Ping Ma

Luyang Fang, Yongkai Chen, Jiazhang Cai et al.

Knowledge distillation is a powerful method for model compression, enabling the efficient deployment of complex deep learning models (teachers), including large language models. However, its underlying statistical mechanisms remain unclear, and uncertainty evaluation is often overlooked, especially in real-world scenarios requiring diverse teacher expertise. To address these challenges, we introduce \textit{Multi-Teacher Bayesian Knowledge Distillation} (MT-BKD), where a distilled student model learns from multiple teachers within the Bayesian framework. Our approach leverages Bayesian inference to capture inherent uncertainty in the distillation process. We introduce a teacher-informed prior, integrating external knowledge from teacher models and task-specific training data, offering better generalization, robustness, and scalability. Additionally, an entropy-based weighting mechanism adaptively adjusts each teacher's influence, allowing the student to combine multiple sources of expertise effectively. MT-BKD enhances the interpretability of the student model's learning process, improves predictive accuracy, and provides uncertainty quantification. We validate MT-BKD on both synthetic and real-world tasks, including protein subcellular location prediction and image classification. Our experiments show improved performance and robust uncertainty quantification, highlighting the strengths of our MT-BKD framework.

Rui Xie, Shuyang Bai, Ping Ma

The Internet of Things (IoT) system generates massive high-speed temporally correlated streaming data and is often connected with online inference tasks under computational or energy constraints. Online analysis of these streaming time series data often faces a trade-off between statistical efficiency and computational cost. One important approach to balance this trade-off is sampling, where only a small portion of the sample is selected for the model fitting and update. Motivated by the demands of dynamic relationship analysis of IoT system, we study the data-dependent sample selection and online inference problem for a multi-dimensional streaming time series, aiming to provide low-cost real-time analysis of high-speed power grid electricity consumption data. Inspired by D-optimality criterion in design of experiments, we propose a class of online data reduction methods that achieve an optimal sampling criterion and improve the computational efficiency of the online analysis. We show that the optimal solution amounts to a strategy that is a mixture of Bernoulli sampling and leverage score sampling. The leverage score sampling involves auxiliary estimations that have a computational advantage over recursive least squares updates. Theoretical properties of the auxiliary estimations involved are also discussed. When applied to European power grid consumption data, the proposed leverage score based sampling methods outperform the benchmark sampling method in online estimation and prediction. The general applicability of the sampling-assisted online estimation method is assessed via simulation studies.

Huimin Cheng, Xiaowei Yu, Shushan Wu et al.

Medical images exhibit latent anatomical groupings, such as organs, tissues, and pathological regions, that standard Vision Transformers (ViTs) fail to exploit. While recent work like SBM-Transformer attempts to incorporate such structures through stochastic binary masking, they suffer from non-differentiability, training instability, and the inability to model complex community structure. We present DCMM-Transformer, a novel ViT architecture for medical image analysis that incorporates a Degree-Corrected Mixed-Membership (DCMM) model as an additive bias in self-attention. Unlike prior approaches that rely on multiplicative masking and binary sampling, our method introduces community structure and degree heterogeneity in a fully differentiable and interpretable manner. Comprehensive experiments across diverse medical imaging datasets, including brain, chest, breast, and ocular modalities, demonstrate the superior performance and generalizability of the proposed approach. Furthermore, the learned group structure and structured attention modulation substantially enhance interpretability by yielding attention maps that are anatomically meaningful and semantically coherent.

Haoran Lu, Luyang Fang, Wenxuan Zhong et al.

Multi-agent language systems are often built as hand-designed workflows, where agents are assigned semantic roles and communication protocols are specified in advance. We propose NeuroMAS, a method that first treats a multi-agent language system as a trainable and scalable neural-network-like architecture with LLM agents as nodes and intermediate textual signals as edges. In NeuroMAS, agent nodes are role-free but structure-aware: the topology only determines how information can flow in general, while reinforcement learning training determines how nodes communicate, specialize, and coordinate. This formulation shifts multi-agent design from workflow engineering toward architecture design, where depth, width, connectivity, and growth protocol become scalable sources of capability. Further, we provide a theoretical perspective showing why such modular textual computation is more parameter-efficient when tasks admit hierarchical decompositions. Experiments show that NeuroMAS improves significantly over both inference-time and trained multi-agent baselines. We further find that organizational scaling is path-dependent: larger systems can be challenging to train from scratch, but become feasible when grown progressively from smaller trained systems. These results suggest that learned neural multi-agent systems are a promising scaling axis for LLMs.

