function M = elliptopefactory(n, k)
% Manifold of n-by-n psd matrices of rank k with unit diagonal elements.
%
% function M = elliptopefactory(n, k)
%
% A point X on the manifold is parameterized as YY^T where Y is a matrix of
% size nxk. As such, X is symmetric, positive semidefinite. We restrict to
% full-rank Y's, such that X has rank exactly k. The point X is numerically
% represented by Y (this is more efficient than working with X, which may
% be big). Tangent vectors are represented as matrices of the same size as
% Y, call them Ydot, so that Xdot = Y Ydot' + Ydot Y and diag(Xdot) == 0.
% The metric is the canonical Euclidean metric on Y.
% 
% The diagonal constraints on X (X(i, i) == 1 for all i) translate to
% unit-norm constraints on the rows of Y: norm(Y(i, :)) == 1 for all i.
% The set of such Y's forms the oblique manifold. But because for any
% orthogonal Q of size k it holds that (YQ)(YQ)' = YY', we "group" all
% matrices of the form YQ in an equivalence class. The set of equivalence
% classes is a Riemannian quotient manifold, implemented here.
%
% Note that this geometry formally breaks down at rank-deficient Y's.
% This does not appear to be a major issue in practice when optimization
% algorithms converge to rank-deficient Y's, but convergence theorems no
% longer hold. As an alternative, you may use the oblique manifold (it has
% larger dimension, but does not break down at rank drop.)
%
% The geometry is taken from the 2010 paper:
% M. Journee, P.-A. Absil, F. Bach and R. Sepulchre,
% "Low-Rank Optimization on the Cone of Positive Semidefinite Matrices".
% Paper link: http://www.di.ens.fr/~fbach/journee2010_sdp.pdf
% 
% 
% Please cite the Manopt paper as well as the research paper:
%     @Article{journee2010low,
%       Title   = {Low-rank optimization on the cone of positive semidefinite matrices},
%       Author  = {Journ{\'e}e, M. and Bach, F. and Absil, P.-A. and Sepulchre, R.},
%       Journal = {SIAM Journal on Optimization},
%       Year    = {2010},
%       Number  = {5},
%       Pages   = {2327--2351},
%       Volume  = {20},
%       Doi     = {10.1137/080731359}
%     }
% 
%
% See also: obliquefactory symfixedrankYYfactory spectrahedronfactory

% This file is part of Manopt: www.manopt.org.
% Original author: Bamdev Mishra, July 12, 2013.
% Contributors:
% Change log:
%   July 18, 2013 (NB):
%       Fixed projection operator for rank-deficient Y'Y.
% 
%   Aug.  8, 2013 (NB):
%       No longer using nested functions, to aim at Octave compatibility.
%       Sign error in right hand side of the call to minres corrected.
% 
%   June 24, 2014 (NB):
%       Used code snippets from obliquefactory to speed up projection,
%       retraction, egrad2rgrad and rand: the code now uses bsxfun for this.
% 
%   Apr. 3, 2015 (NB):
%       Replaced trace(A'*B) by A(:)'*B(:) : equivalent but faster.
% 
%   Apr. 17, 2018 (NB):
%       Removed dependency on lyap entirely.
%
%   Sep.  6, 2018 (NB):
%       Removed M.exp() as it was not implemented.

% TODO: modify normalize_rows and project_rows to work without transposes.
% TODO: enhance ehess2rhess to also use bsxfun.
    
    if k < 2
        warning('manopt:elliptopefactory:lowk', ...
                'k should be an integer >= 2. At k = 1, the set is discrete.');
    end
    
    
    M.name = @() sprintf('YY'' quotient manifold of %dx%d psd matrices of rank %d with diagonal elements being 1', n, n, k);
    
    M.dim = @() n*(k-1) - k*(k-1)/2; % Extra -1 is because of the diagonal constraint that
    
    % Euclidean metric on the total space
    M.inner = @(Y, eta, zeta) eta(:)'*zeta(:);
    
    M.norm = @(Y, eta) sqrt(M.inner(Y, eta, eta));
    
