### Abstract

We present a novel method to tackle the multi-class classification problem with sparse grids and show how the computational procedure can be split into an Offline phase (pre-processing) and a very rapid Online phase. For each class of the training data the underlying probability density function is estimated on a sparse grid. The class of a new data point is determined by the values of the density functions at this point. Our classification method can deal with more than two classes in a natural way and it provides a stochastically motivated confidence value which indicates how to rate the respond to a new point. Furthermore, the underlying density estimation method allows us to pre-compute the system matrix and store it in an appropriate format. This so-called Offline/Online splitting of the computational procedure allows an Online phase where only a few matrix vector products are necessary to learn a new, previously unseen training data set. In particular, we do not have to solve a system of linear equations anymore. We show that speed ups by a factor of several hundred are possible. A typical application for such an Offline/Online splitting is cross validation. We present the algorithm and the computational procedure for our classification method, report on the employed density estimation method on sparse grids and show by means of artificial and real world data sets that we obtain competitive results compared to the classical sparse grid classification method based on regression.

Original language | English (US) |
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Title of host publication | Sparse Grids and Applications - Munich 2012 |

Editors | Dirk Pfluger, Jochen Garcke |

Publisher | Springer-Verlag |

Pages | 255-270 |

Number of pages | 16 |

ISBN (Electronic) | 9783319045368 |

State | Published - Jan 1 2014 |

Event | 2nd Workshop on Sparse Grids and Applications, SGA 2012 - Gammarth, Tunisia Duration: Jul 2 2012 → Jul 6 2012 |

### Publication series

Name | Lecture Notes in Computational Science and Engineering |
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Volume | 97 |

ISSN (Print) | 1439-7358 |

### Other

Other | 2nd Workshop on Sparse Grids and Applications, SGA 2012 |
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Country | Tunisia |

City | Gammarth |

Period | 7/2/12 → 7/6/12 |

### Fingerprint

### ASJC Scopus subject areas

- Modeling and Simulation
- Engineering(all)
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Mathematics

### Cite this

*Sparse Grids and Applications - Munich 2012*(pp. 255-270). (Lecture Notes in Computational Science and Engineering; Vol. 97). Springer-Verlag.

**Classification with probability density estimation on sparse grids.** / Peherstorfer, Benjamin; Franzelin, Fabian; Pflüger, Dirk; Bungartz, Hans Joachim.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Sparse Grids and Applications - Munich 2012.*Lecture Notes in Computational Science and Engineering, vol. 97, Springer-Verlag, pp. 255-270, 2nd Workshop on Sparse Grids and Applications, SGA 2012, Gammarth, Tunisia, 7/2/12.

}

TY - GEN

T1 - Classification with probability density estimation on sparse grids

AU - Peherstorfer, Benjamin

AU - Franzelin, Fabian

AU - Pflüger, Dirk

AU - Bungartz, Hans Joachim

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We present a novel method to tackle the multi-class classification problem with sparse grids and show how the computational procedure can be split into an Offline phase (pre-processing) and a very rapid Online phase. For each class of the training data the underlying probability density function is estimated on a sparse grid. The class of a new data point is determined by the values of the density functions at this point. Our classification method can deal with more than two classes in a natural way and it provides a stochastically motivated confidence value which indicates how to rate the respond to a new point. Furthermore, the underlying density estimation method allows us to pre-compute the system matrix and store it in an appropriate format. This so-called Offline/Online splitting of the computational procedure allows an Online phase where only a few matrix vector products are necessary to learn a new, previously unseen training data set. In particular, we do not have to solve a system of linear equations anymore. We show that speed ups by a factor of several hundred are possible. A typical application for such an Offline/Online splitting is cross validation. We present the algorithm and the computational procedure for our classification method, report on the employed density estimation method on sparse grids and show by means of artificial and real world data sets that we obtain competitive results compared to the classical sparse grid classification method based on regression.

AB - We present a novel method to tackle the multi-class classification problem with sparse grids and show how the computational procedure can be split into an Offline phase (pre-processing) and a very rapid Online phase. For each class of the training data the underlying probability density function is estimated on a sparse grid. The class of a new data point is determined by the values of the density functions at this point. Our classification method can deal with more than two classes in a natural way and it provides a stochastically motivated confidence value which indicates how to rate the respond to a new point. Furthermore, the underlying density estimation method allows us to pre-compute the system matrix and store it in an appropriate format. This so-called Offline/Online splitting of the computational procedure allows an Online phase where only a few matrix vector products are necessary to learn a new, previously unseen training data set. In particular, we do not have to solve a system of linear equations anymore. We show that speed ups by a factor of several hundred are possible. A typical application for such an Offline/Online splitting is cross validation. We present the algorithm and the computational procedure for our classification method, report on the employed density estimation method on sparse grids and show by means of artificial and real world data sets that we obtain competitive results compared to the classical sparse grid classification method based on regression.

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M3 - Conference contribution

AN - SCOPUS:84927647167

T3 - Lecture Notes in Computational Science and Engineering

SP - 255

EP - 270

BT - Sparse Grids and Applications - Munich 2012

A2 - Pfluger, Dirk

A2 - Garcke, Jochen

PB - Springer-Verlag

ER -