Factorial learning, finding a statistically independent representation of a sensory “image”—a factorial code—is applied here to solve multilayer supervised learning problems that have traditionally required backpropagation. This lends support to Barlow's argument for factorial sensory processing, by demonstrating how it can solve actual pattern recognition problems. Two techniques for supervised factorial learning are explored, one of which gives a novel distributed solution requiring only positive examples. Also, a new nonlinear technique for factorial learning is introduced that uses neural networks based on almost reversible cellular automata. Due to the special functional connectivity of these networks—which resemble some biological microcircuits—learning requires only simple local algorithms. Also, supervised factorial learning is shown to be a viable alternative to backpropagation. One significant advantage is the existence of a measure for the performance of intermediate learning stages.