Supervised learning and systems with excess degrees of freedom

WHEN DISTINCT OUTPUTS OF AN ADAPTIVE SYSTEM HAVE EQUIVALENT EFFECTS ON THE ENVIRONMENT, THE PROBLEM OF FINDING APPROPRIATE ACTIONS GIVEN DESIRED RESULTS IS ILL-POSED. FOR SUPERVISED LEARNING ALGORITHMS, THE ILL-POSEDNESS OF SUCH "INVERSE LEARNING PROBLEMS" IMPLIES A CERTAIN FLEXIBILITY---DURING TRAINING, THERE ARE IN GENERAL MANY POSSIBLE TARGET VECTORS CORRESPONDING TO EACH INPUT VECTOR. TO ALLOW SUPERVISED LEARNING ALGORITHMS TO MAKE USE OF THIS FLEXIBILITY, THE CURRENT PAPER CONSIDERS HOW TO SPECIFY TARGETS BY SETS OF CONSTRAINTS, RATHER THAN AS PARTICULAR VECTORS. TWO CLASSES OF CONSTRAINTS ARE DISTINGUISHED---`CONFIGURATIONAL'' CONSTRAINTS, WHICH DEFINE REGIONS OF OUTPUT SPACE IN WHICH AN OUTPUT VECTOR MUST LIE, AND `TEMPORAL'' CONSTRAINTS, WHICH DEFINE RELATIONSHIPS BETWEEN OUTPUTS PRODUCED AT DIFFER- ENT POINTS IN TIME. LEARNING ALGORITHMS MINIMIZE A COST FUNCTION THAT CON- TAINS TERMS FOR BOTH KINDS OF CONSTRAINTS. THIS APPROACH TO INVERSE LEARN- ING IS ILLUSTRATED BY A ROBOTICS APPLICATION IN WHICH A NETWORK FINDS TRA- JECTORIES OF INVERSE KINEMATIC SOLUTIONS FOR MANIPULATORS WITH EXCESS DEGREES OF FREEDOM.