Lipschitz Based Polynomial Interpolation for Stable Incomplete Data Reconstruction
DOI:
https://doi.org/10.30605/proximal.v9i3.9607Keywords:
Incomplete Data Reconstruction, Lipschitz Continuity, Noisy Data, Numerical Stability, Polynomial InterpolationAbstract
Incomplete data reconstruction may experience excessive fluctuations when the available observations become sparse or irregular and are contaminated by noise, making global polynomial interpolation sensitive to local variations. This study proposes a Lipschitz-based polynomial interpolation approach that combines classical Lagrange interpolation with a gradient-limiting mechanism, which constrains local changes between consecutive points within a prescribed Lipschitz bound. Unlike reconstruction approaches based on optimization or iterative regularization, the proposed mechanism is applied as a simple post-interpolation stabilization process without requiring model training or repeated optimization. The method was evaluated using synthetic nonlinear data with Gaussian noise under missing-data rates of 30%, 40%, and 50%. The results reveal a trade-off between reconstruction accuracy and smoothness. While classical interpolation achieved lower RMSE and MAE values, the Lipschitz-based method produced lower Total Variation (TV) values in most scenarios. These results indicate that gradient limiting effectively controls local fluctuations and produces smoother reconstructions at the expense of reduced pointwise accuracy. Therefore, the proposed approach provides a simple and interpretable stabilization mechanism for incomplete and noisy data reconstruction when oscillation control is a primary concern.
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