inline
typename detail::enable_if< is_floating_point< T >, T >::type
invert( mat< T, 1 >& m )
{
if ( std::fabs( m.elem( 0 ) ) <= std::numeric_limits< T >::epsilon() )
{
return 0;
}
T r = m.elem( 0 );
m.elem( 0 ) = static_cast< T >( 1 ) / m.elem( 0 );
return r;
}
inline
typename detail::enable_if< is_floating_point< T >, T >::type
invert( mat< T, 2 >& m )
{
T det = m.elem( 0 ) * m.elem( 3 ) - m.elem( 2 ) * m.elem( 1 );
if ( std::fabs( det ) <= std::numeric_limits< T >::epsilon() )
{
return 0;
}
std::swap( m.elem( 0 ), m.elem( 3 ) );
m.elem( 1 ) = -m.elem( 1 );
m.elem( 2 ) = -m.elem( 2 );
m /= det;
return det;
}
inline
typename detail::enable_if< is_floating_point< T >, T >::type
invert( mat< T, 3 >& m )
{
mat< T, 3 > res;
res.elem( 0 ) = m.elem( 4 ) * m.elem( 8 ) - m.elem( 5 ) * m.elem( 7 );
res.elem( 1 ) = m.elem( 2 ) * m.elem( 7 ) - m.elem( 1 ) * m.elem( 8 );
res.elem( 2 ) = m.elem( 1 ) * m.elem( 5 ) - m.elem( 2 ) * m.elem( 4 );
res.elem( 3 ) = m.elem( 5 ) * m.elem( 6 ) - m.elem( 3 ) * m.elem( 8 );
res.elem( 4 ) = m.elem( 0 ) * m.elem( 8 ) - m.elem( 2 ) * m.elem( 6 );
res.elem( 5 ) = m.elem( 2 ) * m.elem( 3 ) - m.elem( 0 ) * m.elem( 5 );
res.elem( 6 ) = m.elem( 3 ) * m.elem( 7 ) - m.elem( 4 ) * m.elem( 6 );
res.elem( 7 ) = m.elem( 1 ) * m.elem( 6 ) - m.elem( 0 ) * m.elem( 7 );
res.elem( 8 ) = m.elem( 0 ) * m.elem( 4 ) - m.elem( 1 ) * m.elem( 3 );
T d = m.elem( 0 ) * res.elem( 0 )
+ m.elem( 1 ) * res.elem( 3 )
+ m.elem( 2 ) * res.elem( 6 );
if ( std::fabs( d ) <= std::numeric_limits< T >::epsilon() )
{
return 0;
}
T invd = static_cast< T >( 1 ) / d;
m = res;
m *= invd;
return d;
}
inline
typename detail::enable_if< is_floating_point< T >, T >::type
invert( mat< T, 4 >& m )
{
mat< T, 4 > res;
T t1[ 6 ] =
{
m.elem( 2 ) * m.elem( 7 ) - m.elem( 6 ) * m.elem( 3 ),
m.elem( 2 ) * m.elem( 11 ) - m.elem( 10 ) * m.elem( 3 ),
m.elem( 2 ) * m.elem( 15 ) - m.elem( 14 ) * m.elem( 3 ),
m.elem( 6 ) * m.elem( 11 ) - m.elem( 10 ) * m.elem( 7 ),
m.elem( 6 ) * m.elem( 15 ) - m.elem( 14 ) * m.elem( 7 ),
m.elem( 10 ) * m.elem( 15 ) - m.elem( 14 ) * m.elem( 11 )
};
res.elem( 0 ) = m.elem( 5 ) * t1[ 5 ] - m.elem( 9 ) * t1[ 4 ] + m.elem( 13 ) * t1[ 3 ];
res.elem( 1 ) = m.elem( 9 ) * t1[ 2 ] - m.elem( 13 ) * t1[ 1 ] - m.elem( 1 ) * t1[ 5 ];
res.elem( 2 ) = m.elem( 13 ) * t1[ 0 ] - m.elem( 5 ) * t1[ 2 ] + m.elem( 1 ) * t1[ 4 ];
res.elem( 3 ) = m.elem( 5 ) * t1[ 1 ] - m.elem( 1 ) * t1[ 3 ] - m.elem( 9 ) * t1[ 0 ];
res.elem( 4 ) = m.elem( 8 ) * t1[ 4 ] - m.elem( 4 ) * t1[ 5 ] - m.elem( 12 ) * t1[ 3 ];
res.elem( 5 ) = m.elem( 0 ) * t1[ 5 ] - m.elem( 8 ) * t1[ 2 ] + m.elem( 12 ) * t1[ 1 ];
res.elem( 6 ) = m.elem( 4 ) * t1[ 2 ] - m.elem( 12 ) * t1[ 0 ] - m.elem( 0 ) * t1[ 4 ];
res.elem( 7 ) = m.elem( 0 ) * t1[ 3 ] - m.elem( 4 ) * t1[ 1 ] + m.elem( 8 ) * t1[ 0 ];
T t2[ 6 ] =
{
