The Computational Complexity of Densest Region Detection

We investigate the computational complexity of the task of detecting dense regions of an unknown distribution from un-labeled samples of this distribution. We introduce a formal learning model for this task that uses a hypothesis class as its ‘anti-overfitting’ mechanism. The learning task in our model can be reduced to a combinatorial optimization problem. We can show that for some constants, depending on the hypothesis class, these problems are NP-hard to approximate to within these constant factors. We go on and introduce a new criterion for the success of approximate optimization geometric problems. The new criterion requires that the algorithm competes with hypotheses only on the points that are separated by some margin µ from their boundaries. Quite surprisingly, we discover that for each of the two hypothesis classes that we investigate, there is a ‘critical value’ of the margin parameter µ. For any value below the critical value the problems are NP-hard to approxima...

• Type: Conference
• Year: 2000
• Where: COLT
• Authors: Shai Ben-David, Nadav Eiron, Hans-Ulrich Simon
• Keywords: COLT 2000, Formal Learning Model, Hypothesis Class, Machine Learning, Optimization Geometric Problems