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

- 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.

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

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