# Spectral-graph Based Classifications: Linear Regression for Classification and Normalized Radial Basis Function Network

## TL;DR

This paper reveals how spectral graph theory underpins supervised classification methods like linear regression and RBF networks, providing insights into error-overfitting tradeoffs, regularization, and practical basis selection for improved model performance.

## Contribution

It introduces a spectral graph theory framework for supervised classification, deriving bounds and insights for linear regression and RBF networks, and proposes a basis selection strategy for practical implementation.

## Key findings

- Spectral risk of nRBFN is bounded by number of classes and basis size.
- Regularization controls overfitting by balancing fitting error and spectral risk.
- nRBFN achieves comparable performance to SVM with easier parameter tuning.

## Abstract

Spectral graph theory has been widely applied in unsupervised and semi-supervised learning. In this paper, we find for the first time, to our knowledge, that it also plays a concrete role in supervised classification. It turns out that two classifiers are inherently related to the theory: linear regression for classification (LRC) and normalized radial basis function network (nRBFN), corresponding to linear and nonlinear kernel respectively. The spectral graph theory provides us with a new insight into a fundamental aspect of classification: the tradeoff between fitting error and overfitting risk. With the theory, ideal working conditions for LRC and nRBFN are presented, which ensure not only zero fitting error but also low overfitting risk. For quantitative analysis, two concepts, the fitting error and the spectral risk (indicating overfitting), have been defined. Their bounds for nRBFN and LRC are derived. A special result shows that the spectral risk of nRBFN is lower bounded by the number of classes and upper bounded by the size of radial basis. When the conditions are not met exactly, the classifiers will pursue the minimum fitting error, running into the risk of overfitting. It turns out that $\ell_2$-norm regularization can be applied to control overfitting. Its effect is explored under the spectral context. It is found that the two terms in the $\ell_2$-regularized objective are one-one correspondent to the fitting error and the spectral risk, revealing a tradeoff between the two quantities. Concerning practical performance, we devise a basis selection strategy to address the main problem hindering the applications of (n)RBFN. With the strategy, nRBFN is easy to implement yet flexible. Experiments on 14 benchmark data sets show the performance of nRBFN is comparable to that of SVM, whereas the parameter tuning of nRBFN is much easier, leading to reduction of model selection time.