Fast Laplacian Eigenmaps for dimensionality reduction.

# Fast Laplacian Eigenmaps in python

Open-source Laplacian Eigenmaps for dimensionality reduction of large data in python. Comes with an wrapper for NMSlib to compute approximate-nearest-neighbors. Performs several times faster than the default scikit-learn implementation.

# Installation

You'll need NMSlib for using this package properly. Installing it with no binaries is recommended if your CPU supports advanced instructions (it problably does):

pip3 install --no-binary :all: nmslib


Along with requirements:

pip3 install numpy pandas scipy scikit-learn


Then you can install this package with pip:

pip3 install fastlapmap


# Usage

See the following example with the handwritten digits data. Here, I visually compare results from the scikit-learn Laplacian Eigenmaps implementation to those from my implementation.

Note that this implementation contains two similarity-learning algorithms: anisotropic diffusion maps and fuzzy simplicial sets.

# Import libraries
import numpy as np
import matplotlib.pyplot as plt
from sklearn.manifold import SpectralEmbedding
from fastlapmap import LapEigenmap

data = digits.data

# Define hyperparameters
N_EIGS=2
N_NEIGHBORS=10
N_JOBS=10

sk_se = SpectralEmbedding(n_components=N_EIGS, n_neighbors=N_NEIGHBORS, n_jobs=N_JOBS).fit_transform(data)

flapmap_diff = LapEigenmap(data, n_eigs=2, similarity='diffusion', norm_laplacian=True, k=N_NEIGHBORS, n_jobs=N_JOBS)
flapmap_fuzzy = LapEigenmap(data, n_eigs=2, similarity='fuzzy', norm_laplacian=True, k=N_NEIGHBORS, n_jobs=N_JOBS)

fig, (ax1, ax2, ax3) = plt.subplots(1, 3)
fig.suptitle('Handwritten digits data:', fontsize=24)
ax1.scatter(sk_se[:, 0], sk_se[:, 1], c=digits.target, cmap='Spectral', s=5)
ax1.set_title('Sklearn\'s Laplacian Eigenmaps', fontsize=20)
ax2.scatter(flapmap_diff[:, 0], flapmap_diff[:, 1], c=digits.target, cmap='Spectral', s=5)
ax2.set_title('Fast Laplacian Eigenmaps with diffusion harmonics', fontsize=20)
ax3.scatter(flapmap_fuzzy[:, 0], flapmap_fuzzy[:, 1], c=digits.target, cmap='Spectral', s=5)
ax3.set_title('Fast Laplacian Eigenmaps with fuzzy simplicial sets', fontsize=20)
plt.show()


As we can see, results are nearly identical.

# Parameters

data : numpy.ndarray, pandas.DataFrame or scipy.sparse.csr_matrix Input data. By default will use nmslib for approximate nearest-neighbors, which works both on numpy arrays and sparse matrices (faster and cheaper option). Alternatively, users can provide a precomputed affinity matrix by stating metric='precomputed'.

n_eigs : int (optional, default 10). Number of eigenvectors to decompose the graph Laplacian into.

k : int (optional, default 10). Number of k-nearest-neighbors to use when computing affinities.

metric : str (optional, default 'euclidean'). which metric to use when computing neighborhood distances. Defaults to 'euclidean'. Accepted metrics include: -'sqeuclidean' -'euclidean' -'l1' -'lp' - requires setting the parameter p - equivalent to minkowski distance -'cosine' -'angular' -'negdotprod' -'levenshtein' -'hamming' -'jaccard' -'jansen-shan'

M : int (optional, default 10). defines the maximum number of neighbors in the zero and above-zero layers during HSNW (Hierarchical Navigable Small World Graph). However, the actual default maximum number of neighbors for the zero layer is 2*M. A reasonable range for this parameter is 5-100. For more information on HSNW, please check https://arxiv.org/abs/1603.09320. HSNW is implemented in python via NMSlib. Please check more about NMSlib at https://github.com/nmslib/nmslib.

efC : int (optional, default 20). A 'hnsw' parameter. Increasing this value improves the quality of a constructed graph and leads to higher accuracy of search. However this also leads to longer indexing times. A reasonable range for this parameter is 10-500.

efS : int (optional, default 100). A 'hnsw' parameter. Similarly to efC, increasing this value improves recall at the expense of longer retrieval time. A reasonable range for this parameter is 10-500.

n_jobs : int (optional, default 1) How many threads to use in approximate-nearest-neighbors computation.

similarity : str (optional, default 'diffusion'). Which algorithm to use for similarity learning. Options are diffusion harmonics ('diffusion') , fuzzy simplicial sets ('fuzzy') and continuous k-nearest-neighbors ('cknn').

norm_laplacian : bool (optional, default True). Whether to renormalize the graph Laplacian.

return_evals : bool (optional, default False). Whether to also return the eigenvalues in a tuple of eigenvectors, eigenvalues. Defaults to False.

verbose : bool (optional, default False). Whether to report information on the current progress of the algorithm.

# Benchmark

See the runtime comparison between this implementation and scikit-learn:

## Load benchmark function:
from fastlapmap.benchmark import runtime_benchmark

data = digits.data

# Define hyperparameters
N_EIGS = 2
N_NEIGHBORS = 10
N_JOBS = 10
SIZES = [1000, 5000, 10000, 25000, 50000, 100000]
N_RUNS = 3

runtime_benchmark(data,
n_eigs=N_EIGS,
n_neighbors=N_NEIGHBORS,
n_jobs=N_JOBS,
sizes=SIZES,
n_runs=N_RUNS)


As you can see, the diffusion harmoics model is the fastest, followed closely by fuzzy simplicial sets. Both outperform scikit-learn default implementation and escalate linearly with sample size.

## Project details

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