For extreme low-bit quantization of large language models (LLMs), Double Binary Factorization (DBF) is attractive as it enables efficient inference without sacrificing accuracy. However, the scaling parameters of DBF are too restrictive; after factoring out signs, all rank components share the same magnitude profile, resulting in performance saturation. We propose Multi-envelope DBF (MDBF), which retains a shared pair of 1-bit sign bases but replaces the single envelope with a rank-$l$ envelope. By sharing sign matrices among envelope components, MDBF effectively maintains a binary carrier and utilizes the limited memory budget for magnitude expressiveness. We also introduce a closed-form initialization and an alternating refinement method to optimize MDBF. Across the LLaMA and Qwen families, MDBF enhances perplexity and zero-shot accuracy over previous binary formats at matched bits per weight while preserving the same deployment-friendly inference primitive.

We introduce two nonlinear sufficient dimension reduction methods for regressions with tensor-valued predictors. Our goal is two-fold: the first is to preserve the tensor structure when performing dimension reduction, particularly the meaning of the tensor modes, for improved interpretation; the second is to substantially reduce the number of parameters in dimension reduction, thereby achieving model parsimony and enhancing estimation accuracy. Our two tensor dimension reduction methods echo the two commonly used tensor decomposition mechanisms: one is the Tucker decomposition, which reduces a larger tensor to a smaller one; the other is the CP-decomposition, which represents an arbitrary tensor as a sequence of rank-one tensors. We developed the Fisher consistency of our methods at the population level and established their consistency and convergence rates. Both methods are easy to implement numerically: the Tucker-form can be implemented through a sequence of least-squares steps, and the CP-form can be implemented through a sequence of singular value decompositions. We investigated the finite-sample performance of our methods and showed substantial improvement in accuracy over existing methods in simulations and two data applications.

Spatial boundaries, such as ecological transitions or climatic regime interfaces, capture steep environmental gradients, and shifts in their structure can signal emerging environmental changes. Quantifying uncertainty in spatial boundary locations and formally testing for temporal shifts remains challenging, especially when boundaries are derived from noisy, gridded environmental data. We present a unified framework that combines heteroskedastic Gaussian process (GP) regression with a scaled Maximum Absolute Difference (MAD) Global Envelope Test (GET) to estimate spatial boundary curves and assess whether they evolve over time. The heteroskedastic GP provides a flexible probabilistic reconstruction of boundary lines, capturing spatially varying mean structure and location specific variability, while the test offers a rigorous hypothesis testing tool for detecting departures from expected boundary behaviors. Simulation studies show that the proposed method achieves the correct size under the null and high power for detecting local boundary shifts. Applying our framework to the Sahel Sahara transition zone, using annual Koppen Trewartha climate classifications from 1960 to 1989, we find no statistically significant decade scale changes in the arid and semi arid or semi arid and non arid interfaces. However, the method successfully identifies localized boundary shifts during the extreme drought years of 1983 and 1984, consistent with climate studies documenting regional anomalies in these interfaces during that period.

Assessing risk of bias (RoB) in randomized controlled trials is essential for trustworthy evidence synthesis, but the process is resource-intensive and prone to variability across reviewers. Large language models (LLMs) offer a route to automation, but existing methods rely on manually engineered prompts that are difficult to reproduce, generalize, or evaluate. This study introduces a programmable RoB assessment pipeline that replaces ad-hoc prompt design with structured, code-based optimization using DSPy and its GEPA module. GEPA refines LLM reasoning through Pareto-guided search and produces inspectable execution traces, enabling transparent replication of every step in the optimization process. We evaluated the method on 100 RCTs from published meta-analyses across seven RoB domains. GEPA-generated prompts were applied to both open-weight models (Mistral Small 3.1 with GPT-oss-20b) and commercial models (GPT-5 Nano and GPT-5 Mini). In domains with clearer methodological reporting, such as Random Sequence Generation, GEPA-generated prompts performed best, with similar results for Allocation Concealment and Blinding of Participants, while the commercial model performed slightly better overall. We also compared GEPA with three manually designed prompts using Claude 3.5 Sonnet. GEPA achieved the highest overall accuracy and improved performance by 30%-40% in Random Sequence Generation and Selective Reporting, and showed generally comparable, competitively aligned performance in the other domains relative to manual prompts. These findings suggest that GEPA can produce consistent and reproducible prompts for RoB assessment, supporting the structured and principled use of LLMs in evidence synthesis.

The simplest nontrivial geometric concept is perhaps the halfspace.

Recent works proposed amortizing the cost by learning generalized wave functions across different structures and compounds instead of solving each problem independently.

Failures in satellite components are costly and challenging to address, often requiring significant human and material resources. Embedding a hybrid AI-based system for fault detection directly in the satellite can greatly reduce this burden by allowing earlier detection. However, such systems must operate with extremely high reliability. To ensure this level of dependability, we employ the formal verification tool Marabou to verify the local robustness of the neural network models used in the AI-based algorithm. This tool allows us to quantify how much a model's input can be perturbed before its output behavior becomes unstable, thereby improving trustworthiness with respect to its performance under uncertainty.

Both of these results improve upon prior work [ Bian et al., 2017a, b,

The simplest nontrivial geometric concept is perhaps the halfspace.

We focus on monotonicity constraints, which are common and require that the function's output increases with increasing values of specific input features.