def _fit(self, X):
Ri = self.reference_old
Rf = self.reference_new
A = sqrtm(Ri)
B = sqrtm(np.dot(invsqrtm(Ri), np.dot(Rf, invsqrtm(Ri))))
C = invsqrtm(Ri)
W = np.dot(A, np.dot(B, C))
self.transporter_ = W
def egrad_function_pair_rie(M, M_tilde, Q):
M_tilde_invsqrt = invsqrtm(M_tilde)
M_sqrt = sqrtm(M)
term_aux = np.dot(Q, np.dot(M, Q.T))
term_aux = np.dot(M_tilde_invsqrt, np.dot(term_aux, M_tilde_invsqrt))
return 4 * np.dot(np.dot(M_tilde_invsqrt, logm(term_aux)), np.dot(
M_sqrt, Q))
def untangent_space(T, Cref):
"""Project a set of Tangent space vectors in the manifold according to the given reference point Cref
Parameters
----------
T: {array-like} ,The Tangent space , shape= ( NWindows X Mfeatures(*(Mfeatures-1)/2))
Cref: {array-like} ,The reference covariance matrix (Mfeatures X Mfeatures )
Returns
----------
covmats: {array-like} ,(Mfeatures X Mfeatures) SPD Matrice
"""
Nt, Nd = T.shape
Ne = (1+np.sqrt(1+8*Nd))/2
C12 = sqrtm(Cref)
idx = np.triu_indices_from(Cref,0)
covmats = np.zeros((Nt, Ne, Ne))
covmats[:, idx[0], idx[1]] = T
for i in range(Nt):
covmats[i] = np.diag(np.diag(covmats[i])) + np.triu(
covmats[i], 0) + np.triu(covmats[i], 0).T
covmats[i] = expm(covmats[i])
covmats[i] = np.dot(np.dot(C12, covmats[i]), C12)
return covmats
def _transform(self, X):
W = self.transporter_
Ri = self.reference_old
Rf = self.reference_new
Nt = X.shape[0]
# detect which kind of input : tangent vectors or cov matrices
if self.tangent_old:
# if tangent vectors are given, transform them back to covs
# (easier to have tg vectors in the form of symmetric matrices later)
X = untangent_space(X, Ri)
# transform covariances to their tangent vectors with respect to Ri
# (these tangent vectors are in the form of symmetric matrices)
eta_i = np.zeros(X.shape)
Ri_sqrt = sqrtm(Ri)
Ri_invsqrt = invsqrtm(Ri)
for i in range(Nt):
Li = logm(np.dot(Ri_invsqrt, np.dot(X[i], Ri_invsqrt)))
eta_i[i, :, :] = np.dot(Ri_sqrt, np.dot(Li, Ri_sqrt))
# multiply the tangent vectors by the transport matrix W
eta_f = np.zeros(X.shape)
for i in range(Nt):
eta_f[i, :, :] = np.dot(W, np.dot(eta_i[i], W.T))
# transform tangent vectors to covariance matrices with respect to Rf
Xnew = np.zeros(X.shape)
Rf_sqrt = sqrtm(Rf)
Rf_invsqrt = invsqrtm(Rf)
for i in range(Nt):
Ef = expm(np.dot(Rf_invsqrt, np.dot(eta_f[i], Rf_invsqrt)))
Xnew[i, :, :] = np.dot(Rf_sqrt, np.dot(Ef, Rf_sqrt))
# transform back to tangent vectors (flat form, not sym matrix) if needed
if self.tangent_new:
Xnew = tangent_space(Xnew, Rf)
return Xnew
def mean_riemann_custom(covmats, mean_args):
"""
A custom version of pyriemann.utils.mean.mean_riemann to handle singular matrices
and I/O with regards to reducing samples classes.
For function doc refer to the doc of pyriemann.utils.mean.mean_riemann.
"""
# Taking arguments
tol, maxiter, init, sample_weight = mean_args
# init
sample_weight = _get_sample_weight(sample_weight, covmats)
Nt, Ne, Ne = covmats.shape
if init is None:
C = np.mean(covmats, axis=0)
else:
C = init
k = 0
nu = 1.0
tau = np.finfo(np.float64).max
crit = np.finfo(np.float64).max
# stop when J<10^-9 or max iteration = 50
while (crit > tol) and (k < maxiter) and (nu > tol):
k = k + 1
C12 = sqrtm(C)
Cm12 = invsqrtm(C)
J = np.zeros((Ne, Ne))
for index in range(Nt):
tmp = np.dot(np.dot(Cm12, covmats[index, :, :]), Cm12)
with warnings.catch_warnings():
warnings.filterwarnings('error')
try:
J += sample_weight[index] * logm(tmp)
except RuntimeWarning:
pass
crit = np.linalg.norm(J, ord='fro')
h = nu * crit
C = np.dot(np.dot(C12, expm(nu * J)), C12)
if h < tau:
nu = 0.95 * nu
tau = h
else:
nu = 0.5 * nu
return C
def mean_riemann(covmats,
tol=10e-9,
maxiter=50,
init=None,
u_prime=lambda x: 1):
Nt, Ne, Ne = covmats.shape
if init is None:
C = np.mean(covmats, axis=0)
