Unequal Covariance Awareness for Fisher Discriminant Analysis and Its Variants in Classification

Fisher Discriminant Analysis (FDA) is one of the essential tools for feature extraction and classification. In addition, it motivates the development of many improved techniques based on the FDA to adapt to different problems or data types. However, none of these approaches make use of the fact that the assumption of equal covariance matrices in FDA is usually not satisfied in practical situations. Therefore, we propose a novel classification rule for the FDA that accounts for this fact, mitigating the effect of unequal covariance matrices in the FDA. Furthermore, since we only modify the classification rule, the same can be applied to many FDA variants, improving these algorithms further. Theoretical analysis reveals that the new classification rule allows the implicit use of the class covariance matrices while increasing the number of parameters to be estimated by a small amount compared to going from FDA to Quadratic Discriminant Analysis. We illustrate our idea via experiments, which show the superior performance of the modified algorithms based on our new classification rule compared to the original ones.

FinNet: Solving Time-Independent Differential Equations with Finite Difference Neural Network

In recent years, deep learning approaches for partial differential equations have received much attention due to their mesh-freeness and other desirable properties. However, most of the works so far concentrated on time-dependent nonlinear differential equations. In this work, we analyze potential issues with the well-known Physic Informed Neural Network for differential equations that are not time-dependent. This analysis motivates us to introduce a novel technique, namely FinNet, for solving differential equations by incorporating finite difference into deep learning. Even though we use a mesh during the training phase, the prediction phase is mesh-free. We illustrate the effectiveness of our method through experiments on solving various equations.