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opt.py
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opt.py
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import cvxpy as cp
import numpy as np
from scipy.linalg import khatri_rao
# Imported here to be able to call getattr(opt, func_name)
from opt_lls import estH_llsscp
VERB = False
def filter_coefs_id(X, Y, Spows):
y = Y.flatten(order='F')
R_minus1 = Spows.shape[0] # NOTE: S_pows expected to be R-1xNxN tensor
# Create matrix Theta
Theta_T = np.zeros((R_minus1+1, X.size))
Theta_T[0,:] = X.flatten(order='F')
for r in range(R_minus1):
Theta_T[r+1,:] = (Spows[r] @ X).flatten(order='F')
Theta = Theta_T.T
return np.linalg.pinv(Theta) @ y
def graph_powers_id(X, Y, Sn, h, Spows, lambd, gamma, beta, inc_gamma, verb=False):
N = Sn.shape[0]
R = Spows.shape[0] + 1
# NOTE: recall h has R entries but Spows R-1 matrices (S^0 is ignored)
for r in range(1, R):
# Compute auxiliary matrix Zr
Zr = Y - h[0]*X
for i in range(1,R):
if i == r:
continue
Zr -= h[i] * Spows[i-1,:,:] @ X
Sr = cp.Variable((N,N), PSD=True) if r % 2 == 0 else cp.Variable((N,N), symmetric=True)
# Sr = cp.Variable((N,N), symmetric=True)
ls_loss = cp.sum_squares(Zr - h[r] * Sr @ X)
comm_los1 = cp.sum_squares(Sr - Spows[r-2,:,:] @ Spows[0,:,:]) if r > 1 else 0
comm_los2 = cp.sum_squares(Spows[r,:,:] - Sr @ Spows[0,:,:]) if r < (R - 1) else 0
sparsity_loss = cp.sum(Sr) if r == 1 else 0
distance_loss = cp.sum(cp.abs(Sr - Sn)) if r == 1 else 0
constraints = [Sr >= 0, cp.diag(Sr) == 0] if r == 1 else [Sr >= 0]
obj = ls_loss + lambd*distance_loss + beta*sparsity_loss + gamma*(comm_los1 + comm_los2)
prob = cp.Problem(cp.Minimize(obj), constraints)
try:
prob.solve()
except cp.SolverError:
if verb:
print("WARNING: Could not find optimal S -- Solver Error")
try:
prob.solve(solver=cp.ECOS, verbose=False)
if verb:
print("Solver error fixed")
except cp.SolverError as e:
if verb:
print("A second solver error")
print(e)
return None
if prob.status in ["optimal", "optimal_inaccurate"]:
Spows[r-1,:,:] = Sr.value
else:
if verb:
print(f"WARNING: problem status: {prob.status}")
return None
return Spows
# TODO: add option for stationarity, add early stopping
def robust_gfid_powersS(X, Y, R, Sn, params, max_iters=20, th=1e-3, patience=4, h_true=None,
S_true=None, verb=False):
"""
"""
lambd, gamma, beta, inc_gamma = params
N, M = X.shape
Spows_prev = np.zeros((R-1, N, N))
Spows_prev[0,:,:] = Sn
# S_prev = Sn
Spows = Spows_prev # np.copy(Spows_prev)
err = []
count_es = 0
min_err = np.inf
norm_h = (h_true**2).sum() if h_true is not None else 0
norm_S = (N*(N-1) / 2) if S_true is not None else 0
for i in range(max_iters):
# Filter identification problem
h = filter_coefs_id(X, Y, Spows)
# Graph identification
Spows = graph_powers_id(X, Y, Sn, h, Spows_prev, lambd, gamma, beta, inc_gamma)
Spows = Spows_prev if Spows is None else Spows
# if h_true is not None and S_true is not None:
# # Early stopping is performed with variables error
# err_h = ((h - h_true)**2).sum() / norm_h
# err_S = ((Spows[0,:,:] - S_true)**2).sum() / norm_S
# err.append(err_h + err_S)
# else:
# # Early stopping is performed with objective funtion
# ls_loss = ((Y - H@X)**2).sum()
# s_loss = np.abs(S-Sn).sum()
# commut_loss = ((S@H - H@S)**2).sum()
# commut_cy_loss = ((Cy@H - H@Cy)**2).sum()
# err.append(ls_loss + lambd*s_loss + gamma*commut_loss)
if inc_gamma:
gamma = inc_gamma*gamma
Spows_prev = Spows
return h, Spows
def filter_id(Y, X, S, gamma, delta, Cy, verb=VERB):
"""
Performs the filter identification step of the robust filter identification algorithm.
