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A leaky recursive least squares (LRLS) algorithm obtained by a criterion of the ridge regression with the exponential weighting factor was recently proposed by one of the authors. On the other hand, an optimization criterion for improving the method of total least squares (TLS) has been proposed by Chandrasekaran et al. In this work, it is expressed that there is a case where the equation obtained by the criterion of the LRLS algorithm is identical to one obtained by the extended criterion of Chandrasekaran et al. In addition, some implementations of the LRLS filter by using the method for updating the eigendecomposition of rank-one matrix updates, or by using the leaky least mean square (LLMS) algorithm, are introduced to decrease the computational complexity of the LRLS algorithm. Moreover, by means of computer experiments, it is shown that the LRLS and the LLMS algorithms yield more precise estimation parameters than the RLS algorithm when the method of Chandrasekaran et al. is more useful than that of LS and TLS. Besides, it is demonstrated that the LLMS algorithm can be effectively introduced into a noise reduction system for noisy speech signals to support the theoretical results in this work.