Home > manopt > manifolds > symfixedrank > sympositivedefinitefactory.m

# sympositivedefinitefactory

## PURPOSE

Manifold of n-by-n symmetric positive definite matrices with

## SYNOPSIS

function M = sympositivedefinitefactory(n)

## DESCRIPTION

 Manifold of n-by-n symmetric positive definite matrices with
the bi-invariant geometry.

function M = sympositivedefinitefactory(n)

A point X on the manifold is represented as a symmetric positive definite
matrix X (nxn). Tangent vectors are symmetric matrices of the same size
(but not necessarily definite).

The Riemannian metric is the bi-invariant metric, described notably in
Chapter 6 of the 2007 book "Positive definite matrices"
by Rajendra Bhatia, Princeton University Press.

The retraction / exponential map involves expm (the matrix exponential).
If too large a vector is retracted / exponentiated (e.g., a solver tries
to make too big a step), this may result in NaN's in the returned point,
which most likely would lead to NaN's in the cost / gradient / ... and
will result in failure of the optimization. For trustregions, this can be
controlled by setting options.Delta0 and options.Delta_bar, to prevent
too large steps.

Note also that many of the functions involve solving linear systems in X
(a point on the manifold), taking matrix exponentals and logarithms, etc.
It could therefore be beneficial to do some precomputation on X (an
eigenvalue decomposition for example) and store both X and the
preprocessing in a structure. This would require modifying the present
factory to work with such structures to represent both points and tangent
vectors. We omit this in favor of simplicity, but it may be good to keep
this in mind if efficiency becomes an issue in your application.

## CROSS-REFERENCE INFORMATION

This function calls:
• hashmd5 Computes the MD5 hash of input data.
• matrixlincomb Linear combination function for tangent vectors represented as matrices.
This function is called by:

## SOURCE CODE

0001 function M = sympositivedefinitefactory(n)
0002 % Manifold of n-by-n symmetric positive definite matrices with
0003 % the bi-invariant geometry.
0004 %
0005 % function M = sympositivedefinitefactory(n)
0006 %
0007 % A point X on the manifold is represented as a symmetric positive definite
0008 % matrix X (nxn). Tangent vectors are symmetric matrices of the same size
0009 % (but not necessarily definite).
0010 %
0011 % The Riemannian metric is the bi-invariant metric, described notably in
0012 % Chapter 6 of the 2007 book "Positive definite matrices"
0013 % by Rajendra Bhatia, Princeton University Press.
0014 %
0015 %
0016 % The retraction / exponential map involves expm (the matrix exponential).
0017 % If too large a vector is retracted / exponentiated (e.g., a solver tries
0018 % to make too big a step), this may result in NaN's in the returned point,
0019 % which most likely would lead to NaN's in the cost / gradient / ... and
0020 % will result in failure of the optimization. For trustregions, this can be
0021 % controlled by setting options.Delta0 and options.Delta_bar, to prevent
0022 % too large steps.
0023 %
0024 %
0025 % Note also that many of the functions involve solving linear systems in X
0026 % (a point on the manifold), taking matrix exponentals and logarithms, etc.
0027 % It could therefore be beneficial to do some precomputation on X (an
0028 % eigenvalue decomposition for example) and store both X and the
0029 % preprocessing in a structure. This would require modifying the present
0030 % factory to work with such structures to represent both points and tangent
0031 % vectors. We omit this in favor of simplicity, but it may be good to keep
0032 % this in mind if efficiency becomes an issue in your application.
0033
0034 % This file is part of Manopt: www.manopt.org.
0035 % Original author: Bamdev Mishra, August 29, 2013.
0036 % Contributors: Nicolas Boumal
0037 % Change log:
0038 %
0039 %   March 5, 2014 (NB)
0040 %       There were a number of mistakes in the code owing to the tacit
0041 %       assumption that if X and eta are symmetric, then X\eta is
0042 %       symmetric too, which is not the case. See discussion on the Manopt
0043 %       forum started on Jan. 19, 2014. Functions norm, dist, exp and log
0044 %       were modified accordingly. Furthermore, they only require matrix
0045 %       inversion (as well as matrix log or matrix exp), not matrix square
0046 %       roots or their inverse.
0047 %
0048 %   July 28, 2014 (NB)
0049 %       The dim() function returned n*(n-1)/2 instead of n*(n+1)/2.
0050 %       Implemented proper parallel transport from Sra and Hosseini (not
0051 %       used by default).
0052 %       Also added symmetrization in exp and log (to be sure).
0053 %
0054 %   April 3, 2015 (NB):
0055 %       Replaced trace(A*B) by a faster equivalent that does not compute
0056 %       the whole product A*B, for inner product, norm and distance.
0057 %
0058 %   May 23, 2017 (NB):
0059 %       As seen in a talk of Wen Huang at the SIAM Optimization Conference
0060 %       today, replaced the retraction of this factory (which was simply
0061 %       equal to the exponential map) with a simpler, second-order
0062 %       retraction. That this retraction is second order can be verified
0063 %       numerically with checkretraction(sympositivedefinitefactory(5));
0064 %       Notice that, for this retraction, it would be cheap to evaluate for
0065 %       many values of t, that is, it is cheap to retract many points along
0066 %       the same tangent direction. This could in principle be exploited to
0067 %       speed up line-searches.
0068
0069     symm = @(X) .5*(X+X');
0070
0071     M.name = @() sprintf('Symmetric positive definite geometry of %dx%d matrices', n, n);
0072
0073     M.dim = @() n*(n+1)/2;
0074
0075     % Helpers to avoid computing full matrices simply to extract their trace
0076     vec     = @(A) A(:);
0077     trinner = @(A, B) vec(A')'*vec(B);  % = trace(A*B)
0078     trnorm  = @(A) sqrt(trinner(A, A)); % = sqrt(trace(A^2))
0079
0080     % Choice of the metric on the orthonormal space is motivated by the
0081     % symmetry present in the space. The metric on the positive definite
0082     % cone is its natural bi-invariant metric.
0083     % The result is equal to: trace( (X\eta) * (X\zeta) )
0084     M.inner = @(X, eta, zeta) trinner(X\eta, X\zeta);
0085
0086     % Notice that X\eta is *not* symmetric in general.
0087     % The result is equal to: sqrt(trace((X\eta)^2))
0088     % There should be no need to take the real part, but rounding errors
0089     % may cause a small imaginary part to appear, so we discard it.
0090     M.norm = @(X, eta) real(trnorm(X\eta));
0091
0092     % Same here: X\Y is not symmetric in general.
0093     % Same remark about taking the real part.
0094     M.dist = @(X, Y) real(trnorm(real(logm(X\Y))));
0095
0096
0097     M.typicaldist = @() sqrt(n*(n+1)/2);
0098
0099
0100     M.egrad2rgrad = @egrad2rgrad;
0101     function eta = egrad2rgrad(X, eta)
0102         eta = X*symm(eta)*X;
0103     end
0104
0105
0106     M.ehess2rhess = @ehess2rhess;
0107     function Hess = ehess2rhess(X, egrad, ehess, eta)
0108         % Directional derivatives of the Riemannian gradient
0109         Hess = X*symm(ehess)*X + 2*symm(eta*symm(egrad)*X);
0110
0111         % Correction factor for the non-constant metric
0112         Hess = Hess - symm(eta*symm(egrad)*X);
0113     end
0114
0115
0116     M.proj = @(X, eta) symm(eta);
0117
0118     M.tangent = M.proj;
0119     M.tangent2ambient = @(X, eta) eta;
0120
0121     M.retr = @retraction;
0122     function Y = retraction(X, eta, t)
0123         if nargin < 3
0124             teta = eta;
0125         else
0126             teta = t*eta;
0127         end
0128         % The symm() call is mathematically unnecessary but numerically
0129         % necessary.
0130         Y = symm(X + teta + .5*teta*(X\teta));
0131     end
0132
0133     M.exp = @exponential;
0134     function Y = exponential(X, eta, t)
0135         if nargin < 3
0136             t = 1.0;
0137         end
0138         % The symm() and real() calls are mathematically not necessary but
0139         % are numerically necessary.
0140         Y = symm(X*real(expm(X\(t*eta))));
0141     end
0142
0143     M.log = @logarithm;
0144     function H = logarithm(X, Y)
0145         % Same remark regarding the calls to symm() and real().
0146         H = symm(X*real(logm(X\Y)));
0147     end
0148
0149     M.hash = @(X) ['z' hashmd5(X(:))];
0150
0151     % Generate a random symmetric positive definite matrix following a
0152     % certain distribution. The particular choice of a distribution is of
0153     % course arbitrary, and specific applications might require different
0154     % ones.
0155     M.rand = @random;
0156     function X = random()
0157         D = diag(1+rand(n, 1));
0158         [Q, R] = qr(randn(n)); %#ok<NASGU>
0159         X = Q*D*Q';
0160     end
0161
0162     % Generate a uniformly random unit-norm tangent vector at X.
0163     M.randvec = @randomvec;
0164     function eta = randomvec(X)
0165         eta = symm(randn(n));
0166         nrm = M.norm(X, eta);
0167         eta = eta / nrm;
0168     end
0169
0170     M.lincomb = @matrixlincomb;
0171
0172     M.zerovec = @(X) zeros(n);
0173
0174     % Poor man's vector transport: exploit the fact that all tangent spaces
0175     % are the set of symmetric matrices, so that the identity is a sort of
0176     % vector transport. It may perform poorly if the origin and target (X1
0177     % and X2) are far apart though. This should not be the case for typical
0178     % optimization algorithms, which perform small steps.
0179     M.transp = @(X1, X2, eta) eta;
0180
0181     % For reference, a proper vector transport is given here, following
0182     % work by Sra and Hosseini: "Conic geometric optimisation on the
0183     % manifold of positive definite matrices", to appear in SIAM J. Optim.
0184     % in 2015; also available here: http://arxiv.org/abs/1312.1039
0185     % This will not be used by default. To force the use of this transport,
0186     % execute "M.transp = M.paralleltransp;" on your M returned by the
0187     % present factory.
0188     M.paralleltransp = @parallel_transport;
0189     function zeta = parallel_transport(X, Y, eta)
0190         E = sqrtm((Y/X));
0191         zeta = E*eta*E';
0192     end
0193
0194     % vec and mat are not isometries, because of the unusual inner metric.
0195     M.vec = @(X, U) U(:);
0196     M.mat = @(X, u) reshape(u, n, n);
0197     M.vecmatareisometries = @() false;
0198
0199 end

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