NIPS Workshop on Algebraic and Combinatorial Methods in Machine Learning, Whistler 2008

Toric Modification on Mixture Models

author: Sumio Watanabe, Tokyo Institute of Technology
author: Keisuke Yamazaki, Tokyo Institute of Technology
published: Dec. 20, 2008, recorded: December 2008, views: 156

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Slides

Slides
0:00	Toric Modification on Machine Learning
0:13	Agenda
0:16	Agenda (1)
0:22	What is the generalization error
1:44	Algebraic geometry connected to learning theory in the Bayes method (1)
2:10	Algebraic geometry connected to learning theory in the Bayes method (2)
2:24	Algebraic geometry connected to learning theory in the Bayes method (3)
2:32	The zeta function has an important role for the connection (1)
3:31	The zeta function has an important role for the connection (2)
4:03	The largest pole of the zeta function determines the generalization error
4:47	Agenda (2)
4:51	Calculation of the zeta function requires well-formed H(w)
6:22	The resolution of singularities with blow-ups is an iterative method (1)
7:10	The resolution of singularities with blow-ups is an iterative method (2)
7:31	The bottleneck is the iterative method
7:42	Toric modification is a systematic method for the resolution of singularities
8:04	Agenda (3)
8:08	Newton diagram is a convex hull in the exponent space
8:56	The borders determine a set of vectors in the dual space
9:37	Add some vectors subdividing the spanned area
10:08	Selected vectors construct the resolution map
11:38	Non-degenerate Kullback divergence
12:30	Toric modification is "systematic"
13:13	Toric modification can be an effective plug-in method
13:53	Agenda (4)
13:57	An application to a mixture model
14:43	The Newton diagram of the mixture
15:20	The resolution map based on the toric modification
15:34	The generalization error of the mixture of binomial distributions
16:09	Agenda (5)
16:11	Summary
16:58	Thank you
17:20	- Questions
17:56	- Questions
18:37	- Questions
20:53	- Questions

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Description

In the Bayes estimation, it was pointed out that resolution of singularity provides an algorithm to elucidate the generalization performance of learning machines. However, there is no effective procedure to find the resolution map. This presentation proposes a new method to find it based on the toric modification, using Newton diagram. By the proposed method, learning curves of several hierarchical models are clarified.