def radius(K):
"""evaluate the radius of the MEB (Minimum Enclosing Ball) of examples in
feature space.
Parameters
----------
K : (n,n) ndarray,
the kernel that represents the data.
Returns
-------
r : np.float64,
the radius of the minimum enclosing ball of examples in feature space.
"""
check_squared(K)
n = K.shape[0]
P = 2 * matrix(K)
p = -matrix([K[i, i] for i in range(n)])
G = -spdiag([1.0] * n)
h = matrix([0.0] * n)
A = matrix([1.0] * n).T
b = matrix([1.0])
solvers.options['show_progress'] = False
sol = solvers.qp(P, p, G, h, A, b)
return np.sqrt(abs(sol['primal objective']))
def frobenius(K):
"""return the frobenius-norm of the kernel as input.
Parameters
----------
K : (n,n) ndarray,
the kernel that represents the data.
Returns
-------
t : np.float64,
the frobenius-norm of *K*
"""
check_squared(K)
return np.sqrt(np.sum(K**2))
def trace(K):
"""return the trace of the kernel as input.
Parameters
----------
K : (n,n) ndarray,
the kernel that represents the data.
Returns
-------
t : np.float64,
the trace of *K*
"""
check_squared(K)
return sum([K[i, i] for i in range(K.shape[0])])
def spectral_ratio(K, norm=True):
"""return the spectral ratio of the kernel as input.
Parameters
----------
K : (n,n) ndarray,
the kernel that represents the data.
norm : bool=True,
True if we want the normalized spectral ratio.
Returns
-------
t : np.float64,
the spectral ratio of *K*, normalized iif *norm=True*
"""
check_squared(K)
n = K.shape[0]
c = trace(K) / frobenius(K)
return (c - 1) / (np.sqrt(n) - 1) if norm else c
def kernel_centering(K):
"""move a squared kernel at the center of axis
Parameters
----------
K : (n,n) ndarray,
the squared kernel matrix.
Returns
-------
Kc : ndarray,
the centered version of *K*.
"""
K = check_squared(K)
N = K.shape[0]
I = np.ones(K.shape)
C = np.diag(np.ones(N)) - (1.0 / N * I)
Kc = np.dot(np.dot(C, K), C)
return Kc
def tracenorm(K):
"""normalize the trace of a squared kernel matrix
Parameters
----------
K : (n,n) ndarray,
the squared kernel matrix.
Returns
-------
Kt : ndarray,
the trace-normalized version of *K*.
Notes
-----
In trace-normalization, the kernel is divided by the average of the diagonal.
"""
K = check_squared(K)
trn = trace(K) / K.shape[0]
return K / trn
def kernel_normalization(K):
"""normalize a squared kernel matrix
Parameters
----------
K : (n,n) ndarray,
the squared kernel matrix.
Returns
-------
Kn : ndarray,
the normalized version of *K*.
Notes
-----
Given a kernel K, the normalized version is defines as:
.. math:: \hat{k}(x,z) = \frac{k(x,z)}{\sqrt{k(x,x)\cdot k(z,z)}}
"""
K = check_squared(K)
n = K.shape[0]
d = np.array([[K[i, i] for i in range(n)]])
Kn = K / np.sqrt(np.dot(d.T, d))
return Kn