m.elem( 0 ) * m.elem( 5 ) - m.elem( 4 ) * m.elem( 1 ),
m.elem( 0 ) * m.elem( 9 ) - m.elem( 8 ) * m.elem( 1 ),
m.elem( 0 ) * m.elem( 13 ) - m.elem( 12 ) * m.elem( 1 ),
m.elem( 4 ) * m.elem( 9 ) - m.elem( 8 ) * m.elem( 5 ),
m.elem( 4 ) * m.elem( 13 ) - m.elem( 12 ) * m.elem( 5 ),
m.elem( 8 ) * m.elem( 13 ) - m.elem( 12 ) * m.elem( 9 )
};
res.elem( 8 ) = m.elem( 7 ) * t2[ 5 ] - m.elem( 11 ) * t2[ 4 ] + m.elem( 15 ) * t2[ 3 ];
res.elem( 9 ) = m.elem( 11 ) * t2[ 2 ] - m.elem( 15 ) * t2[ 1 ] - m.elem( 3 ) * t2[ 5 ];
res.elem( 10 ) = m.elem( 15 ) * t2[ 0 ] - m.elem( 7 ) * t2[ 2 ] + m.elem( 3 ) * t2[ 4 ];
res.elem( 11 ) = m.elem( 7 ) * t2[ 1 ] - m.elem( 3 ) * t2[ 3 ] - m.elem( 11 ) * t2[ 0 ];
res.elem( 12 ) = m.elem( 10 ) * t2[ 4 ] - m.elem( 6 ) * t2[ 5 ] - m.elem( 14 ) * t2[ 3 ];
res.elem( 13 ) = m.elem( 2 ) * t2[ 5 ] - m.elem( 10 ) * t2[ 2 ] + m.elem( 14 ) * t2[ 1 ];
res.elem( 14 ) = m.elem( 6 ) * t2[ 2 ] - m.elem( 14 ) * t2[ 0 ] - m.elem( 2 ) * t2[ 4 ];
res.elem( 15 ) = m.elem( 2 ) * t2[ 3 ] - m.elem( 6 ) * t2[ 1 ] + m.elem( 10 ) * t2[ 0 ];
T d =
m.elem( 0 ) * res.elem( 0 ) + m.elem( 4 ) * res.elem( 1 ) +
m.elem( 8 ) * res.elem( 2 ) + m.elem( 12 ) * res.elem( 3 );
if ( std::fabs( d ) <= std::numeric_limits< T >::epsilon() )
{
return 0;
}
T invd = static_cast< T >( 1 ) / d;
m = res;
m *= invd;
return d;
}
//[[Rcpp::export]]
Rcpp::List nnmf(const mat & A, const unsigned int k, mat W, mat H, umat Wm, umat Hm,
const vec & alpha, const vec & beta, const unsigned int max_iter, const double rel_tol,
const int n_threads, const int verbose, const bool show_warning, const unsigned int inner_max_iter,
const double inner_rel_tol, const int method, unsigned int trace)
{
/******************************************************************************************************
* Non-negative Matrix Factorization(NNMF) using alternating scheme
* ----------------------------------------------------------------
* Description:
* Decompose matrix A such that
* A = W H
* Arguments:
* A : Matrix to be decomposed
* W, H : Initial matrices of W and H, where ncol(W) = nrow(H) = k. # of rows/columns of W/H could be 0
* Wm, Hm : Masks of W and H, s.t. masked entries are no-updated and fixed to initial values
* alpha : [L2, angle, L1] regularization on W (non-masked entries)
* beta : [L2, angle, L1] regularization on H (non-masked entries)
* max_iter : Maximum number of iteration
* rel_tol : Relative tolerance between two successive iterations, = |e2-e1|/avg(e1, e2)
* n_threads : Number of threads (openMP)
* verbose : Either 0 = no any tracking, 1 == progression bar, 2 == print iteration info
* show_warning : If to show warning if targeted `tol` is not reached
* inner_max_iter : Maximum number of iterations passed to each inner W or H matrix updating loop
* inner_rel_tol : Relative tolerance passed to inner W or H matrix updating loop, = |e2-e1|/avg(e1, e2)
