Score Function

Conformal prediction provides distribution-free uncertainty quantification under exchangeability. However, this assumption is violated by label shift, where the marginal distribution of labels changes while the conditional distribution of inputs given labels remains stable. Under such shifts, standard conformal procedures no longer maintain their intended coverage behavior. Existing approaches address this via importance weighting. They pair the reweighting with residual-based nonconformity scores that ignore predictive uncertainty. The resulting intervals have uniform width. Bayesian conformal methods produce adaptive intervals by leveraging predictive distributions. They evaluate conformity under the source predictive, which is misaligned with the target domain under label shift. We propose the \emph{Label-Shift-Adjusted Bayesian Score} (LSA score), a nonconformity score derived from a posterior predictive tilting identity. This identity shows that the target predictive is an importance-weighted transformation of the source predictive. We use it to derive a direct correction to the Bayesian score. We evaluate the method on molecular property prediction under controlled label shift. The LSA score consistently yields shorter intervals than residual-based and source-based Bayesian scores. Coverage in the target domain remains comparable. Under stronger shift, all methods incur some coverage loss due to pseudo-label-based density-ratio estimation. The LSA score is defined for any source predictive with a tractable log-density. We instantiate it with Bayesian Ridge Regression, where the correction admits a closed form.

A document modeled as a discrete sequence of tokens can be thought of as being generated from a composition of texts from different domains; a README file, for example, moves between prose, code, and configuration. When such a document is corrupted and only frozen domain experts are available, restoring it requires deciding both what is missing and which expert to trust at each position, at test time and without region labels or a trained router. We introduce evidence-aligned local composition, which infers a soft, position-wise weighting over the experts from the marginal evidence of the corrupted observation under a given corruption model, estimating the evidence from the experts' own denoising losses and smoothing the weights across positions. Because the weighting is soft, it recovers a mixture when the true composition is mixed and concentrates on one expert when that suffices. Across a categorical simulator, byte-level experts, and experts fine-tuned from a $1.3$B discrete flow-matching model, the inferred weights track the true regions at $0.85$ field accuracy on naturally mixed scientific documents, and at $0.98$ on constructed mixtures whose regions are lexically disjoint. Restoration improves over a single global weight when the experts are genuinely distinct and reduces to it when they converge, tracking a measure of expert separation.

Maximum likelihood (ML) estimation is a principled and statistically efficient approach for learning probabilistic models. However, for unnormalized models, ML estimation requires evaluating the partition function and differentiating through it, which may not always be tractable. Score matching provides a practically viable alternative that circumvents this obstacle by fitting the score in a way that eliminates dependence on the normalizing constant. We derive the generalized score matching objective on a convex subset of $\mathbb{R}^{d}$ constructively starting from Minimum Probability Flow (MPF) learning, and show how classical score matching as well as domain-adapted variants for non-negative data arise naturally within the proposed framework. We show that the resulting objective is a {\it proper local scoring rule} of second-order, which provides the theoretical guarantee that the true density is recovered when the objective is minimized. Furthermore, for a model belonging to the exponential family, we establish convexity of the objective together with consistency of the finite-sample estimator under standard regularity conditions. Our derivation sheds new light on the scope and applicability of generalized score matching in various problem settings. We compare generalized score matching-based estimators on constrained domains, where the partition function is analytically intractable. We provide experimental results on parameter estimation for model densities belonging to the exponential family defined over convex subsets of $\mathbb{R}^{d}$, and a generative modeling use-case to demonstrate broader applicability of the proposed generalized score matching framework.

Many NLP tasks, such as summarization and extractive question answering, reduce to retrieving relevant content from documents under two constraints: coverage, retaining enough pertinent information to achieve some goal, and conciseness, removing as much irrelevant information as possible. Conformal prediction methods have been used to guarantee coverage, and must be optimized for conciseness through design of a score function. State-of-the-art scoring functions use hand-engineered LLM prompts asking the model to rate the importance of content, but manual prompt engineering is labor-intensive and task-specific. We introduce the Conformal Relevance framework which uses in-context learning example curation and ensembling to create a score function which maintains coverage while improving conciseness with minimal manual input. We demonstrate this framework's application on seven NLP tasks, and also theoretically study the impact of diversity for ensembled conformal scores, giving a complementarity condition that characterizes when ensembling improves worst-case sentence scores, and a saturation bound on ensemble improvement.

As AI systems are increasingly used to draft radiology reports, reliably evaluating their clinical quality remains a critical challenge. Large language model (LLM)-based metrics are now the best-correlated with radiologist judgment, yet they output a single opaque score that neither a clinician nor a model builder can easily interpret or audit. We introduce RadMatch, a multi-stage, LLM-based metric that decomposes report comparison into a structured finding-level matching with significance-aware scoring and error characterization across seven clinical attribute dimensions (status, location, severity, morphology, certainty, longitudinal comparison, and measurement). The main score is the actionable-error count, both interpretable and auditable. Candidate findings are graded correct, partial, or incorrect, and unmatched findings are counted as missed or hallucinated. Triage and actionable safety recall/precision and per-subset views add complementary, deployment-oriented lenses. Across two expert benchmarks, RadMatch is the most clinically aligned metric, matching inter-radiologist agreement on ReXVal and more than doubling the best prior metric on the harder RadEvalExpert. Relying only on few-shot prompting, it is designed to extend to other modalities and anatomies. We will release RadMatch as open-source code with an interactive dashboard for inspecting results.

Minimum Bayes Risk (MBR) decoding enables high-quality text generation by selecting the hypothesis that maximizes a utility metric over sampled pseudo-references. However, it is highly susceptible to metric overfitting: it can irregularly inflate the chosen utility metric at the direct expense of other unoptimized evaluation metrics. To mitigate this, we introduce SVD-MBR, which frames the pairwise utility matrix as a noisy information signal. By computing a low-rank approximation via Singular Value Decomposition (SVD) and retaining only the top-$k$ components, we effectively decouple true consensus from metric noise. Experiments demonstrate that SVD-MBR successfully regularizes decoding, yielding substantial gains across a range of generalized metrics. Furthermore, we reveal that this denoising is metric-dependent: neural metrics encode a robust low-rank consensus ideal for SVD, whereas surface-level metrics struggle to separate signal from metric noise.

In pointwise document reranking, Chain-of-Thought models typically underperform direct scoring models. While existing diagnostics attribute this to inferior classification, score polarization, or calibration breakdown, whether targeted training can bridge this gap remains unclear. Our empirical study first confirms that this gap is stable across scales up to 32B parameters, ruling out model and data capacity confounders. We then apply stress tests utilizing reinforcement learning, fine-grained supervision, and architectural decoupling to explicitly repair these deviations. Although these interventions improve classification accuracy and absolute scores, the relative ranking gap persists. These findings suggest that, within the pointwise scoring paradigm, routing continuous relevance semantics through discrete text constrains ranking signal resolution, revealing a bottleneck that is stable and difficult to overcome under current standard methods, rather than an easily resolvable training bias.

