Comparing Embeddings vs. RMetric Stability

In [1]:

import numpy as np
seeds = [20251106, 20251108]

import numpy as np seeds = [20251106, 20251108]

In [2]:

import pandas as pd

N = 1500
K = 2
noise = 0.0

X_noisy = pd.read_csv(f"./baselines/data/swiss_noise_{noise}.csv").values
t = X_noisy[:, 3]
X_noisy = X_noisy[:, :3]
X = X_noisy

import pandas as pd N = 1500 K = 2 noise = 0.0 X_noisy = pd.read_csv(f"./baselines/data/swiss_noise_{noise}.csv").values t = X_noisy[:, 3] X_noisy = X_noisy[:, :3] X = X_noisy

Data Generation¶

Here's a plot of the swiss roll data.

In [3]:

import matplotlib.pyplot as plt
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

sc = ax.scatter(X_noisy[:, 0], X_noisy[:, 1], X_noisy[:, 2], c=t, s=20, alpha=0.7, cmap='viridis')
plt.colorbar(sc, ax=ax, label='t')
plt.show()

import matplotlib.pyplot as plt fig = plt.figure(figsize=(10, 8)) ax = fig.add_subplot(111, projection='3d') sc = ax.scatter(X_noisy[:, 0], X_noisy[:, 1], X_noisy[:, 2], c=t, s=20, alpha=0.7, cmap='viridis') plt.colorbar(sc, ax=ax, label='t') plt.show()

In [4]:

import altair as alt
import pandas as pd

def plot_swiss_emb(X_emb, t):
    df_emb = pd.DataFrame(X_emb, columns=['x', 'y'])
    df_emb["t"] = t
    return alt.Chart(df_emb).mark_circle(size=60).encode(
        x='x',
        y='y',
        color='t'
    ).properties(width=400, height=300)

import altair as alt import pandas as pd def plot_swiss_emb(X_emb, t): df_emb = pd.DataFrame(X_emb, columns=['x', 'y']) df_emb["t"] = t return alt.Chart(df_emb).mark_circle(size=60).encode( x='x', y='y', color='t' ).properties(width=400, height=300)

$t$-SNE Distortions¶

In [5]:

from sklearn.manifold import TSNE
tsne = TSNE(n_components=2, perplexity=100, random_state=seeds[0], learning_rate='auto') # set to init='random' for more variation
Z1 = tsne.fit_transform(X_noisy)

tsne = TSNE(n_components=2, perplexity=100, random_state=seeds[1], learning_rate='auto')
Z2 = tsne.fit_transform(X_noisy)

from sklearn.manifold import TSNE tsne = TSNE(n_components=2, perplexity=100, random_state=seeds[0], learning_rate='auto') # set to init='random' for more variation Z1 = tsne.fit_transform(X_noisy) tsne = TSNE(n_components=2, perplexity=100, random_state=seeds[1], learning_rate='auto') Z2 = tsne.fit_transform(X_noisy)

Apply procrustes to align the two embeddings.

In [6]:

from scipy.linalg import orthogonal_procrustes

Z1 = Z1 - Z1.mean(axis=0)
Z2 = Z2 - Z2.mean(axis=0)
R, scale = orthogonal_procrustes(Z2, Z1)
Z2 = Z2 @ R

from scipy.linalg import orthogonal_procrustes Z1 = Z1 - Z1.mean(axis=0) Z2 = Z2 - Z2.mean(axis=0) R, scale = orthogonal_procrustes(Z2, Z1) Z2 = Z2 @ R

In [7]:

plots = [
    plot_swiss_emb(Z1, t),
    plot_swiss_emb(Z2, t)
]

[display(p) for p in plots]

plots = [ plot_swiss_emb(Z1, t), plot_swiss_emb(Z2, t) ] [display(p) for p in plots]

Out[7]:

[None, None]

