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        "author": "Christian Grussler",
        "author_email": "christian.grussler@eng.cam.ac.uk",
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        "description": "# LRIPy\nPython code for Low-rank optimization by Low-Rank Inducing Norms as well as non-convex Douglas-Rachford. \n\n## Purpose:\nLow-rank rank inducing norms and non-convex Proximal Splitting Algoriths attempt to find exact rank/cardinality-r solutions to minimization problems with convex loss functions, i.e., avoiding of regularzation heuristics. LRIPy provides Python implementations for the proximal mappings of the low-rank inducing Frobenius and Spectral norms, as well as, their epi-graph projections and non-convex counter parts.\n\n## Literature:\n\n### Low-rank inducing norms: \n* [Rank Reduction with Convex Constraints](https://lup.lub.lu.se/search/publication/54cb814f-59fe-4bc9-a7ef-773cbcf06889)\n* [Low-rank Inducing Norms with Optimality Interpretations](https://arxiv.org/abs/1612.03186)\n* [Low-rank Optimization with Convex Constraints](https://arxiv.org/abs/1606.01793)\n* [The Use of the r* Heuristic in Covariance Completion Problems](http://www.control.lth.se/index.php?mact=ReglerPublicationsB,cntnt01,showpublication,0&cntnt01LUPid=a61669c7-29b9-41ee-82da-9c825b08f8d8&cntnt01returnid=60)\n* [On optimal low-rank approximation of non-negative matirces](http://lup.lub.lu.se/search/ws/files/21812505/2015cdcGrusslerRantzer.pdf)\n\n### Non-convex counter parts:\n* [Local Convergence of Proximal Splittinge Methods for Rank Constrained Problems](https://arxiv.org/abs/1710.04248)\n\n## Installation\n\nThe easiest way to install the package is to run ``pip install lripy``. To install the package from source, run ``python setup.py install`` in the main folder.  \n\n## Documentation\nIn the following it holds that\n* for the low-rank inducing Frobenius norm: p = 2\n* for the low-rank inducing Spectral norm:  p = 'inf'\n\n### Examples\nThere are two examples in the \"example\" folder:\n\n1. Exact Matrix Completion\n2. Low-rank approximation with Hankel constraint\n\n### Optimization\n\nLRIPy contains Douglas-Rachford splitting implementations for \"Exact Matrix Completion\" and \"Low-rank Hankel Approximation\", both with low-rank inducing norms, as well as, non-convex Douglas-Rachford splitting. It is easy to modify these functions for other constraints! \n\n#### Exact Matrix completion\n\nLet N be a matrix and Index be a binary matrix of the same size, where the ones indicate the known entries N. We attempt to find a rank-r completion M:\n\n```\n# Import the Douglas-Rachford Completion function:\n\nfrom lripy import drcomplete\n\n# Low-rank inducing norms with Douglas-Rachford splitting:\n\nM = drcomplete(N,Index,r,p)[0]\n\n# Non-convex Douglas-Rachford splitting:\n\nM = drcomplete(N,Index,r,p,solver = 'NDR')[0]\n```\n\n#### Low-rank Hankel Approximation\n\nLet H be a matrix. We attempt to find a rank-r Hankel approximation M that minimizes the Frobenius norm:\n\n```\n# Import the Douglas-Rachford Hankel Approximation function:\n\nfrom lripy import drhankelapprox\n\n# Low-rank inducing norms with Douglas-Rachford splitting:\n\nM = drhankelapprox(H,r)[0]\n\n# Non-convex Douglas-Rachford splitting:\n\nM = drhankelapprox(H,r,solver = 'NDR')[0]\n```\n\n### Proximal Mappings\nLRIPy provides Python implemenations for the proximal mappings to the low-rank inducing Frobenius and Spectral norm as well as their epi-graph projections and non-convex counter parts.\n\n#### Low-rank inducing Spectral and Frobenius norms: \n\nProximal mapping of the low-rank inducing norms at Z with parameter r and scaling factor gamma:\n```\nX = proxnormrast(Z,r,p,gamma)[0]\n```\n#### Squared Low-rank inducing Spectral and Frobenius norms: \nProximal mapping of the SQUARED low-rank inducing norms at Z with parameter r and scaling factor gamma:\n```\nX = proxnormrast_square(Z,r,p,gamma)[0]\n```\n#### Projection onto the epi-graph of the low-rank inducing norms: \nProjection of (Z,zv) on the epi-graph of the low-rank inducing norms with parameter r and scaling factor gamma:\n```\nX,xv = projrast(Z,zv,r,p,gamma)[0:2]\n```\n\n#### Non-convex proximal mappings for Frobenius and Spectral norm: \n\nNon-convex proximal mapping of at Z with parameter r and scaling factor gamma:\n```\nX = proxnonconv(Z,r,p,gamma)\n```\n#### Non-convex proximal mappings for squared Frobenius and Spectral norm:\nNon-convex proximal mapping for the SQUARED norms at Z with parameter r and scaling factor gamma:\n```\nX = proxnonconv_square(Z,r,p,gamma)\n```\n\n\n",
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