UnitaryCoherence( DistMatrix<F>& U )
{
#ifndef RELEASE
CallStackEntry entry("UnitaryCoherence");
#endif
typedef BASE(F) R;
const Grid& grid = U.Grid();
const Int n = U.Height();
const Int r = U.Width();
// Z := U U' in n^2 r work
DistMatrix<F> Z( grid );
Herk( UPPER, NORMAL, F(1), U, Z );
// Now make Z explicitly Hermitian so that our job is easier
MakeHermitian( UPPER, Z );
// Compute the maximum column two-norm squared
const Int localWidth = Z.LocalWidth();
const Int localHeight = Z.LocalHeight();
std::vector<R> normsSquared( localWidth );
for( Int jLocal=0; jLocal<localWidth; ++jLocal )
{
const R localNorm =
blas::Nrm2( localHeight, Z.LockedBuffer(0,jLocal), 1 );
normsSquared[jLocal] = localNorm*localNorm;
}
mpi::AllReduce( &normsSquared[0], localWidth, grid.ColComm() );
R maxLocalNormSquared =
*std::max_element( normsSquared.begin(), normsSquared.end() );
const R maxNormSquared =
mpi::AllReduce( maxLocalNormSquared, mpi::MAX, grid.RowComm() );
return (n*maxNormSquared)/r;
}
void QP
( const SparseMatrix<Real>& A,
const Matrix<Real>& B,
Matrix<Real>& X,
const qp::direct::Ctrl<Real>& ctrl )
{
DEBUG_CSE
const Int n = A.Width();
const Int k = B.Width();
SparseMatrix<Real> Q, AHat;
Matrix<Real> bHat, c;
Herk( LOWER, ADJOINT, Real(1), A, Q );
MakeHermitian( LOWER, Q );
Zeros( AHat, 0, n );
Zeros( bHat, 0, 1 );
Zeros( X, n, k );
Matrix<Real> y, z;
for( Int j=0; j<k; ++j )
{
auto x = X( ALL, IR(j) );
auto b = B( ALL, IR(j) );
Zeros( c, n, 1 );
Multiply( ADJOINT, Real(-1), A, b, Real(0), c );
El::QP( Q, AHat, bHat, c, x, y, z, ctrl );
}
}
UnitaryCoherence( Matrix<F>& U )
{
#ifndef RELEASE
CallStackEntry entry("UnitaryCoherence");
#endif
typedef BASE(F) R;
const Int n = U.Height();
const Int r = U.Width();
// Z := U U' in n^2 r work
Matrix<F> Z;
Herk( UPPER, NORMAL, F(1), U, Z );
// Now make Z explicitly Hermitian so that our job is easier
MakeHermitian( UPPER, Z );
// Compute the maximum column two-norm
R maxColNorm = 0;
for( Int j=0; j<n; ++j )
{
const R colNorm = blas::Nrm2( n, Z.LockedBuffer(0,j), 1 );
maxColNorm = std::max( colNorm, maxColNorm );
}
return (n*maxColNorm*maxColNorm)/r;
}
QDWHInfo QDWH( Matrix<F>& A, Matrix<F>& P, const QDWHCtrl& ctrl )
{
EL_DEBUG_CSE
Matrix<F> ACopy( A );
auto info = QDWH( A, ctrl );
Zeros( P, A.Height(), A.Height() );
Trrk( LOWER, NORMAL, NORMAL, F(1), A, ACopy, F(0), P );
MakeHermitian( LOWER, P );
return info;
}
inline void HermitianSVD
( UpperOrLower uplo, DistMatrix<F>& A,
DistMatrix<BASE(F),VR,STAR>& s, DistMatrix<F>& U, DistMatrix<F>& V )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSVD");
#endif
#ifdef HAVE_PMRRR
typedef BASE(F) R;
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s, V );
