Conventionally, multimedia researchers have adhered to the maxim “There is no data like more data”. However, the new General Data Protection Regulation, which came into force in Europe in 2018, has caused the data science world to shift its perspective. If there are constraints on how data can be gathered and used, then researchers must adapt themselves to design effective algorithms and systems under those constraints. In this talk, we look at how “data greed” has characterized past research on analyzing multimedia content and discover why it is time to revisit our assumption that more data will lead to better performance of algorithms. The talk covers examples of research on data dropping and data bleaching that make the case for the potential of data minimization, and motivation the need for future work in that direction. Finally, we point out that techniques focusing on how to perturb or remove information in multimedia data have a potential to protect the privacy of users, without requiring them to stop sharing their images and videos online.