def __init__(self, input_dim, connections):
"""Create a generic switchboard.
The input and output dimension as well as dtype have to be fixed
at initialization time.
Keyword arguments:
input_dim -- Dimension of the input data (number of connections).
connections -- 1d Array or sequence with an entry for each output
connection, containing the corresponding index of the
input connection.
"""
# check connections for inconsistencies
if len(connections) == 0:
err = "Received empty connection list."
raise SwitchboardException(err)
if numx.nanmax(connections) >= input_dim:
err = ("One or more switchboard connection "
"indices exceed the input dimension.")
raise SwitchboardException(err)
# checks passed
self.connections = numx.array(connections)
output_dim = len(connections)
super(Switchboard, self).__init__(input_dim=input_dim,
output_dim=output_dim)
# try to invert connections
if self.input_dim == self.output_dim and len(
numx.unique(self.connections)) == self.input_dim:
self.inverse_connections = numx.argsort(self.connections)
else:
self.inverse_connections = None
def _execute(self, x):
#----------------------------------------------------
# similar algorithm to that within self.stop_training()
# refer there for notes & comments on code
#----------------------------------------------------
N = self.data.shape[0]
Nx = x.shape[0]
W = numx.zeros((Nx, N), dtype=self.dtype)
k, r = self.k, self.r
d_out = self.output_dim
Q_diag_idx = numx.arange(k)
for row in range(Nx):
#find nearest neighbors of x in M
M_xi = self.data - x[row]
nbrs = numx.argsort((M_xi**2).sum(1))[:k]
M_xi = M_xi[nbrs]
#find corrected covariance matrix Q
Q = mult(M_xi, M_xi.T)
if r is None and k > d_out:
sig2 = (svd(M_xi, compute_uv=0))**2
r = numx.sum(sig2[d_out:])
Q[Q_diag_idx, Q_diag_idx] += r
if r is not None:
Q[Q_diag_idx, Q_diag_idx] += r
#solve for weights
w = self._refcast(numx_linalg.solve(Q, numx.ones(k)))
w /= w.sum()
W[row, nbrs] = w
#multiply weights by result of SVD from training
return numx.dot(W, self.training_projection)
def _execute(self, x):
#----------------------------------------------------
# similar algorithm to that within self.stop_training()
# refer there for notes & comments on code
#----------------------------------------------------
N = self.data.shape[0]
Nx = x.shape[0]
W = numx.zeros((Nx, N), dtype=self.dtype)
k, r = self.k, self.r
d_out = self.output_dim
Q_diag_idx = numx.arange(k)
for row in range(Nx):
#find nearest neighbors of x in M
M_xi = self.data-x[row]
nbrs = numx.argsort( (M_xi**2).sum(1) )[:k]
M_xi = M_xi[nbrs]
#find corrected covariance matrix Q
Q = mult(M_xi, M_xi.T)
if r is None and k > d_out:
sig2 = (svd(M_xi, compute_uv=0))**2
r = numx.sum(sig2[d_out:])
Q[Q_diag_idx, Q_diag_idx] += r
if r is not None:
Q[Q_diag_idx, Q_diag_idx] += r
#solve for weights
w = self._refcast(numx_linalg.solve(Q , numx.ones(k)))
w /= w.sum()
W[row, nbrs] = w
#multiply weights by result of SVD from training
return numx.dot(W, self.training_projection)
def __init__(self, input_dim, connections):
"""Create a generic switchboard.
The input and output dimension as well as dtype have to be fixed
at initialization time.
Keyword arguments:
input_dim -- Dimension of the input data (number of connections).
connections -- 1d Array or sequence with an entry for each output
connection, containing the corresponding index of the
input connection.
