A retrieval algorithm that uses a statistical strategy based on dimension reduction is proposed. The methodology and details of the implementation of the new algorithm are presented and discussed. The algorithm has been applied to high resolution spectra measured by the Infrared Atmospheric Sounding Interferometer instrument to retrieve 5 atmospheric profiles of temperature, water vapour and ozone. The performance of the inversion strategy has been assessed by comparing the retrieved profiles to the ones obtained by co-locating in space and time profiles from the European Centre for Medium-Range Weather Forecasts analysis.