Neural Collapse is a phenomenon that helps identify sparse and low rank structures in deep classifiers. Recent work has extended the definition of neural collapse to regression problems, albeit only measuring the phenomenon at the last layer. In this paper, we establish that Neural Regression Collapse (NRC) also occurs below the last layer across different types of models. We show that in the collapsed layers of neural regression models, features lie in a subspace that corresponds to the target dimension, the feature covariance aligns with the target covariance, the input subspace of the layer weights aligns with the feature subspace, and the linear prediction error of the features is close to the overall prediction error of the model. In addition to establishing Deep NRC, we also show that models that exhibit Deep NRC learn the intrinsic dimension of low rank targets and explore the necessity of weight decay in inducing Deep NRC. This paper provides a more complete picture of the simple structure learned by deep networks in the context of regression.

Understanding how biomarker distributions evolve over time is a central challenge in digital health and chronic disease monitoring. In diabetes, changes in the distribution of glucose measurements can reveal patterns of disease progression and treatment response that conventional summary measures miss. Motivated by a 26-week clinical trial comparing the closed-loop insulin delivery system t:slim X2 with standard therapy in children with type 1 diabetes, we propose a probabilistic framework to model the continuous-time evolution of time-indexed distributions using continuous glucose monitoring data (CGM) collected every five minutes. We represent the glucose distribution as a Gaussian mixture, with time-varying mixture weights governed by a neural ODE. We estimate the model parameter using a distribution-matching criterion based on the maximum mean discrepancy. The resulting framework is interpretable, computationally efficient, and sensitive to subtle temporal distributional changes. Applied to CGM trial data, the method detects treatment-related improvements in glucose dynamics that are difficult to capture with traditional analytical approaches.

Background: Determining an adequate sample size is essential for developing reliable and generalisable clinical prediction models, yet practical guidance on selecting appropriate methods remains limited. Existing analytical and simulation-based approaches often rely on restrictive assumptions and focus on mean-based criteria. We present and validate pmsims, an R package that uses Gaussian process surrogate modelling to provide a flexible and computationally efficient simulation-based framework for sample size determination across diverse prediction settings. Methods: We conducted a comprehensive simulation study with two aims. First, we compared three search engines implemented in pmsims: a Gaussian process-based adaptive method, a deterministic bisection method, and a hybrid approach, across binary, continuous, and survival outcomes. Second, we benchmarked the best-performing pmsims engine against existing analytical (pmsampsize) and simulation-based (samplesizedev) methods, evaluating recommended sample sizes, computational time, and achieved performance on large independent validation datasets. Results: The Gaussian process-based method consistently produced the most stable sample size estimates, particularly in low-signal, high-dimensional settings. In benchmarking, pmsims achieved performance close to prespecified targets across all outcome types, matching simulation-based approaches and outperforming analytical methods in more challenging scenarios. Conclusions: pmsims provides an efficient and flexible framework for principled sample size planning in clinical prediction modelling, requiring fewer model evaluations than non-adaptive simulation approaches.

Privacy and algorithmic fairness have become two central issues in modern machine learning. Although each has separately emerged as a rapidly growing research area, their joint effect remains comparatively under-explored. In this paper, we systematically study the joint impact of differential privacy and fairness on classification in a federated setting, where data are distributed across multiple servers. Targeting demographic disparity constrained classification under federated differential privacy, we propose a two-step algorithm, namely FDP-Fair. In the special case where there is only one server, we further propose a simple yet powerful algorithm, namely CDP-Fair, serving as a computationally-lightweight alternative. Under mild structural assumptions, theoretical guarantees on privacy, fairness and excess risk control are established. In particular, we disentangle the source of the private fairness-aware excess risk into a) intrinsic cost of classification, b) cost of private classification, c) non-private cost of fairness and d) private cost of fairness. Our theoretical findings are complemented by extensive numerical experiments on both synthetic and real datasets, highlighting the practicality of our designed algorithms.

Deep neural networks (DNNs), particularly those using Rectified Linear Unit (ReLU) activation functions, have achieved remarkable success across diverse machine learning tasks, including image recognition, audio processing, and language modeling. Despite this success, the non-convex nature of DNN loss functions complicates optimization and limits theoretical understanding. In this paper, we highlight how recently developed convex equivalences of ReLU NNs and their connections to sparse signal processing models can address the challenges of training and understanding NNs. Recent research has uncovered several hidden convexities in the loss landscapes of certain NN architectures, notably two-layer ReLU networks and other deeper or varied architectures. This paper seeks to provide an accessible and educational overview that bridges recent advances in the mathematics of deep learning with traditional signal processing, encouraging broader signal processing applications.

