t-SNE 與 UMAP：漂亮的群集圖能證明什麼？

p(j|i) = exp(-||x_i-x_j||² / (2 sigma_i²))
         / sum_{k != i} exp(-||x_i-x_k||² / (2 sigma_i²))

p_ij = (p(j|i) + p(i|j)) / (2n),  i != j

H_i = -sum_j p(j|i) log p(j|i)
perplexity_i = exp(H_i)

q_ij = (1 + ||y_i-y_j||²)^(-1)
       / sum_{k != l} (1 + ||y_k-y_l||²)^(-1)

loss = KL(P || Q) = sum_{i != j} p_ij log(p_ij / q_ij)

w(i -> j) = exp(-max(0, d(i,j) - rho_i) / sigma_i)

q_ij = 1 / (1 + a ||y_i-y_j||^(2b))
loss = -sum_{i<j} [w_ij log(q_ij) + (1-w_ij) log(1-q_ij)]

numpy==1.26.3
scipy==1.14.0
scikit-learn==1.5.1
umap-learn==0.5.6
pynndescent==0.5.13
numba==0.60.0
llvmlite==0.43.0
joblib==1.4.2
threadpoolctl==3.5.0
tqdm==4.66.4

python3 -m venv .venv
source .venv/bin/activate
# 將上方版本清單另存為requirements.txt
python -m pip install -r requirements.txt
# 將下方程式另存為neighborhood_embeddings.py
python neighborhood_embeddings.py

import numpy as np
import sklearn
import numba
import pynndescent
import umap
from sklearn.datasets import load_digits
from sklearn.decomposition import PCA
from sklearn.manifold import TSNE, trustworthiness
from threadpoolctl import threadpool_limits

# Descriptive exploration of a fixed400-row sample, not a held-out prediction.
# Digit labels are not used for sampling, PCA or either embedding.
X, _ = load_digits(return_X_y=True)
indices = np.sort(np.random.default_rng(42).choice(len(X), 400, replace=False))
with threadpool_limits(limits=1):
    base = PCA(n_components=20, svd_solver="full").fit_transform(X[indices])
    tsne_options = dict(n_components=2, perplexity=30, max_iter=500,
                        init="pca", learning_rate="auto",
                        random_state=42, n_jobs=1)
    umap_options = dict(n_components=2, n_neighbors=15, min_dist=0.1,
                        metric="euclidean", random_state=42,
                        n_jobs=1, n_epochs=300)
    t1 = TSNE(**tsne_options).fit_transform(base)
    t2 = TSNE(**tsne_options).fit_transform(base)
    u1 = umap.UMAP(**umap_options).fit_transform(base)
    u2 = umap.UMAP(**umap_options).fit_transform(base)
    print(f"numpy={np.__version__} sklearn={sklearn.__version__} "
          f"umap={umap.__version__} pynndescent={pynndescent.__version__} "
          f"numba={numba.__version__}")
    print(f"exploration rows={len(base)} original_features=64 PCA_features=20")
    for name, embedding, repeated in [("TSNE", t1, t2), ("UMAP", u1, u2)]:
        print(f"{name} shape={embedding.shape} finite={np.isfinite(embedding).all()} "
              f"same_environment_repeat_equal={np.array_equal(embedding, repeated)}")
        print(f"{name} trustworthiness_k10_in_PCA_space="
              f"{trustworthiness(base, embedding, n_neighbors=10):.6f}")

uniform = np.full(3, 1 / 3)
entropy = -np.sum(uniform * np.log(uniform))
print(f"three_uniform_neighbors perplexity={np.exp(entropy):.6f}")
a, b = 0.6, 0.5
print(f"fuzzy_union_0.6_0.5={a + b - a * b:.6f}")

numpy=1.26.3 sklearn=1.5.1 umap=0.5.6 pynndescent=0.5.13 numba=0.60.0
exploration rows=400 original_features=64 PCA_features=20
TSNE shape=(400, 2) finite=True same_environment_repeat_equal=True
TSNE trustworthiness_k10_in_PCA_space=0.985141
UMAP shape=(400, 2) finite=True same_environment_repeat_equal=True
UMAP trustworthiness_k10_in_PCA_space=0.980451
three_uniform_neighbors perplexity=3.000000
fuzzy_union_0.6_0.5=0.800000

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  contributors may be used to endorse or promote products derived from
  this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.