norm

PURPOSE

SYNOPSIS

DESCRIPTION

NORM Norm of a TT/MPS tensor.
   norm(X) computes the Frobenius norm of the TT/MPS tensor X.

   norm(X, SAFE) with SAFE=true computes the Frobenius norm of the TT/MPS tensor X
       with reorthogonalization to increase the accuracy

   See also INNERPROD

CROSS-REFERENCE INFORMATION

• innerprod INNERPROD Inner product between two TT/MPS tensors.
• norm NORM Norm of a TT/MPS tensor.
• orthogonalize ORTHOGONALIZE Orthogonalize tensor.
• innerprod INNERPROD Inner product between two TT/MPS tensors.
• norm NORM Norm of a TT/MPS block-mu tensor.
• orthogonalize ORTHOGONALIZE Orthogonalize TT/MPS Block-mu tensor.
• orthogonalize Orthonormalizes a basis of tangent vectors in the Manopt framework.

• PCA_stochastic Example of stochastic gradient algorithm in Manopt on a PCA problem.
• dominant_invariant_subspace Returns an orthonormal basis of the dominant invariant p-subspace of A.
• dominant_invariant_subspace_complex Returns a unitary basis of the dominant invariant p-subspace of A.
• doubly_stochastic_denoising Find a doubly stochastic matrix closest to a given matrix, in Frobenius norm.
• generalized_eigenvalue_computation Returns orthonormal basis of the dominant invariant p-subspace of B^-1 A.
• generalized_procrustes Rotationally align clouds of points (generalized Procrustes problem)
• low_rank_matrix_completion Given partial observation of a low rank matrix, attempts to complete it.
• low_rank_tensor_completion Given partial observation of a low rank tensor, attempts to complete it.
• low_rank_tensor_completion_TT Example file for the manifold encoded in fixedTTrankfactory.
• low_rank_tensor_completion_embedded Given partial observation of a low rank tensor (possibly including noise),
• radio_interferometric_calibration Returns the gain matrices of N stations with K receivers.
• shapefit_smoothed ShapeFit formulation for sensor network localization from pair directions
• sparse_pca Sparse principal component analysis based on optimization over Stiefel.
• truncated_svd Returns an SVD decomposition of A truncated to rank p.
• using_gpu Manopt example on how to use GPU with manifold factories that allow it.
• using_gpu_AD Manopt example on how to use GPU to compute the egrad and the ehess via AD.
• complexcirclefactory Returns a manifold struct to optimize over unit-modulus complex numbers.
• realphasefactory Returns a manifold struct to optimize over phases of fft's of real signals
• essentialfactory Manifold structure to optimize over the space of essential matrices.
• essential_distMinAnglePair_discontinuityDistance
• centeredmatrixfactory Linear manifold struct. for optimization over matrices with centered cols
• euclideancomplexfactory Returns a manifold struct to optimize over complex matrices.
• euclideanfactory Returns a manifold struct to optimize over real matrices.
• euclideansparsefactory Returns a manifold struct to optimize over real matrices with given sparsity pattern.
• shapefitfactory Linear manifold structure for optimization over the ShapeFit search space
• skewsymmetricfactory Returns a manifold struct to optimize over k skew-symmetric matrices of size n
• symmetricfactory Returns a manifold struct to optimize over k symmetric matrices of size n
• fixedrankembeddedfactory Manifold struct to optimize fixed-rank matrices w/ an embedded geometry.
• fixedrankfactory_2factors Manifold of m-by-n matrices of rank k with balanced quotient geometry.
• fixedrankfactory_2factors_preconditioned Manifold of m-by-n matrices of rank k with two factor quotient geometry.
• grassmanncomplexfactory Returns a manifold struct to optimize over the set of subspaces in C^n.
• grassmannfactory Returns a manifold struct to optimize over the space of vector subspaces.