Yongkai Chen, Wenxuan Zhong, Ping Ma

Deep learning excels at prediction but often lacks finite-sample guarantees and calibrated uncertainty; RKHS (Reproducing Kernel Hilbert Space)-based methods provide those guarantees but struggle to adapt in high dimensions. We propose Wahkon, a deep RKHS superposition network that unifies Kolmogorov's superposition principle with RKHS regularization in the smoothing-spline tradition of Wahba. This yields a finite-dimensional deep representer theorem that makes training tractable and provides explicit layerwise complexity control. We show the penalized estimator is exactly the MAP (maximum a posteriori) estimate under a hierarchical Gaussian-process prior, extending the spline/GP duality to deep compositions. Using metric-entropy arguments, we establish minimax-optimal convergence rates under mild smoothness and clarify how depth and width trade off with regularity. Empirically, Wahkon outperforms multilayer perceptrons, Neural Tangent Kernels, and Kolmogorov--Arnold Networks across simulation benchmarks and a single-cell CITE-seq study. By unifying Kolmogorov's superposition principle with RKHS regularization, Wahkon delivers accuracy, interpretability, and statistical rigor in a single framework.

Ehsan Latif, Yifan Zhou, Shuchen Guo et al.

As artificial intelligence (AI) continues to advance, it demonstrates capabilities comparable to human intelligence, with significant potential to transform education and workforce development. This study evaluates OpenAI o1-preview's ability to perform higher-order cognitive tasks across 14 dimensions, including critical thinking, systems thinking, computational thinking, design thinking, metacognition, data literacy, creative thinking, abstract reasoning, quantitative reasoning, logical reasoning, analogical reasoning, and scientific reasoning. We used validated instruments like the Ennis-Weir Critical Thinking Essay Test and the Biological Systems Thinking Test to compare the o1-preview's performance with human performance systematically. Our findings reveal that o1-preview outperforms humans in most categories, achieving 150% better results in systems thinking, computational thinking, data literacy, creative thinking, scientific reasoning, and abstract reasoning. However, compared to humans, it underperforms by around 25% in logical reasoning, critical thinking, and quantitative reasoning. In analogical reasoning, both o1-preview and humans achieved perfect scores. Despite these strengths, the o1-preview shows limitations in abstract reasoning, where human psychology students outperform it, highlighting the continued importance of human oversight in tasks requiring high-level abstraction. These results have significant educational implications, suggesting a shift toward developing human skills that complement AI, such as creativity, abstract reasoning, and critical thinking. This study emphasizes the transformative potential of AI in education and calls for a recalibration of educational goals, teaching methods, and curricula to align with an AI-driven world.

Luyang Fang, Xiaowei Yu, Jiazhang Cai et al.

The exponential growth of Large Language Models (LLMs) continues to highlight the need for efficient strategies to meet ever-expanding computational and data demands. This survey provides a comprehensive analysis of two complementary paradigms: Knowledge Distillation (KD) and Dataset Distillation (DD), both aimed at compressing LLMs while preserving their advanced reasoning capabilities and linguistic diversity. We first examine key methodologies in KD, such as task-specific alignment, rationale-based training, and multi-teacher frameworks, alongside DD techniques that synthesize compact, high-impact datasets through optimization-based gradient matching, latent space regularization, and generative synthesis. Building on these foundations, we explore how integrating KD and DD can produce more effective and scalable compression strategies. Together, these approaches address persistent challenges in model scalability, architectural heterogeneity, and the preservation of emergent LLM abilities. We further highlight applications across domains such as healthcare and education, where distillation enables efficient deployment without sacrificing performance. Despite substantial progress, open challenges remain in preserving emergent reasoning and linguistic diversity, enabling efficient adaptation to continually evolving teacher models and datasets, and establishing comprehensive evaluation protocols. By synthesizing methodological innovations, theoretical foundations, and practical insights, our survey charts a path toward sustainable, resource-efficient LLMs through the tighter integration of KD and DD principles.

Wei Ruan, Yanjun Lyu, Jing Zhang et al.

