A model of noise reduction for signal processing and other optimization tasks is introduced. Each noise source puts a symmetric constraint on the space of the signal vector within a tolerance bound. When the number of noise sources increases sequences of transitions take place, causing the solution space to vanish. We find that the transition from an extended solution space to a shrunk space is retarded because of the symmetry of the constraints, in contrast with 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.