"""
# check connections for inconsistencies
if len(connections) == 0:
err = "Received empty connection list."
raise SwitchboardException(err)
if numx.nanmax(connections) >= input_dim:
err = ("One or more switchboard connection "
"indices exceed the input dimension.")
raise SwitchboardException(err)
# checks passed
self.connections = numx.array(connections)
output_dim = len(connections)
super(Switchboard, self).__init__(input_dim=input_dim,
output_dim=output_dim)
# try to invert connections
if (self.input_dim == self.output_dim and
len(numx.unique(self.connections)) == self.input_dim):
self.inverse_connections = numx.argsort(self.connections)
else:
self.inverse_connections = None
def _inverse(self, x):
"""Take the mean of overlapping values."""
n_y_cons = numx.bincount(self.connections) # n. connections to y_i
y_cons = numx.argsort(self.connections) # x indices for y_i
y = numx.zeros((len(x), self.input_dim))
i_x_counter = 0 # counter for processed x indices
i_y = 0 # current y index
while True:
n_cons = n_y_cons[i_y]
if n_cons > 0:
y[:, i_y] = old_div(
numx.sum(x[:, y_cons[i_x_counter:i_x_counter + n_cons]],
axis=1), n_cons)
i_x_counter += n_cons
if i_x_counter >= self.output_dim:
break
i_y += 1
return y
def _inverse(self, x):
"""Take the mean of overlapping values."""
n_y_cons = numx.bincount(self.connections) # n. connections to y_i
y_cons = numx.argsort(self.connections) # x indices for y_i
y = numx.zeros((len(x), self.input_dim))
i_x_counter = 0 # counter for processed x indices
i_y = 0 # current y index
while True:
n_cons = n_y_cons[i_y]
if n_cons > 0:
y[:,i_y] = old_div(numx.sum(x[:,y_cons[i_x_counter:
i_x_counter + n_cons]],
axis=1), n_cons)
i_x_counter += n_cons
if i_x_counter >= self.output_dim:
break
i_y += 1
return y
def _stop_training(self):
Cumulator._stop_training(self)
if self.verbose:
msg = ('training LLE on %i points'
' in %i dimensions...' %
(self.data.shape[0], self.data.shape[1]))
print msg
# some useful quantities
M = self.data
N = M.shape[0]
k = self.k
r = self.r
# indices of diagonal elements
W_diag_idx = numx.arange(N)
Q_diag_idx = numx.arange(k)
if k > N:
err = ('k=%i must be less than or '
'equal to number of training points N=%i' % (k, N))
raise TrainingException(err)
# determines number of output dimensions: if desired_variance
# is specified, we need to learn it from the data. Otherwise,
# it's easy
learn_outdim = False
if self.output_dim is None:
if self.desired_variance is None:
self.output_dim = self.input_dim
else:
learn_outdim = True
# do we need to automatically determine the regularization term?
auto_reg = r is None
# determine number of output dims, precalculate useful stuff
if learn_outdim:
Qs, sig2s, nbrss = self._adjust_output_dim()
# build the weight matrix
#XXX future work:
#XXX for faster implementation, W should be a sparse matrix
W = numx.zeros((N, N), dtype=self.dtype)
if self.verbose:
print ' - constructing [%i x %i] weight matrix...' % W.shape
for row in range(N):
if learn_outdim:
Q = Qs[row, :, :]
nbrs = nbrss[row, :]
else:
# -----------------------------------------------
# find k nearest neighbors
# -----------------------------------------------
M_Mi = M - M[row]
nbrs = numx.argsort((M_Mi**2).sum(1))[1:k + 1]
M_Mi = M_Mi[nbrs]
# compute covariance matrix of distances
Q = mult(M_Mi, M_Mi.T)
# -----------------------------------------------
# compute weight vector based on neighbors
# -----------------------------------------------
#Covariance matrix may be nearly singular:
# add a diagonal correction to prevent numerical errors
if auto_reg:
# automatic mode: correction is equal to the sum of
# the (d_in-d_out) unused variances (as in deRidder &
# Duin)
if learn_outdim:
sig2 = sig2s[row, :]
else:
sig2 = svd(M_Mi, compute_uv=0)**2
r = numx.sum(sig2[self.output_dim:])
Q[Q_diag_idx, Q_diag_idx] += r
else:
# Roweis et al instead use "a correction that
# is small compared to the trace" e.g.:
# r = 0.001 * float(Q.trace())
# this is equivalent to assuming 0.1% of the variance is unused
Q[Q_diag_idx, Q_diag_idx] += r * Q.trace()
#solve for weight
# weight is w such that sum(Q_ij * w_j) = 1 for all i
# XXX refcast is due to numpy bug: floats become double
w = self._refcast(numx_linalg.solve(Q, numx.ones(k)))
w /= w.sum()
#update row of the weight matrix
W[nbrs, row] = w
if self.verbose:
msg = (' - finding [%i x %i] null space of weight matrix\n'
' (may take a while)...' % (self.output_dim, N))
print msg
self.W = W.copy()
#to find the null space, we need the bottom d+1
# eigenvectors of (W-I).T*(W-I)
#Compute this using the svd of (W-I):
W[W_diag_idx, W_diag_idx] -= 1.
