def plot_eigenvalues_set(fx, nx, limits=(-6, 6), nsamples=1000, fname=None):
x = torch.rand(nsamples, nx,
dtype=torch.float) * (limits[1] - limits[0]) + limits[0]
Astars, *_ = lpv_batched(fx, x)
Astars = Astars.detach().numpy()
eigvals = compute_eigenvalues(Astars)
plot_eigenvalues(eigvals, fname=fname)
return eigvals
def plot_phase_portrait_hidim(fx,
nx,
limits=(-6, 6),
nsamples=1000,
fname=None):
x = torch.rand(nsamples, nx,
dtype=torch.float) * (limits[1] - limits[0]) + limits[0]
Astars, *_ = lpv_batched(fx, x)
XY = torch.empty(nsamples, 2)
UV = torch.empty(nsamples, 2)
for i, (Astar, x_sample) in enumerate(zip(Astars, x)):
z_sample = torch.matmul(Astar, x_sample)
u, s, vh = torch.svd(Astar)
# vh = u[:2, ...]
# # TODO Jan: I believe we want to use right singular vectors
# # for projection onto principal components
vh = vh[:2, ...]
XY[i, ...] = torch.matmul(vh, x_sample)
UV[i, ...] = torch.matmul(vh, z_sample)
X, Y = XY.T.detach().numpy()
U, V = UV.T.detach().numpy()
# plot vector field
fig, ax = plt.subplots()
ax.quiver(X,
Y,
U,
V,
angles="uv",
pivot="mid",
width=0.002,
headwidth=4,
headlength=5)
ax.set_aspect(1)
if fname is not None:
plt.savefig(fname)
plt.close()
return [ax]
def plot_Jacobian_norms(fx, nx, limits=(-6, 6), nsamples=10, fname=None):
x = torch.rand(nsamples, nx,
dtype=torch.float) * (limits[1] - limits[0]) + limits[0]
Astars, *_ = lpv_batched(fx, x)
Astars_np = Astars.detach().numpy()
Astars_mean = np.mean(Astars_np, 0)
Astars_var = np.var(Astars_np, 0)
Astars_min = np.min(Astars_np, 0)
Astars_max = np.max(Astars_np, 0)
fig, ax = plt.subplots(1, 2)
im1 = ax[0].imshow(Astars_mean,
vmin=abs(Astars_mean).min(),
vmax=abs(Astars_mean).max(),
cmap=plt.cm.CMRmap)
ax[0].set_title('mean($A^{\star}}$)')
divider = make_axes_locatable(ax[0])
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(im1, cax=cax)
im2 = ax[1].imshow(Astars_var,
vmin=abs(Astars_var).min(),
vmax=abs(Astars_var).max(),
cmap=plt.cm.CMRmap)
ax[1].set_title('var($A^{\star}}$)')
divider = make_axes_locatable(ax[1])
cax = divider.append_axes("right", size="5%", pad=0.05)
fig.colorbar(im2, cax=cax)
# im3 = ax[1,0].imshow(Astars_min, vmin=abs(Astars_min).min(), vmax=abs(Astars_min).max(), cmap=plt.cm.CMRmap)
# ax[1, 0].set_title('min($A^{\star}}$)')
# fig.colorbar(im3, ax=ax[1,0])
# im4 = ax[1,1].imshow(Astars_max, vmin=abs(Astars_max).min(), vmax=abs(Astars_max).max(), cmap=plt.cm.CMRmap)
# ax[1, 1].set_title('max($A^{\star}}$)')
# fig.colorbar(im4, ax=ax[1,1])
fig.tight_layout()
if fname is not None:
plt.savefig(fname)
plt.close()
return Astars_mean, Astars_var, Astars_min, Astars_max
def plot_singular_values(fx, nx, limits=(-6, 6), nsamples=10, fname=None):
x = torch.rand(nsamples, nx,
dtype=torch.float) * (limits[1] - limits[0]) + limits[0]
Astars, *_ = lpv_batched(fx, x)
fig, ax = plt.subplots()
S = []
for i, Astar in enumerate(Astars):
_, s, _ = torch.svd(Astar, compute_uv=False)
S.append(s.T.detach().numpy())
plt.plot(np.arange(1, nx + 1, 1), s.T.detach().numpy())
plt.grid(True)
plt.xlabel('$k$')
plt.ylabel('$\sigma_k$')
ax.set_yscale("log")
if fname is not None:
plt.savefig(fname)
plt.close()
return [ax], S
},
)
for act_name, bias in params:
act = activations[act_name]
fx.nonlin = nn.ModuleList([act() for _ in fx.nonlin])
combo_string = f"{linmap}_x{nx}_({sigmin},{sigmax})_{act_name}_{nlayers}l_{'real' if real else 'complex'}{'_bias' if bias else ''}"
print(combo_string)
grid_x, grid_y = torch.meshgrid(
torch.arange(-6, 6, 0.1),
