Neighbourhood Components Analysis

A PyTorch implementation of Neighbourhood Components Analysis by J. Goldberger, G. Hinton, S. Roweis, R. Salakhutdinov.

NCA learns a linear transformation of the dataset such that the expected leave-one-out performance of kNN in the transformed space is maximized.


# instantiate nca object and initialize with
# an identity matrix
nca = NCA(dim=2, init="identity")

# fit an nca model to a dataset
# normalize the input data before
# running the optimization
nca.train(X, y, batch_size=64, normalize=True)

# apply the learned linear map to the data
X_nca = nca(X)

Dimensionality Reduction

We generate a 3-D dataset where the first 2 dimensions are concentric rings and the third dimension is Gaussian noise. We plot the result of PCA, LDA and NCA with 2 components.

Notice how PCA has failed to project out the noise, a result of a high noise variance in the third dimension. LDA also struggles to recover the concentric pattern since the classes themselves are not linearly separable.


We compute the classification error, computation time and storage cost of two algorithms:

  • kNN (k = 5) on the raw 784 dimensional MNIST dataset
  • kNN (k = 5) on a learned 32 dimensional NCA projection of the MNIST dataset
Method NCA + kNN Raw kNN
Time 2.37s 155.25s
Storage 6.40 Mb 156.8 Mb
Error 3.3% 2.8%