Autoencoder能否学习极坐标转换？环形2D数据实验遇困

一组2D数据近似分布在圆形上，存在一个由角度参数化的低维流形可最优描述该数据。尝试了多种模型版本均未成功，实验代码如下：

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import tensorflow as tf

from tensorflow.keras import layers, losses
from tensorflow.keras.models import Model
from tensorflow.keras.optimizers import Adam

class DeepAutoEncoder(Model):
  def __init__(self, dim_data: int, num_hidden: int,
               num_comp: int,
               activation: str = 'linear'):
    super(DeepAutoEncoder, self).__init__()
    self.encoder = tf.keras.Sequential([
      layers.Dense(num_hidden, activation=activation),
      layers.Dense(num_comp, activation=activation),
    ])

    self.decoder = tf.keras.Sequential([
      layers.Dense(num_hidden, activation=activation),
      layers.Dense(dim_data, activation='linear')
    ])

  def call(self, x):
    encoded = self.encoder(x)
    decoded = self.decoder(encoded)
    return decoded



# Data
num_obs = 1000
np.random.seed(1238)
e = np.random.randn(num_obs, 1)
t = np.linspace(0, 2*np.pi, num_obs)
x = 1 * np.cos(t)
y = np.sin(t) + 0.2*e[:, 0]
X = np.column_stack((x, y))

num_comp = 1
activations = ['linear', 'sigmoid']
ae = {a: None for a in activations}
for act in activations:
    ae[act] = DeepAutoEncoder(dim_data=2, num_comp=num_comp,
                              num_hidden=3, activation=act)
    ae[act].compile(optimizer=Adam(learning_rate=0.01),
                    loss='mse')
    ae[act].build(input_shape=(None, 2))
    ae[act].summary()
    history = ae[act].fit(X, X, epochs=200,
                          batch_size=32,
                          shuffle=True)
    ae[act].summary()
    plt.plot(history.history["loss"], label=act)
    plt.legend()

f, axs = plt.subplots(2, 2)
for i, a in enumerate(activations):
    axs[0, i].plot(x, y, '.', c='k')
    z = ae[a].encoder(X)
    # x_ae = ae[a].decoder(ae[a].encoder(X))
    x_ae = ae[a](X)
    axs[0, i].plot(x_ae[:, 0], x_ae[:, 1], '.')
    # axs[0, i].plot(x_pca[:, 0], x_pca[:, 1], '.', c='C3')
    axs[1, i].plot(z)
    axs[0, i].axis('equal')
    axs[0, i].set(title=a)

重构数据效果如图：

推测失败原因是sigmoid(W * z + b)变换与将潜变量映射回原空间所需的非线性矩阵[[cos(theta) sin(theta)], [-sin(theta) sin(theta)]]差距较大，恳请各位提供思路，非常感谢！

内容的提问来源于stack exchange，提问作者deckard