How to do feature selection for clustering and implement it in python?
Often people confuse between unsupervised feature selection (UFS) and dimensionality reduction (DR) algorithms as the same. For instance, a famous DR algorithm is Principal Component Analysis (PCA) which is often confused as a UFS method! Researchers have suggested that PCA is a feature extraction algorithm and not feature selection because it transforms the original feature set into a subset of interrelated transformed features, which are difficult to emulate (Abdi &amp; Williams, 2010).
An UFS approach present in literature is Principal Feature Analysis PFA. The way it works is given as;
Steps:

Compute the sample covariance matrix or correlation matrix,
Compute the Principal components and eigenvalues of the Covariance or Correlation matrix A.
Choose the subspace dimension n, we get new matrix A_n, the vectors Vi are the rows of A_n.
Cluster the vectors |Vi|, using K-Means
For each cluster, find the corresponding vector Vi which is closest to the mean of the cluster.

A possible python implementation of PFA is given below;
from sklearn.decomposition import PCA
from sklearn.cluster import KMeans
from collections import defaultdict
from sklearn.metrics.pairwise import euclidean_distances
from sklearn.preprocessing import StandardScaler
import pandas as pd
# create some dummy data
df = pd.DataFrame({'num_legs': [2, 4, 8, 0],
                   'num_wings': [2, 0, 0, 0],
                   'num_specimen_seen': [10, 2, 1, 8]},
                  index=['falcon', 'dog', 'spider', 'fish'])
print(df)
class PFA(object):
    def __init__(self, n_features, q=None):
        self.q = q
        self.n_features = n_features

    def fit(self, X):
        if not self.q:
            self.q = X.shape[1]

        sc = StandardScaler()
        X = sc.fit_transform(X)

        pca = PCA(n_components=self.q).fit(X) # calculation Cov matrix is embeded in PCA
        A_q = pca.components_.T

        kmeans = KMeans(n_clusters=self.n_features).fit(A_q)
        clusters = kmeans.predict(A_q)
        cluster_centers = kmeans.cluster_centers_

        dists = defaultdict(list)
        for i, c in enumerate(clusters):
            dist = euclidean_distances([A_q[i, :]], [cluster_centers[c, :]])[0][0]
            dists[c].append((i, dist))

        self.indices_ = [sorted(f, key=lambda x: x[1])[0][0] for f in dists.values()]
        self.features_ = X[:, self.indices_]
        
# Usage
pfa = PFA(n_features=3)
pfa.fit(df)
# To get the transformed matrix
x = pfa.features_
print(x)
# To get the column indices of the kept features
column_indices = pfa.indices_

Results
        num_legs  num_wings  num_specimen_seen
falcon         2          2                 10
dog            4          0                  2
spider         8          0                  1
fish           0          0                  8
[[-0.50709255  1.73205081  1.23942334]
 [ 0.16903085 -0.57735027 -0.84802649]
 [ 1.52127766 -0.57735027 -1.10895772]
 [-1.18321596 -0.57735027  0.71756088]]

Reference
Abdi, H., &amp; Williams, L. J. (2010). Principal component analysis. Wiley interdisciplinary reviews: computational statistics, 2(4), 433-459.