torch.arange(-6, 6, 0.1),
)
X = torch.stack((grid_x.flatten(), grid_y.flatten())).T
with torch.no_grad():
Astars, *_ = lpv_batched(fx, X)
Astars = torch.unbind(Astars, dim=0)
eigvals.append(compute_eigenvalues(Astars))
plot_nlayers.append(nlayers)
if i == len(combos) - 1 or (i < len(combos) - 1
and combos[i][0] != combos[i + 1][0]):
plt.figure(figsize=(5, 4))
for nl, eigval_list in zip(plot_nlayers, eigvals):
eigval_list = np.array(eigval_list).flatten()
sns.kdeplot(eigval_list,
log_scale=True,
label=f"Layers = {nl}")
plt.tight_layout()
plt.legend()
def lin_regions(nx, layers, maps, activations, outdir="./plots_region"):
for nlayers in layers:
for (linmap, sigmin, sigmax, real) in maps:
torch.manual_seed(408)
np.random.seed(408)
fx = blocks.MLP(
nx,
nx,
nonlin=nn.Identity,
linear_map=slim.linear.maps[linmap],
hsizes=[nx] * nlayers,
bias=False,
linargs={
"sigma_min": sigmin,
"sigma_max": sigmax,
"real": real,
},
)
for (act_name, act) in activations:
fx.nonlin = nn.ModuleList([act() for _ in fx.nonlin])
for bias in [False]:
combo_string = f"{linmap}_x{nx}_({sigmin},{sigmax})_{act_name}_{nlayers}l_{'real' if real else 'complex'}{'_bias' if bias else ''}"
print(combo_string)
if nx == 2:
plot_astar_phase_portrait(
fx,
x_lims=(-6, 6),
y_lims=(-6, 6),
step=0.5,
use_bias=bias,
initial_states=initial_states,
fname=os.path.join(outdir, f"phase_{combo_string}.png"),
)
grid_x, grid_y = torch.meshgrid(
torch.arange(-6, 6.1, 0.1),
torch.arange(-6, 6.1, 0.1),
)
X = torch.stack((grid_x.flatten(), grid_y.flatten())).T
else:
X = torch.arange(-6, 6.1, 0.1).unsqueeze(-1).expand(-1, nx)
Astars, Astar_b, _, _, _ = lpv_batched(fx, X)
Astars = Astars.detach().numpy()
# plot Anorms
Anorms = compute_norms(Astars)
Anorm_mat = np.reshape(Anorms, grid_x.shape)
fig1, ax1 = plt.subplots()
im1 = ax1.imshow(Anorm_mat, vmin=abs(Anorm_mat).min(), vmax=abs(Anorm_mat).max(), cmap=PALETTE, origin='lower',
extent=[X.min(), X.max(), X.min(), X.max()])#, interpolation="bilinear")
fig1.colorbar(im1, ax=ax1)
im1.set_clim(0., 2.)
# ax1.set_title('Metric: 'r'$\Vert A^* \Vert$')
fname1 = os.path.join(outdir, f"norm_region_{combo_string}.pdf")
plt.savefig(fname1)
# plot dominant eigenvalues
eigvals = compute_eigenvalues(Astars)
dom_eigs = [np.absolute(eigs).max() for eigs in eigvals]
dom_eigs_mat = np.reshape(dom_eigs, grid_x.shape)
fig2, ax2 = plt.subplots()
vmin = abs(dom_eigs_mat).min()
vmax = abs(dom_eigs_mat).max()
im2 = ax2.imshow(dom_eigs_mat, vmin=0, vmax=2,
cmap=PALETTE, origin='lower',
extent=[X.min(), X.max(), X.min(), X.max()])#, interpolation="bilinear")
fig2.colorbar(im2, ax=ax2)
im2.set_clim(0., 2.)
#ax2.set_title('Metric: 'r'$\| \lambda_1 \|$')
fname2 = os.path.join(outdir, f"dom_eig_region_{combo_string}.pdf")
plt.savefig(fname2)
# plot sum of absolute eigenvalues
sum_eigs = [np.absolute(eigs).sum() for eigs in eigvals]
sum_eigs_mat = np.reshape(sum_eigs, grid_x.shape)
fig3, ax3 = plt.subplots()
im3 = ax3.imshow(sum_eigs_mat, vmin=abs(sum_eigs_mat).min(), vmax=abs(sum_eigs_mat).max(), cmap=PALETTE, origin='lower',
extent=[X.min(), X.max(), X.min(), X.max()])#, interpolation="bilinear")
# im3 = ax3.imshow(sum_eigs_mat, vmin=0, vmax=0.3,
# cmap=plt.cm.CMRmap, origin='lower',
# extent=[X.min(), X.max(), X.min(), X.max()], interpolation="bilinear")
fig3.colorbar(im3, ax=ax3)
im3.set_clim(0., 2.)
# ax3.set_title('Metric: 'r'$\sum_{i=1}^n{\| \lambda_i \|}$')
fname3 = os.path.join(outdir, f"sum_eig_region_{combo_string}.pdf")
plt.savefig(fname3)
plt.close('all')