Lossy compression algorithms trade bits for quality, aiming at reducing as much as possible the bitrate needed to represent the original source (or set of sources), while preserving the source quality. In this letter, we propose a novel paradigm of compression algorithms, aimed at minimizing the information loss perceived by the final user instead of the actual source quality loss, under compression rate constraints. As main contributions, we first introduce the concept of perceived information (PI), which reflects the information perceived by a given user experiencing a data collection, and which is evaluated as the volume spanned by the sources features in a personalized latent space. We then formalize the rate-PI optimization problem and propose an algorithm to solve this compression problem. Finally, we validate our algorithm against benchmark solutions with simulation results, showing the gain in taking into account users' preferences while also maximizing the perceived information in the feature domain.