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Added comparison of gradient ratios and hessian condition numbers
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| import os | ||
| import numpy as np | ||
| import pandas as pd | ||
| from scipy.spatial.transform import Rotation | ||
| import jax | ||
| import jax.numpy as jnp | ||
| import matplotlib.pyplot as plt | ||
| import matplotlib.colors as matcolors | ||
| from matplotlib.ticker import FormatStrFormatter | ||
| import seaborn as sns | ||
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| from hitchhiking_rotations import HITCHHIKING_ROOT_DIR | ||
| from mpl_toolkits.mplot3d import Axes3D | ||
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| #import lovely_jax as lj | ||
| #lj.monkey_patch() | ||
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| N = int(1e3) # Number of randomly sampled rotations and predicted matrices | ||
| plot_frames = False # Plot frames before/after SVD/GSO transform | ||
| plot_frames_with_grads = True | ||
| plot_grads = False # Plot ratio between gradients entries | ||
| plot_ratios = False # Plot 2D scatter plot of gradients | ||
| plot_condnums = False # Plot condition numbers of Hessian matrices and max eigen values | ||
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| rot = Rotation.random(N) # generate N random rotations | ||
| rotmats = jnp.array(rot.as_matrix()) | ||
| #predmats = rotmats + 1e-1 * jax.random.normal(key=jax.random.PRNGKey(42), shape=(N, 3, 3)) | ||
| predmats = jax.random.uniform(key=jax.random.PRNGKey(42), shape=(N, 3, 3), minval=-2., maxval=2.) | ||
| #predmats = 2*jax.random.normal(key=jax.random.PRNGKey(1), shape=(N, 3, 3)) | ||
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| @jax.jit | ||
| def gso(m: jnp.ndarray) -> jnp.ndarray: | ||
| """ Gram-Schmidt orthogonalization from 6D input. | ||
| Source: Google research - https://github.com/google-research/google-research/blob/193eb9d7b643ee5064cb37fd8e6e3ecde78737dc/special_orthogonalization/utils.py#L93-L115 | ||
| """ | ||
| x = m[:, 0] | ||
| y = m[:, 1] | ||
| xn = x / jnp.linalg.norm(x, axis=0) | ||
| z = jnp.cross(xn, y) | ||
| zn = z / jnp.linalg.norm(z, axis=0) | ||
| yn = jnp.cross(zn, xn) | ||
| return jnp.c_[xn, yn, zn] | ||
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| @jax.jit | ||
| def svd(m: jnp.ndarray) -> jnp.ndarray: | ||
| """ Maps 3x3 matrices onto SO(3) via symmetric orthogonalization. | ||
| Source: Google research - https://github.com/google-research/google-research/blob/193eb9d7b643ee5064cb37fd8e6e3ecde78737dc/special_orthogonalization/utils.py#L93-L115 | ||
| """ | ||
| """ | ||
| m = jax.lax.cond(jnp.linalg.matrix_rank(m) < 3, | ||
| true_fun=lambda x: x + jnp.eye(3) * 1e-10, | ||
| false_fun=lambda x: x, | ||
| operand=m) | ||
| """ | ||
| U, _, Vh = jnp.linalg.svd(m, full_matrices=False) | ||
| det = jnp.linalg.det(jnp.matmul(U, Vh)) | ||
| return jnp.matmul(jnp.c_[U[:, :-1], U[:, -1] * det], Vh) | ||
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| gso_vmap = jax.vmap(gso) | ||
| pred_gso = gso_vmap(predmats) | ||
| rot_gso = gso_vmap(rotmats) | ||
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| svd_vmap = jax.vmap(svd) | ||
| pred_svd = svd_vmap(predmats) | ||
| rot_svd = svd_vmap(rotmats) | ||
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| def plot_matrix(ax, mat, color, label, offset=jnp.zeros(3,)): | ||
| for i in range(len(mat)): | ||
| if i == 0: | ||
| ax.quiver(offset[0][i], offset[1][i], offset[2][i], | ||
| mat[0][i], mat[1][i], mat[2][i], | ||
| color=color, label=f'{label}') | ||
| else: | ||
| ax.quiver(offset[0][i], offset[1][i], offset[2][i], | ||
| mat[0][i], mat[1][i], mat[2][i], | ||
| color=color) | ||
| ax.text(mat[0][i], mat[1][i], mat[2][i], f'$e_{i + 1}$', color='black') | ||
