Processing high-volume, streaming data is increasingly common in modern statistics and machine learning, where batch-mode algorithms are often impractical because they require repeated passes over the full dataset. This has motivated incremental stochastic estimation methods, including the incremental stochastic Expectation-Maximization (EM) algorithm formulated via stochastic approximation. In this work, we revisit and analyze an incremental stochastic variant of the Majorization-Minimization (MM) algorithm, which generalizes incremental stochastic EM as a special case. Our approach relaxes key EM requirements, such as explicit latent-variable representations, enabling broader applicability and greater algorithmic flexibility. We establish theoretical guarantees for the incremental stochastic MM algorithm, proving consistency in the sense that the iterates converge to a stationary point characterized by a vanishing gradient of the objective. We demonstrate these advantages on a softmax-gated mixture of experts (MoE) regression problem, for which no stochastic EM algorithm is available. Empirically, our method consistently outperforms widely used stochastic optimizers, including stochastic gradient descent, root mean square propagation, adaptive moment estimation, and second-order clipped stochastic optimization. These results support the development of new incremental stochastic algorithms, given the central role of softmax-gated MoE architectures in contemporary deep neural networks for heterogeneous data modeling. Beyond synthetic experiments, we also validate practical effectiveness on two real-world datasets, including a bioinformatics study of dent maize genotypes under drought stress that integrates high-dimensional proteomics with ecophysiological traits, where incremental stochastic MM yields stable gains in predictive performance.

We study grokking, the onset of generalization long after overfitting, in a classical ridge regression setting. We prove end-to-end grokking results for learning over-parameterized linear regression models using gradient descent with weight decay. Specifically, we prove that the following stages occur: (i) the model overfits the training data early during training; (ii) poor generalization persists long after overfitting has manifested; and (iii) the generalization error eventually becomes arbitrarily small. Moreover, we show, both theoretically and empirically, that grokking can be amplified or eliminated in a principled manner through proper hyperparameter tuning. To the best of our knowledge, these are the first rigorous quantitative bounds on the generalization delay (which we refer to as the "grokking time") in terms of training hyperparameters. Lastly, going beyond the linear setting, we empirically demonstrate that our quantitative bounds also capture the behavior of grokking on non-linear neural networks. Our results suggest that grokking is not an inherent failure mode of deep learning, but rather a consequence of specific training conditions, and thus does not require fundamental changes to the model architecture or learning algorithm to avoid.

The empirical success of deep learning is often attributed to deep networks' ability to exploit hierarchical structure in data, constructing increasingly complex features across layers. Yet despite substantial progress in deep learning theory, most optimization results sill focus on networks with only two or three layers, leaving the theoretical understanding of hierarchical learning in genuinely deep models limited. This leads to a natural question: can we prove that deep networks, trained by gradient-based methods, can efficiently exploit hierarchical structure? In this work, we consider Random Hierarchy Models -- a hierarchical context-free grammar introduced by arXiv:2307.02129 and conjectured to separate deep and shallow networks. We prove that, under mild conditions, a deep convolutional network can be efficiently trained to learn this function class. Our proof builds on a general observation: if intermediate layers can receive clean signal from the labels and the relevant features are weakly identifiable, then layerwise training each individual layer suffices to hierarchically learn the target function.

Imputing missing node features in graphs is challenging, particularly under high missing rates. Existing methods based on latent representations or global diffusion often fail to produce reliable estimates, and may propagate errors across the graph. We propose FSD-CAP, a two-stage framework designed to improve imputation quality under extreme sparsity. A fractional diffusion operator adjusts propagation sharpness based on local structure. In the second stage, imputed features are refined using class-aware propagation, which incorporates pseudo-labels and neighborhood entropy to promote consistency. We evaluated FSD-CAP on multiple datasets. With 99 .5% of features missing across five benchmark datasets, FSD-CAP achieves average accuracies of 80 .06% For link prediction under the same setting, it reaches AUC scores of 91. Furthermore, FSD-CAP demonstrates superior performance on both large-scale and heterophily datasets when compared to other models. Graph Neural Networks (GNNs) are widely used for learning from graph-structured data, with successful applications in social networks (Bian et al., 2020), biology (Li et al., 2022), and recommendation systems (He et al., 2020). GNN architectures(Chen et al., 2023; Chien et al., 2020) always assume nodal features are fully observed, allowing information to be aggregated effectively from neighboring nodes. In practice, this assumption often fails. Node attributes are frequently missing due to privacy constraints, sensor failures, or incomplete data collection. High missing rates disrupt the message-passing process and significantly degrade model performance. A variety of methods have been proposed for imputing missing features, including statistical estimators (Srebro et al., 2004), machine learning models (Chen & Guestrin, 2016), and generative approaches (Vincent et al., 2008). Recent work has shifted toward deep learning techniques that model the distribution of node attributes. These include latent space models that align observed features with learned embeddings (Chen et al., 2020; Y oo et al., 2022), and GNN-based architectures designed to operate on incomplete inputs (Taguchi et al., 2021). These approaches, which rely on correlations in both feature and graph structure, are effective under moderate missing rates but experience significant performance degradation as sparsity increases, ultimately falling below simple baselines like zero-filling or mean imputation in highly incomplete settings(Y ou et al., 2020).