#XXX future work:
#XXX use of upcoming ARPACK interface for bottom few eigenvectors
#XXX of a sparse matrix will significantly increase the speed
#XXX of the next step
if self.svd:
sig, U = nongeneral_svd(W.T, range=(2, self.output_dim + 1))
else:
# the following code does the same computation, but uses
# symeig, which computes only the required eigenvectors, and
# is much faster. However, it could also be more unstable...
WW = mult(W, W.T)
# regularizes the eigenvalues, does not change the eigenvectors:
WW[W_diag_idx, W_diag_idx] += 0.1
sig, U = symeig(WW, range=(2, self.output_dim + 1), overwrite=True)
self.training_projection = U
def _stop_training(self):
Cumulator._stop_training(self)
k = self.k
M = self.data
N = M.shape[0]
if k > N:
err = ('k=%i must be less than'
' or equal to number of training points N=%i' % (k, N))
raise TrainingException(err)
if self.verbose:
print 'performing HLLE on %i points in %i dimensions...' % M.shape
# determines number of output dimensions: if desired_variance
# is specified, we need to learn it from the data. Otherwise,
# it's easy
learn_outdim = False
if self.output_dim is None:
if self.desired_variance is None:
self.output_dim = self.input_dim
else:
learn_outdim = True
# determine number of output dims, precalculate useful stuff
if learn_outdim:
Qs, sig2s, nbrss = self._adjust_output_dim()
d_out = self.output_dim
#dp = d_out + (d_out-1) + (d_out-2) + ...
dp = d_out * (d_out + 1) / 2
if min(k, N) <= d_out:
err = ('k=%i and n=%i (number of input data points) must be'
' larger than output_dim=%i' % (k, N, d_out))
raise TrainingException(err)
if k < 1 + d_out + dp:
wrn = ('The number of neighbours, k=%i, is smaller than'
' 1 + output_dim + output_dim*(output_dim+1)/2 = %i,'
' which might result in unstable results.' %
(k, 1 + d_out + dp))
_warnings.warn(wrn, MDPWarning)
#build the weight matrix
#XXX for faster implementation, W should be a sparse matrix
W = numx.zeros((N, dp * N), dtype=self.dtype)
if self.verbose:
print ' - constructing [%i x %i] weight matrix...' % W.shape
for row in range(N):
if learn_outdim:
nbrs = nbrss[row, :]
else:
# -----------------------------------------------
# find k nearest neighbors
# -----------------------------------------------
M_Mi = M - M[row]
nbrs = numx.argsort((M_Mi**2).sum(1))[1:k + 1]
#-----------------------------------------------
# center the neighborhood using the mean
#-----------------------------------------------
nbrhd = M[nbrs] # this makes a copy
nbrhd -= nbrhd.mean(0)
#-----------------------------------------------
# compute local coordinates
# using a singular value decomposition
#-----------------------------------------------
U, sig, VT = svd(nbrhd)
nbrhd = U.T[:d_out]
del VT
#-----------------------------------------------
# build Hessian estimator
#-----------------------------------------------
Yi = numx.zeros((dp, k), dtype=self.dtype)
ct = 0
for i in range(d_out):
Yi[ct:ct + d_out - i, :] = nbrhd[i] * nbrhd[i:, :]
ct += d_out - i
Yi = numx.concatenate(
[numx.ones((1, k), dtype=self.dtype), nbrhd, Yi], 0)
#-----------------------------------------------
# orthogonalize linear and quadratic forms
# with QR factorization
# and make the weights sum to 1
#-----------------------------------------------