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| def plot_matrices(ax, r_list, labels, off_list=None): | ||
| ax.set(xlim=(-1.25, 1.25), ylim=(-1.25, 1.25), zlim=(-1.25, 1.25)) | ||
| colors = ['r', 'g', 'b', 'y', 'm', 'c'] | ||
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| if off_list is None: | ||
| off_list = [jnp.zeros((3,3)) for _ in range(len(r_list))] | ||
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| for i in range(len(r_list)): | ||
| plot_matrix(ax, r_list[i], colors[i], label=labels[i], offset=off_list[i]) | ||
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| ax.set_xlabel('X') | ||
| ax.set_ylabel('Y') | ||
| ax.set_zlabel('Z') | ||
| ax.set_box_aspect([1, 1, 1]) | ||
| ax.legend() | ||
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| # PLOT frames to check that GSO and SVD layers are working | ||
| if plot_frames: | ||
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| for r00, r01, r02, r10, r11, r12 in zip(rotmats, rot_svd, rot_gso, predmats, pred_svd, pred_gso): | ||
| fig = plt.figure() | ||
| ax1 = fig.add_subplot(121, projection='3d', proj_type="ortho") | ||
| plot_matrices(ax1, [r00, r01, r02], ["rotmat", "rot_svd", "rot_gso"]) | ||
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| ax2 = fig.add_subplot(122, projection='3d', proj_type="ortho") | ||
| plot_matrices(ax2, [r10, r11, r12], ["predmat", "pred_svd", "pred_gso"]) | ||
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| plt.show() | ||
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| ############################################################################### | ||
| # DEFINE GRADIENTS AND HESSIANS | ||
| ############################################################################### | ||
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| def norm(mat1: jnp.ndarray, mat2: jnp.ndarray) -> jnp.ndarray: | ||
| return jnp.linalg.norm(mat1.flatten() - mat2.flatten()) | ||
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| def norm_gso(predmat_vec, rotmat): | ||
| return norm(rotmat, gso(predmat_vec.reshape(3,2))) | ||
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| def norm_svd(predmat_vec, rotmat): | ||
| return norm(rotmat, svd(predmat_vec.reshape(3,3))) | ||
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| def hess_gso(rotmat: jnp.ndarray, predmat_vec: jnp.ndarray) -> jnp.ndarray: | ||
| return jax.hessian(norm_gso)(predmat_vec, rotmat) | ||
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| def hess_svd(rotmat: jnp.ndarray, predmat_vec: jnp.ndarray) -> jnp.ndarray: | ||
| return jax.hessian(norm_svd)(predmat_vec, rotmat) | ||
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| ############################################################################### | ||
| # COMPUTE GRADIENTS | ||
| ############################################################################### | ||
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| loss_svd = jax.vmap(norm, (0, 0))(rotmats, pred_svd) | ||
| loss_gso = jax.vmap(norm,(0, 0))(rotmats, pred_gso) | ||
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| grads_gso = jax.vmap(jax.grad(norm_gso),(0, 0))(predmats[:,:,:2].reshape(N,6), rotmats) | ||
| grads_svd = jax.vmap(jax.grad(norm_svd),(0, 0))(predmats.reshape(N,9), rotmats) | ||
| gradnorm_gso = jnp.linalg.norm(grads_gso, axis=-1) | ||
| gradnorm_svd = jnp.linalg.norm(grads_svd, axis=-1) | ||
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| gradnorm1_gso = jnp.linalg.norm(grads_gso[:, 0:3], axis=-1) | ||
| gradnorm2_gso = jnp.linalg.norm(grads_gso[:, 3:6], axis=-1) | ||
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| gradnorm1_svd = jnp.linalg.norm(grads_svd[:, 0:3], axis=-1) | ||
| gradnorm2_svd = jnp.linalg.norm(grads_svd[:, 3:6], axis=-1) | ||
| gradnorm3_svd = jnp.linalg.norm(grads_svd[:, 6:9], axis=-1) | ||