With the rapid advancements in large language model (LLM) technology and the emergence of bioinformatics-specific language models (BioLMs), there is a growing need for a comprehensive analysis of the current landscape, computational characteristics, and diverse applications. This survey aims to address this need by providing a thorough review of BioLMs, focusing on their evolution, classification, and distinguishing features, alongside a detailed examination of training methodologies, datasets, and evaluation frameworks. We explore the wide-ranging applications of BioLMs in critical areas such as disease diagnosis, drug discovery, and vaccine development, highlighting their impact and transformative potential in bioinformatics. We identify key challenges and limitations inherent in BioLMs, including data privacy and security concerns, interpretability issues, biases in training data and model outputs, and domain adaptation complexities. Finally, we highlight emerging trends and future directions, offering valuable insights to guide researchers and clinicians toward advancing BioLMs for increasingly sophisticated biological and clinical applications.

Ehsan Latif, Luyang Fang, Ping Ma et al.

This study proposes a method for knowledge distillation (KD) of fine-tuned Large Language Models (LLMs) into smaller, more efficient, and accurate neural networks. We specifically target the challenge of deploying these models on resource-constrained devices. Our methodology involves training the smaller student model (Neural Network) using the prediction probabilities (as soft labels) of the LLM, which serves as a teacher model. This is achieved through a specialized loss function tailored to learn from the LLM's output probabilities, ensuring that the student model closely mimics the teacher's performance. To validate the performance of the KD approach, we utilized a large dataset, 7T, containing 6,684 student-written responses to science questions and three mathematical reasoning datasets with student-written responses graded by human experts. We compared accuracy with state-of-the-art (SOTA) distilled models, TinyBERT, and artificial neural network (ANN) models. Results have shown that the KD approach has 3% and 2% higher scoring accuracy than ANN and TinyBERT, respectively, and comparable accuracy to the teacher model. Furthermore, the student model size is 0.03M, 4,000 times smaller in parameters and x10 faster in inferencing than the teacher model and TinyBERT, respectively. The significance of this research lies in its potential to make advanced AI technologies accessible in typical educational settings, particularly for automatic scoring.

Haoran Lu, Luyang Fang, Ruidong Zhang et al.

Due to the remarkable capabilities and growing impact of large language models (LLMs), they have been deeply integrated into many aspects of society. Thus, ensuring their alignment with human values and intentions has emerged as a critical challenge. This survey provides a comprehensive overview of practical alignment techniques, training protocols, and empirical findings in LLM alignment. We analyze the development of alignment methods across diverse paradigms, characterizing the fundamental trade-offs between core alignment objectives. Our analysis shows that while supervised fine-tuning enables basic instruction-following, preference-based methods offer more flexibility for aligning with nuanced human intent. We discuss state-of-the-art techniques, including Direct Preference Optimization (DPO), Constitutional AI, brain-inspired methods, and alignment uncertainty quantification (AUQ), highlighting their approaches to balancing quality and efficiency. We review existing evaluation frameworks and benchmarking datasets, emphasizing limitations such as reward misspecification, distributional robustness, and scalable oversight. We summarize strategies adopted by leading AI labs to illustrate the current state of practice. We conclude by outlining open problems in oversight, value pluralism, robustness, and continuous alignment. This survey aims to inform both researchers and practitioners navigating the evolving landscape of LLM alignment.

Luyang Fang, Ehsan Latif, Haoran Lu et al.

Automatic scoring of student responses enhances efficiency in education, but deploying a separate neural network for each task increases storage demands, maintenance efforts, and redundant computations. To address these challenges, this paper introduces the Gromov-Wasserstein Scoring Model Merging (GW-SMM) method, which merges models based on feature distribution similarities measured via the Gromov-Wasserstein distance. Our approach begins by extracting features from student responses using individual models, capturing both item-specific context and unique learned representations. The Gromov-Wasserstein distance then quantifies the similarity between these feature distributions, identifying the most compatible models for merging. Models exhibiting the smallest pairwise distances, typically in pairs or trios, are merged by combining only the shared layers preceding the classification head. This strategy results in a unified feature extractor while preserving separate classification heads for item-specific scoring. We validated our approach against human expert knowledge and a GPT-o1-based merging method. GW-SMM consistently outperformed both, achieving a higher micro F1 score, macro F1 score, exact match accuracy, and per-label accuracy. The improvements in micro F1 and per-label accuracy were statistically significant compared to GPT-o1-based merging (p=0.04, p=0.01). Additionally, GW-SMM reduced storage requirements by half without compromising much accuracy, demonstrating its computational efficiency alongside reliable scoring performance.

