Let the buffer size be ��. Buffer-N replaces http://www.selleckchem.com/products/Dasatinib.html the oldest example (xt?��(1,2), yt?��, ��t?��) in the buffer with the new arrived example (xt(1,2), yt, ��t) on each learning round, which means St = t ? �� + 1, t ? �� + 2,��, t.Buffer-L. This buffering strategy replaces the oldest unlabeled point in the buffer with the incoming point while keeping labeled points. The oldest labeled point is evicted from the buffer only when it is filled with labeled points.Figure 2 shows the set of the associated coefficient vectors which are used to ascend the dual function on each learning round for different choices of St. Essentially, different choices of St construct different QP problems on each learning round.Figure 2Four choices of St to update multiple dual coefficient vectors.
The horizontal thin line on each learning round represents the whole training data sequence, while the thick boxes represent the set of examples whose dual coefficient vectors are used to …5.3. Sparse Approximations for Kernel RepresentationIn practice, kernel functions are always used to find a linear classifier, like SVM. Our online coregularization framework only contains the product of two points, so we can easily introduce the kernel function into our framework. If we note K the kernel matrix such thatKij=��(xi)?��(xj),(36)xi can be replaced by ��(xi) in our framework. Therefore, we can rewrite (19) as��t(1)=1��1��i=1t(��iyi(��i1)t+(1?��i)(��i0)t��c)��(1)(xi(1))=��i=1t(��i(1))t��(1)(xi(1)),��t(2)=1��2��i=1t(��i��yi(��i2)t?(1?��i)(��i0)t��c)��(2)(xi(2))=��i=1t(��i(2))t��(2)(xi(2)).
(37)Unfortunately, our previous derived online coregularization algorithms with kernel functions have to store the example sequence up to the current round, and the stored matrix size is t �� t (worst case). For practical purpose, we present two approaches to sparsify the kernel representation of boundaries on each learning round. Absolute Threshold. To construct a sparse representation for the boundaries, absolute threshold discards the examples whose associated coefficients ��(1,2) are close to zero. Let �� > 0 denote the absolute threshold. When an arrived example xi(1,2) would not be used to update the boundaries in further learning process, xi(1,2) is discarded if |(��i(1,2))t| < ��. The examples whose indices are in St cannot be discarded on round t since they would be used to ascend the dual function.
k Maximal Coefficients (k-MC). Another way to sparsify the kernel representation is to keep the examples of which the absolute value of ��(1,2) is the Batimastat first k maximum. Similar as the absolute threshold, k-MC does not discard the examples in St which would be used to ascend the dual function on round t. Based on this sparse approximation, the stored matrix size on round t reduces to (k + sizeof(St)) �� (k + sizeof(St)).