In [8]:

from distortions.geometry import Geometry, bind_metric, local_distortions, neighborhoods
from distortions.visualization import dplot
from anndata import AnnData
from sklearn.neighbors import NearestNeighbors

def distortion_plot(Z, X, t, n_neighbors=40, geom_radius=1, threshold=0.1, outlier_factor=2):
    adata = AnnData(X=X)
    nn = NearestNeighbors(n_neighbors=n_neighbors, metric="euclidean").fit(X)
    knn_graph = nn.kneighbors_graph(X, mode="distance")
    adata.obsp["distances"] = knn_graph
    adata.obsm["X_tsne"] = Z

    geom_radius = 0.5 * np.mean(adata.obsp["distances"].data)
    geom = Geometry(affinity_kwds={"radius": geom_radius}, adjacency_kwds={"n_neighbors": n_neighbors})
    H, Hvv, Hs = local_distortions(Z, X, geom)
    hq = np.quantile(Hs, 0.99)
    Hs[Hs > hq] = hq
    embedding = bind_metric(Z, Hvv, Hs)
    embedding["t"] = t

    N = neighborhoods(adata, threshold=threshold, outlier_factor=outlier_factor, embed_key="X_tsne")

    pal = ['#B776A6', '#BAC4A2']
    plot = dplot(embedding, height=350, width=450)\
        .mapping(x="embedding_0", y="embedding_1", color="t")\
        .inter_edge_link(N=N, strokeWidth=.2, opacity=0.9, threshold=10, stroke="#363E59", highlightColor="#363E59", backgroundOpacity=0.6)\
        .geom_ellipse(radiusMin=1, radiusMax=10)\
        .labs(x='t-SNE 1', y='t-SNE 2')\
        .scale_color(scheme=pal)

    return plot, H, embedding, N

from distortions.geometry import Geometry, bind_metric, local_distortions, neighborhoods from distortions.visualization import dplot from anndata import AnnData from sklearn.neighbors import NearestNeighbors def distortion_plot(Z, X, t, n_neighbors=40, geom_radius=1, threshold=0.1, outlier_factor=2): adata = AnnData(X=X) nn = NearestNeighbors(n_neighbors=n_neighbors, metric="euclidean").fit(X) knn_graph = nn.kneighbors_graph(X, mode="distance") adata.obsp["distances"] = knn_graph adata.obsm["X_tsne"] = Z geom_radius = 0.5 * np.mean(adata.obsp["distances"].data) geom = Geometry(affinity_kwds={"radius": geom_radius}, adjacency_kwds={"n_neighbors": n_neighbors}) H, Hvv, Hs = local_distortions(Z, X, geom) hq = np.quantile(Hs, 0.99) Hs[Hs > hq] = hq embedding = bind_metric(Z, Hvv, Hs) embedding["t"] = t N = neighborhoods(adata, threshold=threshold, outlier_factor=outlier_factor, embed_key="X_tsne") pal = ['#B776A6', '#BAC4A2'] plot = dplot(embedding, height=350, width=450)\ .mapping(x="embedding_0", y="embedding_1", color="t")\ .inter_edge_link(N=N, strokeWidth=.2, opacity=0.9, threshold=10, stroke="#363E59", highlightColor="#363E59", backgroundOpacity=0.6)\ .geom_ellipse(radiusMin=1, radiusMax=10)\ .labs(x='t-SNE 1', y='t-SNE 2')\ .scale_color(scheme=pal) return plot, H, embedding, N

In [9]:

import time
start_time = time.time()
display(distortion_plot(Z1, X_noisy, t)[0])
elapsed = time.time() - start_time
with open(f"baselines/data/runtime_dist_{noise}.txt", "w") as f:
    f.write(str(elapsed))

import time start_time = time.time() display(distortion_plot(Z1, X_noisy, t)[0]) elapsed = time.time() - start_time with open(f"baselines/data/runtime_dist_{noise}.txt", "w") as f: f.write(str(elapsed))

dplot(dataset=[{'embedding_0': -17.553518295288086, 'embedding_1': -4.967970371246338, 'x0': -0.97378945252753…