Yingchuan Zhang, Terry Ma, Wenxuan Zhong et al.

Large language models (LLMs) achieve higher accuracy on challenging reasoning tasks by scaling test-time compute through multiple trajectory sampling. However, standard aggregation methods like majority voting or individual confidence-based filtering face a fundamental "blind spot": they evaluate each trace in isolation. As problems scale in difficulty, models often generate hallucinated paths that exhibit misleadingly high confidence, causing the true solution to be suppressed by a narrow margin in traditional voting. We ask: can we enable traces to "peer-review" each other to resolve these near-miss errors? We introduce Packet-Conditioned Revision (PACER), a training-free, inference-only framework that enables reasoning traces to revise their conclusions through a structured coordination step. After a preliminary screening of generated traces, PACER constructs a compact consensus packet containing (i) unique candidate answers, (ii) their aggregated confidence scores, and (iii) representative reasoning summaries for each candidate answer. Individual traces then perform a targeted self-review conditioned on this packet, allowing them to identify specific logical junctions where they diverged from the broader consensus and pivot if their original reasoning is found to be flawed. Final predictions are obtained via confidence-weighted voting over these revised trajectories. On challenging competitive math benchmarks such as AIME and BRUMO, PACER matches or exceeds the accuracy of 256-sample majority voting, significantly outperforming raw ensemble baselines by transforming simple consensus into a collaborative logical refinement process.

Xilin Gong, Yongkai Chen, Shushan Wu et al.

Echocardiogram is the most commonly used imaging modality in cardiac assessment duo to its non-invasive nature, real-time capability, and cost-effectiveness. Despite its advantages, most clinical echocardiograms provide only two-dimensional views, limiting the ability to fully assess cardiac anatomy and function in three dimensions. While three-dimensional echocardiography exists, it often suffers from reduced resolution, limited availability, and higher acquisition costs. To overcome these challenges, we propose a deep learning framework S2MNet that reconstructs continuous and high-fidelity 3D heart models by integrating six slices of routinely acquired 2D echocardiogram views. Our method has three advantages. First, our method avoid the difficulties on training data acquasition by simulate six of 2D echocardiogram images from corresponding slices of a given 3D heart mesh. Second, we introduce a deformation field-based method, which avoid spatial discontinuities or structural artifacts in 3D echocardiogram reconstructions. We validate our method using clinically collected echocardiogram and demonstrate that our estimated left ventricular volume, a key clinical indicator of cardiac function, is strongly correlated with the doctor measured GLPS, a clinical measurement that should demonstrate a negative correlation with LVE in medical theory. This association confirms the reliability of our proposed 3D construction method.

Lijie Hu, Tianhao Huang, Huanyi Xie et al.

Concept Bottleneck Models (CBMs) have garnered increasing attention due to their ability to provide concept-based explanations for black-box deep learning models while achieving high final prediction accuracy using human-like concepts. However, the training of current CBMs is heavily dependent on the precision and richness of the annotated concepts in the dataset. These concept labels are typically provided by experts, which can be costly and require significant resources and effort. Additionally, concept saliency maps frequently misalign with input saliency maps, causing concept predictions to correspond to irrelevant input features - an issue related to annotation alignment. To address these limitations, we propose a new framework called SSCBM (Semi-supervised Concept Bottleneck Model). Our SSCBM is suitable for practical situations where annotated data is scarce. By leveraging joint training on both labeled and unlabeled data and aligning the unlabeled data at the concept level, we effectively solve these issues. We proposed a strategy to generate pseudo labels and an alignment loss. Experiments demonstrate that our SSCBM is both effective and efficient. With only 10% labeled data, our model's concept and task accuracy on average across four datasets is only 2.44% and 3.93% lower, respectively, compared to the best baseline in the fully supervised learning setting.

Yijun Yan, Jinchang Ren, Barry Harrison et al.

