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This study delves into the intricate dynamics of trained deep neural networksand their relationships with network parameters. Trained networks predominantlycontinue training in a single direction, known as the drift mode. This driftmode can be explained by the quadratic potential model of the loss function,suggesting a slow exponential decay towards the potential minima. We unveil acorrelation between Hessian eigenvectors and network weights. Thisrelationship, hinging on the magnitude of eigenvalues, allows us to discernparameter directions within the network. Notably, the significance of thesedirections relies on two defining attributes: the curvature of their potentialwells (indicated by the magnitude of Hessian eigenvalues) and their alignmentwith the weight vectors. Our exploration extends to the decomposition of weightmatrices through singular value decomposition. This approach proves practicalin identifying critical directions within the Hessian, considering both theirmagnitude and curvature. Furthermore, our examination showcases theapplicability of principal component analysis in approximating the Hessian,with update parameters emerging as a superior choice over weights for thispurpose. Remarkably, our findings unveil a similarity between the largestHessian eigenvalues of individual layers and the entire network. Notably,higher eigenvalues are concentrated more in deeper layers. Leveraging theseinsights, we venture into addressing catastrophic forgetting, a challenge ofneural networks when learning new tasks while retaining knowledge from previousones. By applying our discoveries, we formulate an effective strategy tomitigate catastrophic forgetting, offering a possible solution that can beapplied to networks of varying scales, including larger architectures.

Hessian Eigenvectors and Principal Component Analysis of Neural Network Weight Matrices