In [10]:

distortion_data = [
    distortion_plot(Z1, X_noisy, t),
    distortion_plot(Z2, X_noisy, t)
]

[display(p[0]) for p in distortion_data]

distortion_data = [ distortion_plot(Z1, X_noisy, t), distortion_plot(Z2, X_noisy, t) ] [display(p[0]) for p in distortion_data]

dplot(dataset=[{'embedding_0': -17.553518295288086, 'embedding_1': -4.967970371246338, 'x0': -0.97378945252753…

dplot(dataset=[{'embedding_0': -17.55352020263672, 'embedding_1': -4.967971324920654, 'x0': -0.973802412845542…

Out[10]:

[None, None]

In [11]:

save_dir = "/Users/krissankaran/Desktop/collaborations/distortions-project/distortions-dev/paper/figures/"
#distortion_data[0][0].save(f"{save_dir}/swiss_roll_baseline_no_interact_{noise}.svg")
#distortion_data[0][0].save(f"{save_dir}/swiss_roll_baseline_interact_{noise}.svg")

save_dir = "/Users/krissankaran/Desktop/collaborations/distortions-project/distortions-dev/paper/figures/" #distortion_data[0][0].save(f"{save_dir}/swiss_roll_baseline_no_interact_{noise}.svg") #distortion_data[0][0].save(f"{save_dir}/swiss_roll_baseline_interact_{noise}.svg")

Neighbor Distance Preservation¶

In [12]:

from scipy.spatial.distance import cdist
from sklearn.neighbors import NearestNeighbors

# Compute pairwise distances in embedding spaces, then the ratio matrix
D1 = cdist(Z1, Z1)
D2 = cdist(Z2, Z2)
R = D1 / D2
R_inv = D2 / D1

# Compute nearest neighbors in original space X
n_neighbors = 15
nn = NearestNeighbors(n_neighbors=n_neighbors, metric='euclidean').fit(X)
knn_indices = nn.kneighbors(X, return_distance=False)

# Build mask M: M[i, j] = 1 if j is among i's nearest neighbors (excluding self)
n = X.shape[0]
M = np.zeros((n, n), dtype=int)
for i in range(n):
    for j in knn_indices[i][1:]:  # skip self (first neighbor)
        M[i, j] = 1

# For each row, compute variance of R[i, j] over j where M[i, j] == 1
# Compute analogous V_inv for R_inv.
V = np.array([np.var(R[i][M[i]==1]) for i in range(n)])
V_inv = np.array([np.var(R_inv[i][M[i]==1]) for i in range(n)])
V_max = np.maximum(V, V_inv)

from scipy.spatial.distance import cdist from sklearn.neighbors import NearestNeighbors # Compute pairwise distances in embedding spaces, then the ratio matrix D1 = cdist(Z1, Z1) D2 = cdist(Z2, Z2) R = D1 / D2 R_inv = D2 / D1 # Compute nearest neighbors in original space X n_neighbors = 15 nn = NearestNeighbors(n_neighbors=n_neighbors, metric='euclidean').fit(X) knn_indices = nn.kneighbors(X, return_distance=False) # Build mask M: M[i, j] = 1 if j is among i's nearest neighbors (excluding self) n = X.shape[0] M = np.zeros((n, n), dtype=int) for i in range(n): for j in knn_indices[i][1:]: # skip self (first neighbor) M[i, j] = 1 # For each row, compute variance of R[i, j] over j where M[i, j] == 1 # Compute analogous V_inv for R_inv. V = np.array([np.var(R[i][M[i]==1]) for i in range(n)]) V_inv = np.array([np.var(R_inv[i][M[i]==1]) for i in range(n)]) V_max = np.maximum(V, V_inv)

In [13]:

alt.data_transformers.enable("vegafusion")

