Rotationally align clouds of points (generalized Procrustes problem) function X = generalized_procrustes(A_measure) The input is a 3D matrix A_measure of size nxmxN. Each of the N slices A_measure(:, :, i) is a cloud of m points in R^n. These clouds are assumed to be (noisy) rotated versions of a reference cloud Atrue. This algorithm tries to find the optimal rotations to apply to the individual clouds such that they will match each other as much as possible following a least-squares cost. The output A is an estimate of the cloud Atrue (up to rotation). The output R is a 3D matrix of size nxnxN containing the rotation matrices such that R(:, :, i) * A is approximately equal to A_measure(:, :, i).

- euclideanfactory Returns a manifold struct to optimize over real matrices.
- randrot Generates uniformly random rotation matrices.
- rotationsfactory Returns a manifold structure to optimize over rotation matrices.
- norm NORM Norm of a TT/MPS tensor.
- norm NORM Norm of a TT/MPS block-mu tensor.
- trustregions Riemannian trust-regions solver for optimization on manifolds.
- hessianspectrum Returns the eigenvalues of the (preconditioned) Hessian at x.
- multiprod Matrix multiply 2-D slices of N-D arrays
- multitransp Transpose the matrix slices of an N-D array (no complex conjugate)
- productmanifold Returns a structure describing a product manifold M = M1 x M2 x ... x Mn.

- function [f, store] = cost(X, store)
- function [g, store] = grad(X, store)
- function [h, store] = hess(X, Xdot, store)

