on the number of mixture components is needed. To the best of our knowledge, no previous method for Gaussian mixture model decomposition simultaneously enjoys all these features. We also prove an upper bound, which cannot be improved by a global constant, for the distance from any mode of a Gaussian mixture model to the set of corresponding means. For mixtures of spherical Gaussians with common variance $$\sigma ^2$$ σ 2 , the bound takes the simple form $$\sqrt{n}\sigma $$ n σ . We evaluate our method on one- and two-dimensional signals. Finally, we discuss the relation between clustering and signal decomposition, and compare our method to the baseline expectation maximization algorithm.

A New Approach to Detection of Changes in Multidimensional Patterns

In the paper we develop an algorithm based on the Parzen kernel estimate for detection of sudden changes in 3-dimensional shapes which happen along the edge curves. Such problems commonly arise in various areas of computer vision, e.g., in edge detection, bioinformatics and processing of satellite imagery. In many engineering problems abrupt change detection may help in fault protection e.g. the jump detection in functions describing the static and dynamic properties of the objects in mechanical systems. We developed an algorithm for detecting abrupt changes which is nonparametric in nature and utilizes Parzen regression estimates of multivariate functions and their derivatives. In tests we apply this method, particularly but not exclusively, to the functions of two variables.