Many kinds of data have structure along one or more axes: words in a sentence, pixels in an image, nodes in a tree, frames in audio, or cells in a 3D volume. Along one axis, order matters: "the dog bit the man" is different from "the man bit the dog." Across independent axes, however, neither composition nor movement should depend on the order of axes: in an image, composing right then down should give the same result as composing down then right, and moving right then down should describe the same relative position as moving down then right. We develop a framework for modeling this kind of multi-axis structure. Each data item carries its content together with a small transformation for each axis. A path connecting two positions defines a journey; the journey operator is the product of per-axis transformations along that path, governing both how data composes along the path and how relative position is described. When the transformations are fixed, our framework recovers Rotary Position Embedding (RoPE) and its multi-dimensional variants. When they depend on the data, the model gains a content-adaptive positional inductive bias. We show exactly when these paths are well-defined: both composition and movement across axes are path-independent precisely when the axis transformations commute. We also prove that, under the stated toral-frame symmetry, cocycle, bilinearity, and norm-preservation assumptions, the resulting pairwise scoring rule must take the form of block-wise rotations, explaining why RoPE-like methods arise naturally. Finally, we use this theory to design JoFormer, a model for value aggregation, and relate it to attention and state-space models (SSMs). Initial experiments across vision, language, and length generalization suggest that these inductive biases can have observable consequences in practice.

Score-based diffusion models have achieved remarkable empirical success in generative modeling, yet their approximation-theoretic foundations remain incomplete. In particular, although classical universal approximation theorems guarantee that neural networks can approximate score functions, it remains unclear whether such approximation guarantees translate into approximation of the probability distributions generated by reverse diffusion processes. In this paper, we establish a rigorous quantitative connection between these two notions. Specifically, we prove that if a neural network approximates the true score function sufficiently accurately, then the probability distribution generated by the corresponding reverse diffusion model is close to the target data distribution in Kullback-Leibler (KL) divergence, up to an irreducible mismatch between the terminal distribution of the forward diffusion process and the prior used to initialize the reverse process. More precisely, we derive an explicit upper bound on the distribution approximation error in terms of the score approximation error, the diffusion noise schedule, and the terminal prior mismatch. Our analysis combines Hornik's universal approximation theorem, Girsanov's theorem on path space, and the data processing inequality for relative entropy. Complementary to recent work that studies score approximation under finite-sample statistical settings and structural assumptions on the data distribution, our work develops an approximation-theoretic analysis based on classical neural network approximation theory. The resulting theorem provides a simple and explicit guarantee linking neural network approximation of score functions to approximation of the probability distributions generated by reverse diffusion models.

Real-time score following from sheet images remains chal- lenging because the model must process streaming au- dio while resolving highly repetitive visual patterns un- der strict latency constraints. Recent image-based meth- ods have attempted to use multi-resolution prediction by simultaneously predicting the positions of the active sys- tem, bar, and note. However, their predictions across these different levels of notation are independent, which makes the predictions unstable and introduces unnecessary ex- tra search space for bar- and note-level predictions. Most existing methods also lack mechanisms to recover from score discontinuities, such as repeats, da capo (D.C.), or coda jumps. This paper proposes CODA, to the best of our knowledge, the first real-time score following system that addresses both gaps. CODA explicitly exploits the cascaded structure of music scores: it first selects the ac- tive system, then the active bar within it, and finally the active note within the selected bar. This enforces pre- diction consistency across resolutions. A silence-driven break mode enables recovery from arbitrary score discon- tinuities without requiring knowledge of the repeat struc- ture. Evaluated on the Multimodal Sheet Music Dataset (MSMD) piano benchmarks, CODA achieves state-of-the- art tracking accuracy and discontinuity-recovery perfor- mance under real-time throughput. Code is available at https://github.com/ValleyC/CODA.

Score Entropy Discrete Diffusion (SEDD) parameterizes discrete reverse processes with unconstrained positive score ratios. While positivity guarantees nonnegative reverse jump rates, it does not ensure Bayes realizability: ratios at a noisy state need not be jointly induced by any clean-token posterior under the forward kernel. The score-entropy loss has the correct population optimum but does not enforce this constraint away from it. In a trained pure-uniform SEDD checkpoint, roughly one quarter of complete score vectors violate the coordinate box, while more than half lie inside it yet remain materially incompatible with any valid posterior. Such violations can produce negative pre-normalization weights in finite-step sampling. Projecting raw scores onto the bridge polytope removes all observed negative weights and improves external generative PPL from $203.6$ to $175.1$ without changing the sampler. We introduce \emph{mean-to-score} (M2S), which predicts a clean-token posterior mean and converts it to the score through an exact kernel-dependent linear map. The construction applies to any known coordinate-wise continuous-time Markov chain (CTMC) satisfying a mild support condition. For uniform corruption, it maps the probability simplex onto the bridge polytope; for absorbing-mask corruption, the resulting objective recovers MD4 exactly. In a controlled 28.4M-parameter CIFAR-10 comparison, M2S lowers test BPD from $3.173$ to $3.129$ and FID-50k from $\CifarSEDDFID$ to $\CifarMtwoSFID$. A 170M-parameter M2S model trained on about 262B OpenWebText token slots outperforms the evaluated pure-uniform SEDD, GIDD, and Neural CTMC checkpoints at every tested sampling budget, reaching generative PPL $143.3$ at 128 steps versus $183.6$ for the strongest pure-uniform baseline.

Scalar metrics are often used to evaluate clusterings against known classes, but they can obscure a fundamental trade-off: clusterings should be informative about class labels while avoiding unnecessary fragmentation. Here we describe normalized scores of cluster homogeneity and parsimony that quantify this trade-off. These scores build on the information bottleneck principle, modified to not reward lossy compression. We show by example and mathematical proof that our definitions of these scores have the intuitive property of varying monotonically under cluster refinement in contrast to related proposals. Extending the information-theoretic framework beyond Shannon entropies, we furthermore derive set-matching and pair-based counterparts of the homogeneity and parsimony scores. These unify commonly used evaluation criteria and show that, in the pair-based setting, the homogeneity-parsimony trade-off recovers the receiver operating characteristic of binary classifiers. We demonstrate the framework's utility for feature selection and algorithm comparison, illustrating how considering scores jointly can clarify clustering operating points and identify Pareto-optimal solutions.

