def nesterov_subproblem_H(V, W, H, min_pgrad, n_maxiter=1000):
a = 1
WtW = W.T @ W
WtV = W.T @ V
dFH = dFnorm_H(WtV, WtW, H)
norm_dFpH_2 = dH_projected_norm2(dFH, H)
if np.sqrt(norm_dFpH_2) < min_pgrad:
return H, min_pgrad / 10, norm_dFpH_2
L = np.linalg.norm(WtW, ord=2)
Y = H.copy()
for i in range(n_maxiter):
H_next = project(Y - 1 / L * dFnorm_H(WtV, WtW, Y))
a_next = (1 + np.sqrt(4 * (a**2) + 1)) / 2
Y = H_next + (a - 1) / a_next * (H_next - H)
a = a_next
H = H_next
dFH = dFnorm_H(WtV, WtW, H)
norm_dFpH_2 = dH_projected_norm2(dFH, H)
if np.sqrt(norm_dFpH_2) < min_pgrad:
break
return H, min_pgrad, norm_dFpH_2
def factorize_Fnorm(V,
inner_dim,
max_steps,
epsilon=0,
time_limit=np.inf,
record_errors=False,
W_init=None,
H_init=None):
W, H = update_empty_initials(V, inner_dim, W_init, H_init)
W_mask = W == 0
H_mask = H == 0
err = norm_Frobenius(V - W @ H)
start_time = get_time()
time = get_time() - start_time
errors = [(time, err)]
dFWt = dFnorm_H(H @ V.T, H @ H.T, W.T)
dFH = dFnorm_H(W.T @ V, W.T @ W, H)
norm_dFpWt_2 = dH_projected_norm2(dFWt, W.T)
norm_dFpH_2 = dH_projected_norm2(dFH, H)
pgrad_norm = np.sqrt(norm_dFpWt_2 + norm_dFpH_2)
min_pgrad_main = epsilon * pgrad_norm
min_pgrad_W = max(1e-3, epsilon) * pgrad_norm
min_pgrad_H = min_pgrad_W
for i in count():
if i >= max_steps:
break
if pgrad_norm < min_pgrad_main:
break
if time > time_limit:
break
W, min_pgrad_W, norm_dFpWt_2 = \
nesterov_subproblem_H(V.T, H.T, W.T, min_pgrad_W)
W = W.T
W[W_mask] = 0
H, min_pgrad_H, norm_dFpH_2 = \
nesterov_subproblem_H(V, W, H, min_pgrad_H)
H[H_mask] = 0
err = norm_Frobenius(V - W @ H)
time = get_time() - start_time
if record_errors:
errors.append((time, err))
pgrad_norm = np.sqrt(norm_dFpWt_2 + norm_dFpH_2)
if record_errors:
return W, H, np.array(errors)
else:
return W, H
def factorize_Fnorm(V,
inner_dim,
n_steps=10000,
epsiolon=1e-6,
record_errors=False,
W_init=None,
H_init=None):
W, H = update_empty_initials(V, inner_dim, W_init, H_init)
dFWt = dFnorm_H(H @ V.T, H @ H.T, W.T)
dFH = dFnorm_H(W.T @ V, W.T @ W, H)
norm_dFpWt_2 = dH_projected_norm2(dFWt, W.T)
norm_dFpH_2 = dH_projected_norm2(dFH, H)
pgrad_norm = np.sqrt(norm_dFpWt_2 + norm_dFpH_2)
min_pgrad_main = epsiolon * pgrad_norm
min_pgrad_W = max(1e-3, epsiolon) * pgrad_norm
min_pgrad_H = min_pgrad_W
start_time = get_time()
err = norm_Frobenius(V - W @ H)
errors = [(get_time() - start_time, err)]
for i in range(n_steps):
if pgrad_norm < min_pgrad_main:
break
W, l_W, min_pgrad_W, norm_dFpWt_2 = \
dtpnn_subproblem_H(V.T, H.T, W.T, min_pgrad_H)
W = W.T
H, l_H, min_pgrad_H, norm_dFpH_2 = \
dtpnn_subproblem_H(V, W, H, min_pgrad_W)
err = norm_Frobenius(V - W @ H)
if record_errors:
errors.append((get_time() - start_time, err))
pgrad_norm = np.sqrt(norm_dFpWt_2 + norm_dFpH_2)
if record_errors:
return W, H, np.array(errors)
else:
return W, H
def dtpnn_subproblem_H(V, W, H, min_pgrad, start_lambda=1):
l = start_lambda
WtV = W.T @ V
WtW = W.T @ W
dF = dFnorm_H(WtV, WtW, H)
H_new, l = dtpnn_subproblem_step(WtW, H, dF, l)
H = H_new
dF = dFnorm_H(WtV, WtW, H)
norm_dFpH_2 = dH_projected_norm2(dF, H)
return H, l, min_pgrad, norm_dFpH_2