Classification by estimating the cumulative distribution function for small data

TL;DR

This paper introduces a novel classification approach for small data by estimating the cumulative distribution function through Fredholm equations, leading to improved interpretability and effectiveness in small data scenarios.

Contribution

It proposes a new risk estimation framework based on distribution estimation and introduces the $ ext{ extepsilon}$-$L_{1} ext{VSVM}$ model for small data classification.

Findings

The proposed method effectively classifies small data sets.

Experimental results validate the interpretability of the new model.

The cumulative distribution function indicator improves classification validity.

Abstract

In this paper, we study the classification problem by estimating the conditional probability function of the given data. Different from the traditional expected risk estimation theory on empirical data, we calculate the probability via Fredholm equation, this leads to estimate the distribution of the data. Based on the Fredholm equation, a new expected risk estimation theory by estimating the cumulative distribution function is presented. The main characteristics of the new expected risk estimation is to measure the risk on the distribution of the input space. The corresponding empirical risk estimation is also presented, and an $\varepsilon$-insensitive $L_{1}$ cumulative support vector machines ($\varepsilon$-$L_{1}VSVM$) is proposed by introducing an insensitive loss. It is worth mentioning that the classification models and the classification evaluation indicators based on the new mechanism are different from the traditional one. Experimental results show the effectiveness of the proposed $\varepsilon$-$L_{1}VSVM$ and the corresponding cumulative distribution function indicator on validity and interpretability of small data classification.