It must already be Sunday Morning somewhere. This one has been bugging me for a little while. What is the relationship between

- This question on LinkedIn: Is there a two-dimensional equivalent of compressed sensing?
- The use of Matlab's
function to vectorize images in simple compressive sensing*reshape* - Putting in different columns the different frames of a video in order to perform Robust PCA
- Group Sparsity
- Low Rank SVD
- Graph wavelets as dictionaries
- Analysis operator leanring
- Recommender systems
- The need for decomposiing a measurement matrix into a series of matrices with specific features
- hyperspectral measurements
- Super Fast FFTs ?

Those were all issues that popped up on my radar screen this week and I think the beginning of a good answer is the quantized tensor train (QTT) format. Say what now ?

Two presentations might be of interest for a bird's eye view

- Tensor trains, TT and QTT formats by Ivan Oseledets
- Introduction to Tensor Numerical Methods in Scientiﬁc Computing by Boris N. Khoromskij

and several attendant papers:

- TT-GMRES: on solution to a linear system in the structured tensor format by Sergey V. Dolgov
- Solution of linear systems and matrix inversion in the TT-format by Sergey Dolgov, and Ivan Oseledets
- Constructive representation of functions in tensor formats by Ivan Oseledets
- Superfast Fourier Transform Using QTT Approximation by Sergey Dolgov, Boris N. Khoromskij and Dmitry Savostyanov.

Let us note that the Full-to-TT compression, the TTε recompression and the TT–rounding algorithms make heavy use of SVD and QR algorithms for matrices.which would seem to be a good candidate for insertion of robust PCA and randomized PCA work. There is also a toolbox to dwell into it:

TT-Toolbox 2.2.New in Version 2.2

TT-Toolbox (TT=Tensor Train) Version 2.2

TT(Tensor Train) format is an efficient way for low-parametric representation of high-dimensional tensors. The TT-Toolbox is a MATLAB implementation of basic operations with

tensors in TT-format. It includes:

- tt_tensor and tt_matrix classes for storing vectors and operators
- Basic linear algebra subroutines (addition, matrix-by-vector product, elementwise multiplication and many others) using standard MATLAB syntax, linear complexity in the dimension, reshape function
- Fast rounding procedure with a prescribed accuracy
- Advanced approximation and solution techniques:
- Approximate solution of linear systems and eigenvalue problems
- Cross methods to approximate “black-box” tensors
- Wavelet tensor train decomposition
- Construction of basic operators and functions (Laplace operator, function of a TT-tensor)
- Computation of maximal and minimal elements of a tensor and several others

- Better documentation
- Mixed QTT-Tucker format (qtt_tucker class)
- reshape function for a TT-tensor/TT-matrix
- dmrg_cross method for black-box tensor approximation
- Convolution in QTT-format

Thanks Laurent for bringing this to my attention.

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