* method : Integer of 1, 2, 3 or 4, which encodes methods
* : 1 = sequential coordinate-wise minimization using square loss
* : 2 = Lee's multiplicative update with square loss, which is re-scaled gradient descent
* : 3 = sequentially quadratic approximated minimization with KL-divergence
* : 4 = Lee's multiplicative update with KL-divergence, which is re-scaled gradient descent
* trace : A positive integer, error will be checked very 'trace' iterations. Computing WH can be very expansive,
* : so one may not want to check error A-WH every single iteration
* Return:
* A list (Rcpp::List) of
* W, H : resulting W and H matrices
* mse_error : a vector of mean square error (divided by number of non-missings)
* mkl_error : a vector (length = number of iterations) of mean KL-distance
* target_error : a vector of loss (0.5*mse or mkl), plus constraints
* average_epoch : a vector of average epochs (one complete swap over W and H)
* Author:
* Eric Xihui Lin <*****@*****.**>
* Version:
* 2015-12-11
******************************************************************************************************/
unsigned int n = A.n_rows;
unsigned int m = A.n_cols;
//int k = H.n_rows; // decomposition rank k
unsigned int N_non_missing = n*m;
if (trace < 1) trace = 1;
unsigned int err_len = (unsigned int)std::ceil(double(max_iter)/double(trace)) + 1;
vec mse_err(err_len), mkl_err(err_len), terr(err_len), ave_epoch(err_len);
// check progression
bool show_progress = false;
if (verbose == 1) show_progress = true;
Progress prgrss(max_iter, show_progress);
double rel_err = rel_tol + 1;
double terr_last = 1e99;
uvec non_missing;
bool any_missing = !A.is_finite();
if (any_missing)
{
non_missing = find_finite(A);
N_non_missing = non_missing.n_elem;
mkl_err.fill(mean((A.elem(non_missing)+TINY_NUM) % log(A.elem(non_missing)+TINY_NUM) - A.elem(non_missing)));
}
else
mkl_err.fill(mean(mean((A+TINY_NUM) % log(A+TINY_NUM) - A))); // fixed part in KL-dist, mean(A log(A) - A)
if (Wm.empty())
Wm.resize(0, n);
else
inplace_trans(Wm);
if (Hm.empty())
Hm.resize(0, m);
if (W.empty())
{
W.randu(k, n);
W *= 0.01;
if (!Wm.empty())
W.elem(find(Wm > 0)).fill(0.0);
}
else
inplace_trans(W);
if (H.empty())
{
H.randu(k, m);
H *= 0.01;
if (!Hm.empty())
H.elem(find(Hm > 0)).fill(0.0);
}
if (verbose == 2)
{
Rprintf("\n%10s | %10s | %10s | %10s | %10s\n", "Iteration", "MSE", "MKL", "Target", "Rel. Err.");
Rprintf("--------------------------------------------------------------\n");
}
int total_raw_iter = 0;
unsigned int i = 0;
unsigned int i_e = 0; // index for error checking
for(; i < max_iter && std::abs(rel_err) > rel_tol; i++)
{
Rcpp::checkUserInterrupt();
prgrss.increment();
if (any_missing)
{
// update W
total_raw_iter += update_with_missing(W, H, A.t(), Wm, alpha, inner_max_iter, inner_rel_tol, n_threads, method);
