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V = VL_HOMKERMAP(X, N) computes a 2*N+1 dimensional approximated kernel map for the Chi2 kernel. X is an array of data points. Each point is expanded into a vector of dimension 2*N+1 and saved to the output V. The expanded feature vectors are stacked along the first dimension, so that the output array V has the same dimensions of the input array X except for the first one, which is 2*N+1 times larger.

The function accepts the following options:

Kernel KCHI2

One of KCHI2 (Chi2 kernel), KINTERS (intersection kernel), KJS (Jensen-Shannon kernel). The 'Kernel' option name can be omitted, i.e. VL_HOMKERMAP(..., 'kernel', 'kchi2') has the same effect of VL_HOMKERMAP(..., 'kchi2').

Period [automatically tuned]

Set the period of the kernel specturm. The approximation is based on periodicizing the kernel specturm. If not specified, the period is automatically set based on the heuristic described in [2].

Window [RECTANGULAR]

Set the window used to truncate the spectrum before The window can be either RECTANGULAR or UNIFORM window. See [2] and the API documentation for details.

Gamma [1]

Set the homogeneity degree of the kernel. The standard kernels are 1-homogeneous, but sometimes smaller values perform better in applications. See [2] for details.

Example

The following code results in approximatively the same similarities matrices between points X and Y:

  x = rand(10,1) ;
  y = rand(10,100) ;
  psix = vl_homkermap(x, 3) ;
  psiy = vl_homkermap(y, 3) ;
  figure(1) ; clf ;
  ker = vl_alldist(x, y, 'kchi2') ;
  ker_ = psix' * psiy ;
  plot([ker ; ker_]') ;

Note

The homogeneous kernels K(X,Y) are normally defined for non-negative data only. VL_HOMKERMAP defines them for both positive and negative data by using the definition SIGN(X)SIGN(Y)K(ABS(X),ABS(Y)) -- note that other extensions are possible as well (see [2]).

REFERENCES

[1] A. Vedaldi and A. Zisserman `Efficient Additive Kernels via Explicit Feature Maps', Proc. CVPR, 2010.

[2] A. Vedaldi and A. Zisserman `Efficient Additive Kernels via Explicit Feature Maps', PAMI, 2011 (submitted).

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