if k >= 1 + d_out + dp:
Q, R = numx_linalg.qr(Yi.T)
w = Q[:, d_out + 1:d_out + 1 + dp]
else:
q, r = _mgs(Yi.T)
w = q[:, -dp:]
S = w.sum(0) #sum along columns
#if S[i] is too small, set it equal to 1.0
# this prevents weights from blowing up
S[numx.where(numx.absolute(S) < 1E-4)] = 1.0
#print w.shape, S.shape, (w/S).shape
#print W[nbrs, row*dp:(row+1)*dp].shape
W[nbrs, row * dp:(row + 1) * dp] = w / S
#-----------------------------------------------
# To find the null space, we want the
# first d+1 eigenvectors of W.T*W
# Compute this using an svd of W
#-----------------------------------------------
if self.verbose:
msg = (' - finding [%i x %i] '
'null space of weight matrix...' % (d_out, N))
print msg
#XXX future work:
#XXX use of upcoming ARPACK interface for bottom few eigenvectors
#XXX of a sparse matrix will significantly increase the speed
#XXX of the next step
if self.svd:
sig, U = nongeneral_svd(W.T, range=(2, d_out + 1))
Y = U * numx.sqrt(N)
else:
WW = mult(W, W.T)
# regularizes the eigenvalues, does not change the eigenvectors:
W_diag_idx = numx.arange(N)
WW[W_diag_idx, W_diag_idx] += 0.01
sig, U = symeig(WW, range=(2, self.output_dim + 1), overwrite=True)
Y = U * numx.sqrt(N)
del WW
del W
#-----------------------------------------------
# Normalize Y
#
# Alternative way to do it:
# we need R = (Y.T*Y)^(-1/2)
# do this with an SVD of Y del VT
# Y = U*sig*V.T
# Y.T*Y = (V*sig.T*U.T) * (U*sig*V.T)
# = V * (sig*sig.T) * V.T
# = V * sig^2 V.T
# so
# R = V * sig^-1 * V.T
# The code is:
# U, sig, VT = svd(Y)
# del U
# S = numx.diag(sig**-1)
# self.training_projection = mult(Y, mult(VT.T, mult(S, VT)))
#-----------------------------------------------
if self.verbose:
print ' - normalizing null space...'
C = sqrtm(mult(Y.T, Y))
self.training_projection = mult(Y, C)
def _adjust_output_dim(self):
# this function is called if we need to compute the number of
# output dimensions automatically; some quantities that are
# useful later are pre-calculated to spare precious time
if self.verbose:
print ' - adjusting output dim:'
#otherwise, we need to compute output_dim
# from desired_variance
M = self.data
k = self.k
N, d_in = M.shape
m_est_array = []
Qs = numx.zeros((N, k, k))
sig2s = numx.zeros((N, d_in))
nbrss = numx.zeros((N, k), dtype='i')
for row in range(N):
#-----------------------------------------------
# find k nearest neighbors
#-----------------------------------------------
M_Mi = M - M[row]
nbrs = numx.argsort((M_Mi**2).sum(1))[1:k + 1]
M_Mi = M_Mi[nbrs]
# compute covariance matrix of distances
Qs[row, :, :] = mult(M_Mi, M_Mi.T)
nbrss[row, :] = nbrs
#-----------------------------------------------
# singular values of M_Mi give the variance:
# use this to compute intrinsic dimensionality
# at this point
#-----------------------------------------------
sig2 = (svd(M_Mi, compute_uv=0))**2
sig2s[row, :sig2.shape[0]] = sig2
#-----------------------------------------------
# use sig2 to compute intrinsic dimensionality of the
# data at this neighborhood. The dimensionality is the
# number of eigenvalues needed to sum to the total
# desired variance
#-----------------------------------------------
sig2 /= sig2.sum()
S = sig2.cumsum()
m_est = S.searchsorted(self.desired_variance)