// Redistribute the singular values into an [MR,* ] distribution
const Grid& grid = A.Grid();
DistMatrix<R,MR,STAR> s_MR_STAR( grid );
s_MR_STAR.AlignWith( V.DistData() );
s_MR_STAR = s;
// Set the singular values to the absolute value of the eigenvalues
const Int numLocalVals = s.LocalHeight();
for( Int iLoc=0; iLoc<numLocalVals; ++iLoc )
{
const R sigma = s.GetLocal(iLoc,0);
s.SetLocal(iLoc,0,Abs(sigma));
}
// Copy V into U (flipping the sign as necessary)
U.AlignWith( V );
U.ResizeTo( V.Height(), V.Width() );
const Int localHeight = V.LocalHeight();
const Int localWidth = V.LocalWidth();
for( Int jLoc=0; jLoc<localWidth; ++jLoc )
{
const R sigma = s_MR_STAR.GetLocal( jLoc, 0 );
F* UCol = U.Buffer( 0, jLoc );
const F* VCol = V.LockedBuffer( 0, jLoc );
if( sigma >= 0 )
for( Int iLoc=0; iLoc<localHeight; ++iLoc )
UCol[iLoc] = VCol[iLoc];
else
for( Int iLoc=0; iLoc<localHeight; ++iLoc )
UCol[iLoc] = -VCol[iLoc];
}
#else
U = A;
MakeHermitian( uplo, U );
SVD( U, s, V );
#endif // ifdef HAVE_PMRRR
}
inline void HermitianSVD
( UpperOrLower uplo, Matrix<F>& A, Matrix<BASE(F)>& s )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSVD");
#endif
#if 1
// Grab the eigenvalues of A
HermitianEig( uplo, A, s );
// Set the singular values to the absolute value of the eigenvalues
for( Int i=0; i<s.Height(); ++i )
s.Set(i,0,Abs(s.Get(i,0)));
#else
MakeHermitian( uplo, A );
SVD( A, s );
#endif
}
inline void
HPSDCholesky( UpperOrLower uplo, DistMatrix<R>& A )
{
#ifndef RELEASE
CallStackEntry entry("HPSDCholesky");
#endif
HPSDSquareRoot( uplo, A );
MakeHermitian( uplo, A );
if( uplo == LOWER )
{
LQ( A );
MakeTriangular( LOWER, A );
}
else
{
QR( A );
MakeTriangular( UPPER, A );
}
}
QDWHInfo
QDWH
( AbstractDistMatrix<F>& APre,
AbstractDistMatrix<F>& PPre,
const QDWHCtrl& ctrl )
{
EL_DEBUG_CSE
DistMatrixReadWriteProxy<F,F,MC,MR> AProx( APre );
DistMatrixWriteProxy<F,F,MC,MR> PProx( PPre );
auto& A = AProx.Get();
auto& P = PProx.Get();
DistMatrix<F> ACopy( A );
auto info = QDWH( A, ctrl );
Zeros( P, A.Height(), A.Height() );
Trrk( LOWER, NORMAL, NORMAL, F(1), A, ACopy, F(0), P );
MakeHermitian( LOWER, P );
return info;
}
void QP
( const DistSparseMatrix<Real>& A,
const DistMultiVec<Real>& B,
DistMultiVec<Real>& X,
const qp::direct::Ctrl<Real>& ctrl )
{
DEBUG_CSE
const Int m = A.Height();
const Int n = A.Width();
const Int k = B.Width();
mpi::Comm comm = A.Comm();
DistSparseMatrix<Real> Q(comm), AHat(comm);
DistMultiVec<Real> bHat(comm), c(comm);
Herk( LOWER, ADJOINT, Real(1), A, Q );
MakeHermitian( LOWER, Q );
Zeros( AHat, 0, n );
Zeros( bHat, 0, 1 );
Zeros( X, n, k );
DistMultiVec<Real> q(comm), y(comm), z(comm);
auto& qLoc = q.Matrix();
auto& XLoc = X.Matrix();
auto& BLoc = B.LockedMatrix();