    M.typicaldist = @() 10*k;
    
    M.proj = @projection;
    
    M.tangent = M.proj;
    M.tangent2ambient = @(Y, eta) eta;
    
    M.retr = @retraction;
    
    M.egrad2rgrad = @egrad2rgrad;
    
    M.ehess2rhess = @ehess2rhess;
    
    % Notice that the hash of two equivalent points will be different...
    M.hash = @(Y) ['z' hashmd5(Y(:))];
    
    M.rand = @() random(n, k);
    
    M.randvec = @randomvec;
    
    M.lincomb = @matrixlincomb;
    
    M.zerovec = @(Y) zeros(n, k);
    
    M.transp = @(Y1, Y2, d) projection(Y2, d);
    
    M.vec = @(Y, u_mat) u_mat(:);
    M.mat = @(Y, u_vec) reshape(u_vec, [n, k]);
    M.vecmatareisometries = @() true;
    
end

% Given a matrix X, returns the same matrix but with each column scaled so
% that they have unit 2-norm.
% See obliquefactory.
function X = normalize_rows(X)
    X = X';
    norms = sqrt(sum(X.^2, 1));
    X = bsxfun(@times, X, 1./norms);
    X = X';
end

% Orthogonal projection of each row of H to the tangent space at the
% corresponding row of X, seen as a point on a sphere.
% See obliquefactory.
function PXH = project_rows(X, H)
    X = X';
    H = H';
    % Compute the inner product between each vector H(:, i) with its root
    % point X(:, i), that is, X(:, i).' * H(:, i). Returns a row vector.
    inners = sum(X.*H, 1);
    % Subtract from H the components of the H(:, i)'s that are parallel to
    % the root points X(:, i).
    PXH = H - bsxfun(@times, X, inners);
    PXH = PXH';
end


% Projection onto the tangent space, i.e., on the tangent space of
% ||Y(i, :)|| = 1
function etaproj = projection(Y, eta)
    eta = project_rows(Y, eta);

    % Projection onto the horizontal space
    YtY = Y'*Y;
    SS = YtY;
    AS = Y'*eta - eta'*Y;
    Omega = lyapunov_symmetric(SS, AS);
        
    % It does not seem necessary to enforce skew-symmetry numerically.
    % Omega = (Omega-Omega')/2;
    
    etaproj = eta - Y*Omega;
end

% Retraction
function Ynew = retraction(Y, eta, t)
    if nargin < 3
        t = 1.0;
    end
    Ynew = Y + t*eta;
    Ynew = normalize_rows(Ynew);
end

% Euclidean gradient to Riemannian gradient conversion.
% We only need the ambient space projection: the remainder of the
% projection function is not necessary because the Euclidean gradient must
% already be orthogonal to the vertical space.
function rgrad = egrad2rgrad(Y, egrad)
    rgrad = project_rows(Y, egrad);
end

% Euclidean Hessian to Riemannian Hessian conversion.
% TODO: speed this function up using bsxfun.
function Hess = ehess2rhess(Y, egrad, ehess, eta)
    k = size(Y, 2);

    % Directional derivative of the Riemannian gradient
    scaling_grad = sum((egrad.*Y), 2); % column vector of size n
    scaling_grad_repeat = scaling_grad*ones(1, k);

    Hess = ehess - scaling_grad_repeat.*eta;

    scaling_hess = sum((eta.*egrad) + (Y.*ehess), 2);
    scaling_hess_repeat = scaling_hess*ones(1, k);
    % directional derivative of scaling_grad_repeat
    Hess = Hess - scaling_hess_repeat.*Y;

    % Project on the horizontal space
    Hess = projection(Y, Hess);
end

% Random point generation on the manifold
function Y = random(n, k)
    Y = randn(n, k);
    Y = normalize_rows(Y);
end

% Random vector generation at Y
function eta = randomvec(Y)
    eta = randn(size(Y));
    eta = projection(Y, eta);
    nrm = norm(eta, 'fro');
    eta = eta / nrm;
end