Peat, a crucial component in whisky production, imparts distinctive and irreplaceable flavours to the final product. However, the extraction of peat disrupts ancient ecosystems and releases significant amounts of carbon, contributing to climate change. This paper aims to address this issue by conducting a feasibility study on enhancing peat use efficiency in whisky manufacturing through non-destructive analysis using hyperspectral imaging. Results show that shot-wave infrared (SWIR) data is more effective for analyzing peat samples and predicting total phenol levels, with accuracies up to 99.81%.

Hongyi Qian, Baohui Wang, Ping Ma et al.

Dataset shift is common in credit scoring scenarios, and the inconsistency between the distribution of training data and the data that actually needs to be predicted is likely to cause poor model performance. However, most of the current studies do not take this into account, and they directly mix data from different time periods when training the models. This brings about two problems. Firstly, there is a risk of data leakage, i.e., using future data to predict the past. This can result in inflated results in offline validation, but unsatisfactory results in practical applications. Secondly, the macroeconomic environment and risk control strategies are likely to be different in different time periods, and the behavior patterns of borrowers may also change. The model trained with past data may not be applicable to the recent stage. Therefore, we propose a method based on adversarial validation to alleviate the dataset shift problem in credit scoring scenarios. In this method, partial training set samples with the closest distribution to the predicted data are selected for cross-validation by adversarial validation to ensure the generalization performance of the trained model on the predicted samples. In addition, through a simple splicing method, samples in the training data that are inconsistent with the test data distribution are also involved in the training process of cross-validation, which makes full use of all the data and further improves the model performance. To verify the effectiveness of the proposed method, comparative experiments with several other data split methods are conducted with the data provided by Lending Club. The experimental results demonstrate the importance of dataset shift in the field of credit scoring and the superiority of the proposed method.

Cheng Meng, Yuan Ke, Jingyi Zhang et al.

This paper studies the estimation of large-scale optimal transport maps (OTM), which is a well-known challenging problem owing to the curse of dimensionality. Existing literature approximates the large-scale OTM by a series of one-dimensional OTM problems through iterative random projection. Such methods, however, suffer from slow or none convergence in practice due to the nature of randomly selected projection directions. Instead, we propose an estimation method of large-scale OTM by combining the idea of projection pursuit regression and sufficient dimension reduction. The proposed method, named projection pursuit Monge map (PPMM), adaptively selects the most ``informative'' projection direction in each iteration. We theoretically show the proposed dimension reduction method can consistently estimate the most ``informative'' projection direction in each iteration. Furthermore, the PPMM algorithm weakly convergences to the target large-scale OTM in a reasonable number of steps. Empirically, PPMM is computationally easy and converges fast. We assess its finite sample performance through the applications of Wasserstein distance estimation and generative models.

Cheng Meng, Jun Yu, Jingyi Zhang et al.

Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport. We study the asymptotic properties of POTD and show that in the cases when the class labels contain no error, POTD estimates the SDR subspace exclusively. Empirical studies show POTD outperforms most of the state-of-the-art linear dimension reduction methods.

Jingyi Zhang, Wenxuan Zhong, Ping Ma

Optimal transport has been one of the most exciting subjects in mathematics, starting from the 18th century. As a powerful tool to transport between two probability measures, optimal transport methods have been reinvigorated nowadays in a remarkable proliferation of modern data science applications. To meet the big data challenges, various computational tools have been developed in the recent decade to accelerate the computation for optimal transport methods. In this review, we present some cutting-edge computational optimal transport methods with a focus on the regularization-based methods and the projection-based methods. We discuss their real-world applications in biomedical research.

Ping Ma, Xinlian Zhang, Xin Xing et al.

The statistical analysis of Randomized Numerical Linear Algebra (RandNLA) algorithms within the past few years has mostly focused on their performance as point estimators. However, this is insufficient for conducting statistical inference, e.g., constructing confidence intervals and hypothesis testing, since the distribution of the estimator is lacking. In this article, we develop an asymptotic analysis to derive the distribution of RandNLA sampling estimators for the least-squares problem. In particular, we derive the asymptotic distribution of a general sampling estimator with arbitrary sampling probabilities. The analysis is conducted in two complementary settings, i.e., when the objective of interest is to approximate the full sample estimator or is to infer the underlying ground truth model parameters. For each setting, we show that the sampling estimator is asymptotically normally distributed under mild regularity conditions. Moreover, the sampling estimator is asymptotically unbiased in both settings. Based on our asymptotic analysis, we use two criteria, the Asymptotic Mean Squared Error (AMSE) and the Expected Asymptotic Mean Squared Error (EAMSE), to identify optimal sampling probabilities. Several of these optimal sampling probability distributions are new to the literature, e.g., the root leverage sampling estimator and the predictor length sampling estimator. Our theoretical results clarify the role of leverage in the sampling process, and our empirical results demonstrate improvements over existing methods.