It estimates H using cvx, not the analytical solution.
"""
H = cp.Variable(S.shape, symmetric=True)
ls_loss = cp.sum_squares(Y - H@X)
commut_loss = cp.sum_squares(H@S - S@H)
commut_cy_loss = cp.sum_squares(H@Cy - Cy@H)
obj = ls_loss + gamma*commut_loss + delta*commut_cy_loss
prob = cp.Problem(cp.Minimize(obj))
try:
prob.solve()
except cp.SolverError:
if verb:
print("WARNING: Could not find optimal H -- Solver Error")
try:
prob.solve(verbose=False)
if verb:
print("Solver error fixed")
except:
if verb:
print("A second solver error")
return None
if prob.status in ["optimal", "optimal_inaccurate"]:
return H.value
if verb:
print(f"WARNING: problem status: {prob.status}")
return None
def graph_id(Sn, H, Cy, lambd, gamma, delta, verb=VERB):
"""
Performs the filter identification step of the robust filter identification algorithm
"""
S = cp.Variable(H.shape, symmetric=True)
s_loss = cp.sum(cp.abs(S - Sn))
commut_loss = cp.sum_squares(H@S - S@H)
commut_cy_loss = cp.sum_squares(S@Cy - Cy@S)
# TODO: add sparsity loss
obj = lambd*s_loss + gamma*commut_loss + delta*commut_cy_loss
constraints = [S >= 0, cp.diag(S) == 0]
prob = cp.Problem(cp.Minimize(obj), constraints)
try:
prob.solve()
except cp.SolverError:
if verb:
print("WARNING: Could not find optimal S -- Solver Error")
try:
prob.solve(solver=cp.ECOS, verbose=False)
if verb:
print("Solver error fixed")
except cp.SolverError as e:
if verb:
print("A second solver error")
print(e)
return None
if prob.status in ["optimal", "optimal_inaccurate"]:
return S.value
else:
if verb:
print(f"WARNING: problem status: {prob.status}")
return None
def graph_id_rew(Sn, H, Cy, W1, W2, lambd, gamma, delta, beta, verb=VERB):
"""
Performs the filter identification step of the robust filter identification algorithm
with the reweighted alternative
"""
N = Sn.shape[0]
S = cp.Variable((N,N), symmetric=True)
sn_loss = cp.sum(cp.multiply(W1, cp.abs(S - Sn)))
s_loss = cp.sum(cp.multiply(W2, cp.abs(S)))
commut_loss = cp.sum_squares(H@S - S@H)
commut_cy_loss = cp.sum_squares(S@Cy - Cy@S)
obj = lambd*sn_loss + beta*s_loss + gamma*commut_loss + delta*commut_cy_loss
constraints = [
S >= 0,
cp.diag(S) == 0
]
prob = cp.Problem(cp.Minimize(obj), constraints)
try:
prob.solve()
except cp.SolverError:
if verb:
print("WARNING: Could not find optimal S -- Solver Error")
try:
prob.solve(verbose=False)
if verb:
print("Solver error fixed")
except:
if verb:
print("A second solver error")
return None
except cp.DCPError:
raise RuntimeError("Could not find optimal S -- DCP Error")
if prob.status in ["optimal", "optimal_inaccurate"]:
S = S.value
else:
if verb:
print(f"WARNING: problem status: {prob.status}")
return None
return S
def estH_iter(X, Y, Sn, Cy, params, max_iters=20, th=1e-3, patience=4, H_true=None, S_true=None):
import warnings
warnings.filterwarnings("ignore")