if m_est > 0:
m_est += (self.desired_variance - S[m_est - 1]) / sig2[m_est]
else:
m_est = self.desired_variance / sig2[m_est]
m_est_array.append(m_est)
m_est_array = numx.asarray(m_est_array)
self.output_dim = int(numx.ceil(numx.median(m_est_array)))
if self.verbose:
msg = (' output_dim = %i'
' for variance of %.2f' %
(self.output_dim, self.desired_variance))
print msg
return Qs, sig2s, nbrss
def _stop_training(self):
Cumulator._stop_training(self)
if self.verbose:
msg = ('training LLE on %i points'
' in %i dimensions...' % (self.data.shape[0],
self.data.shape[1]))
print msg
# some useful quantities
M = self.data
N = M.shape[0]
k = self.k
r = self.r
# indices of diagonal elements
W_diag_idx = numx.arange(N)
Q_diag_idx = numx.arange(k)
if k > N:
err = ('k=%i must be less than or '
'equal to number of training points N=%i' % (k, N))
raise TrainingException(err)
# determines number of output dimensions: if desired_variance
# is specified, we need to learn it from the data. Otherwise,
# it's easy
learn_outdim = False
if self.output_dim is None:
if self.desired_variance is None:
self.output_dim = self.input_dim
else:
learn_outdim = True
# do we need to automatically determine the regularization term?
auto_reg = r is None
# determine number of output dims, precalculate useful stuff
if learn_outdim:
Qs, sig2s, nbrss = self._adjust_output_dim()
# build the weight matrix
#XXX future work:
#XXX for faster implementation, W should be a sparse matrix
W = numx.zeros((N, N), dtype=self.dtype)
if self.verbose:
print ' - constructing [%i x %i] weight matrix...' % W.shape
for row in range(N):
if learn_outdim:
Q = Qs[row, :, :]
nbrs = nbrss[row, :]
else:
# -----------------------------------------------
# find k nearest neighbors
# -----------------------------------------------
M_Mi = M-M[row]
nbrs = numx.argsort((M_Mi**2).sum(1))[1:k+1]
M_Mi = M_Mi[nbrs]
# compute covariance matrix of distances
Q = mult(M_Mi, M_Mi.T)
# -----------------------------------------------
# compute weight vector based on neighbors
# -----------------------------------------------
#Covariance matrix may be nearly singular:
# add a diagonal correction to prevent numerical errors
if auto_reg:
# automatic mode: correction is equal to the sum of
# the (d_in-d_out) unused variances (as in deRidder &
# Duin)
if learn_outdim:
sig2 = sig2s[row, :]
else:
sig2 = svd(M_Mi, compute_uv=0)**2
r = numx.sum(sig2[self.output_dim:])
Q[Q_diag_idx, Q_diag_idx] += r
else:
# Roweis et al instead use "a correction that
# is small compared to the trace" e.g.:
# r = 0.001 * float(Q.trace())
# this is equivalent to assuming 0.1% of the variance is unused
Q[Q_diag_idx, Q_diag_idx] += r*Q.trace()
#solve for weight
# weight is w such that sum(Q_ij * w_j) = 1 for all i
# XXX refcast is due to numpy bug: floats become double
w = self._refcast(numx_linalg.solve(Q, numx.ones(k)))
w /= w.sum()
#update row of the weight matrix
W[nbrs, row] = w
if self.verbose:
msg = (' - finding [%i x %i] null space of weight matrix\n'
' (may take a while)...' % (self.output_dim, N))
print msg
self.W = W.copy()
#to find the null space, we need the bottom d+1
# eigenvectors of (W-I).T*(W-I)
#Compute this using the svd of (W-I):
W[W_diag_idx, W_diag_idx] -= 1.
