Activities This Week

The problem of computational super-resolution asks to recover fine features of a signal from inaccurate and bandlimited data, using an a-priori model as a regularization. I will describe several situations for which sharp bounds for stable reconstruction are known, depending on signal complexity, noise/uncertainty level, and available data bandwidth. I will also discuss optimal recovery algorithms, and some open questions.

Krieger’s generator theorem shows that any free invertible ergodic measure preserving action $(Y,\mu, S)$ can be modelled by $A^Z$ (equipped with the shift action) provided the natural entropy constraint is satisfied; we call such systems (here it is $A^Z$) universal. Along with Tom Meyerovitch, we establish general specification like conditions under which $Z^d$-dynamical systems are universal. These conditions are general enough to prove that

1) A self-homeomorphism with almost weak specification on a compact metric space (answering a question by Quas and Soo and recovering recent results by David Burguet) 2) Proper colourings of the $Z^d$ lattice with more than two colours and the domino tilings of the $Z^2$ lattice (answering a question by Şahin and Robinson) are universal. Our results also extend to the almost Borel category giving partial answers to some questions by Gao and Jackson.