Regularization via f-Divergence: An Application to Multi-Oxide Spectroscopic Analysis

TL;DR

This paper proposes a novel regularization method based on f-divergence to predict the multi-oxide weights of rock samples based on spectroscopic data collected under Martian conditions, and develops a differentiable f-divergence and incorporates it into the f-divergence regularization, making the network training feasible.

Abstract

In this paper, we address the task of characterizing the chemical composition of planetary surfaces using convolutional neural networks (CNNs). Specifically, we seek to predict the multi-oxide weights of rock samples based on spectroscopic data collected under Martian conditions. We frame this problem as a multi-target regression task and propose a novel regularization method based on f-divergence. The f-divergence regularization is designed to constrain the distributional discrepancy between predictions and noisy targets. This regularizer serves a dual purpose: on the one hand, it mitigates overfitting by enforcing a constraint on the distributional difference between predictions and noisy targets. On the other hand, it acts as an auxiliary loss function, penalizing the neural network when the divergence between the predicted and target distributions becomes too large. To enable backpropagation during neural network training, we develop a differentiable f-divergence and incorporate it into the f-divergence regularization, making the network training feasible. We conduct experiments using spectra collected in a Mars-like environment by the remote-sensing instruments aboard the Curiosity and Perseverance rovers. Experimental results on multi-oxide weight prediction demonstrate that the proposed $f$-divergence regularization performs better than or comparable to standard regularization methods including $L_1$, $L_2$, and dropout. Notably, combining the $f$-divergence regularization with these standard regularization further enhances performance, outperforming each regularization method used independently.

Figures (8)

• Figure 1: True function and various approximation functions are compared, along with their training and test mean squared errors (MSE). The training MSE implicitly quantifies the divergence between the training targets and predictions. Consequently, the function approximation in (a), which maintains an appropriate level of divergence, achieves a smaller test error compared to the approximations in (b) and (c), where the divergences are too small and large, respectively.
• Figure 2: (a) and (b) illustrate two scenarios of samples generated from $p_0$ and $p_1$. Samples generated from $p_0$ and $p_1$ are represented as red and blue nodes and a nearest neighbour graph is constructed over the nodes. Green edges denote the edges connecting nodes from different samples. In (a), the red nodes are farther from the blue nodes, resulting in a smaller cut-edge number, indicating a larger$f$-divergence. In contrast, in (b), the red and blue nodes are closer, leading to a larger cut-edge number, which indicates a smaller$f$-divergence.
• Figure 3: Candidates for $L_2$ regularized curves are highlighted in blue. These curves avoid overfitting to the noisy data. By accounting for the presence of $f$-divergence between noisy data and predictions made by the target function, candidate curves with divergence exceeding a specified threshold ($\gamma$ in \ref{['eq_regularizer']}) are eliminated. This yields a final selected curve that maintains an appropriate $f$-divergence and closely approximates the target function.
• Figure 4: (a) and (b) illustrate two scenarios of predictions (red nodes) and targets (blue nodes) in a fully connected graph, where edge weights are inversely proportional to the distance between nodes. The green edges represent the edges between prediction and target nodes. In (a), the red nodes are farther from the blue nodes, resulting in a smaller sum of green edge weights, indicating a larger$f$-divergence. In contrast, in (b), the red and blue nodes are closer, leading to a larger sum of green edge weights, which indicates a smaller$f$-divergence.
• Figure 5: Differentiable approximation of the cut-edge ratio $t_n$djolonga2017learning.

Theorems & Definitions (3)

• Definition 1
• Theorem 1
• proof