Algebraic topology as an analytic tool to check lossless nature of geometric algorithms

Geometric representation/compression is the process of representing data as points on nonlinear spaces such as manifolds. Geometric representation algorithms for video/image data have gained significant attention in the past decade due to their ability to eliminate redundant information for storage and classification. This work presents the novel idea of using algebraic topology to decide whether a particular geometric com-pression algorithm or data representation algorithm is lossless. A necessary condition required for the compression to be lossless is derived by using the shape (topology) of the space on which compressed frames reside. The fundamental group functor is used to formulate the necessary condition as an algebraic condition which is much easier to work with. Finally, I demonstrate how the proposed idea can be utilized in a more specific setting to draw conclusions regarding a geometric algorithm’s lossless nature.