A Panorama of Phase Transitions

The purpose of this chapter is to provide a unified review of a wide variety of phase-transition results in the detection and exact support recovery problems for sparse signals. In order to emphasize ideas, the focus is on the simple case of a high-dimensional sparse signal observed with additive independent Gaussian errors. The classic phase-transition result for the signal detection problem obtained first by Yuri Izmailovich Ingster as well as very recent results on approximate support recovery are reviewed. The unified approach based on the four different risk functionals introduced in Chap. 2 yields a variety of phase-transition results, many of which are new. The optimality and sub-optimality of some popular support estimation procedures are also established. The chapter provides a concise and yet complete account of the phase transitions in the context of sparse signal detection and support estimation in high dimensions.