adaptive sampling

This paper addresses high computational complexity in constrained optimization for one-bit/few-bit signal processing—particularly involving semidefinite and low-rank constraints. We introduce the novel concept of “sample-abundant singularity”: when the number of measurements vastly exceeds classical requirements, high-dimensional nonconvex/nonlinear constraints degenerate into an overdetermined linear feasibility problem. Leveraging the finite-volume property, we theoretically establish that this phenomenon guarantees exact signal recovery. Methodologically, we integrate low-precision quantization models with efficient linear feasibility solvers, eliminating iterative optimization and matrix decomposition. Experiments demonstrate that our framework reduces computational cost by several orders of magnitude while maintaining high accuracy in phase retrieval and covariance estimation. Moreover, it exhibits superior practicality in hardware-constrained settings.

Conventional fixed-rate sampling—e.g., adhering to the Nyquist–Shannon criterion—is suboptimal when prior knowledge about the signal is available, limiting information efficiency in optical spectroscopy, blood analysis, optical displacement metrology, and hyperspectral imaging. This work challenges that paradigm by establishing a prior-informed, information-optimal adaptive measurement framework. We first provide a rigorous proof that classical uniform sampling is optimal *only* under complete ignorance of the signal statistics. Then, we propose a real-time adaptive sampling strategy that jointly leverages Bayesian uncertainty quantification and information-entropy-driven decision making, transforming the measurement system into an autonomous, information-aware agent. Experiments across multiple optical diagnostic tasks demonstrate a 2–5× improvement in measured information gain over conventional methods, while maintaining comparable real-time performance.

For expensive black-box target density sampling, this paper proposes an active sampling method framed within the multi-armed bandit (MAB) paradigm. Unlike conventional approaches that optimize a proposal distribution, our method models sample location selection as a sequential decision-making process, jointly leveraging a Gaussian process surrogate model and space-filling criteria to adaptively select evaluation points with maximal information gain. To our knowledge, this is the first work to directly apply MAB for sampling point scheduling—bypassing explicit distribution modeling and substantially reducing the number of target function evaluations. Experiments demonstrate superior performance over state-of-the-art importance sampling methods on multimodal and heavy-tailed distributions. In Bayesian inference tasks, our approach achieves higher approximation accuracy with significantly fewer evaluations.

This study addresses the fundamental challenge of achieving optimal function approximation under limited evaluation data—a central problem in numerical analysis and machine learning. From the perspective of information-based complexity, the work systematically investigates function recovery under generalized sampling by integrating information-theoretic analysis, optimal recovery theory, and nonlinear, adaptive, and randomized sampling mechanisms. It uncovers intrinsic connections among diverse sampling strategies, characterizes the information-theoretic limits of function approximation given finite data, and proposes efficient algorithms and sampling schemes that approach these limits. The results provide foundational insights and a unified framework for optimal sampling theory.

This work addresses the design of **optimal randomized sampling strategies** for weighted least-squares approximation, extending beyond classical settings restricted to pointwise evaluations and linear approximation spaces to encompass **generalized non-pointwise observations** (e.g., integrals, derivatives) and **nonlinear approximation spaces**. Methodologically, it introduces a systematic generalization of the Christoffel function to the **generalized recovery framework**, yielding a unified theoretical foundation. The resulting sampling scheme achieves near-optimal sample complexity—requiring only $O(n log n)$ measurements for $n$ degrees of freedom—improving upon the classical $O(n^2)$ bound. Theoretical analysis guarantees stable, high-probability reconstruction. The approach integrates tools from approximation theory, randomized sampling design, and numerical linear algebra. Extensive experiments demonstrate its effectiveness and broad applicability in machine learning and scientific computing.

This work addresses a key challenge in compressive sensing: reconciling theoretical performance guarantees with the practical need for deterministic selection of critical sampling rows. The authors propose an optimized Bernoulli sampling scheme that rigorously integrates random and deterministic strategies, explicitly identifying and prioritizing crucial rows in unitary measurement matrices. By incorporating both sparsity and generative prior models, the method yields tighter sample complexity bounds and novel denoising guarantees in theory. Experimental results demonstrate that, in image compressive sensing tasks, the proposed approach achieves significantly higher reconstruction quality compared to conventional sampling methods with and without replacement.

This work addresses the quadratic sample complexity and numerical instability inherent in conventional generalized sampling for infinite-dimensional signal reconstruction, which arise from dependence on specific bases. To overcome these limitations, the authors propose a fully randomized generalized sampling framework that draws samples according to an optimal leverage score distribution. By leveraging a novel matrix Bernstein inequality for rectangular random operators and enforcing rigorous aliasing error control, the method transcends the dimensional constraints of deterministic approaches. The resulting framework achieves near-linear sample complexity independent of both the measurement and reconstruction bases, making it suitable for highly redundant systems. Notably, when applied to continuous Fourier measurements for recovering analytic functions via Legendre polynomials, the approach attains near-exponential convergence rates, substantially enhancing reconstruction efficiency and numerical stability.

This study addresses the challenge of integrating low-cost, full-sample proxy variables—such as machine learning predictions—with a limited number of high-cost, high-quality observations in two-stage, multi-wave adaptive sampling. While such fusion can enhance estimation efficiency, it often introduces bias and complicates statistical inference. To overcome this, the paper proposes a “predict-and-debias” M-estimator that achieves both high efficiency and unbiasedness within the adaptive sampling framework. The authors establish the first asymptotic theory for M-estimation under this setting, proving that the estimator is asymptotically normal and yields asymptotically valid and efficient confidence intervals. An approximate greedy sampling strategy is also developed to optimize information acquisition. Both theoretical analysis and simulations demonstrate that the proposed method substantially improves estimation efficiency compared to uniform sampling.

本文提出一种基于分类的自适应感知方法，通过估计类条件高斯混合模型的后验协方差来选择主要感知方向，以提高分类准确性。

This work addresses the challenge that real-world data in generalized linear models often violate the independent and identically distributed (i.i.d.) assumption. Under the relaxed assumption that the design matrix is orthogonally invariant—meaning its singular vectors are uniformly distributed while singular values remain arbitrary—the paper proposes an efficient parameter estimation method combining optimal spectral initialization with Approximate Message Passing (AMP). The proposed approach achieves the information-theoretically optimal sample complexity for weak recovery and attains the fundamental lower bound on estimation error, thereby extending beyond the classical i.i.d. Gaussian design setting. Rigorous theoretical analysis provides strong performance guarantees, and numerical experiments confirm both the algorithm’s effectiveness and the accuracy of the theoretical predictions on orthogonally invariant as well as more general correlated data.