# Flatten R to a 1D array and create a DataFrame for plotting
R_flat = R.flatten()
R_df = pd.DataFrame({'R': np.log10(R_flat)})

alt.Chart(R_df).mark_bar().encode(
    alt.X('R', bin=alt.Bin(maxbins=100), title='Distance Ratio R (log_{10})'),
    alt.Y('count()', title='Frequency')
).properties(
    width=400, height=250,
    title='Histogram of Distance Ratios log_{10}(R)'
)

alt.data_transformers.enable("vegafusion") # Flatten R to a 1D array and create a DataFrame for plotting R_flat = R.flatten() R_df = pd.DataFrame({'R': np.log10(R_flat)}) alt.Chart(R_df).mark_bar().encode( alt.X('R', bin=alt.Bin(maxbins=100), title='Distance Ratio R (log_{10})'), alt.Y('count()', title='Frequency') ).properties( width=400, height=250, title='Histogram of Distance Ratios log_{10}(R)' )

Out[13]:

In [14]:

from scipy.linalg import fractional_matrix_power, logm

Hs1 = distortion_data[0][1]  # shape: (n, 2, 2)
Hs2 = distortion_data[1][1]
norm = 'fro'
H_instability = np.linalg.norm(Hs1 - Hs2, ord=norm, axis=(1, 2))
Hs1_norm = np.linalg.norm(Hs1, ord=norm, axis=(1, 2))
Hs2_norm = np.linalg.norm(Hs2, ord=norm, axis=(1, 2))

stability_data = pd.DataFrame({
    "v_d": V_max,
    "n_H": H_instability / (Hs1_norm * Hs2_norm),
    "t": t
})

# Compute n_H_det: |det(Hs1[i]^{-1} @ Hs2[i])| for each i
n = Hs1.shape[0]
n_H_det = np.empty(n)
for i in range(n):
    n_H_det[i] = np.abs(np.log(np.linalg.det(Hs1[i])) - np.log(np.linalg.det(Hs2[i])))
stability_data["n_H_det"] = n_H_det

n_H_sim = np.empty(n)
for i in range(n):
    H1_inv_sqrt = fractional_matrix_power(Hs1[i], -0.5)
    sim = H1_inv_sqrt @ Hs2[i] @ H1_inv_sqrt
    sim_log = logm(sim)
    n_H_sim[i] = 0.5 * np.linalg.norm(sim_log, ord='fro')
stability_data["n_H_sim"] = n_H_sim

from scipy.linalg import fractional_matrix_power, logm Hs1 = distortion_data[0][1] # shape: (n, 2, 2) Hs2 = distortion_data[1][1] norm = 'fro' H_instability = np.linalg.norm(Hs1 - Hs2, ord=norm, axis=(1, 2)) Hs1_norm = np.linalg.norm(Hs1, ord=norm, axis=(1, 2)) Hs2_norm = np.linalg.norm(Hs2, ord=norm, axis=(1, 2)) stability_data = pd.DataFrame({ "v_d": V_max, "n_H": H_instability / (Hs1_norm * Hs2_norm), "t": t }) # Compute n_H_det: |det(Hs1[i]^{-1} @ Hs2[i])| for each i n = Hs1.shape[0] n_H_det = np.empty(n) for i in range(n): n_H_det[i] = np.abs(np.log(np.linalg.det(Hs1[i])) - np.log(np.linalg.det(Hs2[i]))) stability_data["n_H_det"] = n_H_det n_H_sim = np.empty(n) for i in range(n): H1_inv_sqrt = fractional_matrix_power(Hs1[i], -0.5) sim = H1_inv_sqrt @ Hs2[i] @ H1_inv_sqrt sim_log = logm(sim) n_H_sim[i] = 0.5 * np.linalg.norm(sim_log, ord='fro') stability_data["n_H_sim"] = n_H_sim

