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 reshape function to vectorize images in simple compressive sensing
- 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 ?
- 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
- Fast solution of multi-dimensional parabolic problems in the TT/QTT-format with initial application to
- 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.
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.