lambd, gamma, delta, inc_gamma = params
N, M = X.shape
S_prev = Sn
H_prev = Sn
S = Sn
err = []
count_es = 0
min_err = np.inf
norm_H = (H_true**2).sum() if H_true is not None else 0
norm_S = (N*(N-1)) if S_true is not None else 0
for i in range(max_iters):
# Filter identification problem
H = filter_id(Y, X, S, gamma, delta, Cy)
H = H_prev if H is None else H
# Graph identification
S = graph_id(Sn, H, Cy, lambd, gamma, delta)
S = S_prev if S is None else S
# Check convergence
if H_true is not None and S_true is not None:
# Early stopping is performed with variables error
err_H = ((H - H_true)**2).sum() / norm_H
err_S = ((S - S_true)**2).sum() / norm_S
err.append(err_H + err_S)
#print(f"estH_iter: {i=}, {err_H=}, {err_S=}, {err[i]=}")
else:
# Early stopping is performed with objective funtion
ls_loss = ((Y - H@X)**2).sum()
s_loss = np.abs(S-Sn).sum()
commut_loss = ((S@H - H@S)**2).sum()
commut_cy_loss = ((Cy@H - H@Cy)**2).sum()
err.append(ls_loss + lambd*s_loss + gamma*commut_loss + delta*commut_cy_loss)
# print(f"Iter: {i} - err: {err[i]}")
if i > 0 and np.abs(err[i] - err[i-1]) < th and err[i] > err[i-1]:
H_min = H
S_min = S
i_min = i
#print(f'\t\tConvergence reached at iteration {i}')
break
if err[i] > min_err:
count_es += 1
else:
min_err = err[i]
H_min = H.copy()
S_min = S.copy()
i_min = i
count_es = 0
if count_es == patience:
#print(f'\t\tES Convergence reached at iteration {i_min}')
break
gamma = inc_gamma*gamma if inc_gamma else gamma
H_prev = H
S_prev = S
return i_min, H_min, S_min
def estH_iter_rew(X, Y, Sn, Cy, params, max_iters=20, th=1e-3, patience=4, H_true=None, S_true=None):
import warnings
warnings.filterwarnings("ignore")
lambd, gamma, delta, beta, inc_gamma = params
N, M = X.shape
S_prev = Sn
H_prev = Sn
S = Sn
err = []
W1 = np.ones((N,N))
W2 = np.ones((N,N))
delta1 = 1e-3
delta2 = 1e-3
count_es = 0
min_err = np.inf
norm_H = (H_true**2).sum() if H_true is not None else 0
norm_S = (N*(N-1) / 2) if S_true is not None else 0
for i in range(max_iters):
# Filter identification problem
H = filter_id(Y, X, S, gamma, delta, Cy)
H = H_prev if H is None else H
# Graph identification
S = graph_id_rew(Sn, H, Cy, W1, W2, lambd, gamma, delta, beta)
S = S_prev if S is None else S
W1 = lambd / (np.abs(S - Sn) + delta1)
W2 = beta / (S + delta2)
if H_true is not None and S_true is not None:
# Early stopping is performed with variables error
err_H = ((H - H_true)**2).sum() / norm_H
err_S = ((S - S_true)**2).sum() / norm_S
err.append(err_H + err_S)
#print(i, err_H, err_S, err[-1])
else:
# Early stopping is performed with objective funtion
ls_loss = ((Y - H@X)**2).sum()
s_loss = np.abs(S-Sn).sum()
commut_loss = ((S@H - H@S)**2).sum()
commut_cy_loss = ((Cy@H - H@Cy)**2).sum()
err.append(ls_loss + lambd*s_loss + gamma*commut_loss + delta*commut_cy_loss)
