To this end, we propose an analogous score function called the "Concretescore",ageneralizationofthe(Stein)scorefordiscretesettings.

Concept Bottleneck Models (CBNMs) are deep learning models that provide interpretability by enforcing a bottleneck layer where predictions are based exclusively on human-understandable concepts. However, this constraint also restricts information flow and often results in reduced predictive accuracy. Concept Sidechannel Models (CSMs) address this limitation by introducing a sidechannel that bypasses the bottleneck and carry additional task-relevant information. While this improves accuracy, it simultaneously compromises interpretability, as predictions may rely on uninterpretable representations transmitted through sidechannels. Currently, there exists no principled technique to control this fundamental trade-off. In this paper, we close this gap. First, we present a unified probabilistic concept sidechannel meta-model that subsumes existing CSMs as special cases. Building on this framework, we introduce the Sidechannel Independence Score (SIS), a metric that quantifies a CSM's reliance on its sidechannel by contrasting predictions made with and without sidechannel information. We propose SIS regularization, which explicitly penalizes sidechannel reliance to improve interpretability. Finally, we analyze how the expressivity of the predictor and the reliance of the sidechannel jointly shape interpretability, revealing inherent trade-offs across different CSM architectures. Empirical results show that state-of-the-art CSMs, when trained solely for accuracy, exhibit low representation interpretability, and that SIS regularization substantially improves their interpretability, intervenability, and the quality of learned interpretable task predictors. Our work provides both theoretical and practical tools for developing CSMs that balance accuracy and interpretability in a principled manner.

Compared to previous score matching algorithms, our method is more scalable to high dimensional data and more stable to optimize.

Diffusion-based inverse problem solvers (DIS) have recently shown outstanding performance in compressed-sensing parallel MRI reconstruction by combining diffusion priors with physical measurement models. However, they typically rely on pre-calibrated coil sensitivity maps (CSMs) and ground truth images, making them often impractical: CSMs are difficult to estimate accurately under heavy undersampling and ground-truth images are often unavailable. We propose Calibration-free Measurement Score-based diffusion Model (C-MSM), a new method that eliminates these dependencies by jointly performing automatic CSM estimation and self-supervised learning of measurement scores directly from k-space data. C-MSM reconstructs images by approximating the full posterior distribution through stochastic sampling over partial measurement posterior scores, while simultaneously estimating CSMs. Experiments on the multi-coil brain fastMRI dataset show that C-MSM achieves reconstruction performance close to DIS with clean diffusion priors -- even without access to clean training data and pre-calibrated CSMs.

Representing probability distributions by the gradient of their density functions has proven effective in modeling a wide range of continuous data modalities.

Representing probability distributions by the gradient of their density functions has proven effective in modeling a wide range of continuous data modalities.

Model calibration seeks to ensure that models produce confidence scores that accurately reflect the true likelihood of their predictions being correct. However, existing calibration approaches are fundamentally tied to datasets of one-hot labels implicitly assuming full certainty in all the annotations. Such datasets are effective for classification but provides insufficient knowledge of uncertainty for model calibration, necessitating the curation of datasets with numerically rich ground-truth confidence values. However, due to the scarcity of uncertain visual examples, such samples are not easily available as real datasets. In this paper, we introduce calibration-aware data augmentation to create synthetic datasets of diverse samples and their ground-truth uncertainty. Specifically, we present \textbf{Calibration-aware Semantic Mixing (CSM)}, a novel framework that generates training samples with mixed class characteristics and annotates them with distinct confidence scores via diffusion models. Based on this framework, we propose calibrated reannotation to tackle the misalignment between the annotated confidence score and the mixing ratio during the diffusion reverse process. Besides, we explore the loss functions that better fit the new data representation paradigm. Experimental results demonstrate that CSM achieves superior calibration compared to the state-of-the-art calibration approaches. Our code is \href{https://github.com/E-Galois/CSM}{available here}.