In [15]:

n = Hs1.shape[0]
HH_sv = np.empty(n)
for i in range(n):
    U, s, Vh = np.linalg.svd(np.linalg.inv(Hs1[i]) @ Hs2[i])
    HH_sv[i] = np.max(np.abs(np.sqrt(s) - 1))
stability_data["HH_sv"] = HH_sv

n = Hs1.shape[0] HH_sv = np.empty(n) for i in range(n): U, s, Vh = np.linalg.svd(np.linalg.inv(Hs1[i]) @ Hs2[i]) HH_sv[i] = np.max(np.abs(np.sqrt(s) - 1)) stability_data["HH_sv"] = HH_sv

In [16]:

N_keys_0 = set([k - 1 for k in distortion_data[0][3].keys()])
stability_data["distorted"] = stability_data.index.isin(N_keys_0)

N_keys_0 = set([k - 1 for k in distortion_data[0][3].keys()]) stability_data["distorted"] = stability_data.index.isin(N_keys_0)

In [17]:

embedding_list = [p[2] for p in distortion_data]
groups = ["seed1", "seed2"]
for emb, group in zip(embedding_list, groups):
    emb["sample"] = embedding_list[0].index
    emb["group"] = group
    emb["V"] = np.log(V_max)

embedding_list = [p[2] for p in distortion_data] groups = ["seed1", "seed2"] for emb, group in zip(embedding_list, groups): emb["sample"] = embedding_list[0].index emb["group"] = group emb["V"] = np.log(V_max)

In [18]:

combined_embedding = pd.concat(embedding_list)
plot_var = dplot(combined_embedding, height = 350, width=450)\
    .mapping(x="embedding_0", y="embedding_1", color="V")\
    .geom_ellipse(opacity=0.9, radiusMin=1, radiusMax=20, stroke=True)\
    .scale_color(stroke=True)\
    .labs(x="UMAP1", y="UMAP2")

plot_group = dplot(combined_embedding, height = 350, width=450)\
    .mapping(x="embedding_0", y="embedding_1", color="group")\
    .geom_ellipse(opacity=0.9, radiusMin=1, radiusMax=20, stroke=True)\
    .scale_color(stroke=True, scheme=["green", "purple"])\
    .labs(x="UMAP1", y="UMAP2")

combined_embedding = pd.concat(embedding_list) plot_var = dplot(combined_embedding, height = 350, width=450)\ .mapping(x="embedding_0", y="embedding_1", color="V")\ .geom_ellipse(opacity=0.9, radiusMin=1, radiusMax=20, stroke=True)\ .scale_color(stroke=True)\ .labs(x="UMAP1", y="UMAP2") plot_group = dplot(combined_embedding, height = 350, width=450)\ .mapping(x="embedding_0", y="embedding_1", color="group")\ .geom_ellipse(opacity=0.9, radiusMin=1, radiusMax=20, stroke=True)\ .scale_color(stroke=True, scheme=["green", "purple"])\ .labs(x="UMAP1", y="UMAP2")

In [19]:

plot_var

plot_var

Out[19]:

dplot(dataset=[{'embedding_0': -17.553518295288086, 'embedding_1': -4.967970371246338, 'x0': -0.97378945252753…

In [20]:

plot_group

plot_group

Out[20]:

dplot(dataset=[{'embedding_0': -17.553518295288086, 'embedding_1': -4.967970371246338, 'x0': -0.97378945252753…

In [21]:

#plot_var.save("/Users/krissankaran/Downloads/v3.svg")
plot_group.save("/Users/krissankaran/Downloads/v4.svg")

#plot_var.save("/Users/krissankaran/Downloads/v3.svg") plot_group.save("/Users/krissankaran/Downloads/v4.svg")

In [22]:

def make_stability_scatter(y, y_title=None, y_scale='log', color='distorted', color_title=None, color_range=None, size=20):
    if y_title is None:
        y_title = y
    if color_title is None:
        color_title = color