# print(f"Iter: {i} - err: {err[i]}")
if i > 0 and np.abs(err[i] - err[i-1]) < th and err[i] > err[i-1]:
H_min = H
S_min = S
i_min = i
#print(f'Convergence reached at iteration {i}')
break
if err[i] > min_err:
count_es += 1
else:
min_err = err[i]
H_min = H.copy()
S_min = S.copy()
i_min = i
count_es = 0
if count_es == patience:
#print(f"Convergence reached at iteration {i_min}", flush=True)
break
if inc_gamma:
gamma = inc_gamma*gamma
H_prev = H
S_prev = S
return i_min, H_min, S_min
def estH_denS(X, Y, Sn, Cy, params, verb=VERB):
import warnings
warnings.filterwarnings("ignore")
N, M = X.shape
gamma, delta = params
# Denoise S
S = cp.Variable((N,N), symmetric=True)
s_loss = cp.sum(cp.abs(S - Sn))
commut_cy_loss = cp.sum_squares(S@Cy - Cy@S)
obj = s_loss + delta*commut_cy_loss
constraints = [
S >= 0,
cp.diag(S) == 0,
#commut_cy_loss <= delta
]
prob = cp.Problem(cp.Minimize(obj), constraints)
try:
prob.solve()
except cp.SolverError:
if verb:
print("estH_denS -- Could not find optimal S -- Solver Error")
try:
prob.solve(verbose=False)
if verb:
print("Solver Error fixed")
except:
if verb:
print("A second solver error")
S = Sn
if prob.status in ["optimal", "optimal_inaccurate"]:
S = S.value
else:
#print("estH_denS -- Could not find optimal S")
S = Sn
# Compute H from S
H = cp.Variable((N,N), symmetric=True)
ls_loss = cp.sum_squares(Y - H@X)
commut_loss = cp.sum_squares(H@S - S@H)
commut_cy_loss = cp.sum_squares(H@Cy - Cy@H)
obj = ls_loss + gamma*commut_loss + delta*commut_cy_loss
#const = [commut_loss <= 0]
prob = cp.Problem(cp.Minimize(obj))#, const)
try:
prob.solve()
except cp.SolverError:
if verb:
print("estH_denS -- Could not find optimal H - SolverError")
if prob.status in ["optimal", "optimal_inaccurate"]:
H = H.value
else:
if verb:
print("estH_denS -- Could not find optimal H")
H = np.zeros((N,N))
ls_loss = ((Y - H@X)**2).sum()
s_loss = np.abs(S-Sn).sum()
commut_loss = ((S@H - H@S)**2).sum()
commut_cy_loss = ((Cy@H - H@Cy)**2).sum()
err = ls_loss + s_loss + gamma*commut_loss + delta*commut_cy_loss
return -1, H, S
def estH_unpertS(X, Y, S, Cy=None, params=None, H_true=None, S_true=None):
"""
Estimation of the graph filter coefficients (h) assuming the GSO S is known.
It uses knowledge only of the eigenvectors of S. The eigenvalues are ignored.
"""
_, eigvecs = np.linalg.eigh(S)
# Z is an MNxN matrix
Z = khatri_rao(X.T @ eigvecs, eigvecs)
h_freq, _, _, _ = np.linalg.lstsq(Z, Y.flatten('F'), rcond=None)
H = eigvecs @ np.diag(h_freq) @ eigvecs.T
return -1, H, S
def estH_ls(X, Y, S, Cy=None, params=None, H_true=None, S_true=None):
"""
Estimation of the H matrix via least squares
"""
H = Y @ np.linalg.pinv(X)
return -1, H, S
def fi_eigval(X, Y, S, K):
"""
Estimation of the graph filter coefficients (h) assuming the GSO S is known.
It uses knowledge of both the eigenvectors and eigenvalues of S.