Recent work has shown that "Vision-Free'' approaches (representing images as text) can be effective for standard image retrieval tasks. However, it remains unclear whether this paradigm can effectively handle a more complex, multimodal task, Composed Image Retrieval (CIR), due to the inherent information loss in textual descriptions. In this paper, we introduce a Vision-Free CIR framework that addresses this challenge through two key techniques: (1) Attribute-Augmented Hybrid Scoring, which compensates for lost visual details via explicit attribute matching, and (2) LLM-Based Reranking, which verifies semantic consistency of top candidates. Experiments on the open-domain CIRR dataset show that our approach outperforms existing Zero-shot CIR methods (44.04% R@1, +8.79%). On FashionIQ, our results highlight the trade-off between semantic reasoning and fine-grained visual matching. Ablation studies reveal that both attribute-augmented scoring and LLM-Based Reranking consistently improve performance.

Molecular docking predicts how a small molecule binds to a protein and is a key bottleneck in drug discovery. Classical scoring functions sum empirical pairwise contacts, blind to quantum-mechanical effects like orbital charge transfer that govern binding specificity. We introduce Q-Score, encoding GNN-predicted orbital donor-acceptor energies into a weighted graph and scoring binding by solving a maximum-weight vertex clique problem via Digitized-Counterdiabatic QAOA. Each interaction anchor maps to one qubit and compatibility constraints become edges. Across 11 protein targets, DC-QAOA recovers the exact optimum on 8 at 10 qubits. On 1000 AI-generated molecules, Q-Score is orthogonal to classical scoring with Spearman rho of 0.05, driven by orbital quality with rho of 0.90, and free of molecular-weight bias, enriching for strong orbital interactions at twice the random rate. DC-QAOA achieves a mean approximation ratio of 0.94 with 52 percent exact. Execution of 1000 circuits on IBM Eagle confirms 6-qubit solvability on NISQ hardware.

Many representation learning problems involve directed relations, such as lexical entailment, sentence entailment, ontology hierarchy, and citation links. Standard Euclidean, cosine, and Mahalanobis heads are symmetric, while generic neural scorers can model directionality but provide limited geometric structure. This paper proposes a role-aware neural convex divergence head for asymmetric representation learning. The head applies source- and target-role projections before evaluating an input-convex neural Bregman divergence, yielding a nonnegative structured score in the role-projected space. We characterize its projected-space identity, source-role convexity, directional-gap decomposition, and Hessian-based local curvature. Experiments on lexical, sentence, ontology, and directed graph benchmarks compare symmetric distances, unstructured asymmetric scorers, order/hyperbolic baselines, plain ICNN-Bregman heads, and the proposed role-aware variant. Across ten random seeds on the main semantic and ontology benchmarks, role-aware projections consistently improve directional accuracy over plain ICNN-Bregman heads while preserving zero observed negative divergence rate. The results also identify a boundary case: on large fixed-feature citation prediction, specialized symmetric or hyperbolic baselines remain stronger in ranking accuracy. Overall, the proposed head is best understood as a structured and interpretable plug-in distance module for tasks where directional relations matter.

Sample-based generative models are increasingly used for probabilistic forecasting in high-stakes decision settings, yet their training objectives are blind to the decision maker's cost structure. These models are commonly trained with strictly proper scoring rules, such as the energy score, which allocate their training signal in proportion to data density, with no awareness of where forecast errors are most costly for downstream decisions. We therefore propose decision-aware training for sample-based generative models, augmenting the energy score objective with a differentiable decision loss that directly penalises the cost incurred by acting on the model's forecast. This combined loss is theoretically grounded, as the decision loss is itself a proper scoring rule. We validate our method on one synthetic and two real-world tasks, showing targeted improvements in cost-sensitive regions while retaining full probabilistic forecasts.

Strictly proper scoring rules identify the true conditional class distribution at population level, but their curvature can alter optimization and finite-sample behavior. We study three multiclass objectives: a class-aware quadratic Bregman score (CAPM), a strongly convex generator with constrained log-cosh ridges (HPG), and an HPG objective with an annealed probability-margin penalty (APMS). CAPM is treated as a structured instance of established quadratic scoring-rule theory. We derive conditional-regret, curvature, range, and logit-gradient bounds for CAPM and HPG, and prove exact penalty-range and conditional-target displacement bounds for APMS. Controlled five-seed experiments use Digits, Wisconsin breast cancer, and synthetic confusion and long-tail problems under clean labels, symmetric and pair-flip corruption, class imbalance, calibration evaluation, input corruption, and first-order adversarial perturbations. The candidates are close to cross-entropy on clean data and show descriptive gains in some noisy-label cells, but the five-seed comparisons are interpreted descriptively rather than as significance evidence. The selected noisy-label baselines perform better on Digits with 40% symmetric label noise, and explicit prior-adjustment methods perform better in the 30:1 synthetic long-tail experiment. Ablations do not show a consistent benefit from the candidate-specific graph, ridge, or margin components. The mathematical analysis establishes the stated properties, and the experiments delimit the empirical evidence; together they do not support a claim of general superiority.

Scoring functions are used to represent the relevance of individual documents. In modern information retrieval or recommendation systems, they are often learned from data and play a pivotal role in ranking sets of documents or items in a way that maximizes utility to a query or user. With the recent interest in algorithmic fairness, the success of scoring has naturally led to methods that learn scores that simultaneously trade off fairness and utility. In this work, we show that in stark contrast with utility-centric objectives, scoring is sub-optimal in achieving all utility-fairness trade-offs. We establish this with a series of counter-examples with a generic fairness formulation. We show that the issue persists whether we have a deterministic scoring function or a randomized one, or whether we measure fairness at the scope of a single query or across multiple queries. On the positive side, we empirically demonstrate that semi-greedy post-processing has the potential to achieve much better trade-offs, often approaching the ideal of exhaustive post-processing in a tractable way.

Synthetic data augmentation is widely used to mitigate class imbalance, but its theoretical effects on score-based classification remain poorly understood. This paper develops a framework for characterizing when synthetic minority augmentation can improve threshold-integrated and threshold-optimized metrics, including AUROC, AUPRC, best-threshold balanced accuracy, and best-threshold $\F_1$ score. We separate the effect of augmentation into two components: a change in effective class weighting and a discrepancy between the synthetic and true minority distributions. Under well-specified score models, the raw estimator already targets the likelihood-ratio ordering, which is population-optimal for the metrics considered. Consequently, augmentation cannot provide a fundamental population-level improvement beyond possible finite-sample variance reduction, and may introduce additional bias through synthetic distributional error. We further establish minimax lower bounds showing that the raw estimator already achieves the optimal metric-regret rate in the well-specified regime. Under misspecification, however, augmentation can play a qualitatively different role: by changing the effective class balance, it can alter the restricted-class projection and correct ranking errors induced by the raw imbalanced objective. We provide explicit improvement bounds quantifying the roles of approximation error, finite-sample estimation error, and synthetic distributional error. Simulation studies corroborate the theory, demonstrating limited gains under well-specification and nontrivial but nonmonotone improvements under misspecification.