for( Int j=0; j<k; ++j )
{
auto xLoc = XLoc( ALL, IR(j) );
auto bLoc = BLoc( ALL, IR(j) );
Zeros( c, n, 1 );
Zeros( q, m, 1 );
qLoc = bLoc;
Multiply( ADJOINT, Real(-1), A, q, Real(0), c );
Zeros( q, n, 1 );
qLoc = xLoc;
El::QP( Q, AHat, bHat, c, q, y, z, ctrl );
xLoc = qLoc;
}
}
void Covariance( const Matrix<F>& D, Matrix<F>& S )
{
DEBUG_CSE
const Int numObs = D.Height();
const Int n = D.Width();
// Compute the average column
Matrix<F> ones, xMean;
Ones( ones, numObs, 1 );
Gemv( TRANSPOSE, F(1)/F(numObs), D, ones, xMean );
// Subtract the mean from each column of D
Matrix<F> DDev( D );
for( Int i=0; i<numObs; ++i )
blas::Axpy
( n, F(-1), xMean.LockedBuffer(), 1, DDev.Buffer(i,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 ValueInt<BASE(F)>
QDWHDivide
( UpperOrLower uplo, DistMatrix<F>& A, DistMatrix<F>& G, bool returnQ=false )
{
DEBUG_ONLY(CallStackEntry cse("herm_eig::QDWHDivide"))
// G := sgn(G)
// G := 1/2 ( G + I )
herm_polar::QDWH( uplo, G );
UpdateDiagonal( G, F(1) );
Scale( F(1)/F(2), G );
// Compute the pivoted QR decomposition of the spectral projection
const Grid& g = A.Grid();
DistMatrix<F,MD,STAR> t(g);
DistMatrix<Int,VR,STAR> p(g);
elem::QR( G, t, p );
// A := Q^H A Q
MakeHermitian( uplo, A );
const Base<F> oneA = OneNorm( A );
if( returnQ )
{
ExpandPackedReflectors( LOWER, VERTICAL, CONJUGATED, 0, G, t );
DistMatrix<F> B(g);
Gemm( ADJOINT, NORMAL, F(1), G, A, B );
Gemm( NORMAL, NORMAL, F(1), B, G, A );
}
else
{
qr::ApplyQ( LEFT, ADJOINT, G, t, A );
qr::ApplyQ( RIGHT, NORMAL, G, t, A );
}
// Return || E21 ||1 / || A ||1 and the chosen rank
auto part = ComputePartition( A );
part.value /= oneA;
return part;
}
inline void HermitianSVD
( UpperOrLower uplo,
Matrix<F>& A, Matrix<BASE(F)>& s, Matrix<F>& U, Matrix<F>& V )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSVD");
#endif
#if 1
typedef BASE(F) R;
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s, V );
// Set the singular values to the absolute value of the eigenvalues
for( Int i=0; i<s.Height(); ++i )
s.Set(i,0,Abs(s.Get(i,0)));
// Copy V into U (flipping the sign as necessary)
const Int n = A.Height();
U.ResizeTo( n, n );
for( Int j=0; j<n; ++j )
{
const R sigma = s.Get( j, 0 );
F* UCol = U.Buffer( 0, j );
const F* VCol = V.LockedBuffer( 0, j );
if( sigma >= 0 )
for( Int i=0; i<n; ++i )
UCol[i] = VCol[i];
else
for( Int i=0; i<n; ++i )
UCol[i] = -VCol[i];
}
#else
U = A;
MakeHermitian( uplo, U );
SVD( U, s, V );
#endif
}
inline void
HPSDCholesky( UpperOrLower uplo, DistMatrix<Complex<R> >& A )
{
#ifndef RELEASE
CallStackEntry entry("HPSDCholesky");
#endif
HPSDSquareRoot( uplo, A );
MakeHermitian( uplo, A );
const Grid& g = A.Grid();
if( uplo == LOWER )
{
DistMatrix<Complex<R>,MD,STAR> t(g);
