Spectral Gap Regularization of Neural Networks

We introduce Fiedler regularization, a novel approach for regularizing neural networks that utilizes spectral/graphical information. Existing regularization methods often focus on penalizing weights in a global/uniform manner that ignores the connectivity structure of the neural network. We propose to use the Fiedler value of the neural network's underlying graph as a tool for regularization. We provide theoretical motivation for this approach via spectral graph theory. We demonstrate several useful properties of the Fiedler value that make it useful as a regularization tool. We provide an approximate, variational approach for faster computation during training. We provide an alternative formulation of this framework in the form of a structurally weighted $\text{L}_1$ penalty, thus linking our approach to sparsity induction. We provide uniform generalization error bounds for Fiedler regularization via a Rademacher complexity analysis. We performed experiments on datasets that compare Fiedler regularization with classical regularization methods such as dropout and weight decay. Results demonstrate the efficacy of Fiedler regularization. This is a journal extension of the conference paper by Tam and Dunson (2020).

Multiscale Graph Comparison via the Embedded Laplacian Distance

We introduce a simple and fast method for comparing graphs of different sizes. Existing approaches are often either limited to comparing graphs with the same number of vertices or are computationally unscalable. We propose the Embedded Laplacian Distance (ELD) for comparing graphs of potentially vastly different sizes. Our approach first projects the graphs onto a common, low-dimensional Laplacian embedding space that respects graphical structure. This reduces the problem to that of comparing point clouds in a Euclidean space. A distance can then be computed efficiently via a natural sliced Wasserstein approach. We show that the ELD is a pseudo-metric and is invariant under graph isomorphism. We provide intuitive interpretations of the ELD using tools from spectral graph theory. We test the efficacy of the ELD approach extensively on both simulated and real data. Results obtained are excellent.

Fiedler Regularization: Learning Neural Networks with Graph Sparsity

We introduce a novel regularization approach for deep learning that incorporates and respects the underlying graphical structure of the neural network. Existing regularization methods often focus on dropping/penalizing weights in a global manner that ignores the connectivity structure of the neural network. We propose to use the Fiedler value of the neural network's underlying graph as a tool for regularization. We provide theoretical support for this approach via spectral graph theory. We demonstrate the convexity of this penalty and provide an approximate, variational approach for fast computation in practical training of neural networks. We provide bounds on such approximations. We provide an alternative but equivalent formulation of this framework in the form of a structurally weighted L1 penalty, thus linking our approach to sparsity induction. We performed experiments on datasets that compare Fiedler regularization with traditional regularization methods such as dropout and weight decay. Results demonstrate the efficacy of Fiedler regularization.