Conformal selection aims to identify test candidates whose unknown responses fall in a target region while controlling the false discovery rate. Existing methods often inherit prediction-oriented nonconformity scores, such as residual or clipped residual scores, from conformal prediction. We argue that the natural score for selection is instead the target-membership probability. This score directly addresses the binary event being selected, and any monotone transform of it gives the Neyman--Pearson oracle ranking at a fixed null selection level. This distinction is irrelevant for mean-monotone targets, where conventional scores induce essentially the same ranking, but becomes important for interval-valued, variance-driven, multimodal, or multi-condition targets, where prediction-oriented scores can be misaligned with selection power. We study membership-score-based conformal selection and isolate one conformal calibration route, Null-Calibrated Conformal Selection (NCCS), which ranks test scores against confirmed non-target calibration examples. Under null exchangeability, NCCS yields finite-sample valid null p-values, which can be combined with BY under arbitrary dependence or with BH under standard positive-dependence conditions. Experiments support the score principle: membership scores match conventional scores on mean-monotone targets, substantially improve over mean-score selection on variance-driven targets, and, when calibrated by NCCS, trade power for finite-sample null validity in rare-target regimes where direct empirical-FDP thresholding can be anti-conservative.

The score matching problem is a central training objective in modern generative modeling, diffusion models, fitting unnormalized statistical models, and inverse problems. A standard approach is to minimize the forward Fisher divergence, where the expectation is taken with respect to the teacher distribution. However, recent results show that even in simple Gaussian mixture model settings, this objective can lead to undesirable and initialization-dependent convergence behavior. In this paper, we study an alternative objective: the reverse Fisher divergence, where the expectation is taken with respect to the student distribution. We analyze gradient descent (GD) for fitting Gaussian mixture models and show that this change in the objective leads to significantly better optimization properties. First, when the teacher distribution is a single Gaussian and the student is a Gaussian mixture model with fixed weights and identity covariances, we prove the global convergence of GD from arbitrary initializations. Second, we extend the analysis to the case where the teacher is also a Gaussian mixture model and prove global convergence guarantees under a global random initialization scheme and a $\widetildeΩ(1)$-separation assumption on the target means. In particular, with high probability, each student component converges near its closest teacher component, and we provide conditions under which the student distribution converges in total variation distance. Our proofs rely on a new Lyapunov-based analysis of the gradient descent dynamics, showing that the reverse Fisher divergence has a much more favorable optimization landscape than the forward Fisher divergence.

Dense retrieval has become the dominant paradigm in information retrieval, in which each document is scored against a query by the inner product of their vector embeddings, and the top-$k$ documents by score are retrieved for this query. However, since each document's score depends solely on the embedding of the query and itself, the retrieval process is oblivious to the content of the entire corpus. Therefore, dense retrieval cannot avoid selecting semantically similar documents from the corpus, which may result in a non-diverse, redundant set of retrieved documents. To this end, we approach retrieval as a joint decoding problem, in which documents are selected as a set with regard to the context of the rest of the corpus. To achieve this, we propose Non-Negative elastic Net (NNN) decoding, which selects documents whose embeddings jointly reconstruct the query embedding as a sparse non-negative linear combination. Our main theoretical result establishes a strict separation between dense retrieval and NNN decoding. For any corpus, every query correctly handled by dense retrieval is also handled by NNN decoding, while on corpora containing correlated documents, NNN decoding additionally handles queries that dense retrieval cannot. Experimental results indicate that applying NNN decoding to frozen embeddings trained for inner-product scoring yields consistent improvements across several benchmarks. Moreover, we introduce an end-to-end training procedure which optimizes the embeddings for NNN decoding, producing significant performance gains surpassing in all metrics and benchmarks compared to dense retrieval. Our work establishes a new paradigm for leveraging dense embeddings in information retrieval, beyond the standard practice of inner-product scoring.

Diffusion and flow-based models learn powerful data priors by training a denoiser to reverse Gaussian corruption. To use this prior to solve a linear inverse problem, one needs to sample from the posterior, but the score that the prior provides is the unconditional score, not the posterior score. Existing methods either steer a fixed pretrained denoiser with approximate measurement-matching corrections, or train a conditional restoration model that abandons the denoising structure of the prior. We derive the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, and show that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance. We turn this identity into Exact Posterior Score (EPS), a denoising training objective that preserves the input/output structure of standard pretraining and can therefore be trained from scratch or fine-tuned from a pretrained denoiser. At inference, EPS uses the same sampler as the underlying backbone, with no likelihood gradients or projections. We evaluate EPS on five linear inverse problems across FFHQ and ImageNet, where it outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics, while using roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.

Driven by their remarkable success in computer vision and inverse problem solving, score-based models are increasingly applied to wireless communications, where they show promise across a range of physical-layer tasks. However, despite this growing interest, the current literature often lacks a rigorous analysis of when score-matching offers a tangible advantage over traditional discriminative learning. This paper aims to address this gap through the use-case of channel estimation, a fundamental inverse problem in wireless systems. We present a theoretically grounded interpretation of score-based channel estimation through the lens of the perception-distortion tradeoff, identifying the conditions where score matching excels as well as its key limitations. In particular, by modeling downstream wireless tasks (e.g., capacity maximization) as functionals of the channel estimation process, we quantify the excess risk incurred by standard distortion-minimization approaches. Extensive numerical results show that under high predictive uncertainty, the large excess risk gap can be offset by score-based estimation, enabling near Bayesian-optimal precoding via the learned posterior, whereas in the low predictive uncertainty regime, discriminative distortion-minimization approaches are preferable due to lower complexity and more efficient use of model capacity.

The most widely used RANSAC variants score candidate models by counting inliers or summing per-point scores that saturate beyond a residual threshold. Every such score requires a user-supplied parameter that is a function of the inlier scale, which must itself be estimated from contaminated data. We remove this dependence by reversing the usual order of inference: rather than estimating the scale and then scoring against it, we marginalize the inlier scale analytically in closed form under a conjugate Inverse-Gamma prior for a fixed inlier partition, then optimize over partitions. A single closed-form expression spans the non-informative Jeffreys limit and informative empirical-Bayes priors, so the same score adapts across data-rich and data-scarce regimes without any change to the algorithm. The proposed RANSAC score is the first in which the inlier scale is genuinely absent from the formula. The score admits O(N log N ) computation via sort-and-sweep. On a benchmark of nearly 70 000 image pairs spanning different two-view estimation problems and both engineered and learned feature pipelines, the proposed score exceeds the state of the art (RANSAC, MSAC, GaU, MAGSAC): it stays nearly flat under threshold miscalibration where baselines degrade, reaches near-optimal accuracy from as few as two validation pairs where baselines need ont he order of 100 times more,. and tightens its prior regularization as validation data grows scarce.

