On the difficulty of designing good classifiers [chapter]

Michelangelo Grigni, Vincent Mirelli, Christos H. Papadimitriou
1996 Lecture Notes in Computer Science  
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
more » ... he 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.
doi:10.1007/3-540-61332-3_161 fatcat:ycr7nx2wkjextmoyw3uj36z2tq