Surprisingly we show that volume sampling can have poor behavior when we require a very accurate approximation - indeed worse than some i.i.d.

In this work, we aim to warm-start BO in the context of hyperparameter optimization (HPO).

We consider a generalization of mixed regression where the response is an additive combination of several mixture components.

Code is publicly available at this url.

This paper presents a tractable algorithm for estimating an unknown Lipschitz function from noisy observations and establishes an upper bound on its convergence rate. The approach extends max-affine methods from convex shape-restricted regression to the more general Lipschitz setting. A key component is a nonlinear feature expansion that maps max-affine functions into a subclass of delta-convex functions, which act as universal ap-proximators of Lipschitz functions while preserving their Lipschitz constants. Leveraging this property, the estimator attains the minimax convergence rate (up to logarithmic factors) with respect to the intrinsic dimension of the data under squared loss and subgaussian distributions in the random design setting. The algorithm integrates adaptive partitioning to capture intrinsic dimension, a penalty-based regularization mechanism that removes the need to know the true Lipschitz constant, and a two-stage optimization procedure combining a convex initialization with local refinement. The framework is also straightforward to adapt to convex shape-restricted regression. Experiments demonstrate competitive performance relative to other theoretically justified methods, including nearest-neighbor and kernel-based regressors.

Such assumptions allow for identification of causal target parameters. For instance, one may be interested in the average variance-weighted treatment effects (Robins et al., 2008; Li et al., 2011),

Heaps' or Herdan's law characterizes the word-type vs. word-token relation by a power-law function, which is concave in linear-linear scale but a straight line in log-log scale. However, it has been observed that even in log-log scale, the type-token curve is still slightly concave, invalidating the power-law relation. At the next-order approximation, we have shown, by twenty English novels or writings (some are translated from another language to English), that quadratic functions in log-log scale fit the type-token data perfectly. Regression analyses of log(type)-log(token) data with both a linear and quadratic term consistently lead to a linear coefficient of slightly larger than 1, and a quadratic coefficient around -0.02. Using the ``random drawing colored ball from the bag with replacement" model, we have shown that the curvature of the log-log scale is identical to a ``pseudo-variance" which is negative. Although a pseudo-variance calculation may encounter numeric instability when the number of tokens is large, due to the large values of pseudo-weights, this formalism provides a rough estimation of the curvature when the number of tokens is small.

Grammatical Error Correction (GEC) is an important aspect of natural language processing. Arabic has a complicated morphological and syntactic structure, posing a greater challenge than other languages. Even though modern neural models have improved greatly in recent years, the majority of previous attempts used individual models without taking into account the potential benefits of combining different systems. In this paper, we present one of the first multi-system approaches for correcting grammatical errors in Arabic, the Arab Enhanced Edit Selection System Complication (ArbESC+). Several models are used to collect correction proposals, which are represented as numerical features in the framework. A classifier determines and implements the appropriate corrections based on these features. In order to improve output quality, the framework uses support techniques to filter overlapping corrections and estimate decision reliability. A combination of AraT5, ByT5, mT5, AraBART, AraBART+Morph+GEC, and Text editing systems gave better results than a single model alone, with F0.5 at 82.63% on QALB-14 test data, 84.64% on QALB-15 L1 data, and 65.55% on QALB-15 L2 data. As one of the most significant contributions of this work, it's the first Arab attempt to integrate linguistic error correction. Improving existing models provides a practical step towards developing advanced tools that will benefit users and researchers of Arabic text processing.

We introduce a highly expressive class of function approximators called Splat Regression Models. Model outputs are mixtures of heterogeneous and anisotropic bump functions, termed splats, each weighted by an output vector. The power of splat modeling lies in its ability to locally adjust the scale and direction of each splat, achieving both high interpretability and accuracy. Fitting splat models reduces to optimization over the space of mixing measures, which can be implemented using Wasserstein-Fisher-Rao gradient flows. As a byproduct, we recover the popular Gaussian Splatting methodology as a special case, providing a unified theoretical framework for this state-of-the-art technique that clearly disambiguates the inverse problem, the model, and the optimization algorithm. Through numerical experiments, we demonstrate that the resulting models and algorithms constitute a flexible and promising approach for solving diverse approximation, estimation, and inverse problems involving low-dimensional data.

Transformer architectures can solve unseen tasks based on input-output pairs in a given prompt due to in-context learning (ICL). Existing theoretical studies on ICL have mainly focused on linear regression tasks, often with i.i.d. inputs. To understand how transformers express ICL when modeling dynamics-driven functions, we investigate Markovian function learning through a structured ICL setup, where we characterize the loss landscape to reveal underlying optimization behaviors. Specifically, we (1) provide the closed-form expression of the global minimizer (in an enlarged parameter space) for a single-layer linear self-attention (LSA) model; (2) prove that recovering transformer parameters that realize the optimal solution is NP-hard in general, revealing a fundamental limitation of one-layer LSA in representing structured dynamical functions; and (3) supply a novel interpretation of a multilayer LSA as performing preconditioned gradient descent to optimize multiple objectives beyond the square loss. These theoretical results are numerically validated using simplified transformers.