This paper reviews how a diverse set of popular data-driven priors commonly used in Bayesian inverse problems can be unified through their respective score functions. By framing these priors under this common perspective, we show that they can benefit from their straightfoward and effective integration into a recently proposed sampling algorithm. The applicability of this common framework is illustrated by considering several data-driven priors, namely regularization-by-denoising, normalizing flow-based priors, score-based generative models, and convex-ridge regularizers. For these four particular priors, the performance of the method is evaluated when conducting image inpainting and single image super-resolution. These results, as well as those obtained when restoring real images acquired in a geological context, demonstrate the efficiency of the method. This unified framework proves versatile enough to handle any posterior distribution defined by a broad class of score function-based priors, beyond the specific cases considered in this paper.

Scientific generative modeling often requires size transfer, where models trained on small systems are evaluated on larger ones. While translation-invariant architectures enable this evaluation, we show that architectural locality alone does not guarantee stable size extrapolation. Instead, stable extrapolation is governed by the quasi-locality of the Gaussian-smoothed score. Through Tweedie's formula, far-away perturbations can influence local score components via posterior covariance, meaning a local model succeeds only if its receptive field covers the smoothed score's response range. We formalize this mechanism, proving a size-uniform comparison theorem for local marginals under reverse diffusion. We also introduce Finite-Depth Local Flow (FDLF), a white-box diagnostic benchmark with exact scores, densities, and controllable response ranges. Empirically, we validate the interplay between spatial mixing, smoothed-score quasi-locality, and model receptive fields. Under spatial mixing, the smoothed score remains quasi-local relative to the receptive field, enabling stable extrapolation. Conversely, when spatial mixing weakens, the score's locality rapidly degrades, causing size transfer to fail.

Proper scoring rules provide a rigorous theoretical basis for the training and evaluation of probabilistic forecasts. However, in the presence of right censoring, the event time is only partially observed, rendering conventional scoring rules inapplicable in their standard form. We propose a framework for proper scoring of right-censored survival outcomes based on a simple idea: first, map the predictive distribution through the censoring mechanism, then apply the underlying proper score on the induced observed-data law. This yields localized scores for fixed censoring times and marginalized scores when the censoring time is random or only partially observed. The resulting construction recovers familiar right-censored likelihood and IPCW-type criteria within a coherent framework, while also yielding right-censored versions of the CRPS, pinball loss, Brier score, and energy score. We show that the marginalized score is proper under conditional independent censoring and strictly proper on the identifiable region. The same principle also leads to censored engression, a sample-based learning objective for multivariate right-censored survival modeling. In experiments, our scores correctly rank the oracle forecast across several censoring regimes, whereas forecast-dependent plug-in weighted scores can exhibit ranking reversals. Censored engression likewise substantially improves over naive training on censored outcomes.

Contrastive audio-language models such as CLAP enable zero-shot audio classification: a sound is labelled by matching its embedding to text prompt embeddings, with no labelled audio. This matching breaks down under acoustic noise, where accuracy and mAP fall by 12-30 percentage points at 0 dB SNR on standard benchmarks. We propose Drift Augmented Scoring (DAS), a small per-class bonus added to the cosine score. The bonus rewards a class when the noisy audio embedding drifts in the direction that the class's noise-conditioned text prompts predict. It is derived from text alone, computed once and cached, and adds a single inner product per class at inference, with no gradients and no test-time batch. On a LAION CLAP backbone, we compare DAS against the four variants of Acevedo et al.'s concurrent method on UrbanSound8K and the full FSD50K eval set, mixing each clip with urban acoustic scene noise across a range of SNRs. DAS improves the metric on every test condition: by +2.60 to +5.75 accuracy points on UrbanSound8K and +1.50 to +1.74 mAP points on FSD50K.

Real-world datasets often contain spurious correlations that are not causally related to the target label. When such correlations dominate the majority of training samples, models tend to rely on them, leading to misclassification of minority samples that do not exhibit the same spurious patterns. While a potential approach is to select subsets of data to better represent the minority samples, this may require access to group labels, which are typically unknown. Furthermore, as we demonstrate, widely used sample scoring functions in the invariant subset or coreset selection literature largely depend on spurious features and therefore fail to accurately capture the importance or difficulty of core, causally relevant features. Accordingly, we propose to mitigate spurious correlations by developing a two-stage sample scoring function that disentangles the learning dynamics of core and spurious features and evaluates their difficulty separately. Based on our proposed metric, we introduce a new algorithm to find and prioritize informative samples both with and without spurious correlations. Extensive experiments demonstrate that a standard ERM model trained on our selected samples achieves superior performance compared to state-of-the-art debiasing techniques, while requiring as little as 10% of the original training data.

Gradient boosted decision trees require a stopping rule to avoid overfitting. The standard rule monitors a validation loss and stops if the loss fails to improve for a fixed patience period. However, the patience parameter has no interpretable scale and validation losses can be noisy or implicitly defined by a user-specified gradient. We propose ScoreStop, a gradient-based early-stopping rule that casts the stopping decision at each iteration as a test of the null hypothesis that the current predictor is the population risk minimizer. We use a functional score test, computed on validation data, with a statistic that is scale-invariant in the update direction, with a known asymptotic distribution under the null. Because our test uses gradients rather than loss values, the same construction applies to implicit losses such as LambdaRank, and data-dependent losses such as Cox regression via influence functions. In synthetic experiments and real-data benchmarks, we show that ScoreStop is competitive with loss-based methods.

InfoNCE is the standard contrastive learning objective, but its softmax form is not only a computational convenience: it also encodes a statistical assumption about how the top-scoring example is selected. Using extreme value theory, we show that this assumption is often misaligned with the normalized embedding setting used in modern contrastive learning. Motivated by this mismatch, we propose \textsc{WEINCE}, a simple modification of InfoNCE that uses anchor-wise online batch statistics to blend the usual softmax logits with an endpoint shortfall correction, adding no trainable parameters. Across five vision benchmarks, \textsc{WEINCE} yields consistent improvements in frozen-feature evaluation. These results show that a more faithful statistical treatment of hard negatives can improve contrastive objectives.