#XXX future work:
#XXX use of upcoming ARPACK interface for bottom few eigenvectors
#XXX of a sparse matrix will significantly increase the speed
#XXX of the next step
if self.svd:
sig, U = nongeneral_svd(W.T, range=(2, self.output_dim+1))
else:
# the following code does the same computation, but uses
# symeig, which computes only the required eigenvectors, and
# is much faster. However, it could also be more unstable...
WW = mult(W, W.T)
# regularizes the eigenvalues, does not change the eigenvectors:
WW[W_diag_idx, W_diag_idx] += 0.1
sig, U = symeig(WW, range=(2, self.output_dim+1), overwrite=True)
self.training_projection = U
def _stop_training(self):
Cumulator._stop_training(self)
k = self.k
M = self.data
N = M.shape[0]
if k > N:
err = ('k=%i must be less than'
' or equal to number of training points N=%i' % (k, N))
raise TrainingException(err)
if self.verbose:
print 'performing HLLE on %i points in %i dimensions...' % M.shape
# determines number of output dimensions: if desired_variance
# is specified, we need to learn it from the data. Otherwise,
# it's easy
learn_outdim = False
if self.output_dim is None:
if self.desired_variance is None:
self.output_dim = self.input_dim
else:
learn_outdim = True
# determine number of output dims, precalculate useful stuff
if learn_outdim:
Qs, sig2s, nbrss = self._adjust_output_dim()
d_out = self.output_dim
#dp = d_out + (d_out-1) + (d_out-2) + ...
dp = d_out*(d_out+1)/2
if min(k, N) <= d_out:
err = ('k=%i and n=%i (number of input data points) must be'
' larger than output_dim=%i' % (k, N, d_out))
raise TrainingException(err)
if k < 1+d_out+dp:
wrn = ('The number of neighbours, k=%i, is smaller than'
' 1 + output_dim + output_dim*(output_dim+1)/2 = %i,'
' which might result in unstable results.'
% (k, 1+d_out+dp))
_warnings.warn(wrn, MDPWarning)
#build the weight matrix
#XXX for faster implementation, W should be a sparse matrix
W = numx.zeros((N, dp*N), dtype=self.dtype)
if self.verbose:
print ' - constructing [%i x %i] weight matrix...' % W.shape
for row in range(N):
if learn_outdim:
nbrs = nbrss[row, :]
else:
# -----------------------------------------------
# find k nearest neighbors
# -----------------------------------------------
M_Mi = M-M[row]
nbrs = numx.argsort((M_Mi**2).sum(1))[1:k+1]
#-----------------------------------------------
# center the neighborhood using the mean
#-----------------------------------------------
nbrhd = M[nbrs] # this makes a copy
nbrhd -= nbrhd.mean(0)
#-----------------------------------------------
# compute local coordinates
# using a singular value decomposition
#-----------------------------------------------
U, sig, VT = svd(nbrhd)
nbrhd = U.T[:d_out]
del VT
#-----------------------------------------------
# build Hessian estimator
#-----------------------------------------------
Yi = numx.zeros((dp, k), dtype=self.dtype)
ct = 0
for i in range(d_out):
Yi[ct:ct+d_out-i, :] = nbrhd[i] * nbrhd[i:, :]
ct += d_out-i
Yi = numx.concatenate([numx.ones((1, k), dtype=self.dtype),
nbrhd, Yi], 0)
#-----------------------------------------------
# orthogonalize linear and quadratic forms
# with QR factorization
# and make the weights sum to 1
#-----------------------------------------------
if k >= 1+d_out+dp:
Q, R = numx_linalg.qr(Yi.T)
w = Q[:, d_out+1:d_out+1+dp]
else:
q, r = _mgs(Yi.T)
w = q[:, -dp:]
S = w.sum(0) #sum along columns
#if S[i] is too small, set it equal to 1.0
