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played a bit around with doxygen latex
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karlnapf committed Jul 12, 2012
1 parent 02496a4 commit 9368244
Showing 1 changed file with 18 additions and 6 deletions.
24 changes: 18 additions & 6 deletions src/shogun/statistics/LinearTimeMMD.h
Expand Up @@ -20,9 +20,16 @@ namespace shogun
class CFeatures;

/** @brief This class implements the linear time Maximum Mean Statistic as
* described in
* Gretton, A., Borgwardt, K. M., Rasch, M. J., Schoelkopf, B., & Smola, A. (2012).
* A Kernel Two-Sample Test. Journal of Machine Learning Research, 13, 671-721.
* described in [1].
* Given two sets of samples \f$\{x_i\}_{i=1}^m\sim p\f$ and
* \f$\{y_i\}_{i=1}^n\sim q\f$
* the (unbiased) statistic is computed as
* \f[
* \text{MMD}_l^2[\mathcal{F},X,Y]=\frac{1}{m(m-1)}\sum_{i=1}^m\sum_{j\neq i}^m
* k(x_i,x_j) + \frac{1}{n(n-1)}\sum_{i=1}^n\sum_{j\neq i}^nk(y_i,y_j)
* - \frac{2}{mn}\sum_{i=1}^m\sum_{j=1}^nk(x_i,y_j)
* \f]
* in a RKHS \f$\mathcal{F}\f$
*
* Along with the statistic comes a method to compute a p-value based on a
* Gaussian approximation of the null-distribution which is also possible in
Expand All @@ -33,6 +40,9 @@ class CFeatures;
*
* Comes with a method for selecting kernel weights, if a combined kernel on
* combined features is used. See optimize_kernel_weights().
*
* [1]: Gretton, A., Borgwardt, K. M., Rasch, M. J., Schoelkopf, B., & Smola, A. (2012).
* A Kernel Two-Sample Test. Journal of Machine Learning Research, 13, 671-721.
*/
class CLinearTimeMMD: public CKernelTwoSampleTestStatistic
{
Expand Down Expand Up @@ -112,11 +122,13 @@ class CLinearTimeMMD: public CKernelTwoSampleTestStatistic
virtual float64_t compute_variance_estimate();

#ifdef HAVE_LAPACK
/** TODO */
/** Selects optimal kernel weights (if the underlying kernel and features)
/** Selects optimal kernel weights (if the underlying kernel and features
* are combined ones) using the ratio of the squared MMD by its standard
* deviation as a criterion, i.e.
* TODO latex
* \f[
* \frac{\text{MMD}_l^2[\mathcal{F},X,Y]}{\sigma_l}
* \f]
* where both expressions are estimated in linear time.
* This comes down to solving a convex program which is quadratic in the
* number of kernels.
*
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