Return a set of points spread out on the sphere. function [X, maxdot] = packing_on_the_sphere(d, n, epsilon, X0) Using optimization on the oblique manifold, that is, the product of spheres, this function returns a set of n points with unit norm in R^d in the form of a matrix X of size nxd, such that the points are spread out on the sphere. Ideally, we would minimize the maximum inner product between any two points X(i, :) and X(j, :), i~=j, but that is a nonsmooth cost function. Instead, we replace the max function by a classical log-sum-exp approximation and (attempt to) solve: min_{X in OB(d, n)} log( .5*sum_{i~=j} exp( xi'*xj/epsilon ) ), with xi = X(:, i) and epsilon is some "diffusion constant". As epsilon goes to zero, the cost function is a sharper approximation of the max function (under some assumptions), but the cost function becomes stiffer and hence harder to optimize. The second output, maxdot, is the maximum inner product between any two points in the returned X. This number is the one we truly are trying to minimize. Notice that this cost function is invariant under rotation of X: f(X) = f(XQ) for all orthogonal Q in O(d). This calls for optimization over the set of symmetric positive semidefinite matrices of size n and rank d with unit diagonal, which can be thought of as the quotient of the oblique manifold OB(d, n) by O(d): See elliptopefactory. This is known as the Thomson or, more specifically, the Tammes problem: http://en.wikipedia.org/wiki/Tammes_problem An interesting page by Neil Sloane collecting best known packings is available here http://neilsloane.com/packings/

- obliquefactory Returns a manifold struct to optimize over matrices w/ unit-norm columns.
- conjugategradient Conjugate gradient minimization algorithm for Manopt.
- hessianspectrum Returns the eigenvalues of the (preconditioned) Hessian at x.

0001 function [X, maxdot] = packing_on_the_sphere(d, n, epsilon, X0) 0002 % Return a set of points spread out on the sphere. 0003 % 0004 % function [X, maxdot] = packing_on_the_sphere(d, n, epsilon, X0) 0005 % 0006 % Using optimization on the oblique manifold, that is, the product of 0007 % spheres, this function returns a set of n points with unit norm in R^d in 0008 % the form of a matrix X of size nxd, such that the points are spread out 0009 % on the sphere. Ideally, we would minimize the maximum inner product 0010 % between any two points X(i, :) and X(j, :), i~=j, but that is a nonsmooth 0011 % cost function. Instead, we replace the max function by a classical 0012 % log-sum-exp approximation and (attempt to) solve: 0013 % 0014 % min_{X in OB(d, n)} log( .5*sum_{i~=j} exp( xi'*xj/epsilon ) ), 0015 % 0016 % with xi = X(:, i) and epsilon is some "diffusion constant". As epsilon 0017 % goes to zero, the cost function is a sharper approximation of the max 0018 % function (under some assumptions), but the cost function becomes stiffer 0019 % and hence harder to optimize. 0020 % 0021 % The second output, maxdot, is the maximum inner product between any two 0022 % points in the returned X. This number is the one we truly are trying to 0023 % minimize. 0024 % 0025 % Notice that this cost function is invariant under rotation of X: 0026 % f(X) = f(XQ) for all orthogonal Q in O(d). 0027 % This calls for optimization over the set of symmetric positive 0028 % semidefinite matrices of size n and rank d with unit diagonal, which can 0029 % be thought of as the quotient of the oblique manifold OB(d, n) by O(d): 0030 % See elliptopefactory. 0031 % 0032 % This is known as the Thomson or, more specifically, the Tammes problem: 0033 % http://en.wikipedia.org/wiki/Tammes_problem 0034 % An interesting page by Neil Sloane collecting best known packings is 0035 % available here http://neilsloane.com/packings/ 0036 0037 % This file is part of Manopt and is copyrighted. See the license file. 0038 % 0039 % Main author: Nicolas Boumal, July 2, 2013 0040 % Contributors: 0041 % 0042 % Change log: 0043 % Aug. 14, 2013 (NB) : Code now compatible to experiment with both the 0044 % obliquefactory and the elliptopefactory. 