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)