    # Split data for overlay
    df_black = stability_data[stability_data[color] == False]
    df_red = stability_data[stability_data[color] == True]

    enc = [
        alt.X('v_d', title="SD of $d_i/d'_i$ ratios among neighbors", scale=alt.Scale(type='log')),
        alt.Y(y, title=y_title, scale=alt.Scale(type=y_scale)),
    ]
    scatter_black = alt.Chart(df_black).mark_circle(size=size, color='black').encode(*enc)
    scatter_red = alt.Chart(df_red).mark_circle(size=size, color='red').encode(*enc)

    line_data = pd.DataFrame({
        'v_d': np.linspace(stability_data['v_d'].min(), stability_data['v_d'].max(), 100)
    })
    line_data[y] = line_data['v_d']
    line = alt.Chart(line_data).mark_line(color='gray', strokeDash=[4,4]).encode(
        x='v_d',
        y=y
    )
    return (scatter_black + scatter_red + line).properties(
        width=400, height=300,
        title=f'Neighbor Preservation vs. Distortion Stability: {y_title}'
    )

def make_stability_scatter(y, y_title=None, y_scale='log', color='distorted', color_title=None, color_range=None, size=20): if y_title is None: y_title = y if color_title is None: color_title = color # Split data for overlay df_black = stability_data[stability_data[color] == False] df_red = stability_data[stability_data[color] == True] enc = [ alt.X('v_d', title="SD of $d_i/d'_i$ ratios among neighbors", scale=alt.Scale(type='log')), alt.Y(y, title=y_title, scale=alt.Scale(type=y_scale)), ] scatter_black = alt.Chart(df_black).mark_circle(size=size, color='black').encode(*enc) scatter_red = alt.Chart(df_red).mark_circle(size=size, color='red').encode(*enc) line_data = pd.DataFrame({ 'v_d': np.linspace(stability_data['v_d'].min(), stability_data['v_d'].max(), 100) }) line_data[y] = line_data['v_d'] line = alt.Chart(line_data).mark_line(color='gray', strokeDash=[4,4]).encode( x='v_d', y=y ) return (scatter_black + scatter_red + line).properties( width=400, height=300, title=f'Neighbor Preservation vs. Distortion Stability: {y_title}' )

In [23]:

stability_data

stability_data

Out[23]:

	v_d	n_H	t	n_H_det	n_H_sim	HH_sv	distorted
0	3.574513e-18	0.001657	11.489719	0.000073	0.000073	0.000074	False
1	3.146408e-14	0.000024	6.187302	0.000005	0.000002	0.000002	False
2	1.894028e-14	0.003019	10.729005	0.000256	0.000101	0.000097	False
3	7.504273e-15	0.000091	7.555652	0.000019	0.000010	0.000012	True
4	2.735290e-15	0.000602	8.282046	0.000035	0.000018	0.000022	False
...	...	...	...	...	...	...	...
935	1.630904e-14	0.000056	13.811979	0.000040	0.000021	0.000021	False
936	3.844885e-14	0.000042	13.368451	0.000010	0.000011	0.000010	False
937	7.932017e-14	0.000026	13.001232	0.000010	0.000004	0.000004	False
938	1.314958e-14	0.000181	12.456238	0.000022	0.000016	0.000016	False
939	1.155698e-14	0.000008	13.616978	0.000002	0.000002	0.000002	False

940 rows × 7 columns

In [24]:

# Frobenius norm of log-matrix similarity
plot_n_H_sim = make_stability_scatter(
    y='n_H_sim',
    y_title="(1/2) * ||\\log H^{-1/2} (H') H^{-1/2}||_{F}",
    color='distorted',
    color_range=['black', 'red'],
    size=20
)

# Absolute log-determinant difference
plot_n_H_det = make_stability_scatter(
    y='n_H_det',
    y_title="|\\log|H| - \\log|H'||",
    color='distorted',
    color_range=['black', 'red'],
    size=20
)

plot_n_H = make_stability_scatter(
    y='n_H',
    y_title="normalized ||H - H'||_{F}",
    color='distorted',
    color_range=['black', 'red'],
    size=20
)