0045 % 0046 % Jan. 7, 2014 (NB) : Added reference to Neil Sloane's page and the 0047 % maxdot output. 0048 % 0049 % June 24, 2014 (NB) : Now shifting exponentials to alleviate numerical 0050 % trouble when epsilon is too small. 0051 % 0052 % Aug. 31, 2021 (XJ) : Added AD to compute the gradient 0053 0054 if ~exist('d', 'var') || isempty(d) 0055 % Dimension of the embedding space: R^d 0056 d = 3; 0057 end 0058 if ~exist('n', 'var') || isempty(n) 0059 % Number n of points to place of the sphere in R^d. 0060 % For example, n=12 yields an icosahedron: 0061 % https://en.wikipedia.org/wiki/Icosahedron 0062 % Notice though that platonic solids are not always optimal. 0063 % Try for example n = 8: you don't get a cube. 0064 n = 24; 0065 end 0066 if ~exist('epsilon', 'var') || isempty(epsilon) 0067 % This value should be as close to 0 as affordable. 0068 % If it is too close to zero, optimization first becomes much 0069 % slower, than simply doesn't work anymore becomes of floating 0070 % point overflow errors (NaN's and Inf's start to appear). 0071 % If it is too large, then log-sum-exp is a poor approximation of 0072 % the max function, and the spread will be less uniform. 0073 % An okay value seems to be 0.01 or 0.001 for example. Note that a 0074 % better strategy than using a small epsilon straightaway is to 0075 % reduce epsilon bit by bit and to warm-start subsequent 0076 % optimization in that way. Trustregions will be more appropriate 0077 % for these fine tunings. 0078 epsilon = 0.0015; 0079 end 0080 0081 % Pick your manifold (the elliptope factory quotients out the global 0082 % rotation invariance of the problem, which is more natural but 0083 % conceptually a bit more complicated --- for usage with the toolbox it 0084 % is the same though: just uncomment the appropriate line). 0085 manifold = obliquefactory(d, n, true); 0086 % manifold = elliptopefactory(n, d); 0087 0088 % Generate a random initial guess if none was given. 0089 if ~exist('X0', 'var') || isempty(X0) 0090 X0 = manifold.rand(); 0091 end 0092 0093 % Define the cost function with caching system used: the store 0094 % structure we receive as input is tied to the input point X. Everytime 0095 % this cost function is called at this point X, we will receive the 0096 % same store structure back. We may modify the store structure inside 0097 % the function and return it: the changes will be remembered for next 0098 % time. 0099 function [f, store] = cost(X, store) 0100 if ~isfield(store, 'ready') 0101 XXt = X*X'; 0102 % Shift the exponentials by the maximum value to reduce 0103 % numerical trouble due to possible overflows. 0104 s = max(max(triu(XXt, 1))); 0105 expXXt = exp((XXt-s)/epsilon); 0106 % Zero out the diagonal 0107 expXXt(1:(n+1):end) = 0; 0108 u = sum(sum(triu(expXXt, 1))); 0109 store.XXt = XXt; 0110 store.s = s; 0111 store.expXXt = expXXt; 0112 store.u = u; 0113 store.ready = true; 0114 end 0115 u = store.u; 0116 s = store.s; 0117 f = s + epsilon*log(u); 0118 end 0119 0120 % Define the gradient of the cost. When the gradient is called at a 0121 % point X for which the cost was already called, the store structure we 0122 % receive remember everything that the cost function stored in it, so 0123 % we can reuse previously computed elements. 0124 function [g, store] = grad(X, store) 0125 if ~isfield(store, 'ready') 0126 [~, store] = cost(X, store); 0127 end 0128 % Compute the Euclidean gradient 0129 eg = store.expXXt*X / store.u; 0130 % Convert to the Riemannian gradient (by projection) 0131 g = manifold.egrad2rgrad(X, eg); 0132 end 0133 0134 % Setup the problem structure with its manifold M and cost+grad 0135 % functions. 