0001 function [A, R] = generalized_procrustes(A_measure) 0002 % Rotationally align clouds of points (generalized Procrustes problem) 0003 % 0004 % function X = generalized_procrustes(A_measure) 0005 % 0006 % The input is a 3D matrix A_measure of size nxmxN. Each of the N slices 0007 % A_measure(:, :, i) is a cloud of m points in R^n. These clouds are 0008 % assumed to be (noisy) rotated versions of a reference cloud Atrue. 0009 % This algorithm tries to find the optimal rotations to apply to the 0010 % individual clouds such that they will match each other as much as 0011 % possible following a least-squares cost. 0012 % 0013 % The output A is an estimate of the cloud Atrue (up to rotation). The 0014 % output R is a 3D matrix of size nxnxN containing the rotation matrices 0015 % such that R(:, :, i) * A is approximately equal to A_measure(:, :, i). 0016 0017 % This file is part of Manopt and is copyrighted. See the license file. 0018 % 0019 % Main author: Nicolas Boumal, July 8, 2013 0020 % Contributors: 0021 % 0022 % Change log: 0023 % 0024 % Xiaowen Jiang Aug. 20, 2021 0025 % Added AD to compute the grad and the hess 0026 0027 if ~exist('A_measure', 'var') 0028 % Generate random data to test the method. 0029 % There are N clouds of m points in R^n. Each of them is a noisy, 0030 % rotated version of a reference cloud A. Rotations are uniformly 0031 % random and noise on each rotated cloud is iid normal with 0032 % standard deviation sigma. 0033 n = 3; 0034 m = 10; 0035 N = 50; 0036 % The reference cloud 0037 Atrue = randn(n, m); 0038 % A 3D matrix containing the N measured clouds 0039 sigma = .3; 0040 A_measure = multiprod(randrot(n, N), Atrue) + sigma*randn(n, m, N); 0041 else 0042 [n, m, N] = size(A_measure); 0043 end 0044 0045 % Construct a manifold structure representing the product of groups of 0046 % rotations with the Euclidean space for A. We optimize simultaneously 0047 % for the reference cloud and for the rotations that affect each of the 0048 % measured clouds. Notice that there is a group invariance because 0049 % there is no way of telling which orientation the reference cloud 0050 % should be in. 0051 tuple.R = rotationsfactory(n, N); 0052 tuple.A = euclideanfactory(n, m); 0053 M = productmanifold(tuple); 0054 0055 % Define the cost function here. Points on the manifold M are 0056 % structures with fields X.A and X.R, containing matrices of sizes 0057 % respectively nxm and nxnxN. The store structure (the caching system) 0058 % is used to keep the residue matrix E in memory, as it is also used in 0059 % the computation of the gradient and of the Hessian. This way, we 0060 % prevent redundant computations. 0061 function [f, store] = cost(X, store) 0062 if ~isfield(store, 'E') 0063 R = X.R; 0064 A = X.A; 0065 store.E = multiprod(R, A) - A_measure; 0066 end 0067 E = store.E; 0068 f = (E(:)'*E(:))/(2*N); 0069 end 0070 0071 % Riemannian gradient of the cost function. 0072 function [g, store] = grad(X, store) 0073 R = X.R; 0074 A = X.A; 0075 if ~isfield(store, 'E') 0076 [~, store] = cost(X, store); 0077 end 0078 E = store.E; 0079 % Compute the Euclidean gradient of the cost wrt the rotations R 0080 % and wrt the cloud A, 0081 egrad.R = multiprod(E, A'/N); 0082 egrad.A = A - mean(multiprod(multitransp(R), A_measure), 3); 0083 % then transform this Euclidean gradient into the Riemannian 0084 % gradient. 0085 g = M.egrad2rgrad(X, egrad); 0086 store.egrad = egrad; 0087 end 0088 0089 % It is not necessary to define the Hessian of the cost. We do it 0090 % mostly to illustrate how to do it and to study the spectrum of the 0091 % Hessian at the solution (see further down). 0092 function [h, store] = hess(X, Xdot, store) 0093 R = X.R; 0094 A = X.A; 0095 % Careful: tangent vectors on the rotation group are represented as 0096 % skew symmetric matrices. To obtain the corresponding vectors in 0097 % the ambient space, we need a little transformation. This 0098 % transformation is typically not needed when we compute the 0099 % formulas for the gradient and the Hessian directly in Riemannian 0100 % form instead of resorting the egrad2rgrad and ehess2rhess. These 0101 % latter tools are convenient for prototyping but are not always 0102 % the most efficient form to execute the computations. 0103 Rdot = tuple.R.tangent2ambient(R, Xdot.R); 0104 Adot = Xdot.A; 0105 if ~isfield(store, 'egrad') 0106 [~, store] = grad(X, store); 0107 end 0108 E = store.E; 0109 egrad = store.egrad; 0110 0111 ehess.R = multiprod(multiprod(Rdot, A) + multiprod(R, Adot), A') + ... 0112 multiprod(E, Adot'); 0113 ehess.R = ehess.R / N; 0114 ehess.A = Adot-mean(multiprod(multitransp(Rdot), A_measure), 3); 0115 0116 h = M.ehess2rhess(X, egrad, ehess, Xdot); 0117 end 0118 0119 % Setup the problem structure with manifold M and cost+grad functions. 0120 problem.M = M; 0121 problem.cost = @cost; 0122 problem.grad = @grad; 0123 problem.hess = @hess; 0124 0125 % An alternative way to compute the gradient and the hessian is to use 0126 % automatic differentiation provided in the deep learning toolbox (slower) 0127 % problem.cost = @cost_AD; 0128 % function f = cost_AD(X) 0129 % R = X.R; 0130 % A = X.A; 0131 % E = multiprod(R, A) - A_measure; 0132 % f = (E(:)'*E(:))/(2*N); 0133 % end 0134 % call manoptAD to prepare AD for the problem structure 0135 % problem = manoptAD(problem); 0136 0137 % For debugging, it's always nice to check the gradient a few times. 0138 % checkgradient(problem); 0139 % pause; 0140 % checkhessian(problem); 0141 % pause; 0142 0143 % Call a solver on our problem. This can probably be much improved if a 0144 % clever initial guess is used instead of a random one. 0145 X = trustregions(problem); 0146 A = X.A; 0147 R = X.R; 0148 0149 % To evaluate the performance of the algorithm, see how well Atrue (the 0150 % reference cloud) matches A (the found cloud). Since the recovery is 0151 % up to rotation, apply Kabsch algorithm (or standard Procrustes), 0152 % i.e., compute the polar factorization to best align Atrue and A. 0153 if exist('Atrue', 'var') 0154 [U, ~, V] = svd(Atrue*A'); 0155 Ahat = (U*V')*A; 0156 fprintf('Registration error: %g.\n', norm(Atrue-Ahat, 'fro')); 0157 end 0158 0159 % Plot the spectrum of the Hessian at the solution found. 0160 % Notice that the invariance of f under a rotation yields dim SO(n), 0161 % that is, n*(n-1)/2 zero eigenvalues in the Hessian spectrum at the 0162 % solution. This indicates that critical points are not isolated and 0163 % can theoretically prevent quadratic convergence. One solution to 0164 % circumvent this would be to fix one rotation arbitrarily. Another 0165 % solution would be to work on a quotient manifold. Both can be 0166 % achieved in Manopt: they simply require a little more work on the 0167 % manifold description side. 0168 if M.dim() <= 512 0169 stairs(sort(hessianspectrum(problem, X))); 0170 title('Spectrum of the Hessian at the solution found.'); 0171 xlabel('Eigenvalue number (sorted)'); 0172 ylabel('Value of the eigenvalue'); 0173 end 0174 0175 end

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