During the talk I will discuss a result connected with Alberti's rank-one theorem for gradients of functions of bounded variations. Namely, I will present elements of our proof of a discrete analog of this theorem for the setting of q-regular (Sobolev) martingales. We derive it from a general description of polar decomposition of vector measures on the boundary of the q-regular infinite tree. This description in turns resembles De Philippis-Rindler structure theorem. Joint work with D. Stolyarov and M. Wojciechowski.