On the Difficulty of Designing Good Classifiers

We consider the problem of designing a near-optimal linear decision tree to classify two given point sets B and W in n. A linear decision tree de nes a polyhedral subdivision of space; it is a classi er if no leaf region contains points from both sets. We show hardness results for computing such a classi er with approximately optimal depth or size in polynomial-time. In particular, we show that unless NP=ZPP, the depth of a classi er cannot be approximated within any constant factor, and that the total number of nodes cannot be approximated within any xed polynomial. Our proof uses a simple connection between this problem and graph coloring, and uses the result of Feige and Kilian on the inapproximability of the chromatic number. We also study the problem of designing a classi er with a single inequality that involves as few variables as possible, and point out certain aspects of the di culty of this problem. Key words. linear decision tree, hardness of approximation, parameterized com...

• Type: Journal
• Year: 2000
• Where: SIAMCOMP
• Authors: Michelangelo Grigni, Vincent Mirelli, Christos H. Papadimitriou
• Keywords: Classi Er, Linear Decision Tree, Near-optimal Linear Decision, SIAMCOMP 2000