Explore the concept of parsimonious representations in data science through this comprehensive lecture by Dr. Armin Eftekhari from the University of Edinburgh. Delve into the importance of exploiting geometric structures hidden within the vast amounts of data produced daily. Begin with an overview of models in data and computational sciences, focusing on the linear subspace model. Examine the band-limited model that revolutionized digital technology and study principal component analysis as a key statistical tool for uncovering linear structures in collected data. Progress to more advanced topics, including nonlinear models, optimization problems, and their mathematical proofs. Gain valuable insights into data analysis techniques and their applications in various fields of science and technology.