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Evaluating a kernel

The module MKLpy.metrics provides several functions to evaluate a kernel, and to compute statistrics and various metrics. Here we show a few of examples, including

  • the margin margin between positive and negative classes in the kernel space;
  • the radius of the Minimum Enclosing Ball (MEB) containing data in the kernel space;
  • the radius/margin ratio;
  • the trace of a kernel matrix;
  • the frobenius norm of a kernel matrix.
from MKLpy import metrics
#we assume K be the kernel matrix and Y be the labels vector
margin = metrics.margin(K,Y)    #works only in binary classification settings
radius = metrics.radius(K)
ratio  = metrics.ratio(K,Y)
trace  = metrics.trace(K)
frob   = metrics.frobenius(K)

Note

K is always a squared kernel matrix, i.e. it is not the kernel computed between test and training examples.

Spectral Ratio

An additional important metric is the Spectral Ratio, that reflects the empirical complexity of a kernel matrix. The Spectral Ratio is defined as the ratio between the trace of a kernel and its Frobenius norm, i.e. \mathcal{C}(K) = \frac{\sum_i K_{i,i}}{\sqrt{\sum_{i,j} K_{i,j}^2}} = \frac{\|K\|_T}{\|K\|_F}.

The high is the spectral ratio, the high is the complexity of the kernel.

from MKLpy import metrics
SR = metrics.spectral_ratio(K, norm=True)

The normalized spectral ratio has range [0,1].

Paper

An exhaustive description of the Spectral Ratio is available in the following paper:
Michele Donini and Fabio Aiolli: "Learning deep kernels in the space of dot product polynomials". Machine Learning (2017)

Alignment

The alignment measures the similarity between two kernels. We have several functions to compute the alignment. These functions, showed in the following example, outputs a score that represents the alignment

from MKLpy import metrics

#produces the alignment between two kernels, K1 and K2
metrics.alignment(K1, K2)

#produces the alignment between the kernel K1 and the identity matrix
metrics.alignment_ID(K1)

#produces the alignment between the kernel K1 and the ideal (or optimal) kernel
metrics.alignment_yy(K1, Y)

where K1 and K2 are two sqared kernel matrices and Y is the binary labels vector

Note

The ideal kernel between two examples outputs 1 if the examples belong to the same class, -1 else.