Estimating Global Input Relevance and Enforcing Sparse Representations with a Scalable Spectral Neural Network Approach

TL;DR

The paper addresses the challenge of identifying globally relevant input features for deep neural networks and enabling sparse representations for explainability. It introduces a spectral parametrization in which a transfer operator $A=\Phi\Lambda\Phi^{-1}$ is optimized, and input relevance is encoded in the eigenvalues $\lambda_i$, allowing a byproduct feature ranking during training and regularization-based sparsification. The approach is validated across independent Gaussian, correlated Gaussian, MNIST, and stellar spectra datasets, showing that nonzero eigenvalues correctly flag relevant inputs, align with domain-discriminative metrics, and enable accurate performance with only a small subset of inputs. Compared to post-hoc methods like SHAP, LRP, and gradient-based explanations, the spectral method yields a fast, globally applicable relevance score and directly enforces sparsity during training, signaling a practical path toward explainable and compressible neural models for scientific applications.

Abstract

In machine learning practice it is often useful to identify relevant input features. Isolating key input elements, ranked according their respective degree of relevance, can help to elaborate on the process of decision making. Here, we propose a novel method to estimate the relative importance of the input components for a Deep Neural Network. This is achieved by leveraging on a spectral re-parametrization of the optimization process. Eigenvalues associated to input nodes provide in fact a robust proxy to gauge the relevance of the supplied entry features. Notably, the spectral features ranking is performed automatically, as a byproduct of the network training, with no additional processing to be carried out. Moreover, by leveraging on the regularization of the eigenvalues, it is possible to enforce solutions making use of a minimum subset of the input components, increasing the explainability of the model and providing sparse input representations. The technique is compared to the most common methods in the literature and is successfully challenged against both synthetic and real data.

Figures (20)

• Figure 1: The linear information flow for a modified network with the inclusion of diagonal entries (eigenvalues) in the triangular matrix $A$. The eigenvalues introduce self-loops (as depicted on the right), and this is at variance with the usual setting without loops (on the left). Notice that the bottom elements of $\textbf{z}^l$ are identical in both cases, yielding exactly the same activation vector upon linear transfer
• Figure 2: Input vector components distributions. Distribution of the values of the 20 components of the input vectors. The distributions for the data belonging to the first (resp. second) class are represented in red (resp. blue). For some components, the two classes are indistinguishable, while in some other cases the two distributions are only partially overlapping.
• Figure 3: The left panel displays the 20 post-training eigenvalues associated to the input layer, for the simple Gaussian dataset. The red dots and the red bars correspond to the average values and the standard deviations of the results obtained by repeating the experiments for 10 independent realizations. In the right panel, the same post-training eigenvalues are reported against a relevance parameter that gauges the distance between the distributions (respectively blue and red, in Fig. \ref{['fig:input_distributions_gauss']}) from which the components are eventually drawn. The distance is defined as the absolute value of the difference between the two means.
• Figure 4: Left column: the distributions of the values of the fourth and fifth components of the input vectors in the correlated Gaussian dataset are shown in the first two panels. In the bottom-left panel, the fifth component is plotted against the corresponding fourth component for some input vectors of the same dataset. Right column: the upper-right panel reports the ten post-training eigenvalues (refereed to the input layer) for the correlated Gaussian dataset. Red dots and red bars quantifies the recorded average values and the associated standard deviations, as obtained by averaging over 10 independent realizations. In the bottom-right panel the same post-training eigenvalues are plotted against the parameter $p_n$, as defined in the main text.
• Figure 5: In panel (a) the distribution of the first $784$ post-training eigenvalues (normalized to their maximum) is reported. Panel (b) shows the input space for the same dataset, where each component (i.e., pixel) is colored according to its relative post-training eigenvalue.