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This paper proposes a novel adaptive filtering scheme named the Krylov-proportionate normalized least-mean-square (KPNLMS) algorithm. KPNLMS exploits the benefits (i.e., fast convergence for sparse unknown systems) of the proportionate NLMS algorithm, but its applications are not limited to sparse unknown systems. A set of orthonormal basis vectors is generated from a certain Krylov sequence. It is proven that the unknown system is sparse with respect to the basis vectors in case of fairly uncorrelated input data. Different adaptation gain is allocated to a coefficient of each basis vector, and the gain is roughly proportional to the absolute value of the corresponding coefficient of the current estimate. KPNLMS enjoys i) fast convergence, ii) linear complexity per iteration, and iii) no use of any a priori information. Numerical examples demonstrate significant advantages of the proposed scheme over the reduced-rank method based on the multistage Wiener filter (MWF) and the transform-domain adaptive filter (TDAF) both in noisy and silent situations.