Tracing sources of epistemic uncertainty in deep learning predictions: homo- and hetero-scedastic linearized estimators

We adapt two classical statistical estimators for quantifying uncertainty to modern deep learning, in order to provide clearer insights into uncertainty attributable to two sources : aleatoric uncertainty, or locally scarce data. Our approach leverages recent advances in approximate Fisher Information Matrices, to enable scaling to actual architectures. Experimental results demonstrate how each test points is differentially impacted by both sources, highlighting the practical utility of our estimators in improving the robustness of real-world applications.

From Uncertainty to Stability and Fidelity: Guiding Sparse-View 3D Gaussian Splatting with Fisher Information

3D Gaussian Splatting (3DGS) has emerged as a promising technique for novel view synthesis. However, 3DGS requires dense input views to achieve high-quality rendering. In sparse-view scenarios, 3DGS often prones to overfitting, resulting in noticeable artifacts and degraded rendering quality. Previous methods explore to address this issue by introducing additional priors (e.g. depth priors) or integrating regularization techniques (e.g. Dropout). However, these methods are often applied without principled guidance. In particular, prior-based augmentation typically samples novel viewpoints randomly, while Dropout-based regularization randomly removes Gaussians. The compounded randomness introduces uncertainty and instability, limiting the fidelity of novel view synthesis. In this paper, we propose a novel method for sparse-view 3DGS that incorporates Fisher Information to quantitatively guide the utilization of geometric priors and regularization. Specifically, our method comprises two key components: (1) Stereo augmentation with Fisher Information. By leveraging Fisher Information, we actively select most informative supporting views and use depth priors to curate reliable pseudo ground truths, which reduces randomness in augmentation and improves stability and rendering fidelity; (2) Uncertainty-aware regularization. We reduce the instability of Dropout-based regularization by using Fisher Information to quantitatively measure the uncertainty of each 3D Gaussian, and adaptively adjust the removal probability, leading to more stable and effective regularization. With these two components, our method effectively mitigates overfitting and improves the stability of optimization in sparse-view 3DGS, resulting in superior rendering fidelity. Extensive experiments show that our method achieves state-of-the-art performance in sparse-view novel view synthesis benchmarks.

Sparsity Curse: Understanding RLVR Model Parameter Space from Model Merging

Reinforcement Learning with Verifiable Reward (RLVR) has emerged as a powerful post-training paradigm that surpasses Supervised Fine-Tuning (SFT) in eliciting reasoning intelligence and resisting catastrophic forgetting. Recent studies further reveal that RLVR induces highly sparse and off-principal parameter updates compared to SFT. This naturally raises the question: does such sparsity make RLVR models more amenable to model merging? If so, model merging would offer a scalable, training-free path to aggregate diverse reasoning capabilities from independently trained RLVR models. Surprisingly, we find the opposite, uncovering a sparsity curse: the sparse RLVR updates are spread farther apart in parameter space, forming near-orthogonal shortcuts that make aggregation inherently fragile. This is likely rooted in the stochasticity of RL optimization and the diversity of emergent reasoning patterns. Unlike SFT models that converge to shared, flat basins and merge naturally, RLVR models suffer severe degradation under standard merging methods. Through systematic empirical analysis of the update geometry, we characterize the mechanisms behind this failure and propose Sensitivity-aware Resolving Merging (SAR-Merging), a merging recipe tailored for the unique structure of RLVR parameter spaces. SAR-Merging resolves conflicts in overlapping update regions via Fisher Information-based sensitivity arbitration, followed by magnitude-aware sparsification and rescaling to preserve fragile reasoning pathways. Experiments on mathematical and coding benchmarks demonstrate that SAR-Merging substantially outperforms existing merging methods on RLVR models, enabling both single-task enhancement and multi-capability fusion.

Effective Dimensionality as an Operator Invariant for Physics-Preserving Constraint Adaptation in Physics-Informed Neural Networks

Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom ($d_{eff}$) in a physics-constrained model. Unlike the classical $d_{eff}$ which measures how many parameter directions are informed by data against a statistical prior, our $d_{eff}$ measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite-dimensional kernel, we show that $d_{eff}$ converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite-dimensional kernel, $d_{eff}$ instead measures the network's finite-dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, $d_{eff}$ also serves as an a priori structural diagnostic. Driving $d_{eff}$ of a well-posed problem to zero certifies that the physics and boundary constraints have absorbed the network's free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre-trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient-based fine-tuning can match or exceed this but needs more wall-clock time and tuning, whereas subspace projection delivers near-equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.