As generative models continue to evolve, detecting AI-generated images remains a critical challenge. While effective detection methods exist, they often lack formal interpretability and may rely on implicit assumptions about fake content, potentially limiting robustness to distributional shifts. In this work, we introduce a rigorous, statistically grounded framework for fake image detection that focuses on producing a probability score interpretable with respect to the real-image population. Our method leverages the strengths of multiple existing detectors by combining training-free statistics. We compute p-values over a range of test statistics and aggregate them using classical statistical ensembling to assess alignment with the unified real-image distribution. This framework is generic, flexible, and training-free, making it well-suited for robust fake image detection across diverse and evolving settings.

Apple could unveil Gemini-powered Siri in Feb. Adobe Photoshop introduced some new features that are rolling out for creators today. As you'd expect from any service operator in this day and age, there's some AI involved. Adobe has improved the tools for Generative Fill, Generative Expand and Remove that are powered by its Firefly generative AI platform. Using these tools for image editing should now produce results in 2K resolution with fewer artifacts and increased detail all while delivering better matches for the provided prompts. The Reference Image option for Generative Fill has also been upgraded to deliver geometry-aware results that better match the scene.

OpenAI's latest product lets you vibe code science Prism is a ChatGPT-powered text editor that automates much of the work involved in writing scientific papers. OpenAI just revealed what its new in-house team, OpenAI for Science, has been up to. The firm has released a free LLM-powered tool for scientists called Prism, which embeds ChatGPT in a text editor for writing scientific papers. The idea is to put ChatGPT front and center inside software that scientists use to write up their work in much the same way that chatbots are now embedded into popular programming editors. Kevin Weil, head of OpenAI for Science, pushes that analogy himself. "I think 2026 will be for AI and science what 2025 was for AI in software engineering," he said at a press briefing yesterday.

Apple could unveil Gemini-powered Siri in Feb. The plan is now out everywhere Google sells its AI services. Google AI Plus, the company's most affordable AI subscription plan, is now rolling out in the US. It will cost you $8 a month for its features, though you can get it for $4 a month for the first two months for a limited time only. AI Plus gives you access to 200GB of storage, as well as access to the Gemini 3 Pro model, Deep Research and Nano Banana Pro inside the Gemini app.

Apple could unveil Gemini-powered Siri in Feb. Prism can edit and format LaTeX. OpenAI is releasing a new app called Prism today, and it hopes it does for science what coding agents like Claude Code and its own Codex platform have done for programming. Prism builds on Crixet, a cloud-based LaTeX platform the company is announcing it acquired today. For the uninitiated, LaTeX is a typesetting system for formatting scientific documents and journals. Nearly the entire scientific community relies on LaTeX, but it can make some tasks, such as drawing diagrams through TikZ commands, time-consuming to do.

PCWorld reports that OpenAI is testing an updated Temporary Chat feature for ChatGPT that retains personal customizations without saving conversations. This hidden setting functions like Incognito Mode, allowing private chats while maintaining user personalization preferences across sessions. AI engineer Tibor Blaho discovered the optional feature, though OpenAI may still store chat copies for up to 30 days for security purposes. OpenAI is testing an update to ChatGPT's "Temporary Chat" feature . Temporary Chat allows you to "have a conversation with a blank slate" where "ChatGPT won't be aware of previous conversations or access memories." It's a lot like Incognito Mode in Chrome, which you can use to "privately" browse the web--except here, it's chatting. The update to Temporary Chat will sort of tweak this, allowing ChatGPT to retain your personal customizations without the conversation itself being saved or affecting your overall account. Temporary chat in ChatGPT has a new, currently hidden option that lets you still use personalization (memory, chat history, style and tone preferences) even though the chat is only temporary ChatGPT web app now also mentions new ChatGPT FinServ plans (Enterprise-like), shopping pic.twitter.com/CaMaeYlCmP