0136 problem.M = manifold; 0137 problem.cost = @cost; 0138 problem.grad = @grad; 0139 0140 % An alternative way to compute the grad is to use automatic 0141 % differentiation provided in the deep learning toolbox (slower) 0142 % Notice that the function triu is not supported for AD so far. 0143 % Replace it with ctriu described in the file manoptADhelp.m 0144 % problem.cost = @cost_AD; 0145 % function f = cost_AD(X) 0146 % XXt = X*X'; 0147 % s = max(max(ctriu(XXt, 1))); 0148 % expXXt = exp((XXt-s)/epsilon); 0149 % expXXt(1:(n+1):end) = 0; 0150 % u = sum(sum(ctriu(expXXt, 1))); 0151 % f = s + epsilon*log(u); 0152 % end 0153 % Call manoptAD to prepare AD for the problem structure 0154 % problem = manoptAD(problem,'egrad'); 0155 0156 % For debugging, it's always nice to check the gradient a few times. 0157 % checkgradient(problem); 0158 % pause; 0159 0160 % Call a solver on our problem with a few options defined. We did not 0161 % specify the Hessian but it is still okay to call trustregion: Manopt 0162 % will approximate the Hessian with finite differences of the gradient. 0163 opts.tolgradnorm = 1e-8; 0164 opts.maxtime = 1200; 0165 opts.maxiter = 1e5; 0166 % X = trustregions(problem, X0, opts); 0167 X = conjugategradient(problem, X0, opts); 0168 0169 % Evaluate the maximum inner product between any two points of X. 0170 XXt = X*X'; 0171 dots = XXt(find(triu(ones(n), 1))); %#ok<FNDSB> 0172 maxdot = max(dots); 0173 0174 % Similarly, even though we did not specify the Hessian, we may still 0175 % estimate its spectrum at the solution. It should reflect the 0176 % invariance of the cost function under a global rotatioon of the 0177 % sphere, which is an invariance under the group O(d) of dimension 0178 % d(d-1)/2 : this translates into d(d-1)/2 zero eigenvalues in the 0179 % spectrum of the Hessian. 0180 % The approximate Hessian is not a linear operator, and is it a 0181 % fortiori not symmetric. The result of this computation is thus not 0182 % reliable. It does display the zero eigenvalues as expected though. 0183 if manifold.dim() < 300 0184 evs = real(hessianspectrum(problem, X)); 0185 figure; 0186 stem(1:length(evs), sort(evs), '.'); 0187 title(['Eigenvalues of the approximate Hessian of the cost ' ... 0188 'function at the solution']); 0189 end 0190 0191 0192 % Show how the inner products X(:, i)'*X(:, j) are distributed. 0193 figure; 0194 hist(real(acos(dots)), 20); 0195 title('Histogram of the geodesic distances'); 0196 0197 % This is the quantity we actually want to minimize. 0198 fprintf('Maximum inner product between two points: %g\n', maxdot); 0199 0200 0201 % Give some visualization if the dimension allows 0202 if d == 2 0203 % For the circle, the optimal solution consists in spreading the 0204 % points with angles uniformly sampled in (0, 2pi). This 0205 % corresponds to the following value for the max inner product: 0206 fprintf('Optimal value for the max inner product: %g\n', cos(2*pi/n)); 0207 figure; 0208 t = linspace(-pi, pi, 201); 0209 plot(cos(t), sin(t), '-', 'LineWidth', 3, 'Color', [152,186,220]/255); 0210 daspect([1 1 1]); 0211 box off; 0212 axis off; 0213 hold on; 0214 plot(X(:, 1), X(:, 2), 'r.', 'MarkerSize', 25); 0215 hold off; 0216 end 0217 if d == 3 0218 figure; 0219 % Plot the sphere 0220 [sphere_x, sphere_y, sphere_z] = sphere(50); 0221 handle = surf(sphere_x, sphere_y, sphere_z); 0222 set(handle, 'FaceColor', [152,186,220]/255); 0223 set(handle, 'FaceAlpha', .5); 0224 set(handle, 'EdgeColor', [152,186,220]/255); 0225 set(handle, 'EdgeAlpha', .5); 0226 daspect([1 1 1]); 0227 box off; 0228 axis off; 0229 hold on; 0230 % Add the chosen points 0231 Y = 1.02*X'; 0232 plot3(Y(1, :), Y(2, :), Y(3, :), 'r.', 'MarkerSize', 25); 0233 % And connect the points which are at minimal distance, 0234 % within some tolerance. 0235 min_distance = real(acos(maxdot)); 0236 connected = real(acos(XXt)) <= 1.20*min_distance; 0237 [Ic, Jc] = find(triu(connected, 1)); 0238 for k = 1 : length(Ic) 0239 i = Ic(k); j = Jc(k); 0240 plot3(Y(1, [i j]), Y(2, [i j]), Y(3, [i j]), 'k-'); 0241 end 0242 hold off; 0243 end 0244 0245 end

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