Diffusion models, though originally designed for generative tasks, have demonstrated impressive self-supervised representation learning capabilities. A particularly intriguing phenomenon in these models is the emergence of unimodal representation dynamics, where the quality of learned features peaks at an intermediate noise level. In this work, we conduct a comprehensive theoretical and empirical investigation of this phenomenon. Leveraging the inherent low-dimensionality structure of image data, we theoretically demonstrate that the unimodal dynamic emerges when the diffusion model successfully captures the underlying data distribution. The unimodality arises from an interplay between denoising strength and class confidence across noise scales. Empirically, we further show that, in classification tasks, the presence of unimodal dynamics reliably reflects the diffusion model's generalization: it emerges when the model generate novel images and gradually transitions to a monotonically decreasing curve as the model begins to memorize the training data.

Selective classifiers improve model reliability by abstaining on inputs the model deems uncertain. However, few practical approaches achieve the gold-standard performance of a perfect-ordering oracle that accepts examples exactly in order of correctness. Our work formalizes this shortfall as the selective-classification gap and present the first finite-sample decomposition of this gap to five distinct sources of looseness: Bayes noise, approximation error, ranking error, statistical noise, and implementation-or shift-induced slack. Crucially, our analysis reveals that monotone post-hoc calibration--often believed to strengthen selective classifiers--has limited impact on closing this gap, since it rarely alters the model's underlying score ranking. Bridging the gap therefore requires scoring mechanisms that can effectively reorder predictions rather than merely rescale them. We validate our decomposition on synthetic two-moons data and on real-world vision and language benchmarks, isolating each error component through controlled experiments. Our results confirm that (i) Bayes noise and limited model capacity can account for substantial gaps, (ii) only richer, feature-aware calibrators meaningfully improve score ordering, and (iii) data shift introduces a separate slack that demands distributionally robust training. Together, our decomposition yields a quantitative error budget as well as actionable design guidelines that practitioners can use to build selective classifiers which approximate ideal oracle behavior more closely.

Ensuring that large language models (LLMs) remain both helpful and harmless poses a significant challenge: fine-tuning on repetitive safety datasets--where unsafe prompts are paired with standard refusal templates--often leads to \emph{false refusals}, in which benign queries are declined. We first quantify this effect, showing that safety data exhibits substantially lower token entropy ($H_{1}\approx9.18$)

The growing importance of mRNA therapeutics and synthetic biology highlights the need for models that capture the latent structure of synonymous codon (different triplets encoding the same amino acid) usage, which subtly modulates translation efficiency and gene expression. While recent efforts incorporate codon-level inductive biases through auxiliary objectives, they often fall short of explicitly modeling the structured relationships that arise from the genetic code's inherent symmetries. We introduce Equi mRNA, the first codon level equivariant mRNA language model that explicitly encodes synonymous codon symmetries as cyclic subgroups of 2D Special Orthogonal matrix ($\mathrm{SO}(2)$). By combining group theoretic priors with an auxiliary equivariance loss and symmetry aware pooling, Equi mRNA learns biologically grounded representations that outperform vanilla baselines across multiple axes. On downstream property prediction tasks including expression, stability, and riboswitch switching Equi mRNA delivers up to $\approx$ 10\% improvements in accuracy. In sequence generation, it produces mRNA constructs that are up to $\approx$ 4$\times$ more realistic under Fréchet BioDistance metrics and $\approx$ 28\% better preserve functional properties compared to vanilla baseline. Interpretability analyses further reveal that learned codon rotation distributions recapitulate known GC content biases and tRNA abundance patterns, offering novel insights into codon usage. Equi mRNA establishes a new biologically principled paradigm for mRNA modeling, with significant implications for the design of next generation therapeutics.

Recalibrating probabilistic classifiers is vital for enhancing the reliability and accuracy of predictive models. Despite the development of numerous recalibration algorithms, there is still a lack of a comprehensive theory that integrates calibration and sharpness (which is essential for maintaining predictive power). In this paper, we introduce the concept of minimum-risk recalibration within the framework of mean-squared-error (MSE) decomposition, offering a principled approach for evaluating and recalibrating probabilistic classifiers. Using this framework, we analyze the uniform-mass binning (UMB) recalibration method and establish a finite-sample risk upper bound of order O(B/n+1/B2) where Bis the number of bins and nis the sample size. By balancing calibration and sharpness, we further determine that the optimal number of bins for UMB scales with n1/3, resulting in a risk bound of approximately O(n 2/3). Additionally, we tackle the challenge of label shift by proposing a two-stage approach that adjusts the recalibration function using limited labeled data from the target domain. Our results show that transferring a calibrated classifier requires significantly fewer target samples compared to recalibrating from scratch. We validate our theoretical findings through numerical simulations, which confirm the tightness of the proposed bounds, the optimal number of bins, and the effectiveness of label shift adaptation.

We develop a geometric framework that links objective accuracy to structural recovery in prototype-based clustering. The analysis is algorithm-agnostic and applies to a broad class of admissible loss functions. We define a clustering condition number that compares within-cluster scale to the minimum loss increase required to move a point across a cluster boundary. When this quantity is small, any solution with a small suboptimality gap must also have a small misclassification error relative to a benchmark partition. The framework also clarifies a fundamental trade-off between robustness and sensitivity to cluster imbalance, leading to sharp phase transitions for exact recovery under different objectives. The guarantees are deterministic and non-asymptotic, and they separate the role of algorithmic accuracy from the intrinsic geometric difficulty of the instance. We further show that errors concentrate near cluster boundaries and that sufficiently deep cluster cores are recovered exactly under strengthened local margins. Together, these results provide a geometric principle for interpreting low objective values as reliable evidence of meaningful clustering structure.

A probabilistic classifier is considered "well calibrated" when its predicted probabilities closely align with the empirical frequencies of the corresponding labels [

From the perspective of statistical learning theory, (1) is rather intriguing. Moreover, they do not provide instance-wise matching lower bounds to verify the tightness of the upper bounds.