Description of orthogonalize

ORTHOGONALIZE Orthogonalize TT/MPS Block-mu tensor.
   X = ORTHOGONALIZE( X ) orthogonalizes all cores of the TTeMPS_block tensor X
   except the supercore at position X.MU. Cores 1...X.MU-1 are left-, cores X.MU+1...end
   are right-orthogonalized.

CROSS-REFERENCE INFORMATION

orthogonalize ORTHOGONALIZE Orthogonalize tensor.

orthogonalize ORTHOGONALIZE Orthogonalize TT/MPS Block-mu tensor.

orthogonalize Orthonormalizes a basis of tangent vectors in the Manopt framework.

low_rank_tensor_completion_TT Example file for the manifold encoded in fixedTTrankfactory.

TTeMPS

gauge_matrices GAUGE_MATRICES Right and left orthogonalization with storage of gauge matrices

norm NORM Norm of a TT/MPS tensor.

norm NORM Norm of a TT/MPS block-mu tensor.

orthogonalize ORTHOGONALIZE Orthogonalize TT/MPS Block-mu tensor.

TTeMPS_tangent_orth

completion Completion for Tensor train but without individual orthogonalization

completion_als ALS Completion

completion_orth RTTC: Riemannian Tensor Train Completion

completion_orth_lambda RTTC: Riemannian Tensor Train Completion

completion_rankincrease

amen_eigenvalue AMEN_EIGENVALUE Calculate p smallest eigenvalues of a TTeMPS operator

block_eigenvalue BLOCK_EIGENVALUE Calculate p smallest eigenvalues of a TTeMPS operator

increaseRank_mod Rank-1 gradient approximation to increase the rank.

alsLinsolve TTeMPS Toolbox.

alsLinsolve_fast TTeMPS Toolbox.

alsLinsolve_rankOne TTeMPS Toolbox.

amen TTeMPS Toolbox.

amen_fast TTeMPS Toolbox.

construct_initial_guess TTeMPS Toolbox.

construct_initial_guess_rankOne TTeMPS Toolbox.

ex_completion_compare_als_riemann This example shows a simple comparison of two different algorithm for tensor completion:

ex_completion_rankadaptive Example script for RANK-ADAPTIVE TENSOR COMPLETION, see Algorithm RTTC described in

example TTeMPS Toolbox.

fixedTTrankfactory Manifold of tensors of fixed Tensor Train (TT) rank, embedded geometry

orthogonalizetwice Orthonormalizes a basis of tangent vectors twice for increased accuracy.

tangentorthobasis Returns an orthonormal basis of tangent vectors in the Manopt framework.

SOURCE CODE

0001 function res = orthogonalize( x )
0002     %ORTHOGONALIZE Orthogonalize TT/MPS Block-mu tensor.
0003     %   X = ORTHOGONALIZE( X ) orthogonalizes all cores of the TTeMPS_block tensor X
0004     %   except the supercore at position X.MU. Cores 1...X.MU-1 are left-, cores X.MU+1...end
0005     %   are right-orthogonalized.
0006 
0007     %   TTeMPS Toolbox.
0008     %   Michael Steinlechner, 2013-2016
0009     %   Questions and contact: michael.steinlechner@epfl.ch
0010     %   BSD 2-clause license, see LICENSE.txt
0011 
0012 
0013     y = TTeMPS_block_to_TTeMPS( x );
0014     yorth = orthogonalize(y, x.mu );
0015     res = TTeMPS_block.TTeMPS_to_TTeMPS_block( yorth, x.mu, x.p );
0016 
0017     %tmp = permute( y.U{mu}, [1 2 4 3] );
0018     %tmp = reshape( tmp, [x.rank(mu), x.size(mu)*x.p, x.rank(mu+1)];
0019 
0020     %X = TTeMPS({x.U{1:mu-1}, tmp, x.U{1:mu+1} );
0021 
0022     
0023 
0024     % left orthogonalization till x.mu (from left)
0025     %for i = 1:x.mu-1
0026     %    [Q,R] = qr( unfold( x.U{i}, 'left' ), 0);
0027     %    x.U{pos} = reshape( Q, [x.rank(pos), x.size(pos), size(Q,2)] );
0028     %    if apply
0029     %        x.U{pos+1} = tensorprod_ttemps( x.U{pos+1}, R, 1);
0030     %    end
0031     %end
0032 
0033     %% right orthogonalization till x.mu (from right)
0034     %for i = x.order:-1:x.mu+1
0035     %
0036     %end
0037 
0038 end
0039

orthogonalize

PURPOSE

SYNOPSIS

DESCRIPTION

CROSS-REFERENCE INFORMATION

SOURCE CODE