This is code to do structured matrix factorizations based on the formulation given in the "Applications" section of the paper:
Haeffele, B., Young, E., and Vidal, R. "Structured low-rank matrix factorization: Optimality, algorithm, and applications to image processing." ICML. 2014.
The code here is a generalization of the formulation given in the paper as it also includes options to include additional 'non-factorized' variables in the optimization and to put non-negative constraints on the factorized variables. This more general formulation is also shown to have similar properties regarding global optimality of local-minima as shown in the following:
Haeffele, Benjamin D., and Rene Vidal. "Global Optimality in Tensor Factorization, Deep Learning, and Beyond." arXiv preprint arXiv:1506.07540 2015.
History: Aug 13, 2015 - First distribution of the code.
FactorTVL1L2_v1.m - The main function.
ToyExample.m - A script with simulated data to show how to run the code.
CalciumImagingExample.m - A script that calls the code with real Calcium imaging data.
Imgs1375.mat - Real Calcium imaging data used in the calcium imaging
example script. The data were taken from awake mouse auditory cortex
transfected with GCaMP5G. The x-axis of the image frame is
approximately 470 microns, and the y-axis is approximately 75 microns.
The data were sampled with a frame rate of 5Hz.
proximalTVL1L2.m - Function to calculate the proximal operator of the sum of the L1, L2, and Total-Variation norms, as given in equation (19) of the above paper.
mexProximalL1TV.cpp - The proximalTVL1L2.m function calls this mex file to calculate the proximal operator of the L1+TV term. The calculation is done by doing coordinate descent on the dual problem and checking the duality gap for the stopping criteria. See below for instuctions on compiling. Note that this mex file only gets called if TV regularization is used, so for applications where TV regularization is not needed it is not necessary to compile this.
mexProximalL1TV.mexw64 - Compiled mex file for 64-bit windows. Other platforms will need to compile this file.
L1TVprox.cpp/h - Library and header files for the mexProximalL1TV.cpp mex function.
GetLatticeIndex.m - Returns the pairs of neighboring indices for an 8-connected 2D lattice (i.e. neighboring pixels of an image of a given size).
MakeFourierD.m - Makes a dictionary of Fourier components as an example for the ToyExample.m script and for fitting background signals in Calcium imaging data.
MakeD.m - Make a dictionary of decaying explonential signals for the use in processing Calcium imaging data.
Compiling the mex file:
The implimentation was purposely written for ease of compilation (i.e. no external libraries, such as LAPACK, are required). Assuming the mex C/C++ compiler is properly configured and that all of the source files are in the current directory, the mex file should compile with the following command:
mex mexProximalL1TV.cpp L1TVprox.cpp
Please send any questions or found bugs to Ben Haeffele firstname.lastname@example.org