Phase Transition in a Noise Reduction Model: Shrinking or Percolation?

A model of noise reduction (NR) for signal processing is introduced. Each noise source puts a symmetric constraint on the space of the signal vector within a tolerable overlap. When the number of noise sources increases, sequences of transitions take place, causing the solution space to vanish. We found that the transition from an extended solution space to a shrunk space is retarded because of the symmetry of the constraints, in contrast to the analogous problem of pattern storage. For low tolerance, the solution space vanishes by volume reduction, whereas for high tolerance, the vanishing becomes more and more like percolation. The model is studied in the replica symmetric, first step and full replica symmetry breaking schemes.