We use ReLU as the activation function.

A.1 Majority Voting The Majority Voting (MV) is the most intuitive algorithm for aggregate LFs' annotations. We omit this case for simplicity. A.3 Snorkel MeTaL The parameters µof Snorkel MeTaL [31] are given by Bayes' theorem we have: pµ(y = c,λ = m) = pµ(λ = m | y = c)p(y = c) = Consider a label model g(L(x),x) F in arbitrary functional class F, e.g., neural network, and having additional dependency on data feature x4, we can still approximate such complicated function with identity function-based label model g W(x)(L(x)) similar to the aforementioned one except that W(x): X RM (C+1) C is a similarly complicated function, e.g., neural network, that maps each data x X to a unique label model parameter W(x). We leave the exploration of more complicated form of label models into future work. B.1 Case 1: identity function We define the loss with reweighted sample as, Instead of employing the decomposing loss function, we introduce a more general influence estimation method - weight-moving Influence, which get ride of the loss decomposition and approximation and is agnostic to the selection of σ() function.

Programmatic Weak Supervision (PWS) aggregates the source votes of multiple weak supervision sources into probabilistic training labels, which are in turn used to train an end model. With its increasing popularity, it is critical to have some tool for users to understand the influence of each component (e.g., the source vote or training data) in the pipeline and interpret the end model behavior. To achieve this, we build on Influence Function (IF) and propose source-aware IF2, which leverages the generation process of the probabilistic labels to decompose the end model's training objective and then calculate the influence associated with each (data, source, class) tuple. These primitive influence score can then be used to estimate the influence of individual component of PWS, such as source vote, supervision source, and training data. On datasets of diverse domains, we demonstrate multiple use cases: (1) interpreting incorrect predictions from multiple angles that reveals insights for debugging the PWS pipeline, (2) identifying mislabeling of sources with a gain of 9%-37% over baselines, and (3) improving the end model's generalization performance by removing harmful components in the training objective (13%-24% better than ordinary IF).

This study presents a conditional flow matching framework for solving physics-constrained Bayesian inverse problems. In this setting, samples from the joint distribution of inferred variables and measurements are assumed available, while explicit evaluation of the prior and likelihood densities is not required. We derive a simple and self-contained formulation of both the unconditional and conditional flow matching algorithms, tailored specifically to inverse problems. In the conditional setting, a neural network is trained to learn the velocity field of a probability flow ordinary differential equation that transports samples from a chosen source distribution directly to the posterior distribution conditioned on observed measurements. This black-box formulation accommodates nonlinear, high-dimensional, and potentially non-differentiable forward models without restrictive assumptions on the noise model. We further analyze the behavior of the learned velocity field in the regime of finite training data. Under mild architectural assumptions, we show that overtraining can induce degenerate behavior in the generated conditional distributions, including variance collapse and a phenomenon termed selective memorization, wherein generated samples concentrate around training data points associated with similar observations. A simplified theoretical analysis explains this behavior, and numerical experiments confirm it in practice. We demonstrate that standard early-stopping criteria based on monitoring test loss effectively mitigate such degeneracy. The proposed method is evaluated on several physics-based inverse problems. We investigate the impact of different choices of source distributions, including Gaussian and data-informed priors. Across these examples, conditional flow matching accurately captures complex, multimodal posterior distributions while maintaining computational efficiency.

The remarkable capability of over-parameterised neural networks to generalise effectively has been explained by invoking a "simplicity bias": neural networks prevent overfitting by initially learning simple classifiers before progressing to

Many real-world datasets contain hidden structure that cannot be detected by simple linear correlations between input features. For example, latent factors may influence the data in a coordinated way, even though their effect is invisible to covariance-based methods such as PCA. In practice, nonlinear neural networks often succeed in extracting such hidden structure in unsupervised and self-supervised learning. However, constructing a minimal high-dimensional model where this advantage can be rigorously analyzed has remained an open theoretical challenge. We introduce a tractable high-dimensional spiked model with two latent factors: one visible to covariance, and one statistically dependent yet uncorrelated, appearing only in higher-order moments. PCA and linear autoencoders fail to recover the latter, while a minimal nonlinear autoencoder provably extracts both. We analyze both the population risk, and empirical risk minimization. Our model also provides a tractable example where self-supervised test loss is poorly aligned with representation quality: nonlinear autoencoders recover latent structure that linear methods miss, even though their reconstruction loss is higher.