Kernel Methods for Implicit Surface Modeling

B. Scholkopf, J. Giesen, S. Spalinger

r92922120 黃邦洪r93922020 楊惠菁r93922038 柯政宏r93944009 劉弘偉

Outline

Introduction Single-Class SVMs Slab SVMs Epsilon-SVR Implement Result

Introduction

Step 1: Given a finite sample points belong to some

hypersurface in Euclidean space R3

The sample can be very large, noisy or sparse

Introduction

Step 2: Use SVM to find an implicit surface

f is the decision function in SVM

The zero set of f is a surface that approximates

the sample points

Single-Class SVM v.s. Slab SVM

3:f

)0(1f

Outline

Introduction Single-Class SVMs Slab SVMs Epsilon-SVR Implement Result

Single-Class SVM

Primary Object function

Concept

Single-Class SVM

Dual Problem

Decision function

Single-Class SVM

SVM TOY example

Outline

Introduction Single-Class SVMs Slab SVMs Epsilon-SVR Implement Result

Slab SVM

Primary object function

Concept

Slab SVM

Lagrangian dual optimization problem

Decision function

Slab SVM

SVM TOY example

Outline

Introduction Single-Class SVMs Slab SVMs Epsilon-SVR Implement Result

Epsilon-SVR

Primary object function

Concept

Epsilon-SVR

Lagrangian dual optimization problem

The approximate function

Outline

Introduction Single-Class SVMs Slab SVMs Epsilon-SVR Implement Result

Implement

Single-Class SVM Use LIBSVM directly

Slab-SVM Modify epsilon-SVR in LIBSVM to slab-SVM The result is parameter-sensitive Modification:

decision function

The result is good but we don’t really know how to explain

Outline

Introduction Single-Class SVMs Slab SVMs Epsilon-SVR Implement Result

Result(1) - One-Class SVM

Bunny Original: 35947 One-Class: 2810

Result(1) - Slab SVM

Bunny Left: 8307。 Right: 1013

Result(1) - Slab SVM

Bunny Predict accuracy: 97.74%

Result(2) - One-Class SVM

Cactus Original: 3337 One-Class: 1617

Result(2) - Slab SVM

Cactus Left: 996 Right: 817

Result(2) - Slab SVM

Cactus Predict accuracy: 96.9434%

Result(3) - One-Class SVM

Dinosaur Original: 56194 One-Class: 44963

Result(3) - Slab SVM

Dinosaur Left: 13005。 Right: 1018

Result(3) - Slab SVM

Dinosaur Predict accuracy: 97.43%

Result(4) - One-Class SVM

Knot Original: 10000 One-Class: 8807

Result(4) - Slab SVM

Knot Left: 1176。 Right: 1036

Result(4) - Slab SVM

Knot Predict accuracy: 96.9%

Result(5) - One-Class SVM

Screwdriver Original: 27152 One-Class:21728

Result(5) - Slab SVM

Screwdriver Left: 6574 Right: 884

Result(5) - Slab SVM

Screwdriver Predict accuracy: 97.1752%

Result(6) - One-Class SVM

Rockerarm Original: 40177 One-Class: 4868

Result(6) - Slab SVM

Rockerarm Left: 2090 Right: 1029

Result(6) - Slab SVM

Rocker arm Predict accuracy: 96.18%

Result(7) - One-Class SVM

Hole Original: 4000 One-Class: 3224

Result(7) - Slab SVM

Hole Left: 648 Right:1039

Result(7) - Slab SVM

Hole Predict accuracy: 93.475%

Q&A

Thank You