Partitioning points by parallel planes

In this paper we consider S-threshold functions defined on not necessarily binary input. By S-threshold function, in an arbitrary dimension we mean a function which can be written as a linear combination of monomials from a predefined set. First, we determine sets of discrete moments which uniquely determine such functions. Based on these, we derive a generic formula for the upper bound of the functions considered. The formula is generic because it works in all dimensions, on any input size, and for any set of monomials used to define certain S-threshold function. Even though the formula is very generic it gives some improvements of the well-known results.

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“…. , x n ) separates the sets f −1 (1)\g −1 (1) and g −1 (1)\ f −1 (1). Finally, Theorem 2.1 establishes the proof.…”

Section: Preliminariesmentioning

confidence: 75%

“…. , x n ) that separates the sets f −1 (1) \ g −1 (1) Now, we define the required auxiliary function:…”

Section: A Examplementioning

confidence: 99%

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On the Number of S-Threshold Functions on Not Necessarily Binary Input

Pantović

Ghilezan

unic

2014

2014 IEEE 44th International Symposium on Multiple-Valued Logic

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“…Multithreshold Entropy Linear Classifier (MELC [5]) has been recently proposed as an information theoretic approach for building model from the multithreshold linear family [1]. It's core idea is to find a linear operator v (with unit norm) such that kernel density estimations of projected classes' training samples maximize the Cauchy-Schwarz Divergence (D CS [9]).…”

Section: Multithreshold Entropy Linear Classifiermentioning

confidence: 99%

Fast optimization of Multithreshold Entropy Linear Classifier

Jozefowicz,

Czarnecki

2015

Preprint

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Multithreshold Entropy Linear Classifier (MELC) is a density based model which searches for a linear projection maximizing the Cauchy-Schwarz Divergence of dataset kernel density estimation. Despite its good empirical results, one of its drawbacks is the optimization speed. In this paper we analyze how one can speed it up through solving an approximate problem. We analyze two methods, both similar to the approximate solutions of the Kernel Density Estimation querying and provide adaptive schemes for selecting a crucial parameters based on user-specified acceptable error. Furthermore we show how one can exploit well known conjugate gradients and L-BFGS optimizers despite the fact that the original optimization problem should be solved on the sphere. All above methods and modifications are tested on 10 real life datasets from UCI repository to confirm their practical usability.

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On data classification by iterative linear partitioning

Anthony

2004

Discrete Applied Mathematics

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Partitioning points by parallel planes

Cited by 7 publications

References 10 publications

On the Number of S-Threshold Functions on Not Necessarily Binary Input

On the Number of S-Threshold Functions on Not Necessarily Binary Input

Fast optimization of Multithreshold Entropy Linear Classifier

On data classification by iterative linear partitioning

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