
\section{Concluding Remarks}
\label{sec:Conclusions}

Our experiments show that an  IPM (with the proposed approach to linear algebra)
can be more efficient than other state-of-the-art methods 
for QP problems arising in the course of training support vector machines, if
the kernel matrix has a low rank (compared to the size of the data) 
or if it can be approximated by a 
low-rank positive semidefinite matrix.
We considered two possible ways of handling linear algebra.
We showed that Sherman-Morrison-Woodbury update may be numerically unstable.
However, if accuracy is not a high priority one might prefer to use the SMW
method over the product form Cholesky factorization, since it is
faster and requires less storage.

For massive data sets which do not fit the memory restrictions, an
IPM may still be the most efficient approach to solve smaller subproblems.
This motivates an effort to embed our approximation technique and 
QP solver in a Chunking/Shrinking meta algorithm. 
Such a scheme is expected to enhance performance in terms
of storage requirement and computational complexity and, 
thus, it will enable to efficiently handle larger chunks.

Finally, if the approximation is not too rough, then the set of active 
constraints (Support Vectors) may be identical to the set 
which correspond to the original optimization problem, though their
values should obviously differ (due to the approximation). 
Hence, one may use the approximation problem just to identify 
the set of active constraints (assuming this set is not too large), and then 
resolve the reduced problem which constructed solely by the corresponding
original (non approximated) SV, and thus obtain the optimal solution.
Working out a bound for the approximation error that will ensure
the identification of the active set is a subject for future research.





