The block-diagonal approximation used in the ROUND step of the Approx-FIRAL algorithm presents several potential limitations. One significant limitation is that this approximation may oversimplify the structure of the Fisher Information matrices, leading to a loss of information about the interactions between different classes. This could result in suboptimal point selection, particularly in cases where class boundaries are complex and not well-represented by block-diagonal structures.
Another limitation is that the block-diagonal approximation may not adequately capture the correlations between features within each class, which can be crucial for accurate classification. This could be particularly problematic in high-dimensional spaces where feature interactions play a significant role in model performance.
To improve this approximation, one approach could be to incorporate a low-rank approximation of the Fisher Information matrices instead of a strict block-diagonal form. This would allow for capturing more complex relationships between classes while still maintaining computational efficiency. Techniques such as Principal Component Analysis (PCA) or Singular Value Decomposition (SVD) could be employed to identify and retain the most informative components of the Fisher Information matrices.
Additionally, hybrid approaches that combine the block-diagonal structure with other forms of regularization or constraints could enhance the robustness of the ROUND step. For instance, incorporating a penalty for deviations from the block-diagonal structure could help maintain the benefits of the approximation while allowing for some flexibility in capturing class interactions.