"""
eigvals, eigvecs = np.linalg.eigh(S)
Psi = np.vander(eigvals, K, increasing=True)
# Z is an MNxL
Z = khatri_rao(X.T @ eigvecs, eigvecs)@Psi
h, _, _, _ = np.linalg.lstsq(Z, Y.flatten('F'), rcond=None)
H = eigvecs @ np.diag(Psi@h) @ eigvecs.T
return H, S, h
def estH_analyticalSol(X, Y, Sn, Cy, params, max_iters=20, th=1e-3):
# Initialization
lambd, gamma, delta, inc_gamma = params
N = X.shape[0]
S_prev = S = Sn
# Precomputing quantities
X_kron = np.kron([email protected], np.eye(N))
Y_kron = np.kron(X, np.eye(N))@Y.flatten(order='F')
diff_H = np.zeros(max_iters - 1)
diff_S = np.zeros(max_iters - 1)
for i in range(max_iters):
# Filter identification step
Z = np.kron(S@S, np.eye(N)) + np.kron(np.eye(N), S@S) - 2*np.kron(S, S)
H = (np.linalg.inv(X_kron + gamma*Z)@Y_kron).reshape((N,N), order='F')
# Graph identification
S = graph_id(Sn, H, Cy, lambd, gamma, delta)
S = S_prev if S is None else S
gamma = inc_gamma*gamma if inc_gamma else gamma
if i == 0:
H_prev = H
S_prev = S
continue
norm_H_prev = np.linalg.norm(H_prev, 'fro')
norm_S_prev = np.linalg.norm(S_prev, 'fro')
diff_H[i-1] = (np.linalg.norm(H - H_prev, 'fro')/norm_H_prev)**2
diff_S[i-1] = (np.linalg.norm(S - S_prev, 'fro')/norm_S_prev)**2
if diff_H[i-1] < th and diff_S[i-1] < th:
print(f'Convergence reached at iteration {i}')
return H, S , diff_H, diff_S
H_prev = H
S_prev = S
return H, S, diff_H, diff_S
# Baselines
def estH_tls_sem(X, Y, Sn, Cy, params, max_iters=20, th=1e-3, patience=4, H_true=None, S_true=None, verb=VERB):
N = Sn.shape[0]
lambd1, lambd2 = params
count_es = 0
min_err = np.inf
norm_H = (H_true**2).sum() if H_true is not None else 0
norm_S = (N*(N-1) / 2) if S_true is not None else 0
Y_aux = Y.copy()
err = []
for i in range(max_iters):
Delta = cp.Variable((N,N), symmetric=True)
ls_loss = cp.sum_squares(Y_aux - (Sn-Delta)@Y_aux - X)
sparsity_loss = cp.sum(cp.abs(Delta))
obj = ls_loss + lambd1*sparsity_loss
const = [cp.diag(Delta) == 0]
prob = cp.Problem(cp.Minimize(obj), const)
try:
prob.solve()
except cp.SolverError:
if verb:
print("estH_tls_sem -- Could not find optimal Delta - SolverError")
if prob.status in ["optimal", "optimal_inaccurate"]:
S = Sn - Delta.value
else:
S = Sn
S_tilde = np.eye(N) - S
H = np.linalg.inv(S_tilde)
# Analytical solution
Y_aux, _, _, _ = np.linalg.lstsq(S_tilde @ S_tilde + lambd2*np.eye(N), S_tilde @ X + lambd2*Y, rcond=None)
if H_true is not None and S_true is not None:
# Early stopping is performed with variables error
err_H = ((H - H_true)**2).sum() / norm_H
err_S = ((S - S_true)**2).sum() / norm_S
err.append(err_H + err_S)
#print(i, err_H, err_S, err[-1])
else:
# Early stopping is performed with objective funtion
ls_loss = ((Y - S@Y - X)**2).sum()
s_loss = np.abs(S).sum()
y_loss = ((Y-Y_aux)**2).sum()
err.append(ls_loss + lambd1*s_loss + lambd2*y_loss)
# print(f"Iter: {i} - err: {err[i]}")
if i > 0 and np.abs(err[i] - err[i-1]) < th and err[i] > err[i-1]:
H_min = H
S_min = S
i_min = i
#print(f'Convergence reached at iteration {i}')
break
if err[i] > min_err:
count_es += 1
else:
min_err = err[i]
H_min = H.copy()
S_min = S.copy()
i_min = i
count_es = 0
if count_es == patience:
#print(f"Convergence reached at iteration {i_min}", flush=True)
break
return i_min, H_min, S_min
########## DEBUG METHODS ##########
def rfi_debug(X, Y, Sn, Cy, params, H_true, S_true, max_iters=20):