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| ratios12_gso = jnp.divide(gradnorm1_gso, gradnorm2_gso) | ||
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| ratios12_svd = jnp.divide(gradnorm1_svd, gradnorm2_svd) | ||
| ratios13_svd = jnp.divide(gradnorm1_svd, gradnorm3_svd) | ||
| ratios23_svd = jnp.divide(gradnorm2_svd, gradnorm3_svd) | ||
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| if plot_frames_with_grads: | ||
| for r0, r_svd, r_gso, g_svd, g_gso in zip(rotmats, pred_svd, pred_gso, grads_svd, grads_gso): | ||
| fig = plt.figure() | ||
| ax = fig.add_subplot(111, projection='3d', proj_type="ortho") | ||
| g_gso = jnp.c_[g_gso.reshape(3,2), np.zeros(3,)] | ||
| r_list = [r0, r_svd, r_gso, -1*g_svd.reshape(3,3), -1*g_gso] | ||
| off_list = [jnp.zeros((3,3)), jnp.zeros((3,3)), jnp.zeros((3,3))] + [r_svd, r_gso] | ||
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| plot_matrices(ax, r_list, | ||
| ["rotmat", "svd", "gso", "grad_svd", "grad_gso"], | ||
| off_list) | ||
| plt.show() | ||
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| ############################################################################### | ||
| # COMPUTE HESSIANS | ||
| ############################################################################### | ||
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| hessmats_gso = jax.vmap(hess_gso, (0, 0))(rotmats, predmats[:,:,:2].reshape(N,6)) | ||
| hessmats_svd = jax.vmap(hess_svd, (0, 0))(rotmats, predmats.reshape(N,9)) | ||
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| eig_gso = jnp.sort(jax.vmap(jnp.linalg.eig)(hessmats_gso)[0], axis=-1) | ||
| eig_svd = jnp.sort(jax.vmap(jnp.linalg.eig)(hessmats_svd)[0], axis=-1) | ||
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| condnums_gso = jnp.divide(jnp.abs(eig_gso[:,-1]), jnp.abs(eig_gso[:,0])) | ||
| condnums_svd = jnp.divide(jnp.abs(eig_svd[:,-1]), jnp.abs(eig_svd[:,0])) | ||
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| ############################################################################### | ||
| # ANALYSE GRADIENTS & HESSIANS | ||
| ############################################################################### | ||
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| df = pd.DataFrame({'loss': np.r_[loss_svd, loss_gso].flatten(), | ||
| 'gradnorm': np.r_[gradnorm_svd, gradnorm_gso].flatten(), | ||
| 'ratios12': np.r_[ratios12_svd, ratios12_gso].flatten(), | ||
| 'ratios13': np.r_[ratios13_svd, [None] * rot_gso.shape[0]].flatten(), | ||
| 'ratios23': np.r_[ratios23_svd, [None] * rot_gso.shape[0]].flatten(), | ||
| 'condnums': np.r_[condnums_svd, condnums_gso].flatten(), | ||
| 'eigmin': np.r_[eig_svd[:,0], eig_gso[:,0]].flatten(), | ||
| 'eigmax': np.r_[eig_svd[:,-1], eig_gso[:,-1]].flatten(), | ||
| }) | ||
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| df['Label'] = ['SVD'] * rot_svd.shape[0] + ['GSO'] * rot_gso.shape[0] | ||
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| def boxplot_labels(labels): | ||
| fig, axs = plt.subplots(1, len(labels), sharey=True) | ||
| for i in range(len(labels)): | ||
| # sns.histplot(data=df, x=labels[i], hue="Label", bins=50, ax=axs[i]) | ||
| sns.boxplot(x="Label", y=labels[i], data=df, ax=axs[i], showfliers=True) | ||
| # sns.violinplot(x="Label", y=labels[i], data=df, ax=axs[i]) | ||
| axs[i].set_xlabel(labels[i]) | ||
| axs[i].set_yscale("log") | ||
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| axs[0].set_ylabel("Count") | ||
| plt.tight_layout() | ||
| plt.show() | ||
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| def plot_2D(labelx, labely, legend=None, plottype="kde"): | ||
| assert len(labelx) == len(labely) == len(legend), \ | ||
| "labelx, labely and legend must have the same length" | ||
| n = len(labelx) | ||
| fig, axs = plt.subplots(2, n, sharex=True, sharey=True) | ||