// update H
total_raw_iter += update_with_missing(H, W, A, Hm, beta, inner_max_iter, inner_rel_tol, n_threads, method);
if (i % trace == 0)
{
const mat & Ahat = W.t()*H;
mse_err(i_e) = mean(square((A - Ahat).eval().elem(non_missing)));
mkl_err(i_e) += mean((-(A+TINY_NUM) % log(Ahat+TINY_NUM) + Ahat).eval().elem(non_missing));
}
}
else
{
// update W
total_raw_iter += update(W, H, A.t(), Wm, alpha, inner_max_iter, inner_rel_tol, n_threads, method);
// update H
total_raw_iter += update(H, W, A, Hm, beta, inner_max_iter, inner_rel_tol, n_threads, method);
if (i % trace == 0)
{
const mat & Ahat = W.t()*H;
mse_err(i_e) = mean(mean(square((A - Ahat))));
mkl_err(i_e) += mean(mean(-(A+TINY_NUM) % log(Ahat+TINY_NUM) + Ahat));
}
}
if (i % trace == 0)
{
ave_epoch(i_e) = double(total_raw_iter)/(n+m);
if (method < 3) // mse based
terr(i_e) = 0.5*mse_err(i_e);
else // KL based
terr(i_e) = mkl_err(i_e);
add_penalty(i_e, terr, W, H, N_non_missing, alpha, beta);
rel_err = 2*(terr_last - terr(i_e)) / (terr_last + terr(i_e) + TINY_NUM );
terr_last = terr(i_e);
if (verbose == 2)
Rprintf("%10d | %10.4f | %10.4f | %10.4f | %10.g\n", i+1, mse_err(i_e), mkl_err(i_e), terr(i_e), rel_err);
total_raw_iter = 0; // reset to 0
++i_e;
}
}
// compute error of the last iteration
if ((i-1) % trace != 0)
{
if (any_missing)
{
const mat & Ahat = W.t()*H;
mse_err(i_e) = mean(square((A - Ahat).eval().elem(non_missing)));
mkl_err(i_e) += mean((-(A+TINY_NUM) % log(Ahat+TINY_NUM) + Ahat).eval().elem(non_missing));
}
else
{
const mat & Ahat = W.t()*H;
mse_err(i_e) = mean(mean(square((A - Ahat))));
mkl_err(i_e) += mean(mean(-(A+TINY_NUM) % log(Ahat+TINY_NUM) + Ahat));
}
ave_epoch(i_e) = double(total_raw_iter)/(n+m);
if (method < 3) // mse based
terr(i_e) = 0.5*mse_err(i_e);
else // KL based
terr(i_e) = mkl_err(i_e);
add_penalty(i_e, terr, W, H, N_non_missing, alpha, beta);
rel_err = 2*(terr_last - terr(i_e)) / (terr_last + terr(i_e) + TINY_NUM );
terr_last = terr(i_e);
if (verbose == 2)
Rprintf("%10d | %10.4f | %10.4f | %10.4f | %10.g\n", i+1, mse_err(i_e), mkl_err(i_e), terr(i_e), rel_err);
++i_e;
}
if (verbose == 2)
{
Rprintf("--------------------------------------------------------------\n");
Rprintf("%10s | %10s | %10s | %10s | %10s\n\n", "Iteration", "MSE", "MKL", "Target", "Rel. Err.");
}
if (i_e < err_len)
{
mse_err.resize(i_e);
mkl_err.resize(i_e);
terr.resize(i_e);
ave_epoch.resize(i_e);
}
if (show_warning && rel_err > rel_tol)
Rcpp::warning("Target tolerance not reached. Try a larger max.iter.");
return Rcpp::List::create(
Rcpp::Named("W") = W.t(),
Rcpp::Named("H") = H,
Rcpp::Named("mse_error") = mse_err,
Rcpp::Named("mkl_error") = mkl_err,
Rcpp::Named("target_error") = terr,
Rcpp::Named("average_epoch") = ave_epoch,
Rcpp::Named("n_iteration") = i
);
}