LQ( A, t );
MakeTriangular( LOWER, A );
}
else
{
DistMatrix<Complex<R>,MD,STAR> t(g);
QR( A, t );
MakeTriangular( UPPER, A );
}
}
void HermitianSVD
( UpperOrLower uplo,
Matrix<F>& A, Matrix<Base<F>>& s, Matrix<F>& U, Matrix<F>& V )
{
DEBUG_ONLY(CallStackEntry cse("HermitianSVD"))
#if 1
// Grab an eigenvalue decomposition of A
HermitianEig( uplo, A, s, V );
// Copy V into U (flipping the sign as necessary)
const Int n = A.Height();
U.Resize( n, n );
for( Int j=0; j<n; ++j )
{
const Base<F> sigma = s.Get( j, 0 );
F* UCol = U.Buffer( 0, j );
const F* VCol = V.LockedBuffer( 0, j );
if( sigma >= 0 )
{
for( Int i=0; i<n; ++i )
UCol[i] = VCol[i];
}
else
{
for( Int i=0; i<n; ++i )
UCol[i] = -VCol[i];
s.Set( j, 0, -sigma );
}
}
// TODO: Descending sort of triplets
#else
U = A;
MakeHermitian( uplo, U );
SVD( U, s, V );
#endif
}
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 HermitianSVD
( UpperOrLower uplo, DistMatrix<F>& A, DistMatrix<BASE(F),VR,STAR>& s )
{
#ifndef RELEASE
CallStackEntry entry("HermitianSVD");
#endif
#ifdef HAVE_PMRRR
typedef BASE(F) R;
// Grab the eigenvalues of A
HermitianEig( uplo, A, s );
// Replace the eigenvalues with their absolute values
const Int numLocalVals = s.LocalHeight();
for( Int iLoc=0; iLoc<numLocalVals; ++iLoc )
{
const R sigma = s.GetLocal(iLoc,0);
s.SetLocal(iLoc,0,Abs(sigma));
}
#else
MakeHermitian( uplo, A );
SVD( A, s );
#endif // ifdef HAVE_PMRRR
}
Int ADMM
( const Matrix<Real>& A,
const Matrix<Real>& b,
const Matrix<Real>& c,
Matrix<Real>& z,
const ADMMCtrl<Real>& ctrl )
{
EL_DEBUG_CSE
// Cache a custom partially-pivoted LU factorization of
// | rho*I A^H | = | B11 B12 |
// | A 0 | | B21 B22 |
// by (justifiably) avoiding pivoting in the first n steps of
// the factorization, so that
// [I,rho*I] = lu(rho*I).
// The factorization would then proceed with
// B21 := B21 U11^{-1} = A (rho*I)^{-1} = A/rho
// B12 := L11^{-1} B12 = I A^H = A^H.
// The Schur complement would then be
// B22 := B22 - B21 B12 = 0 - (A*A^H)/rho.
// We then factor said matrix with LU with partial pivoting and
// swap the necessary rows of B21 in order to implicitly commute
// the row pivots with the Gauss transforms in the manner standard
// for GEPP. Unless A A' is singular, pivoting should not be needed,
// as Cholesky factorization of the negative matrix should be valid.
//
// The result is the factorization
// | I 0 | | rho*I A^H | = | I 0 | | rho*I U12 |,
// | 0 P22 | | A 0 | | L21 L22 | | 0 U22 |
// where [L22,U22] are stored within B22.
Matrix<Real> U12, L21, B22, bPiv;
Adjoint( A, U12 );
L21 = A;
L21 *= 1/ctrl.rho;
Herk( LOWER, NORMAL, -1/ctrl.rho, A, B22 );
MakeHermitian( LOWER, B22 );
// TODO: Replace with sparse-direct Cholesky version?
