Thanks to Igor Carron for pointing me to this workshop – Algebraic Topology and Machine Learning . There is very interesting paper there Persistent Homological Structures in Compressed Sensing and Sparse Likelihood by Moo K. Chung, Hyekyung Lee and Matthew Arnold. The paper is very comprehensive and require only minimal understanding of some algebraic topology concepts (which is exactly where I’m in realation to algebraic topology). Basically it’s application of topological data analysis to compressive sensing. They use such thing as persistent homology and “barcodes”. Before, persistent homology and barcodes were used for such things as extracting solid structure from noisiy point cloud. Barcode is stable to noise dependence of some topological invariants on some parameter. In case of point cloud parameter is the radius of the ball around each point. As radius go from very big to zero topology of union of balls change, and those changes of topology make barcode. Because barcode is stable topological invariant learning barcode is the same as learning topology of solid structure underlying point cloud.
In the paper authors using graphical lasso (glasso) with regularizer to find interdependency between set of sampled variables. However if consider parameter of regularizer as a kind of radius of ball this problem aquire persistent homology and barcode. The correlation matrix is thresholded by and become adjacency matrix of some graph. Barcode is now dependence of topology of that graph on parameter . What is especially spectacular is that to calculate barcode no glasso iteration are needed – barcode obtained by simple thresholding of correlation matrix. Thus barcode easily found and with it topology of correlations of variables. Well, at least that is how I understood the paper.
PS Using this approach for total variation denoising barcode would include dependance of size function from smoothing parameter.