class06

class06 - Manifold Regularization 9.520 Class 06, 27...

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Manifold Regularization 9.520 Class 06, 27 February 2006 Andrea Caponnetto About this class Goal To analyze the limits of learning from examples in high dimensional spaces. To introduce the semi-supervised setting and the use of unlabeled data to learn the in- trinsic geometry of a problem. To define Riemannian Manifolds, Manifold Laplacians, Graph Laplacians. To introduce a new class of algorithms based on Manifold Regularization (LapRLS, LapSVM). Unlabeled data Why using unlabeled data? labeling is often an expensive process semi-supervised learning is the natural setting for hu- man learning Semi-supervised setting u i.i.d. samples drawn on X from the marginal distribution p ( x ) { x 1 , x 2 , . . . , x u } , only n of which endowed with labels drawn from the con- ditional distributions p ( y | x ) { y 1 , y 2 , . . . , y n } . The extra u n unlabeled samples give additional informa- tion about the marginal distribution p ( x ). The importance of unlabeled data Curse of dimensionality and p ( x ) Assume X is the D-dimensional hypercube [0 , 1] D . The worst case scenario corresponds to uniform marginal dis- tribution p ( x ). Two perspectives on curse of dimensionality: As d increases, local techniques (eg nearest neighbors) become rapidly ineffective. Minimax results show that rates of convergence of em- pirical estimators to optimal solutions of known smooth- ness, depend critically on D Curse of dimensionality and k-NN It would seem that with a reasonably large set of train- ing data, we could always approximate the conditional expectation by k-nearest-neighbor averaging. We should be able to find a fairly large set of observa- tions close to any x [0 , 1] D and average them. This approach and our intuition breaks down in high dimensions . Sparse sampling in high dimension Suppose we send out a cubical neighborhood about one vertex to capture a fraction r of the observations. Since this corresponds to a fraction r of the unit volume, the expected edge length will be 1 e D ( r ) = r D . Already in ten dimensions e 10 (0 . 01) = 0 . 63, that is to capture 1% of the data, we must cover 63% of the range of each input variable! No more local neighborhoods! Distance vs volume in high dimensions 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Distance p=1 p=2 p=3 p=10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fraction of Volume Curse of dimensionality and smoothness Assuming that the target function f (in the squared loss case) belongs to the Sobolev space W s 2 ([0 , 1] D ) = { f L 2 ([0 , 1] D ) | 2 s | f ( ) | 2 < + } Z d it is possible to show that s sup IE S ( I [ f S ] I [ f ]) > Cn D ,f W 2 s More smoothness s faster rate of convergence Higher dimension D slower rate of convergence A Distribution-Free Theory of Nonparametric Regression , Gyorfi Intrinsic dimensionality Raw format of natural data is often high dimensional, but in many cases...
View Full Document

This note was uploaded on 11/11/2011 for the course BIO 9.07 taught by Professor Ruthrosenholtz during the Spring '04 term at MIT.

Page1 / 36

class06 - Manifold Regularization 9.520 Class 06, 27...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online