Permutation P2;
LU( B22, P2 );
P2.PermuteRows( L21 );
bPiv = b;
P2.PermuteRows( bPiv );
// Possibly form the inverse of L22 U22
Matrix<Real> X22;
if( ctrl.inv )
{
X22 = B22;
MakeTrapezoidal( LOWER, X22 );
FillDiagonal( X22, Real(1) );
TriangularInverse( LOWER, UNIT, X22 );
Trsm( LEFT, UPPER, NORMAL, NON_UNIT, Real(1), B22, X22 );
}
Int numIter=0;
const Int m = A.Height();
const Int n = A.Width();
Matrix<Real> g, xTmp, y, t;
Zeros( g, m+n, 1 );
PartitionDown( g, xTmp, y, n );
Matrix<Real> x, u, zOld, xHat;
Zeros( z, n, 1 );
Zeros( u, n, 1 );
Zeros( t, n, 1 );
while( numIter < ctrl.maxIter )
{
zOld = z;
// Find x from
// | rho*I A^H | | x | = | rho*(z-u)-c |
// | A 0 | | y | | b |
// via our cached custom factorization:
//
// |x| = inv(U) inv(L) P' |rho*(z-u)-c|
// |y| |b |
// = |rho*I U12|^{-1} |I 0 | |I 0 | |rho*(z-u)-c|
// = |0 U22| |L21 L22| |0 P22'| |b |
// = " " |rho*(z-u)-c|
// | P22' b |
xTmp = z;
xTmp -= u;
xTmp *= ctrl.rho;
xTmp -= c;
y = bPiv;
Gemv( NORMAL, Real(-1), L21, xTmp, Real(1), y );
if( ctrl.inv )
{
Gemv( NORMAL, Real(1), X22, y, t );
y = t;
}
else
{
Trsv( LOWER, NORMAL, UNIT, B22, y );
Trsv( UPPER, NORMAL, NON_UNIT, B22, y );
}
Gemv( NORMAL, Real(-1), U12, y, Real(1), xTmp );
xTmp *= 1/ctrl.rho;
// xHat := alpha*x + (1-alpha)*zOld
xHat = xTmp;
xHat *= ctrl.alpha;
Axpy( 1-ctrl.alpha, zOld, xHat );
// z := pos(xHat+u)
z = xHat;
z += u;
LowerClip( z, Real(0) );
// u := u + (xHat-z)
u += xHat;
u -= z;
const Real objective = Dot( c, xTmp );
// rNorm := || x - z ||_2
t = xTmp;
t -= z;
const Real rNorm = FrobeniusNorm( t );
// sNorm := |rho| || z - zOld ||_2
t = z;
t -= zOld;
const Real sNorm = Abs(ctrl.rho)*FrobeniusNorm( t );
const Real epsPri = Sqrt(Real(n))*ctrl.absTol +
ctrl.relTol*Max(FrobeniusNorm(xTmp),FrobeniusNorm(z));
const Real epsDual = Sqrt(Real(n))*ctrl.absTol +
ctrl.relTol*Abs(ctrl.rho)*FrobeniusNorm(u);
if( ctrl.print )
{
t = xTmp;
LowerClip( t, Real(0) );
t -= xTmp;
const Real clipDist = FrobeniusNorm( t );
cout << numIter << ": "
<< "||x-z||_2=" << rNorm << ", "
<< "epsPri=" << epsPri << ", "
<< "|rho| ||z-zOld||_2=" << sNorm << ", "
<< "epsDual=" << epsDual << ", "
<< "||x-Pos(x)||_2=" << clipDist << ", "
<< "c'x=" << objective << endl;
}
if( rNorm < epsPri && sNorm < epsDual )
break;
++numIter;
}
if( ctrl.maxIter == numIter )
cout << "ADMM failed to converge" << endl;
x = xTmp;
return numIter;
}
void Wigner( ElementalMatrix<F>& A, Int n, F mean, Base<F> stddev )
{
EL_DEBUG_CSE
Gaussian( A, n, n, mean, stddev );
MakeHermitian( LOWER, A );
}
void Wigner( AbstractDistMatrix<F>& A, Int n, F mean, Base<F> stddev )
{
DEBUG_ONLY(CSE cse("Wigner"))
Gaussian( A, n, n, mean, stddev );
MakeHermitian( LOWER, A );
}
