Four methods to implementing NCA which is often used for metric learning and dimension reduction.

• Methods:
  • Two-layer for loop
  • Matrix operation acceleration
  • Gradient expression
  • SciPy.optimize implementation

• nca_naive.py Implementation using two nested for loops to calculate the gradient but very slow.
• nca_matrix.py Uses matrix operations for acceleration, but still has one for loop; slightly faster, but consumes more memory.
• nca_fast.py Uses another form of gradient representation, full matrix operation, no for loop; very fast, less memory footprint.
• nca_scipy.py Uses nca_fast.py method + scipy.optimize optimization package; includes gradient descent and coordinate descent.
• example.py Uses the MNIST dataset for testing.
• usage.py Demonstrates how to use it; since all four implementations are encapsulated into the NCA class and implement the fit, fit_transform, and transform methods similar to PCA, it is very easy to use.

• The first 9 images are the results on MNIST, with the number of classes selected from 2 to 9. Since the original image has 784 dimensions, PCA is first used to reduce the dimensionality to 100 dimensions, and then NCA is used to reduce the dimensionality to 2 dimensions.

• The next 3 images are the results obtained by directly using NCA on the digits (provided by NumPy), breast_cancer and iris datasets to reduce the dimensionality to 2 dimensions.