机器学习的几何观点-LAMDA课件.ppt
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- 机器 学习 几何 观点 LAMDA 课件
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1、A Geometric Perspective on Machine Learning何晓飞浙江大学计算机学院1Machine Learning:the problemf何晓飞Information(training data)f:XYX and Y are usually considered as a Euclidean spaces.2Manifold Learning:geometric perspectiveo The data space may not be a Euclidean space,but a nonlinear manifold.Euclidean distance
2、.f is defined on Euclidean space.ambient dimension geodesic distance.f is defined on nonlinear manifold.manifold dimension.instead3Manifold Learning:the challenges The manifold is unknown!We have only samples!o How do we know M is a sphere or a torus,or else?o How to compute the distance on M?o vers
3、usThis is unknown:This is what we have:?or else?TopologyGeometryFunctional analysis4Manifold Learning:current solutiono Find a Euclidean embedding,and then perform traditional learning algorithms in the Euclidean space.5Simplicity6Simplicity7Simplicity is relative8Manifold-based Dimensionality Reduc
4、tionoGiven high dimensional data sampled from a low dimensional manifold,how to compute a faithful embedding?oHow to find the mapping function?oHow to efficiently find the projective function?fff9A Good Mapping Function o If xi and xj are close to each other,we hope f(xi)and f(xj)preserve the local
5、structure(distance,similarity)o k-nearest neighbor graph:o Objective function:n Different algorithms have different concerns10Locality Preserving ProjectionsPrinciple:if xi and xj are close,then their maps yi and yj are also close.11Locality Preserving ProjectionsPrinciple:if xi and xj are close,the
6、n their maps yi and yj are also close.Mathematical formulation:minimize the integral of the gradient of f.12Locality Preserving ProjectionsPrinciple:if xi and xj are close,then their maps yi and yj are also close.Mathematical formulation:minimize the integral of the gradient of f.Stokes Theorem:13Lo
7、cality Preserving ProjectionsPrinciple:if xi and xj are close,then their maps yi and yj are also close.Mathematical formulation:minimize the integral of the gradient of f.Stokes Theorem:LPP finds a linear approximation to nonlinear manifold,while preserving the local geometric structure.14Manifold o
8、f Face ImagesExpression(Sad Happy)Pose(Right Left)15Manifold of Handwritten DigitsThicknessSlant16o Learning target:o Training Examples:o Linear Regression ModelActive and Semi-Supervised Learning:A Geometric Perspective17Generalization Erroro Goal of RegressionObtain a learned function that minimiz
9、es the generalization error(expected error for unseen test input points).o Maximum Likelihood Estimate18Gauss-Markov TheoremFor a given x,the expected prediction error is:19-4-3-2-10123400.10.20.30.40.50.60.70.8-4-3-2-10123400.10.20.30.40.50.60.70.8Gauss-Markov TheoremFor a given x,the expected pred
10、iction error is:Good!Bad!20Experimental Design MethodsThree most common scalar measures of the size of the parameter(w)covariance matrix:o A-optimal Design:determinant of Cov(w).o D-optimal Design:trace of Cov(w).o E-optimal Design:maximum eigenvalue of Cov(w).Disadvantage:these methods fail to take
11、 into account unmeasured(unlabeled)data points.21Manifold Regularization:Semi-Supervised Settingo Measured(labeled)points:discriminant structureo Unmeasured(unlabeled)points:geometrical structure?22o Measured(labeled)points:discriminant structureo Unmeasured(unlabeled)points:geometrical structure?ra
12、ndom labelingManifold Regularization:Semi-Supervised Setting23o Measured(labeled)points:discriminant structureo Unmeasured(unlabeled)points:geometrical structure?random labelingactive learningactive learning+semi-supervsed learningManifold Regularization:Semi-Supervised Setting24Unlabeled Data to Es
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