else:
C = init
k = 0
nu = 1.0
tau = np.finfo(np.float64).max
crit = np.finfo(np.float64).max
# stop when J<10^-9 or max iteration = 50
while (crit > tol) and (k < maxiter) and (nu > tol):
k = k + 1
C12 = sqrtm(C)
Cm12 = invsqrtm(C)
J = np.zeros((Ne, Ne))
for i in range(Nt):
tmp = (Cm12 @ covmats[i, :, :]) @ Cm12
if type(u_prime(1)) == list:
J += logm(tmp) * u_prime(
distance_riemann(C, covmats[i, :, :])**2)[i] / Nt
else:
J += logm(tmp) * u_prime(
distance_riemann(C, covmats[i, :, :])**2) / Nt
crit = np.linalg.norm(J, ord='fro')
h = nu * crit
C = np.dot(np.dot(C12, expm(nu * J)), C12)
if h < tau:
nu = 0.95 * nu
tau = h
else:
nu = 0.5 * nu
return C
def power_means(C, p):
phi = 0.375/np.abs(p)
K = len(C)
n = C[0].shape[0]
w = np.ones(K)
w = w/(1.0*len(w))
G = np.sum([wk*powm(Ck, p) for (wk,Ck) in zip(w,C)], axis=0)
if p > 0:
X = invsqrtm(G)
else:
X = sqrtm(G)
zeta = 10e-10
test = 10*zeta
while test > zeta:
H = np.sum([wk*powm(np.dot(X, np.dot(powm(Ck, np.sign(p)), X.T)), np.abs(p)) for (wk,Ck) in zip(w,C)], axis=0)
X = np.dot(powm(H, -phi), X)
test = 1.0/np.sqrt(n) * np.linalg.norm(H - np.eye(n))
if p > 0:
P = np.dot(np.linalg.inv(X), np.linalg.inv(X.T))
else:
P = np.dot(X.T, X)
return P
def test_sqrtm():
"""Test matrix square root"""
C = 2 * np.eye(3)
Ctrue = np.sqrt(2) * np.eye(3)
assert_array_almost_equal(sqrtm(C), Ctrue)
def transform_org2opt(source, target_train, target_test):
target_opt_train = {}
target_opt_test = {}
target_opt_train['labels'] = target_train['labels']
target_opt_test['labels'] = target_test['labels']
# get cost matrix
Cs = source['covs']
ys = source['labels']
Ct_train = target_train['covs']
Ct_test = target_test['covs']
M = np.zeros((len(Cs), len(Ct_train)))
for i, Cs_i in enumerate(Cs):
for j, Ct_j in enumerate(Ct_train):
M[i, j] = distance_riemann(Cs_i, Ct_j)**2
# get the transportation plan
mu_s = distribution_estimation_uniform(Cs)
mu_t = distribution_estimation_uniform(Ct_train)
gamma = sinkhorn_lpl1_mm(mu_s, ys, mu_t, M, reg=1.0)
# transport the target matrices (train)
Ct_train_transported = np.zeros(Ct_train.shape)
for j in range(len(Ct_train_transported)):
Ct_train_transported[j] = mean_riemann(Cs, sample_weight=gamma[:, j])
target_opt_train['covs'] = Ct_train_transported
# transport the target matrices (test)
D = np.zeros((len(Ct_test), len(Ct_train)))
for k, Ct_k in enumerate(Ct_test):
for l, Ct_l in enumerate(Ct_train):
D[k, l] = distance_riemann(Ct_k, Ct_l)**2
idx = np.argmin(D, axis=1) # nearest neighbour to each target test matrix
Ct_test_transported = np.zeros(Ct_test.shape)
for i in range(len(Ct_test)):
j = idx[i]
Ci = Ct_test[i]
Ri = Ct_train[j]
Rf = Ct_train_transported[j]
Ri_sqrt = sqrtm(Ri)
Ri_invsqrt = invsqrtm(Ri)
Li = logm(np.dot(Ri_invsqrt, np.dot(Ci, Ri_invsqrt)))
eta_i = np.dot(Ri_sqrt, np.dot(Li, Ri_sqrt))
Ri_Rf = geodesic_riemann(Rf, Ri, alpha=0.5)
Ri_inv = np.linalg.inv(Ri)
eta_f = np.dot(Ri_inv, np.dot(eta_i, Ri_inv))
eta_f = np.dot(Ri_Rf, np.dot(eta_f, Ri_Rf))
Rf_sqrt = sqrtm(Rf)
Rf_invsqrt = invsqrtm(Rf)
Ef = expm(np.dot(Rf_invsqrt, np.dot(eta_f, Rf_invsqrt)))
Ct_test_transported[i] = np.dot(Rf_sqrt, np.dot(Ef, Rf_sqrt))
target_opt_test['covs'] = Ct_test_transported
return source, target_opt_train, target_opt_test
def log_riemann(X, Y):
""" log_X(Y) = X log(X^{-1}Y) = X^{1/2} log(X^{-1/2} Y X^{-1/2}) X^{1/2}"""
Xsqrt = sqrtm(X)
Xinvsqrt = invsqrtm(X)
return Xsqrt @ logm(Xinvsqrt @ Y @ Xinvsqrt) @ Xsqrt
def exp_riemann(X, Y):
""" exp_X(Y) = X exp(X^{-1}Y) = X^{1/2} exp(X^{-1/2} Y X^{-1/2}) X^{1/2}"""
Xsqrt = sqrtm(X)
Xinvsqrt = invsqrtm(X)
return Xsqrt @ expm(Xinvsqrt @ Y @ Xinvsqrt) @ Xsqrt
def test_sqrtm():
"""Test matrix square root"""
C = 2*np.eye(3)
Ctrue = np.sqrt(2)*np.eye(3)
assert_array_almost_equal(sqrtm(C),Ctrue)