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| labels = ["SVD", "GSO"] | ||
| if plottype == "scatter" and legend is not None: | ||
| for i in range(2): | ||
| for j in range(n): | ||
| idx = i*n+j | ||
| if idx < 4: | ||
| points = axs[i, j].scatter(df[labelx[j]][df['Label'] == labels[i]], | ||
| df[labely[j]][df['Label'] == labels[i]], | ||
| c=df[legend[j]][df['Label'] == labels[i]], | ||
| s=20, cmap="Spectral_r", | ||
| norm=matcolors.LogNorm()) # set style options | ||
| axs[i, j].set_xscale('log') | ||
| axs[i, j].set_yscale('log') | ||
| axs[i, j].set_xlabel(labelx[j]) | ||
| axs[i, j].set_ylabel(labely[j]) | ||
| axs[i, j].set_title(f"{labels[i]}") | ||
| plt.colorbar(points, label=legend[j]) | ||
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| elif plottype == "kde": | ||
| for i in range(2): | ||
| for j in range(n): | ||
| idx = i * n + j | ||
| if idx < 4: | ||
| sns.kdeplot(x=df[labelx[j]][df['Label'] == labels[i]], | ||
| y=df[labely[j]][df['Label'] == labels[i]], | ||
| #norm=matcolors.LogNorm(), | ||
| ax=axs[i, j], | ||
| cmap="Spectral_r", #cmap="Reds",'Greens',# | ||
| fill=True, | ||
| levels=30, | ||
| log_scale=(False, True), | ||
| cbar=True, | ||
| clip=((None, None), (None, None)), | ||
| ) | ||
| axs[i, j].set_xlabel(labelx[j]) | ||
| axs[i, j].set_ylabel(labely[j]) | ||
| axs[i, j].set_title(f"{labels[i]}") | ||
| axs[i, j].grid(axis='y', linestyle='--') | ||
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| else: | ||
| axs[i, j].axis('off') | ||
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| plt.show() | ||
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| def plot_2D_paper(): | ||
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| labels = ["GSO", "SVD", "SVD", "SVD"] | ||
| dataname = ["ratios12", "ratios12", "ratios13", "ratios23"] | ||
| datalabels = [r"$\|\nabla_{v_1}L\| / \|\nabla_{v_2}L\|$", | ||
| r"$\|\nabla_{m_1}L\| / \|\nabla_{m_2}L\|$", | ||
| r"$\|\nabla_{m_1}L\| / \|\nabla_{m_3}L\|$", | ||
| r"$\|\nabla_{m_2}L\| / \|\nabla_{m_3}L\|$"] | ||
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| n = len(labels) | ||
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| fig, axs = plt.subplots(1, n, sharey=True) | ||
| axs = axs.ravel() | ||
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| for i in range(n): | ||
| cnt = sns.kdeplot(x=df['loss'][df['Label'] == labels[i]], | ||
| y=df[dataname[i]][df['Label'] == labels[i]], | ||
| #norm=matcolors.LogNorm(), | ||
| ax=axs[i], | ||
| cmap='coolwarm', #'Greens',#"Spectral_r", #cmap="Reds", | ||
| fill=True, | ||
| levels=50, | ||
| log_scale=(False, True), | ||
| #cbar=True, | ||
| clip=((None, None), (None, None)), | ||
| antialiased=True | ||
| ) | ||
| for c in cnt.collections: | ||
| c.set_edgecolor("face") | ||
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| axs[i].set_ylim(0.08, 20) | ||
| axs[i].set_xlabel('L2 loss') | ||
| axs[i].set_ylabel(None) | ||
| axs[i].set_title(datalabels[i]) | ||
| plt.text(.02, .98, labels[i], | ||
| ha='left', va='top', | ||
| fontsize=14, color='black', | ||
| transform=axs[i].transAxes) | ||
| axs[i].grid(axis='y', linestyle='--') | ||
| if i == 0: | ||
| axs[i].set_facecolor("gray") | ||
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| plt.show() | ||
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| if plot_ratios: | ||
| plot_2D_paper() | ||
| #plot_2D(["loss", "loss", "loss"], | ||
| # ["ratios12", "ratios13", "ratios23"], | ||
| # ["condnums", "condnums", "condnums"], | ||
| # plottype="kde") | ||
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| if plot_grads: | ||
| boxplot_labels(["ratios12", "ratios13", "ratios23"]) | ||
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| if plot_condnums: | ||
| boxplot_labels(["condnums", "eigmax"]) |