Understanding Isometrization with a Known Transformation

In [1]:

import random
import numpy as np
import pandas as pd
random.seed(20250409)

N = 500
u = np.random.uniform(0, 1, (N, 2))
u_df = pd.DataFrame(u, columns=['x', 'y'])

import random import numpy as np import pandas as pd random.seed(20250409) N = 500 u = np.random.uniform(0, 1, (N, 2)) u_df = pd.DataFrame(u, columns=['x', 'y'])

In [2]:

from distortions.geometry import bind_metric
from distortions.visualization import dplot

def plot_transformation(f, H, Hvv, Hs, transformation_bw=2, metric_bw=20, width=450, height=450, radiusMin=1, radiusMax=10):
    combined_df = bind_metric(f, Hvv, Hs)
    combined_df["id"] = combined_df.index

    f_df = pd.DataFrame(f, columns=['x', 'y'])
    embed_dict = {}
    for n in range(len(f_df)):
        embed_dict[n] = f_df.iloc[n, :]
        embed_dict[n]["id"] = n

    h_dict = {k: H[k] for k in range(len(H))}
    return dplot(combined_df, width=width, height=height, labelFontSize=14)\
        .mapping(x="embedding_0", y="embedding_1")\
        .geom_ellipse(radiusMin=radiusMin, radiusMax=radiusMax, opacity=0.8)\
        .inter_isometry(metrics=h_dict, otherClasses=["ellipse"], transformation_bw=transformation_bw, metric_bw=metric_bw, opacity=0.4, stroke="#c6c6c6", strokeWidth=1.5)\
        .labs(x="x-axis", y="y-axis")

from distortions.geometry import bind_metric from distortions.visualization import dplot def plot_transformation(f, H, Hvv, Hs, transformation_bw=2, metric_bw=20, width=450, height=450, radiusMin=1, radiusMax=10): combined_df = bind_metric(f, Hvv, Hs) combined_df["id"] = combined_df.index f_df = pd.DataFrame(f, columns=['x', 'y']) embed_dict = {} for n in range(len(f_df)): embed_dict[n] = f_df.iloc[n, :] embed_dict[n]["id"] = n h_dict = {k: H[k] for k in range(len(H))} return dplot(combined_df, width=width, height=height, labelFontSize=14)\ .mapping(x="embedding_0", y="embedding_1")\ .geom_ellipse(radiusMin=radiusMin, radiusMax=radiusMax, opacity=0.8)\ .inter_isometry(metrics=h_dict, otherClasses=["ellipse"], transformation_bw=transformation_bw, metric_bw=metric_bw, opacity=0.4, stroke="#c6c6c6", strokeWidth=1.5)\ .labs(x="x-axis", y="y-axis")

In [3]:

H = np.zeros((N, 2, 2))
Hvv = np.zeros((N, 2, 2))
Hs = np.zeros((N, 2))

for i in range(N):
    H[i] = np.array([[1, 0], [0, 1]])
    Hvv[i] = np.eye(2)
    Hs[i] = [1, 1]

H = np.zeros((N, 2, 2)) Hvv = np.zeros((N, 2, 2)) Hs = np.zeros((N, 2)) for i in range(N): H[i] = np.array([[1, 0], [0, 1]]) Hvv[i] = np.eye(2) Hs[i] = [1, 1]

In [4]:

plots = {}
plots["original"] = plot_transformation(u, H, Hvv, Hs)

plots = {} plots["original"] = plot_transformation(u, H, Hvv, Hs)

In [5]:

f = u.copy()
f[:, 1] = f[:, 1] ** 2

f = u.copy() f[:, 1] = f[:, 1] ** 2

In [6]:

plots["transformed"] = plot_transformation(f, H, Hvv, Hs)

plots["transformed"] = plot_transformation(f, H, Hvv, Hs)

In [7]:

for i in range(N):
    H[i] = np.array([[1, 0], [0, 2 * f[i, 1]]])
    Hvv[i] = np.eye(2)
    Hs[i] = [1, 2 * f[i, 1]]

plots["distortion"] = plot_transformation(f, H, Hvv, Hs)

for i in range(N): H[i] = np.array([[1, 0], [0, 2 * f[i, 1]]]) Hvv[i] = np.eye(2) Hs[i] = [1, 2 * f[i, 1]] plots["distortion"] = plot_transformation(f, H, Hvv, Hs)

In [8]:

#[p.save(f"../paper/figures/{k}.svg") for k, p in plots.items()]

#[p.save(f"../paper/figures/{k}.svg") for k, p in plots.items()]

In [9]:

[display(p) for p in plots.values()]

[display(p) for p in plots.values()]

dplot(dataset=[{'embedding_0': 0.8448678500972645, 'embedding_1': 0.8949533007524632, 'x0': 1.0, 'y0': 0.0, 'x…

dplot(dataset=[{'embedding_0': 0.8448678500972645, 'embedding_1': 0.8009414105277287, 'x0': 1.0, 'y0': 0.0, 'x…

dplot(dataset=[{'embedding_0': 0.8448678500972645, 'embedding_1': 0.8009414105277287, 'x0': 1.0, 'y0': 0.0, 'x…

Out[9]:

[None, None, None]