• grassmanngeneralizedfactory Returns a manifold struct of "scaled" vector subspaces.
• hyperbolicfactory Factory for matrices whose columns live on the hyperbolic manifold
• multinomialfactory Manifold of n-by-m column-stochastic matrices with positive entries.
• obliquecomplexfactory Returns a manifold struct defining complex matrices w/ unit-norm columns.
• obliquefactory Returns a manifold struct to optimize over matrices w/ unit-norm columns.
• positivefactory Manifold of m-by-n matrices with positive entries; scale invariant metric
• rotationsfactory Returns a manifold structure to optimize over rotation matrices.
• unitaryfactory Returns a manifold structure to optimize over unitary matrices.
• spherecomplexfactory Returns a manifold struct to optimize over unit-norm complex matrices.
• spherefactory Returns a manifold struct to optimize over unit-norm vectors or matrices.
• spheresymmetricfactory Returns a manifold struct to optimize over unit-norm symmetric matrices.
• stiefelcomplexfactory Returns a manifold struct. to optimize over complex orthonormal matrices.
• stiefelfactory Returns a manifold structure to optimize over orthonormal matrices.
• stiefelgeneralizedfactory Returns a manifold structure of "scaled" orthonormal matrices.
• stiefelstackedfactory Stiefel(k, d)^m, represented as matrices of size m*d-by-k.
• elliptopefactory Manifold of n-by-n psd matrices of rank k with unit diagonal elements.
• spectrahedronfactory Manifold of n-by-n symmetric positive semidefinite matrices of rank k
• symfixedrankYYfactory Manifold of n-by-n symmetric positive semidefinite matrices of rank k.
• TT_weingarten Weingarten map computation for the fixed TT-rank manifold.
• TTeMPS
• norm NORM Norm of a TT/MPS tensor.
• TTeMPS_block
• norm NORM Norm of a 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.
• RiemannLinsolve Riemannian approx. Newton for linear systems. For more information, we refer to the report
• RiemannPrecondSteep TTeMPS Toolbox.
• alsLinsolve TTeMPS Toolbox.
• alsLinsolve_fast TTeMPS Toolbox.
• alsLinsolve_rankOne TTeMPS Toolbox.
• amen TTeMPS Toolbox.
• amen_fast TTeMPS Toolbox.
• check_precond_laplace 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
• linearsystem_compare Example code for the algorithms described in
• fixedTTrankfactory Manifold of tensors of fixed Tensor Train (TT) rank, embedded geometry
• arc_lanczos Subproblem solver for ARC based on a Lanczos process.
• minimize_cubic_newton Minimize a cubicly regularized quadratic via Newton root finding.
• TRSgep Solves trust-region subproblem by a generalized eigenvalue problem.
• checkmanifold Run a collection of tests on a manifold obtained from a manopt factory
• checkretraction Check the order of agreement of a retraction with an exponential.

SOURCE CODE

0001 function res = norm( x, safe )
0002     %NORM Norm of a TT/MPS tensor.
0003     %   norm(X) computes the Frobenius norm of the TT/MPS tensor X.
0004     %
0005     %   norm(X, SAFE) with SAFE=true computes the Frobenius norm of the TT/MPS tensor X
0006     %       with reorthogonalization to increase the accuracy
0007     %
0008     %   See also INNERPROD
0009     
0010     %   TTeMPS Toolbox.
0011     %   Michael Steinlechner, 2013-2016
0012     %   Questions and contact: michael.steinlechner@epfl.ch
0013     %   BSD 2-clause license, see LICENSE.txt
0014     
0015     if ~exist('safe','var')
0016         safe = true;
0017     end
0018     
0019     if safe
0020         x = orthogonalize(x, x.order );
0021         res = norm( x.U{end}(:) );
0022     else
0023         res = sqrt(innerprod( x, x )); 
0024 
0025         if res < 1e-7
0026             x = orthogonalize(x, x.order );
0027             res = norm( x.U{end}(:) );
0028         end
0029     end
0030     
0031 end