We introduce Score Broadcast and Decorrelation (SBD), a principled framework for broadcast-based credit assignment for general families of differentiable losses. Error broadcast is a biologically plausible alternative to backpropagation that sends output information to hidden layers without weight transport. The Error Broadcast and Decorrelation (EBD) framework, recently introduced for the mean-squared-error (MSE) setting, grounded this mechanism in the stochastic orthogonality of optimal estimators, under which the optimal residual is orthogonal to functions of the input. We generalize that foundation by introducing an orthogonality principle between the output score (the gradient of loss with respect to the final-layer output) and hidden-layer activations, which holds whenever the optimal score has conditional mean zero. This single principle unifies broadcast-based credit assignment across the standard differentiable-loss families, including cross-entropy, Bregman divergences, proper scoring rules, and exponential-family negative log-likelihoods. The framework supplies a theoretical grounding for the three-factor learning rule under general losses, with the neuromodulatory factor derived as the broadcast loss score. We derive the cross-entropy case explicitly, characterize the admissible loss class, and introduce a score vector expansion technique that enriches the broadcast signal while preserving the orthogonality framework. Experiments on CIFAR-10 and Tiny ImageNet show that SBD substantially improves over existing broadcast approaches, with score vector expansion delivering further gains. Overall, this work identifies the loss score as the signal to broadcast, supplies the orthogonality theory and theoretical grounding for the three-factor learning rule from neuroscience, and shows how score vector expansion enriches the decorrelation directions of the resulting objective.

The classic concept of "calibrated forecasts" and its more recent refinement, "calibeating," are defined with respect to the standard quadratic scoring rule. We extend these notions to the class of $\textit{proper}$ scoring rules (for which the best forecast is the true distribution) and define $\textit{proper-calibration}$ and $\textit{proper-calibeating}$ by requiring the errors to converge to zero uniformly over all bounded proper scoring rules. We first establish that calibration always implies proper-calibration, whereas calibeating need not imply proper-calibeating. Second, we show how to guarantee proper-calibeating and proper-multicalibeating. Finally, we demonstrate the equivalence between proper-calibration and universal no regret when best replying to forecasts in decision-making under uncertainty.

We study denoising score matching (DSM) when the latent distribution is supported on a smooth embedded manifold $M \subset \mathbb{R}^D$. Under ambient Gaussian corruption, the tangent denoising target contains a singular normal-fiber noise channel whose variance diverges as $d/σ^2$ as $σ\to 0^+$. We show that conditioning on the nearest-point projection $π(X)$ canonically removes this singularity: the resulting conditional expectation is the unique $L^2$-optimal Rao-Blackwellized predictor of the tangent DSM target among all estimators depending only on the projected observation $π(X)$. We then compute the small-noise expansion of this canonical target and show that it equals the intrinsic Riemannian score up to an explicit order-$σ^2$ correction that decomposes into an intrinsic Tweedie term and an extrinsic curvature term involving the Weingarten and Ricci operators. In the flat case, the construction reduces exactly to ordinary lower-dimensional Gaussian DSM, while on $S^d$ the extrinsic correction simplifies to the scalar factor $(1-d/2)\nabla_M \log q$; this extrinsic $σ^2$ correction cancels identically on $S^2$, though the intrinsic Tweedie term remains.

Score matching is an alternative to maximum likelihood estimation when the normalizing constant is unknown or too costly to evaluate. However, vanilla score matching has shown to be inefficient relative to maximum likelihood estimation for multimodal distributions with well-separated modes, which are commonly encountered in practical applications. We compare a novel diffusion-based denoising score matching estimator (DDSME) to the vanilla score matching estimator (SME) in this scenario. In particular, we prove statistical guarantees for both estimators, showing that the error bound for the vanilla SME worsens when the separation between the modes increases, which can be avoided in case of the DDSME with suitable hyperparameter tuning. This provides a novel theoretical explanation for the superior behavior of diffusion-based score matching over the vanilla version.

Compositional score-based approaches to simulation-based inference (SBI) approximate the posterior over a shared parameter given $n$ independent observations by aggregating individually learned posterior scores: currently, there are two main propositions of such methods (Geffner et al. (2023), Linhart et al. (2026)). As the resulting composite score does not correspond to the score of any distribution along the forward diffusion path of the true multi-observation posterior, sampling from it via a reverse SDE leads to an irreducible bias. Annealed Langevin dynamics provides a principled alternative: it treats the composite score as the genuine score of a sequence of tractable bridging densities and samples from them in succession. When properly tuned, it could lead to a controllable bias. However, its hyperparameters, namely step sizes, the number of steps per level, and the number of annealing levels, have so far been chosen empirically. We derive Wasserstein bounds for annealed Langevin with approximate scores and translate them into explicit decision rules for these hyperparameters that guarantee a prescribed sampling accuracy, while highlighting different theoretical aspects of each composite score formulation. In the Gaussian setting, we obtain closed-form expressions for all relevant quantities and prove that the bridging densities of Linhart et al. (2026) consistently admit larger step sizes and require fewer total Langevin steps than those of Geffner et al. (2023). Furthermore, we show empirically that the tuning obtained in the Gaussian setting generalizes to more complex problems, thus providing a well-understood and theoretically grounded starting point for practitioners using compositional score-based approaches.

Identifying latent variables and the causal structure involving them is essential across various scientific fields. While many existing works fall under the category of constraint-based methods (with e.g. conditional independence or rank deficiency tests), they may face empirical challenges such as testing-order dependency, error propagation, and choosing an appropriate significance level. These issues can potentially be mitigated by properly designed score-based methods, such as Greedy Equivalence Search (GES) (Chickering, 2002) in the specific setting without latent variables. Yet, formulating score-based methods with latent variables is highly challenging. In this work, we develop score-based methods that are capable of identifying causal structures containing causally-related latent variables with identifiability guarantees. Specifically, we show that a properly formulated scoring function can achieve score equivalence and consistency for structure learning of latent variable causal models. We further provide a characterization of the degrees of freedom for the marginal over the observed variables under multiple structural assumptions considered in the literature, and accordingly develop both exact and continuous score-based methods. This offers a unified view of several existing constraint-based methods with different structural assumptions. Experimental results validate the effectiveness of the proposed methods.

A central obstacle in nonlinear Bayesian filtering is representing the belief distribution. Moment-based filters address this by propagating polynomial moments and reconstructing a density from them. Recent work completes the predict-update loop via the maximum-entropy (MaxEnt) principle, but each step requires the partition function and its gradient, both $n$-dimensional integrals whose cost scales exponentially, restricting the demonstrated MaxEnt moment filtering to $n \le 4$. We avoid the partition function entirely by combining score matching with Stein's identity. In our setting, score matching reduces the density fit to a single linear solve whose coefficients are assembled directly from the propagated moments. The same parameters then drive Stein's identity to close the moment hierarchy during prediction and to recover posterior moments after each Bayesian update, keeping the full predict-update loop free of partition function evaluation. The resulting Score Kalman Filter (SKF) reduces to the classical information-form Kalman filter as a special case and performs every step through linear algebra. On nonlinear coupled-oscillator networks, the SKF runs through $n=20$ and reports lower RMSE than the EKF, UKF, EnKF, and particle-filter baselines on the tested synthetic benchmarks.

