Home > manopt > manifolds > euclidean > shapefitfactory.m

shapefitfactory

PURPOSE ^

Linear manifold structure for optimization over the ShapeFit search space

SYNOPSIS ^

function M = shapefitfactory(VJt)

DESCRIPTION ^

 Linear manifold structure for optimization over the ShapeFit search space

 function M = shapefitfactory(VJt)

 Input: VJt is a matrix of size dxn, such that VJt * ones(n, 1) = 0.

 Returns M, a structure describing the Euclidean space of d-by-n matrices
 equipped with the standard Frobenius distance and associated trace inner
 product, as a manifold for Manopt. Matrices on M, denoted by T, have size
 dxn and obey T*ones(n, 1) = 0 (centered columns) and <VJt, T> = 1, where
 <A, B> = Trace(A' * B).

 See this paper: http://arxiv.org/abs/1506.01437
 ShapeFit: Exact location recovery from corrupted pairwise directions, 2015
 Paul Hand, Choongbum Lee, Vladislav Voroninski

 See also: shapefit_smoothed

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SUBFUNCTIONS ^

SOURCE CODE ^

0001 function M = shapefitfactory(VJt)
0002 % Linear manifold structure for optimization over the ShapeFit search space
0003 %
0004 % function M = shapefitfactory(VJt)
0005 %
0006 % Input: VJt is a matrix of size dxn, such that VJt * ones(n, 1) = 0.
0007 %
0008 % Returns M, a structure describing the Euclidean space of d-by-n matrices
0009 % equipped with the standard Frobenius distance and associated trace inner
0010 % product, as a manifold for Manopt. Matrices on M, denoted by T, have size
0011 % dxn and obey T*ones(n, 1) = 0 (centered columns) and <VJt, T> = 1, where
0012 % <A, B> = Trace(A' * B).
0013 %
0014 % See this paper: http://arxiv.org/abs/1506.01437
0015 % ShapeFit: Exact location recovery from corrupted pairwise directions, 2015
0016 % Paul Hand, Choongbum Lee, Vladislav Voroninski
0017 %
0018 % See also: shapefit_smoothed
0019 
0020 % This file is part of Manopt: www.manopt.org.
0021 % Original author: Nicolas Boumal, June 18, 2015.
0022 % Contributors:
0023 % Change log:
0024 %
0025 %   Jan. 25, 2017 (NB):
0026 %       M.tangent = M.proj now, instead of being identity. This is notably
0027 %       necessary so that checkgradient will pick up on gradients that do
0028 %       not lie in the appropriate tangent space.
0029     
0030     [d, n] = size(VJt);
0031 
0032     M.name = @() sprintf('ShapeFit space of size %d x %d', d, n);
0033     
0034     M.dim = @() d*n - d - 1;
0035     
0036     M.inner = @(x, d1, d2) d1(:).'*d2(:);
0037     
0038     M.norm = @(x, d) norm(d, 'fro');
0039     
0040     M.dist = @(x, y) norm(x-y, 'fro');
0041     
0042     M.typicaldist = @() sqrt(d*n);
0043     
0044     M.proj = @(T, U) projection(U);
0045     VJt_normed = VJt / norm(VJt, 'fro');
0046     function PU = projection(U)
0047         % Center the columns
0048         PU = bsxfun(@minus, U, mean(U, 2));
0049         % Remove component along VJt
0050         % Note: these two actions can be executed separately, without
0051         % interference, owing to VJt having centered columns itself.
0052         PU = PU - (VJt_normed(:)'*U(:))*VJt_normed;
0053     end
0054     
0055     M.egrad2rgrad = M.proj;
0056     
0057     M.ehess2rhess = @(x, eg, eh, d) projection(eh);
0058     
0059     M.tangent = M.proj;
0060     
0061     M.exp = @exp;
0062     function y = exp(x, d, t)
0063         if nargin == 3
0064             y = x + t*d;
0065         else
0066             y = x + d;
0067         end
0068     end
0069     
0070     M.retr = M.exp;
0071     
0072     M.log = @(x, y) y-x;
0073 
0074     M.hash = @(x) ['z' hashmd5(x(:))];
0075     
0076     M.randvec = @(x) randvec();
0077     function u = randvec()
0078         u = projection(randn(d, n));
0079         u = u / norm(u, 'fro');
0080     end
0081     
0082     % We exploit the fact that VJt_normed belongs to the manifold
0083     M.rand = @() VJt_normed + randn(1) * randvec();
0084     
0085     M.lincomb = @matrixlincomb;
0086     
0087     M.zerovec = @(x) zeros(d, n);
0088     
0089     M.transp = @(x1, x2, d) d;
0090     
0091     M.pairmean = @(x1, x2) .5*(x1+x2);
0092     
0093     M.vec = @(x, u_mat) u_mat(:);
0094     M.mat = @(x, u_vec) reshape(u_vec, [d, n]);
0095     M.vecmatareisometries = @() true;
0096 
0097 end

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