Predictive inference is a fundamental task in statistics, traditionally addressed using parametric assumptions about the data distribution and detailed analyses of how models learn from data. In recent years, conformal prediction has emerged as a rapidly growing alternative framework that is particularly well suited to modern applications involving high-dimensional data and complex machine learning models. Its appeal stems from being both distribution-free -- relying mainly on symmetry assumptions such as exchangeability -- and model-agnostic, treating the learning algorithm as a black box. Even under such limited assumptions, conformal prediction provides exact finite-sample guarantees, though these are typically of a marginal nature that requires careful interpretation. This paper explains the core ideas of conformal prediction and reviews selected methods. Rather than offering an exhaustive survey, it aims to provide a clear conceptual entry point and a pedagogical overview of the field.

The company's sudden decision to pull the plug on Sora is a sign of deeper trouble. When I opened Sora this morning, I was met with a flood of strange and disturbing AI-generated videos. On OpenAI's video app, I scrolled through fabricated scenes of the Iran war and a barrage of fake Donald Trumps blabbering about Jeffrey Epstein. In my least favorite clip, I watched a man deep-fry an infant. The app lets users create fairly realistic-looking AI-generated clips--including of their own likeness--and then post them on a TikTok-like feed.

Why is Anthropic sponsoring an exhibition about Monet? Shower thoughts are typically best left in the shower. Such as: What might Claude the AI chatbot have to say about Claude Monet? Earlier this month, San Francisco's de Young Museum unveiled its newest exhibition, "Monet and Venice," which is dedicated to the impressionist painter's beautiful and meditative canvases of the floating city. And Anthropic, perhaps having seized on a marketing opportunity, is one of the show's lead sponsors.

Knowledge distillation has emerged as a powerful technique for compressing large language models (LLMs) into efficient, deployable architectures while preserving their advanced capabilities. Recent advances in low-rank knowledge distillation, particularly methods like Low-Rank Clone (LRC), have demonstrated remarkable empirical success, achieving comparable performance to full-parameter distillation with significantly reduced training data and computational overhead. However, the theoretical foundations underlying these methods remain poorly understood. In this paper, we establish a rigorous theoretical framework for low-rank knowledge distillation in language models. We prove that under mild assumptions, low-rank projection preserves the optimization dynamics, yielding explicit convergence rates of $O(1/\sqrt{T})$. We derive generalization bounds that characterize the fundamental trade-off between model compression and generalization capability, showing that the generalization error scales with the rank parameter as $O(r(m+n)/\sqrt{n})$. Furthermore, we provide an information-theoretic analysis of the activation cloning mechanism, revealing its role in maximizing the mutual information between the teacher's and student's intermediate representations. Our theoretical results offer principled guidelines for rank selection, mathematically suggesting an optimal rank $r^* = O(\sqrt{n})$ where $n$ is the sample size. Experimental validation on standard language modeling benchmarks confirms our theoretical predictions, demonstrating that the empirical convergence, rank scaling, and generalization behaviors align closely with our bounds.

We introduce Exponential Family Discriminant Analysis (EFDA), a unified generative framework that extends classical Linear Discriminant Analysis (LDA) beyond the Gaussian setting to any member of the exponential family. Under the assumption that each class-conditional density belongs to a common exponential family, EFDA derives closed-form maximum-likelihood estimators for all natural parameters and yields a decision rule that is linear in the sufficient statistic, recovering LDA as a special case and capturing nonlinear decision boundaries in the original feature space. We prove that EFDA is asymptotically calibrated and statistically efficient under correct specification, and we generalise it to $K \geq 2$ classes and multivariate data. Through extensive simulation across five exponential-family distributions (Weibull, Gamma, Exponential, Poisson, Negative Binomial), EFDA matches the classification accuracy of LDA, QDA, and logistic regression while reducing Expected Calibration Error (ECE) by $2$-$6\times$, a gap that is structural: it persists for all $n$ and across all class-imbalance levels, because misspecified models remain asymptotically miscalibrated. We further prove and empirically confirm that EFDA's log-odds estimator approaches the Cramér-Rao bound under correct specification, and is the only estimator in our comparison whose mean squared error converges to zero. Complete derivations are provided for nine distributions. Finally, we formally verify all four theoretical propositions in Lean 4, using Aristotle (Harmonic) and OpenGauss (Math, Inc.) as proof generators, with all outputs independently machine-checked by AXLE (Axiom).