New multi-linear algebra algorithms for analysis of large and high-dimensional data sets

What is it about?

Combination of low-tensor rank techniques and the Fast Fourier transform (FFT) based methods had turned out to be prominent in accelerating various statistical operations such as Kriging, computing conditional covariance, geostatistical optimal design, and others. However, the approximation of a full tensor by its low-rank format can be computationally formidable. In this work, we incorporate the robust Tensor Train (TT) approximation of covariance matrices and the efficient TT-Cross algorithm into the FFT-based Kriging. It is shown that here the computational complexity of Kriging is reduced to $\mathcal{O}(d r^3 n)$, where $n$ is the mode size of the estimation grid, $d$ is the number of variables (the dimension), and $r$ is the rank of the TT approximation of the covariance matrix. For many popular covariance functions the TT rank $r$ remains stable for increasing $n$ and $d$. The advantages of this approach against those using plain FFT are demonstrated in synthetic and real data examples

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Photo by Shahadat Rahman on Unsplash

Photo by Shahadat Rahman on Unsplash

Why is it important?

Kriging is an interpolation method that makes estimates of unmeasured quantities based on (sparse) scattered measurements. It is widely applied in the estimation of some spatially distributed quantities such as daily moisture, rainfall intensities, temperatures, contaminant concentrations or hydraulic conductivities, etc. Kriging is also used as a surrogate of some complex physical models for the purpose of efficient uncertainty quantification (UQ), in which it estimates the model response under some random perturbation of the parameters. In the first case the estimation grids are usually in two or three dimensions or four dimensions in a space-time Kriging, while in the latter the dimension number could be much larger (equals to the number of uncertain parameters).

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