def test_rbf_kernel_gram_1d():
rng = onp.random.RandomState(123)
n_samples = 100
X = rng.rand(n_samples)
# X
K_sk = rbf_sklearn(X[:, np.newaxis], X[:, np.newaxis], gamma=1.0)
K = gram(rbf_kernel, {"gamma": 1.0}, X, X)
onp.testing.assert_array_almost_equal(K_sk, onp.array(K))
Y = 10 * X + 0.1 * rng.randn(n_samples)
# Y
K_sk = rbf_sklearn(Y[:, np.newaxis], Y[:, np.newaxis], gamma=1.0)
K = gram(rbf_kernel, {"gamma": 1.0}, Y, Y)
onp.testing.assert_array_almost_equal(K_sk, onp.array(K))
# X AND Y
K_sk = rbf_sklearn(X[:, np.newaxis], Y[:, np.newaxis], gamma=1.0)
K = gram(rbf_kernel, {"gamma": 1.0}, X, Y)
onp.testing.assert_array_almost_equal(K_sk, onp.array(K))
def mmd(
X: np.ndarray,
Y: np.ndarray,
kernel: Callable,
params_x: Dict[str, float],
params_y: Dict[str, float],
params_xy: Dict[str, float],
bias: bool = False,
center: bool = False,
) -> float:
"""Maximum Mean Discrepancy
Parameters
----------
X : jax.numpy.ndarray
array-like of shape (n_samples, n_features)
Y : np.ndarray
The data matrix.
Notes
-----
This method is equivalent to the HSIC method.
"""
n_samples, m_samples = X.shape[0], Y.shape[0]
# constants
a00 = 1.0 / (n_samples * (n_samples - 1.0))
a11 = 1.0 / (m_samples * (m_samples - 1.0))
a01 = -1.0 / (n_samples * m_samples)
# kernel matrices
Kx = gram(kernel, params_x, X, X)
Ky = gram(kernel, params_y, Y, Y)
Kxy = gram(kernel, params_xy, X, Y)
if bias:
mmd = np.mean(Kx) + np.mean(Ky) - 2 * np.mean(Kxy)
return np.where(mmd >= 0.0, np.sqrt(mmd), 0.0)
else:
return (
2 * a01 * np.mean(Kxy)
+ a00 * (np.sum(Kx) - n_samples)
+ a11 * (np.sum(Ky) - m_samples)
)
def _hsic_uncentered(
X: np.ndarray,
Y: np.ndarray,
kernel: Callable,
params_x: Dict[str, float],
params_y: Dict[str, float],
) -> float:
"""A method to calculate the uncentered HSIC version"""
# kernel matrix
Kx = gram(kernel, params_x, X, X)
Ky = gram(kernel, params_y, Y, Y)
#
K = np.dot(Kx, Ky.T)
hsic_value = np.mean(K)
return hsic_value
def test_rbf_kernel_cov_2d():
X = onp.random.rand(100, 2)
# sklearn rbf_kernel
K_sk = rbf_sklearn(X, X, gamma=1.0)
K = gram(rbf_kernel, {"gamma": 1.0}, X, X)
onp.testing.assert_array_almost_equal(K_sk, onp.array(K))
def mmd_mi(
X: np.ndarray,
Y: np.ndarray,
kernel: Callable,
params_x: Dict[str, float],
params_y: Dict[str, float],
) -> float:
"""Maximum Mean Discrepancy
Parameters
----------
X : jax.numpy.ndarray
array-like of shape (n_samples, n_features)
Y : np.ndarray
The data matrix.
Notes
-----
This method is equivalent to the HSIC method.
"""
# calculate kernel matrices
Kx = gram(kernel, params_x, X, X)
Ky = gram(kernel, params_y, Y, Y)
# center kernel matrices
Kx = centering(Kx)
Ky = centering(Ky)
# get the expectrations
A = np.mean(Kx * Ky)
B = np.mean(np.mean(Kx, axis=0) * np.mean(Ky, axis=0))
C = np.mean(Kx) * np.mean(Ky)
# calculate the mmd value
mmd_value = A - 2 * B + C
return mmd_value
def test_centering():
n_samples = 100
X = onp.random.rand(n_samples)
# sklearn rbf_kernel
K_sk = rbf_sklearn(X[:, np.newaxis], X[:, np.newaxis], gamma=1.0)
K_sk = KernelCenterer().fit_transform(K_sk)
K = gram(rbf_kernel, {"gamma": 1.0}, X, X)
# H = np.eye(n_samples) - (1.0 / n_samples) * np.ones((n_samples, n_samples))
# K = np.einsum("ij,jk,kl->il", H, K, H)
# K = np.dot(H, np.dot(K, H))
K = centering(K)
onp.testing.assert_array_almost_equal(K_sk, onp.array(K))
def test_rbf_kernel_gram_2d():
rng = onp.random.RandomState(123)
n_samples, n_features = 100, 2
X = onp.random.rand(n_samples, n_features)
# sklearn rbf_kernel
K_sk = rbf_sklearn(X, X, gamma=1.0)
K = covariance_matrix(rbf_kernel, {"gamma": 1.0}, X, X)
onp.testing.assert_array_almost_equal(K_sk, onp.array(K))
Y = 10 * X + 0.1 * rng.randn(n_samples, n_features)
# sklearn rbf_kernel
K_sk = rbf_sklearn(Y, Y, gamma=1.0)
K = gram(rbf_kernel, {"gamma": 1.0}, Y, Y)
onp.testing.assert_array_almost_equal(K_sk, onp.array(K))
def nhsic_cca(
X: np.ndarray,
Y: np.ndarray,
kernel: Callable,
params_x: Dict[str, float],
params_y: Dict[str, float],
epsilon: float = 1e-5,
bias: bool = False,
) -> float:
"""Normalized HSIC (Tangent Kernel Alignment)
A normalized variant of HSIC method which divides by
the HS-Norm of each dataset.
Parameters
----------
X : jax.numpy.ndarray
the input value for one dataset
Y : jax.numpy.ndarray
the input value for the second dataset
kernel : Callable
the kernel function to be used for each of the kernel
calculations
params_x : Dict[str, float]
a dictionary of parameters to be used for calculating the
kernel function for X
params_y : Dict[str, float]
a dictionary of parameters to be used for calculating the
kernel function for Y
Returns
-------
cka_value : float
the normalized hsic value.
Notes
-----
This is a metric that is similar to the correlation, [0,1]
"""
n_samples = X.shape[0]
# kernel matrix
Kx = gram(kernel, params_x, X, X)
Ky = gram(kernel, params_y, Y, Y)
# center kernel matrices
Kx = centering(Kx)
Ky = centering(Ky)
K_id = np.eye(Kx.shape[0])
Kx_inv = np.linalg.inv(Kx + epsilon * n_samples * K_id)
Ky_inv = np.linalg.inv(Ky + epsilon * n_samples * K_id)
Rx = np.dot(Kx, Kx_inv)
Ry = np.dot(Ky, Ky_inv)
hsic_value = np.sum(Rx * Ry)
if bias:
bias = 1 / (Kx.shape[0] ** 2)
else:
bias = 1 / (Kx.shape[0] - 1) ** 2
return bias * hsic_value