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packing_on_the_sphere

PURPOSE ^

Return a set of points spread out on the sphere.

SYNOPSIS ^

function [X, maxdot] = packing_on_the_sphere(d, n, epsilon, X0)

DESCRIPTION ^

 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/

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SUBFUNCTIONS ^

SOURCE CODE ^

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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