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Voice conversion can be formulated as finding a mapping function which transforms the features of the source speaker to those of the target speaker. Gaussian mixture model (GMM)-based conversion is commonly used, but it is subject to overfitting. In this paper, we propose to use partial least squares (PLS)-based transforms in voice conversion. To prevent overfitting, the degrees of freedom in the mapping can be controlled by choosing a suitable number of components. We propose a technique to combine PLS with GMMs, enabling the use of multiple local linear mappings. To further improve the perceptual quality of the mapping where rapid transitions between GMM components produce audible artefacts, we propose to low-pass filter the component posterior probabilities. The conducted experiments show that the proposed technique results in better subjective and objective quality than the baseline joint density GMM approach. In speech quality conversion preference tests, the proposed method achieved 67% preference score against the smoothed joint density GMM method and 84% preference score against the unsmoothed joint density GMM method. In objective tests the proposed method produced a lower Mel-cepstral distortion than the reference methods.