Learning of continuous exponential family distributions with unbounded support remains an important area of research for both theory and applications in high-dimensional statistics. In recent years, score matching has become a widely used method for learning exponential families with continuous variables due to its computational ease when compared against maximum likelihood estimation. However, theoretical understanding of the statistical properties of score matching is still lacking. In this work, we provide a non-asymptotic sample complexity analysis for learning the structure of exponential families of polynomials with score matching. The derived sample bounds show a polynomial dependence on the model dimension. These bounds are the first of its kind, as all prior work has shown only asymptotic bounds on the sample complexity.

For stochastic process models, parameter inference is often severely bottlenecked by computationally expensive likelihood functions. Simulation-based inference (SBI) bypasses this restriction by constructing amortized surrogate likelihoods, but most SBI methods assume a black-box data generating process. While these surrogates are exact in the limit of infinite training data, practical scenarios force a strict tradeoff between model quality and simulation cost. In this work, we loosen the black-box assumption of SBI to improve this tradeoff for structured stochastic process models. Specifically, for neural network likelihood surrogates trained via probabilistic classification, we propose to augment the standard binary cross-entropy loss with exact score information $\nabla_θ\log p(x \mid θ)$ and adaptive weighting based on loss gradients. We evaluate our approach on case studies involving network dynamics and spatial processes, demonstrating that our method improves surrogate quality at a drastically lower computational cost than generating more training data. Notably, in some cases, our approach achieves downstream inference performance equivalent to a 10x increase in training data with less than a 1.1x increase in training time.

Diffusion guidance is a powerful technique that enables controllable and high-fidelity sample generation with diffusion models. At a high level, it modifies the score function by incorporating a guidance term that steers the generative process toward a desired condition. Despite its empirical success, the theoretical properties of diffusion guidance remain largely unexplored, and it is not well understood why it consistently produces high-quality samples. In this work, we explain the effectiveness of diffusion guidance by establishing a robustness of support property. Specifically, we show that, given exact access to the score functions, guided diffusion processes almost always generate samples that remain close to the target support. This property is particularly desirable, as samples that lie off the support are often structurally implausible and may adversely affect downstream tasks. Our analysis covers both Denoising Diffusion Implicit Models (DDIM) and Denoising Diffusion Probabilistic Models (DDPM), and applies to a wide range of discretization schemes induced by exponential integrators. Our results provide a rigorous foundation for understanding why diffusion guidance produces physically meaningful and structurally plausible samples.

Recent advances in generative video models are increasingly driven by post-training and test-time scaling, both of which critically depend on the quality of video reward models (RMs). An ideal reward model should predict accurate rewards that align with human preferences across diverse scenarios. However, existing paradigms face a fundamental dilemma: \textit{Discriminative RMs} regress rewards directly on features extracted by multimodal large language models (MLLMs) without explicit reasoning, making them prone to shortcut learning and heavily reliant on massive data scaling for generalization. In contrast, \textit{Generative RMs} with Chain-of-Thought (CoT) reasoning exhibit superior interpretability and generalization potential, as they leverage fine-grained semantic supervision to internalize the rationales behind human preferences. However, they suffer from inherent optimization bottlenecks due to the coupling of reasoning and scoring within a single autoregressive inference chain. To harness the generalization benefits of CoT reasoning while mitigating the training instability of coupled reasoning and scoring, we introduce DeScore, a training-efficient and generalizable video reward model. DeScore employs a decoupled ``think-then-score'' paradigm: an MLLM first generates an explicit CoT, followed by a dedicated discriminative scoring module consisting of a learnable query token and a regression head that predicts the final reward. DeScore is optimized via a two-stage framework: (1) a discriminative cold start incorporating a random mask mechanism to ensure robust scoring capabilities, and (2) a dual-objective reinforcement learning stage that independently refines CoT reasoning quality and calibrates the final reward, ensuring that higher-quality reasoning directly translates to superior model performance.

In sequential search, alternatives are tested until the true class is found. Standard proper scoring rules like log loss are local, ignoring the ranking of competitors and misaligning model evaluation with search utility. We show that sequential search induces a pairwise structure that overcomes this. By analyzing the expected cost of optimal search under varying testing costs, we derive Pandora's Regret: a closed-form, pairwise-additive, and strictly proper scoring rule. Pandora's Regret both elicits true probabilities and penalizes rank-reversing miscalibrations where distractors outrank the true class. Our construction yields a one-parameter Beta family that balances penalties for rank-swapping versus probability magnitude, while retaining a grounded interpretation as expected search cost. We prove that log loss, accuracy, and macro-F1 rely on implicit decision models misaligned with sequential search. Across 597 MedMNIST models, Pandora-based metrics better predict clinical diagnostic costs than standard alternatives, extending decision-theoretic scoring rule construction to the multiclass setting.

The Hypencoder, proposed by Killingback et al., is a retrieval framework that replaces the fixed inner-product scoring function used in standard bi-encoders with a query-specific neural network (the $q$-net), whose weights are generated by a hypernetwork from the contextualized query embeddings. This design enables more expressive relevance estimation while preserving independent query and document encoding. In this work, we conduct a reproducibility study of the Hypencoder and extend the original analysis in three directions. Our reproduction confirms that the Hypencoder outperforms a similarly trained bi-encoder baseline on in-domain and out-of-domain benchmarks, and that the proposed efficient search algorithm substantially reduces query latency with minimal performance loss. On hard retrieval tasks, we find partial support: the Hypencoder outperforms the baseline on DL-Hard and FollowIR, but not on TREC TOT, where checkpoint incompatibility and fine-tuning sensitivity complicate full verification. Beyond reproduction, we investigate three extensions: (i)~integrating alternative pre-trained encoders into the Hypencoder framework, where we find that performance gains depend on the encoder and fine-tuning strategy; (ii)~comparing query latency against a Faiss-based bi-encoder pipeline, revealing that standard bi-encoder retrieval remains faster under both exhaustive and efficient search settings; and (iii)~evaluating adversarial robustness, where we find that the $q$-net's non-linear scoring does not provide a consistent robustness disadvantage over inner-product scoring. Our code is publicly available at https://github.com/arneeichholtz/Hypencoder-reprod.