# this prevents weights from blowing up
S[numx.where(numx.absolute(S)<1E-4)] = 1.0
#print w.shape, S.shape, (w/S).shape
#print W[nbrs, row*dp:(row+1)*dp].shape
W[nbrs, row*dp:(row+1)*dp] = w / S
#-----------------------------------------------
# To find the null space, we want the
# first d+1 eigenvectors of W.T*W
# Compute this using an svd of W
#-----------------------------------------------
if self.verbose:
msg = (' - finding [%i x %i] '
'null space of weight matrix...' % (d_out, N))
print msg
#XXX future work:
#XXX use of upcoming ARPACK interface for bottom few eigenvectors
#XXX of a sparse matrix will significantly increase the speed
#XXX of the next step
if self.svd:
sig, U = nongeneral_svd(W.T, range=(2, d_out+1))
Y = U*numx.sqrt(N)
else:
WW = mult(W, W.T)
# regularizes the eigenvalues, does not change the eigenvectors:
W_diag_idx = numx.arange(N)
WW[W_diag_idx, W_diag_idx] += 0.01
sig, U = symeig(WW, range=(2, self.output_dim+1), overwrite=True)
Y = U*numx.sqrt(N)
del WW
del W
#-----------------------------------------------
# Normalize Y
#
# Alternative way to do it:
# we need R = (Y.T*Y)^(-1/2)
# do this with an SVD of Y del VT
# Y = U*sig*V.T
# Y.T*Y = (V*sig.T*U.T) * (U*sig*V.T)
# = V * (sig*sig.T) * V.T
# = V * sig^2 V.T
# so
# R = V * sig^-1 * V.T
# The code is:
# U, sig, VT = svd(Y)
# del U
# S = numx.diag(sig**-1)
# self.training_projection = mult(Y, mult(VT.T, mult(S, VT)))
#-----------------------------------------------
if self.verbose:
print ' - normalizing null space...'
C = sqrtm(mult(Y.T, Y))
self.training_projection = mult(Y, C)
def _adjust_output_dim(self):
# this function is called if we need to compute the number of
# output dimensions automatically; some quantities that are
# useful later are pre-calculated to spare precious time
if self.verbose:
print ' - adjusting output dim:'
#otherwise, we need to compute output_dim
# from desired_variance
M = self.data
k = self.k
N, d_in = M.shape
m_est_array = []
Qs = numx.zeros((N, k, k))
sig2s = numx.zeros((N, d_in))
nbrss = numx.zeros((N, k), dtype='i')
for row in range(N):
#-----------------------------------------------
# find k nearest neighbors
#-----------------------------------------------
M_Mi = M-M[row]
nbrs = numx.argsort((M_Mi**2).sum(1))[1:k+1]
M_Mi = M_Mi[nbrs]
# compute covariance matrix of distances
Qs[row, :, :] = mult(M_Mi, M_Mi.T)
nbrss[row, :] = nbrs
#-----------------------------------------------
# singular values of M_Mi give the variance:
# use this to compute intrinsic dimensionality
# at this point
#-----------------------------------------------
sig2 = (svd(M_Mi, compute_uv=0))**2
sig2s[row, :sig2.shape[0]] = sig2
#-----------------------------------------------
# use sig2 to compute intrinsic dimensionality of the
# data at this neighborhood. The dimensionality is the
# number of eigenvalues needed to sum to the total
# desired variance
#-----------------------------------------------
sig2 /= sig2.sum()
S = sig2.cumsum()
m_est = S.searchsorted(self.desired_variance)
if m_est > 0:
m_est += (self.desired_variance-S[m_est-1])/sig2[m_est]
else:
m_est = self.desired_variance/sig2[m_est]
m_est_array.append(m_est)
m_est_array = numx.asarray(m_est_array)
self.output_dim = int( numx.ceil( numx.median(m_est_array) ) )
if self.verbose:
msg = (' output_dim = %i'
' for variance of %.2f' % (self.output_dim,
self.desired_variance))
print msg
return Qs, sig2s, nbrss