Linear Discriminant Analysis inverse transform

Question

I try to use Linear Discriminant Analysis from scikit-learn library, in order to perform dimensionality reduction on my data which has more than 200 features. But I could no

Answer 1

The inverse of the LDA does not necessarily make sense beause it loses a lot of information.

For comparison, consider the PCA. Here we get a coefficient matrix that is used to transform the data. We can do dimensionality reduction by stripping rows from the matrix. To get the inverse transform, we first invert the full matrix and then remove the columns corresponding to the removed rows.

The LDA does not give us a full matrix. We only get a reduced matrix that cannot be directly inverted. It is possible to take the pseudo inverse, but this is much less efficient than if we had the full matrix at our disposal.

Consider a simple example:

C = np.ones((3, 3)) + np.eye(3)  # full transform matrix
U = C[:2, :]  # dimensionality reduction matrix
V1 = np.linalg.inv(C)[:, :2]  # PCA-style reconstruction matrix
print(V1)
#array([[ 0.75, -0.25],
#       [-0.25,  0.75],
#       [-0.25, -0.25]])

V2 = np.linalg.pinv(U)  # LDA-style reconstruction matrix
print(V2)
#array([[ 0.63636364, -0.36363636],
#       [-0.36363636,  0.63636364],
#       [ 0.09090909,  0.09090909]])

If we have the full matrix we get a different inverse transform (V1) than if we simple invert the transform (V2). That is because in the second case we lost all information about the discarded components.

You have been warned. If you still want to do the inverse LDA transform, here is a function:

import matplotlib.pyplot as plt

from sklearn import datasets
from sklearn.decomposition import PCA
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis

from sklearn.utils.validation import check_is_fitted
from sklearn.utils import check_array, check_X_y

import numpy as np


def inverse_transform(lda, x):
    if lda.solver == 'lsqr':
        raise NotImplementedError("(inverse) transform not implemented for 'lsqr' "
                                  "solver (use 'svd' or 'eigen').")
    check_is_fitted(lda, ['xbar_', 'scalings_'], all_or_any=any)

    inv = np.linalg.pinv(lda.scalings_)

    x = check_array(x)
    if lda.solver == 'svd':
        x_back = np.dot(x, inv) + lda.xbar_
    elif lda.solver == 'eigen':
        x_back = np.dot(x, inv)

    return x_back


iris = datasets.load_iris()

X = iris.data
y = iris.target
target_names = iris.target_names

lda = LinearDiscriminantAnalysis()
Z = lda.fit(X, y).transform(X)

Xr = inverse_transform(lda, Z)

# plot first two dimensions of original and reconstructed data
plt.plot(X[:, 0], X[:, 1], '.', label='original')
plt.plot(Xr[:, 0], Xr[:, 1], '.', label='reconstructed')
plt.legend()

You see, the result of the inverse transform does not have much to do with the original data (well, it's possible to guess the direction of the projection). A considerable part of the variation is gone for good.