# Initialization
lambd, gamma, delta, inc_gamma = params
N, M = X.shape
S_prev = S = Sn
# Precomputing quantities
norm_H = np.linalg.norm(H_true,'fro')
norm_S = np.linalg.norm(S_true, 'fro')
X_kron = np.kron([email protected], np.eye(N))
Y_kron = np.kron(X, np.eye(N))@Y.flatten(order='F')
err_S = np.zeros(max_iters)
err_H = np.zeros(max_iters)
diff_S = np.zeros(max_iters-1)
diff_H = np.zeros(max_iters-1)
for i in range(max_iters):
# Filter identification step
Z = np.kron(S@S, np.eye(N)) + np.kron(np.eye(N), S@S) - 2*np.kron(S, S)
H = (np.linalg.inv(X_kron + gamma*Z)@Y_kron).reshape((N,N), order='F')
# Graph identification
S = graph_id(Sn, H, Cy, lambd, gamma, delta)
if S is None:
print(f"rfi_debug - Iter {i} - S is None")
S = S_prev
if inc_gamma:
gamma = inc_gamma*gamma
err_H[i] = (np.linalg.norm(H - H_true, 'fro')/norm_H)**2
err_S[i] = (np.linalg.norm(S - S_true, 'fro')/norm_S)**2
if i > 0:
diff_H[i-1] = ((H-H_prev)**2).sum()
diff_S[i-1] = ((S-S_prev)**2).sum()
H_prev = H
S_prev = S
return H, S, err_H, err_S, diff_H, diff_S
########## DEBUG METHODS ##########
def rfi_cvx_debug(X, Y, Sn, Cy, params, H_true, S_true, max_iters=20):
# Initialization
lambd, gamma, delta, inc_gamma = params
N, M = X.shape
S_prev = S = Sn
H_prev = Sn
# Precomputing quantities
norm_H = np.linalg.norm(H_true,'fro')
norm_S = np.linalg.norm(S_true, 'fro')
err_S = np.zeros(max_iters)
err_H = np.zeros(max_iters)
diff_S = np.zeros(max_iters-1)
diff_H = np.zeros(max_iters-1)
for i in range(max_iters):
# Filter identification problem
H = filter_id(Y, X, S, gamma, delta, Cy)
H = H_prev if H is None else H
# Graph identification
S = graph_id(Sn, H, Cy, lambd, gamma, delta)
if S is None:
print(f"rfi_cvx_debug - Iter {i} - S is None")
S = S_prev
if inc_gamma:
gamma = inc_gamma*gamma
err_H[i] = (np.linalg.norm(H - H_true, 'fro')/norm_H)**2
err_S[i] = (np.linalg.norm(S - S_true, 'fro')/norm_S)**2
if i > 0:
diff_H[i-1] = ((H-H_prev)**2).sum()
diff_S[i-1] = ((S-S_prev)**2).sum()
H_prev = H
S_prev = S
return H, S, err_H, err_S, diff_H, diff_S
########## DEBUG METHODS ##########
def rfi_rew_debug(X, Y, Sn, Cy, params, H_true, S_true, max_iters=20):
# Initialization
lambd, gamma, delta, beta, inc_gamma = params
N, M = X.shape
S_prev = S = Sn
H_prev = Sn
# Precomputing quantities
norm_H = np.linalg.norm(H_true,'fro')
norm_S = np.linalg.norm(S_true, 'fro')
W1 = np.ones((N,N))
W2 = np.ones((N,N))
delta1 = 1e-3
delta2 = 1e-3
norm_H = (H_true**2).sum() if H_true is not None else 0
norm_S = np.linalg.norm(S_true, 'fro') if S_true is not None else 0
err_S = np.zeros(max_iters)
err_H = np.zeros(max_iters)
diff_S = np.zeros(max_iters-1)
diff_H = np.zeros(max_iters-1)
for i in range(max_iters):
# Filter identification problem
H = filter_id(Y, X, S, gamma, delta, Cy)
H = H_prev if H is None else H
# Graph identification
S = graph_id_rew(Sn, H, Cy, W1, W2, lambd, gamma, delta, beta)
if S is None:
print(f"rfi_rew_debug - Iter {i} - S is None")
S = S_prev
W1 = lambd / (np.abs(S - Sn) + delta1)
W2 = beta / (S + delta2)
if inc_gamma:
gamma = inc_gamma*gamma
err_H[i] = (np.linalg.norm(H - H_true, 'fro')/norm_H)**2
err_S[i] = (np.linalg.norm(S - S_true, 'fro')/norm_S)**2
if i > 0:
diff_H[i-1] = ((H-H_prev)**2).sum()
diff_S[i-1] = ((S-S_prev)**2).sum()
H_prev = H
S_prev = S
return H, S, err_H, err_S, diff_H, diff_S