In continuously monitored quantum systems, the feedback protocol of García-Pintos, Liu, and Gorshkov reshapes the arrow of time: a Hamiltonian $H_{\mathrm{meas}} = r A / τ$ applied with gain $X$ tilts the distribution of measurement trajectories, with $X < -2$ producing statistically time-reversed outcomes. Why this specific Hamiltonian achieves reversal, and how the mechanism relates to score-based diffusion models in machine learning, has remained unexplained. We compute the functional derivative of the log path probability of the quantum trajectory distribution directly in density-matrix space. Combining Girsanov's theorem applied to the measurement record, Fréchet differentiation on the Banach space of trace-class operators, and Kähler geometry on the pure-state projective manifold, we prove that $δ\log P_F / δρ= r A / τ= H_{\mathrm{meas}}$. The García-Pintos feedback Hamiltonian is the score function of the quantum trajectory distribution -- exactly the object Anderson's reverse-time diffusion theorem requires for trajectory reversal. The identification extends to multi-qubit systems with independent measurement channels, where the score is a sum of local operators. Two consequences follow. First, the feedback gain $X$ generates a continuous one-parameter family of path measures (for feedback-active Hamiltonians with $[H, A] \neq 0$), with $X = -2$ recovering the backward process in leading-order linearization -- a structure absent from classical diffusion, where reversal is binary. Second, the score identification enables machine learning (ML) score estimation methods -- denoising score matching, sliced score matching -- to replace the analytic formula when its idealizations (unit efficiency, zero delay, Gaussian noise) fail in real experiments.

Graph-based Retrieval-Augmented Generation (GraphRAG) enhances LLMs by structuring corpus into graphs to facilitate multi-hop reasoning. While recent lightweight approaches reduce indexing costs by leveraging Named Entity Recognition (NER), they rely strictly on structural co-occurrence, failing to capture latent semantic connections between disjoint entities. To address this, we propose EHRAG, a lightweight RAG framework that constructs a hypergraph capturing both structure and semantic level relationships, employing a hybrid structural-semantic retrieval mechanism. Specifically, EHRAG constructs structural hyperedges based on sentence-level co-occurrence with lightweight entity extraction and semantic hyperedges by clustering entity text embeddings, ensuring the hypergraph encompasses both structural and semantic information. For retrieval, EHRAG performs a structure-semantic hybrid diffusion with topic-aware scoring and personalized pagerank (PPR) refinement to identify the top-k relevant documents. Experiments on four datasets show that EHRAG outperforms state-of-the-art baselines while maintaining linear indexing complexity and zero token consumption for construction. Code is available at https://github.com/yfsong00/EHRAG.

Punctuation restoration improves ASR (Automatic Speech Recognition) readability. However streaming ASR requires online decisions with limited future context. In streaming ASR, the system predicts punctuation incrementally, which makes generation-based approaches prone to latency and alignment failures under boundary-wise evaluation. This paper proposes a non-autoregressive scoring method (no free-form generation) that preserves the input transcript and makes a decision at each word boundary. Our method compares punctuation insertion hypotheses against a no-insertion baseline under a bounded K-subword-token lookahead, and calibrates decisions using a weight α and a validation-calibrated threshold τ (no parameter updates during inference). On IWSLT 2017, our scoring method achieves a 4-class macro F1 of 0.893 in the no fine-tuning setting (validation-calibrated, K=2) and 0.937 after fine-tuning (K=2), outperforming the prompt-based baseline (0.566) and a fine-tuned ELECTRA baseline (0.913) under the same lookahead budget. We analyze the impact of the lookahead budget through ablation studies on K.

We study the theoretical behavior of denoising score matching--the learning task associated to diffusion models--when the data distribution is supported on a low-dimensional manifold and the score is parameterized using a random feature neural network. We derive asymptotically exact expressions for the test, train, and score errors in the high-dimensional limit. Our analysis reveals that, for linear manifolds the sample complexity required to learn the score function scales linearly with the intrinsic dimension of the manifold, rather than with the ambient dimension. Perhaps surprisingly, the benefits of low-dimensional structure starts to diminish once we have a non-linear manifold. These results indicate that diffusion models can benefit from structured data; however, the dependence on the specific type of structure is subtle and intricate.

Proper scoring rules are essential for evaluating probabilistic forecasts. We propose a simple algebraic rearrangement of the Yates covariance decomposition of the Brier score into three independently non-negative terms: a variance mismatch term, a correlation deficit term, and a calibration-in-the-large term. This rearrangement makes the optimality conditions for perfect forecasting transparent: the optimal forecast must simultaneously match the variance of outcomes, achieve perfect positive correlation with outcomes, and match the mean of outcomes. Any deviation from these conditions results in a positive contribution to the Brier score.

Sampling based on score diffusions has led to striking empirical results, and has attracted considerable attention from various research communities. It depends on availability of (approximate) Stein score functions for various levels of additive noise. We show how in some generality, the availability of scores allows the general problem to be ``reduced'' to sampling from an adaptively constructed sequence of $K$ strongly log-concave (SLC) sub-problems. The reduction is simple, constructive and algorithm-independent, so that any SLC sampler can be used as a subroutine. Various bounds on score-based sampling complexity follow directly: for instance, high-accuracy SLC samplers yield $\tilde{\mathcal{O}}(K \sqrt{d} \operatorname{polylog}(1/\varepsilon))$ guarantees for accuracy $\varepsilon$ in dimension $d$, where randomized midpoint SLC schemes yield $\tilde{\mathcal{O}}(K d^{1/3} \operatorname{poly}(1/\varepsilon))$ guarantees. When the original distribution itself is SLC, we prove that $K \leq 1 + \log_2(κ)$, thereby obtaining the first efficient procedure with logarithmic dependence on condition number $κ$; for general distributions, the quantity $K$ depends on the geometry of score Hessian across the trajectory. Our analysis is direct and simple, involving techniques and insights complementary to those in standard analyses of discretized diffusions.

Diffusion models achieve remarkable generation quality, yet face a fundamental challenge known as memorization, where generated samples can replicate training samples exactly. We develop a theoretical framework to explain this phenomenon by showing that the empirical score function (the score function corresponding to the empirical distribution) is a weighted sum of the score functions of Gaussian distributions, in which the weights are sharp softmax functions. This structure causes individual training samples to dominate the score function, resulting in sampling collapse. In practice, approximating the empirical score function with a neural network can partially alleviate this issue and improve generalization. Our theoretical framework explains why: In training, the neural network learns a smoother approximation of the weighted sum, allowing the sampling process to be influenced by local manifolds rather than single points. Leveraging this insight, we propose two novel methods to further enhance generalization: (1) Noise Unconditioning enables each training sample to adaptively determine its score function weight to increase the effect of more training samples, thereby preventing single-point dominance and mitigating collapse. (2) Temperature Smoothing introduces an explicit parameter to control the smoothness. By increasing the temperature in the softmax weights, we naturally reduce the dominance of any single training sample and mitigate memorization. Experiments across multiple datasets validate our theoretical analysis and demonstrate the effectiveness of the proposed methods in improving generalization while maintaining high generation quality.