Given the regularized kernel matrix , the new regularized kernel matrix can be constructed by removing the first row and column of , referred to as , and adding kernels of the new data as the last row and column:
Calculating the inverse kernel matrix is done in two steps, using the two inversion formulas derived in appendices A.1 and A.2. Note that these formulas do not calculate the inverse matrices explicitly, but rather derive them from known matrices maintaining an overall time and memory complexity of of the algorithm.
First, given and , the inverse of the matrix is calculated according to Eq. (12). Then can be calculated applying the matrix inversion formula from Eq. 11, based on the knowledge of and . The complete algorithm is summarized in Alg. (1).