void Covariance
( const ElementalMatrix<F>& DPre, ElementalMatrix<F>& SPre )
{
DEBUG_CSE
DistMatrixReadProxy<F,F,MC,MR>
DProx( DPre );
DistMatrixWriteProxy<F,F,MC,MR>
SProx( SPre );
auto& D = DProx.GetLocked();
auto& S = SProx.Get();
const Grid& g = D.Grid();
const Int numObs = D.Height();
// Compute the average column
DistMatrix<F> ones(g), xMean(g);
Ones( ones, numObs, 1 );
Gemv( TRANSPOSE, F(1)/F(numObs), D, ones, xMean );
DistMatrix<F,MR,STAR> xMean_MR(g);
xMean_MR.AlignWith( D );
xMean_MR = xMean;
// Subtract the mean from each column of D
DistMatrix<F> DDev( D );
for( Int iLoc=0; iLoc<DDev.LocalHeight(); ++iLoc )
blas::Axpy
( DDev.LocalWidth(), F(-1),
xMean_MR.LockedBuffer(), 1,
DDev.Buffer(iLoc,0), DDev.LDim() );
// Form S := 1/(numObs-1) DDev DDev'
Herk( LOWER, ADJOINT, Base<F>(1)/Base<F>(numObs-1), DDev, S );
Conjugate( S );
MakeHermitian( LOWER, S );
}
inline void ALM
( const ElementalMatrix<F>& MPre,
ElementalMatrix<F>& LPre,
ElementalMatrix<F>& SPre,
const RPCACtrl<Base<F>>& ctrl )
{
DistMatrixReadProxy<F,F,MC,MR>
MProx( MPre );
DistMatrixWriteProxy<F,F,MC,MR>
LProx( LPre ),
SProx( SPre );
auto& M = MProx.GetLocked();
auto& L = LProx.Get();
auto& S = SProx.Get();
typedef Base<F> Real;
const Int m = M.Height();
const Int n = M.Width();
const int commRank = mpi::Rank( M.Grid().Comm() );
// If tau is unspecified, set it to 1/sqrt(max(m,n))
const Base<F> tau =
( ctrl.tau <= Real(0) ? Real(1) / sqrt(Real(Max(m,n))) : ctrl.tau );
if( ctrl.tol <= Real(0) )
LogicError("tol cannot be non-positive");
const Base<F> tol = ctrl.tol;
const double startTime = mpi::Time();
DistMatrix<F> Y( M );
NormalizeEntries( Y );
const Real twoNorm = TwoNorm( Y );
const Real maxNorm = MaxNorm( Y );
const Real infNorm = maxNorm / tau;
const Real dualNorm = Max( twoNorm, infNorm );
Y *= F(1)/dualNorm;
// If beta is unspecified, set it to 1 / 2 || sign(M) ||_2
Base<F> beta =
( ctrl.beta <= Real(0) ? Real(1) / (2*twoNorm) : ctrl.beta );
const Real frobM = FrobeniusNorm( M );
const Real maxM = MaxNorm( M );
if( ctrl.progress && commRank == 0 )
cout << "|| M ||_F = " << frobM << "\n"
<< "|| M ||_max = " << maxM << endl;
Zeros( L, m, n );
Zeros( S, m, n );
Int numIts=0, numPrimalIts=0;
DistMatrix<F> LLast( M.Grid() ), SLast( M.Grid() ), E( M.Grid() );
while( true )
{
++numIts;
Int rank, numNonzeros;
while( true )
{
++numPrimalIts;
LLast = L;
SLast = S;
// ST_{tau/beta}(M - L + Y/beta)
S = M;
S -= L;
Axpy( F(1)/beta, Y, S );
SoftThreshold( S, tau/beta );
numNonzeros = ZeroNorm( S );
// SVT_{1/beta}(M - S + Y/beta)
L = M;
L -= S;
Axpy( F(1)/beta, Y, L );
if( ctrl.usePivQR )
rank = SVT( L, Real(1)/beta, ctrl.numPivSteps );
else
rank = SVT( L, Real(1)/beta );
LLast -= L;
SLast -= S;
const Real frobLDiff = FrobeniusNorm( LLast );
const Real frobSDiff = FrobeniusNorm( SLast );
if( frobLDiff/frobM < tol && frobSDiff/frobM < tol )
{
if( ctrl.progress && commRank == 0 )
cout << "Primal loop converged: "
<< mpi::Time()-startTime << " total secs" << endl;
break;
}
else
{
if( ctrl.progress && commRank == 0 )
cout << " " << numPrimalIts
<< ": \n"
<< " || Delta L ||_F / || M ||_F = "
<< frobLDiff/frobM << "\n"
<< " || Delta S ||_F / || M ||_F = "
<< frobSDiff/frobM << "\n"
<< " rank=" << rank
<< ", numNonzeros=" << numNonzeros