Xin Xing, Zuofeng Shang, Pang Du et al.

We consider the problem of comparing probability densities between two groups. A new probabilistic tensor product smoothing spline framework is developed to model the joint density of two variables. Under such a framework, the probability density comparison is equivalent to testing the presence/absence of interactions. We propose a penalized likelihood ratio test for such interaction testing and show that the test statistic is asymptotically chi-square distributed under the null hypothesis. Furthermore, we derive a sharp minimax testing rate based on the Bernstein width for nonparametric two-sample tests and show that our proposed test statistics is minimax optimal. In addition, a data-adaptive tuning criterion is developed to choose the penalty parameter. Simulations and real applications demonstrate that the proposed test outperforms the conventional approaches under various scenarios.

HaiYing Wang, Rong Zhu, Ping Ma

For massive data, the family of subsampling algorithms is popular to downsize the data volume and reduce computational burden. Existing studies focus on approximating the ordinary least squares estimate in linear regression, where statistical leverage scores are often used to define subsampling probabilities. In this paper, we propose fast subsampling algorithms to efficiently approximate the maximum likelihood estimate in logistic regression. We first establish consistency and asymptotic normality of the estimator from a general subsampling algorithm, and then derive optimal subsampling probabilities that minimize the asymptotic mean squared error of the resultant estimator. An alternative minimization criterion is also proposed to further reduce the computational cost. The optimal subsampling probabilities depend on the full data estimate, so we develop a two-step algorithm to approximate the optimal subsampling procedure. This algorithm is computationally efficient and has a significant reduction in computing time compared to the full data approach. Consistency and asymptotic normality of the estimator from a two-step algorithm are also established. Synthetic and real data sets are used to evaluate the practical performance of the proposed method.

Rong Zhu, Ping Ma, Michael W. Mahoney et al.

A significant hurdle for analyzing large sample data is the lack of effective statistical computing and inference methods. An emerging powerful approach for analyzing large sample data is subsampling, by which one takes a random subsample from the original full sample and uses it as a surrogate for subsequent computation and estimation. In this paper, we study subsampling methods under two scenarios: approximating the full sample ordinary least-square (OLS) estimator and estimating the coefficients in linear regression. We present two algorithms, weighted estimation algorithm and unweighted estimation algorithm, and analyze asymptotic behaviors of their resulting subsample estimators under general conditions. For the weighted estimation algorithm, we propose a criterion for selecting the optimal sampling probability by making use of the asymptotic results. On the basis of the criterion, we provide two novel subsampling methods, the optimal subsampling and the predictor- length subsampling methods. The predictor-length subsampling method is based on the L2 norm of predictors rather than leverage scores. Its computational cost is scalable. For unweighted estimation algorithm, we show that its resulting subsample estimator is not consistent to the full sample OLS estimator. However, it has better performance than the weighted estimation algorithm for estimating the coefficients. Simulation studies and a real data example are used to demonstrate the effectiveness of our proposed subsampling methods.

Ping Ma, Michael W. Mahoney, Bin Yu

One popular method for dealing with large-scale data sets is sampling. For example, by using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows/columns of data matrices to reduce the data size before performing computations on the subproblem. This method has been successful in improving computational efficiency of algorithms for matrix problems such as least-squares approximation, least absolute deviations approximation, and low-rank matrix approximation. Existing work has focused on algorithmic issues such as worst-case running times and numerical issues associated with providing high-quality implementations, but none of it addresses statistical aspects of this method. In this paper, we provide a simple yet effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model with a fixed number of predictors. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms. A detailed empirical evaluation of existing leverage-based methods as well as these two new methods is carried out on both synthetic and real data sets. The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage-based algorithms and that the new algorithms achieve improved performance.