# Singular value difference
plot_HH_sv = make_stability_scatter(
    y='HH_sv',
    y_title="\max(|\sqrt{\lambda_1}-1|,|\sqrt{\lambda_2}-1|)",
    color='distorted',
    color_range=['black', 'red'],
    size=20
)

# Frobenius norm of log-matrix similarity plot_n_H_sim = make_stability_scatter( y='n_H_sim', y_title="(1/2) * ||\\log H^{-1/2} (H') H^{-1/2}||_{F}", color='distorted', color_range=['black', 'red'], size=20 ) # Absolute log-determinant difference plot_n_H_det = make_stability_scatter( y='n_H_det', y_title="|\\log|H| - \\log|H'||", color='distorted', color_range=['black', 'red'], size=20 ) plot_n_H = make_stability_scatter( y='n_H', y_title="normalized ||H - H'||_{F}", color='distorted', color_range=['black', 'red'], size=20 ) # Singular value difference plot_HH_sv = make_stability_scatter( y='HH_sv', y_title="\max(|\sqrt{\lambda_1}-1|,|\sqrt{\lambda_2}-1|)", color='distorted', color_range=['black', 'red'], size=20 )

In [25]:

plot_n_H_sim.save("/Users/krissankaran/Downloads/n_H_sim.png")
plot_n_H_det.save("/Users/krissankaran/Downloads/n_H_det.png")
plot_HH_sv.save("/Users/krissankaran/Downloads/HH_sv.png")
plot_n_H.save("/Users/krissankaran/Downloads/n_H.png")

plot_n_H_sim.save("/Users/krissankaran/Downloads/n_H_sim.png") plot_n_H_det.save("/Users/krissankaran/Downloads/n_H_det.png") plot_HH_sv.save("/Users/krissankaran/Downloads/HH_sv.png") plot_n_H.save("/Users/krissankaran/Downloads/n_H.png")

In [26]:

def make_t_scatter(metric, title, scale_type='log', size=40):
    df_plot = stability_data.sort_values("distorted", ascending=True)
    return alt.Chart(df_plot).mark_circle(size=size).encode(
        x=alt.X(
            't',
            title='Swiss Roll Parameter',
            scale=alt.Scale(domain=[df_plot['t'].min(), df_plot['t'].max()])
        ),
        y=alt.Y(metric, scale=alt.Scale(type=scale_type), title = title),
        color=alt.Color('distorted:N', title='Distorted', scale=alt.Scale(domain=[False, True], range=['black', 'red']))
    ).properties(width=400, height=300)

def make_t_scatter(metric, title, scale_type='log', size=40): df_plot = stability_data.sort_values("distorted", ascending=True) return alt.Chart(df_plot).mark_circle(size=size).encode( x=alt.X( 't', title='Swiss Roll Parameter', scale=alt.Scale(domain=[df_plot['t'].min(), df_plot['t'].max()]) ), y=alt.Y(metric, scale=alt.Scale(type=scale_type), title = title), color=alt.Color('distorted:N', title='Distorted', scale=alt.Scale(domain=[False, True], range=['black', 'red'])) ).properties(width=400, height=300)

In [27]:

make_t_scatter('n_H_sim', "(1/2) * ||\\log H^{-1/2} (H') H^{-1/2}||_{F}")

make_t_scatter('n_H_sim', "(1/2) * ||\\log H^{-1/2} (H') H^{-1/2}||_{F}")

Out[27]:

In [28]:

make_t_scatter('HH_sv', "max(|\sqrt{\lambda_1}-1|,|\sqrt{\lambda_2}|)")

make_t_scatter('HH_sv', "max(|\sqrt{\lambda_1}-1|,|\sqrt{\lambda_2}|)")

Out[28]:

In [ ]: