## SoftwareOn this page we provide software created by our group.## Kuramoto-Vicsek Consensus-Based OptimizationIn the two papers "Consensus-Based Optimization on the Sphere I: Well-Posedness and Mean-Field Limit", 2020 and "Consensus-based optimization on the sphere II: Convergence to global mininizers and machine learning", 2020, by Massimo Fornasier, Hui Huang, Lorenzo Pareschi and Philippe Sünnen, a novel algorithm for the minimization of a non-convex cost functional is proposed and analysed. The algorithm is based on a consensus-based approach, that is, many particles are placed on the graph of the cost functional and move randomly according to a coupled system of stochastic differential equations. Eventually the particles find consensus at the location where they believe the minimum of the cost functional to be. The MATLAB code of the method, including simple to use sample scripts, can be downloaded from GitHub, see [ KV-CBO ]## Alternating Tikhonov regularization and LASSO for Recovery of Low-Rank Matrices with Sparse DecompositionsIn “A-T-LAS_{2,1}: A Multi-Penalty Approach to Compressed Sensing of Low-Rank Matrices with Sparse Decompositions” (M. Fornasier, J. Maly, and V. Naumova^{}), a novel algorithm for recovering sparse and low-rank matrices from compressed sensing measurements is proposed and analyzed. The algorithm is built on alternating minimization of a non-convex multi-penalty functional. A minimal example code (MATLAB) including the implementation of the algorithm and all simulations presented in the paper can be downloaded here.
ATLAS Version 1.0 (2018-01-19) [.zip ]
## Harmonic Mean Iteratively Reweighted Least Squares for Low-Rank Matrix RecoveryIn “Harmonic Mean Iteratively Reweighted Least Squares for Low-Rank Matrix Recovery” (C. Kümmerle, J. Sigl) [ .pdf ], the authors propose and analyze a new iteratively reweighted least squares (IRLS) algorithm, called harmonic mean iteratively reweighted least squares (HM-IRLS), for the recovery of low-rank matrices from incomplete linear observations. A minimal example code (MATLAB) including an implementation of the algorithm and a test file demonstrating its performance in the matrix completion setting can be downloaded here. HM-IRLS Version 1.0 (2017-03-14) [.zip ]## Mean-field control hierarchyIn "Mean-field control hierarchy" (G. Albi^{}, Y. Choi ^{}, M. Fornasier, D. Kalise ) [ .pdf ] In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrained by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We makes available the code to reproduce tests, which compares the hierarchy of controls.
Dedicated page: [Complementary Materials]
BOLTZBELL Version 1.0 (2017-02-14) [ TBA .zip ]
## Damping Noise Folding and Enhanced Support RecoveryIn "Damping Noise Folding and Enhanced Support Recovery in Compressed Sensing" (M. Fornasier, M. Artina, S. Peter) [ .pdf ] we investigate the ability of classical methods for compressed sensing to exactly recover the support and to approximate well the significant entries of a signal when non-sparse noise on the signal is present. We proposed new decoding methods significantly outperforming known algorithms in the literature and we performed several numerical results by means of the software which you can download here. You can use (and also extend) this software to create your own tests with various parameter settings and methods. For further details see the README file. DNFESR Version 1.0 (2014-01-10) [ .zip ]## Quasilinear Compressed SensingIn "Quasilinear Compressed Sensing" (M. Ehler, M. Fornasier, J. Sigl) [ .pdf ] we extend the concept of classical compressed sensing for linear measurements to the quasilinear case. We formulate natural generalizations of the well-known Restricted Isometry Property (RIP) towards nonlinear measurements, which allow us to prove both unique identifiability of sparse signals as well as the convergence of recovery algorithms to compute them effciently. We propose decoding strategies that are generalized versions of greedy and iterative soft- and hard-thresholding algorithms and performed numerical experiments by means of the software which you can download here. You can use (and also extend) this software to create your own tests with various parameter settings and methods. For further details see the README files in the folders for the different algorithm types. QLCS1.0 (2014-11-03) [ .zip ]## Low-rank matrix recovery via iteratively reweighted least squares minimizationIn "Low rank matrix recovery via iteratively reweighted least squares minimization" (M. Fornasier, H. Rauhut, R. Ward) [ .pdf ] we present and analyze an efficient implementation of an iteratively reweighted least squares algorithm for recovering a matrix from a small number of linear measurements. The software was created by Stephan Worm. IRLS-M Version 1.0 (2011) [ .zip ]## Iteratively Reweighted Least Squares Minimization for Sparse RecoveryIn "Iteratively Reweighted Least Squares Minimization for Sparse Recovery" (I. Daubechies, R. DeVore, M. Fornasier, C.-S. Güntürk) [ .pdf ] we study an alternative method of determining the solution of a sparse recovery problem, as the limit of an iteratively reweighted least squares (IRLS) algorithm. In the software, we implement the method. IRLS Version 1.0 (2010) [ .zip ]## Domain Decomposition for Total Variation MinimizationFor software concerning the papers "Subspace correction methods for total variation and l1-minimization" (M. Fornasier and C. Schoenlieb) [ .pdf ], "Domain decomposition methods for compressed sensing" (M. Fornasier, A. Langer and C.-B. Schönlieb) [ .pdf ], and "A convergent overlapping domain decomposition method for total variation minimization" (M. Fornasier, A. Langer and C.-B. Schoenlieb) [ .pdf ], we refer to the software site, maintained by Carola Schönlieb. [ link (status 2014-03-14) ] |