<｜begin▁of▁sentence｜>### Instruction:
Given a numerical dataset with a potentially low-rank structure, how can a machine learning practitioner determine the appropriate number of dimensions to embed the data for downstream tasks, using the profile likelihood method as described by Zhu and Godsie?
### Response:
import numpy as np
from graspologic.embed import select_dimension
from sklearn.datasets import make_low_rank_matrix

# Generate a synthetic low-rank dataset with ambient dimension of 100
# and a rank of 5
n_samples = 200
n_features = 100
rank = 5
X = make_low_rank_matrix(n_samples=n_samples, n_features=n_features, effective_rank=rank, random_state=42)

# Compute the singular values and select the optimal dimension
n_elbows = 2
elbows, sing_vals = select_dimension(X, n_elbows=n_elbows)

# Print the optimal embedding dimensions
print(f'Optimal embedding dimensions: {elbows}')
print(f'Singular values at elbows: {sing_vals}')
<|EOT|>