<< ", " << mpi::Time()-startTime << " total secs"
<< endl;
}
}
// E := M - (L + S)
E = M;
E -= L;
E -= S;
const Real frobE = FrobeniusNorm( E );
if( frobE/frobM <= tol )
{
if( ctrl.progress && commRank == 0 )
cout << "Converged after " << numIts << " iterations and "
<< numPrimalIts << " primal iterations with rank="
<< rank << ", numNonzeros=" << numNonzeros << " and "
<< "|| E ||_F / || M ||_F = " << frobE/frobM
<< ", " << mpi::Time()-startTime << " total secs"
<< endl;
break;
}
else if( numIts >= ctrl.maxIts )
{
if( ctrl.progress && commRank == 0 )
cout << "Aborting after " << numIts << " iterations and "
<< mpi::Time()-startTime << " total secs" << endl;
break;
}
else
{
if( ctrl.progress && commRank == 0 )
cout << numPrimalIts << ": || E ||_F / || M ||_F = "
<< frobE/frobM << ", rank=" << rank
<< ", numNonzeros=" << numNonzeros << ", "
<< mpi::Time()-startTime << " total secs"
<< endl;
}
// Y := Y + beta E
Axpy( beta, E, Y );
beta *= ctrl.rho;
}
}
inline void ADMM
( const ElementalMatrix<F>& MPre,
ElementalMatrix<F>& LPre,
ElementalMatrix<F>& SPre,
const RPCACtrl<Base<F>>& ctrl )
{
DistMatrixReadProxy<F,F,MC,MR>
MProx( MPre );
DistMatrixWriteProxy<F,F,MC,MR>
LProx( LPre ),
SProx( SPre );
auto& M = MProx.GetLocked();
auto& L = LProx.Get();
auto& S = SProx.Get();
typedef Base<F> Real;
const Int m = M.Height();
const Int n = M.Width();
const int commRank = mpi::Rank( M.Grid().Comm() );
// If tau is not specified, then set it to 1/sqrt(max(m,n))
const Base<F> tau =
( ctrl.tau <= Real(0) ? Real(1)/sqrt(Real(Max(m,n))) : ctrl.tau );
if( ctrl.beta <= Real(0) )
LogicError("beta cannot be non-positive");
if( ctrl.tol <= Real(0) )
LogicError("tol cannot be non-positive");
const Base<F> beta = ctrl.beta;
const Base<F> tol = ctrl.tol;
const double startTime = mpi::Time();
DistMatrix<F> E( M.Grid() ), Y( M.Grid() );
Zeros( Y, m, n );
const Real frobM = FrobeniusNorm( M );
const Real maxM = MaxNorm( M );
if( ctrl.progress && commRank == 0 )
cout << "|| M ||_F = " << frobM << "\n"
<< "|| M ||_max = " << maxM << endl;
Zeros( L, m, n );
Zeros( S, m, n );
Int numIts = 0;
while( true )
{
++numIts;
// ST_{tau/beta}(M - L + Y/beta)
S = M;
S -= L;
Axpy( F(1)/beta, Y, S );
SoftThreshold( S, tau/beta );
const Int numNonzeros = ZeroNorm( S );
// SVT_{1/beta}(M - S + Y/beta)
L = M;
L -= S;
Axpy( F(1)/beta, Y, L );
Int rank;
if( ctrl.usePivQR )
rank = SVT( L, Real(1)/beta, ctrl.numPivSteps );
else
rank = SVT( L, Real(1)/beta );
// E := M - (L + S)
E = M;
E -= L;
E -= S;
const Real frobE = FrobeniusNorm( E );
if( frobE/frobM <= tol )
{
if( ctrl.progress && commRank == 0 )
cout << "Converged after " << numIts << " iterations "
<< " with rank=" << rank
<< ", numNonzeros=" << numNonzeros << " and "
<< "|| E ||_F / || M ||_F = " << frobE/frobM
<< ", and " << mpi::Time()-startTime << " total secs"
<< endl;
break;
}
else if( numIts >= ctrl.maxIts )
{
if( ctrl.progress && commRank == 0 )
cout << "Aborting after " << numIts << " iterations and "
<< mpi::Time()-startTime << " total secs"
<< endl;
break;
}
else
{
if( ctrl.progress && commRank == 0 )
cout << numIts << ": || E ||_F / || M ||_F = "
<< frobE/frobM << ", rank=" << rank
<< ", numNonzeros=" << numNonzeros
<< ", " << mpi::Time()-startTime << " total secs"
<< endl;
}
// Y := Y + beta E
Axpy( beta, E, Y );
}
}
