bilinear form estimation

This study addresses the accurate estimation and inference of smooth functionals of mean parameters in Banach spaces under high-dimensional, unstructured settings. The authors propose a cross-fitting estimator based on a single sample split, which achieves asymptotic normality without requiring structural assumptions such as sparsity, provided the dimension satisfies \(d \log^2(en) = o(n)\). By leveraging non-asymptotic moment bounds and Berry–Esseen-type inequalities, they obtain sharp characterizations of the estimation error. Furthermore, the method enables polynomial-time computation for a broad class of matrix functionals and, in high-dimensional Euclidean settings, simultaneously attains statistical optimality and computational scalability.

This study addresses the tensorization and approximation of multivariate functions. We construct the first standardized benchmark suite of multivariate functions—including nonsmooth, symmetric, and irrational types—and systematically convert them into high-dimensional tensors. A comprehensive evaluation framework is proposed to quantitatively assess tensor-based surrogate models—including the multivariate Loewner framework (mLF), rational approximation, and tensor neural networks—across accuracy, computational efficiency, and hyperparameter robustness. We provide a novel in-depth analysis of mLF’s applicability boundaries, accompanied by reproducible implementations. Additionally, we develop a unified evaluation protocol and practical guidelines. The outcomes constitute an interdisciplinary tensor approximation toolchain (integrated into MDSPACK), supporting model selection and algorithmic improvement. This work establishes a reusable, open benchmarking infrastructure for scientific computing and surrogate modeling.

This paper addresses complex causal problems—such as interference—that resist conventional experimental design, by proposing a unified functional-space framework for causal inference. Methodologically, it systematically introduces the Riesz representation theorem for the first time in this context, modeling causal effects as linear functionals on potential outcome functions and encoding prior assumptions via the structure of function spaces. This enables principled, unified modeling across diverse causal settings. Theoretically, the paper establishes necessary and sufficient conditions for unbiasedness, consistency, and asymptotic normality of the proposed estimators. Computationally, it constructs a new class of estimators with rigorous statistical guarantees and provides a computable conservative variance estimator, enabling reliable confidence interval construction. Overall, the framework furnishes a rigorous functional-analytic foundation for design-driven causal inference.

Parameter estimation in high-dimensional structured generalized linear models suffers from low efficiency, particularly under realistic design matrices exhibiting anisotropy and strong correlations. Method: This paper introduces a novel spectral estimation framework based on Approximate Message Passing (AMP). Contribution/Results: We provide the first exact asymptotic characterization of spectral estimators under correlated Gaussian designs. We identify a universally optimal covariance-adaptive preprocessing strategy, partially resolving a long-standing conjecture on optimal spectral estimation for rotationally invariant models. Theoretically and empirically, our approach substantially reduces sample complexity and achieves provably statistically optimal estimation accuracy—outperforming existing heuristic methods on canonical designs from computational imaging and genomics.

This paper addresses the problem of estimating functionals of an unknown target function under a structure-agnostic setting—where no specific structural assumptions (e.g., Hölder smoothness) are imposed on the nuisance function, and only a generic convergence rate for nuisance estimation is assumed. Methodologically, it introduces the first formal framework for structure-agnostic estimation, operating under three simultaneous constraints: weak regularity conditions, compatibility with general-purpose nuisance estimators, and sample splitting. Theoretically, it establishes, for the first time, the essential optimality of first-order debiased estimators in this setting. Through minimax lower bound analysis, higher-order perturbation theory, and a unified debiasing framework, the paper precisely characterizes the optimal convergence rate and quantifies the fundamental trade-off between incorporating structural priors and improving estimation efficiency. These results provide foundational theoretical support for nonparametric and semiparametric inference.

This study addresses the lack of a unified formulation for scalar, multivariate, and functional regression models, which obscures their intrinsic connections. By leveraging an integral operator defined with respect to general measures, the authors propose a unified framework that subsumes all three regression types as special cases of the same operator under different input and output measures. This framework reveals classical regression forms as measure-dependent manifestations of a single operator, clarifies discretized modeling as operator estimation under specific measures, and explains the efficacy of vectorized multivariate regression in linear settings. Theoretically, the authors prove that discrete representations correspond exactly to operator evaluations under discrete measures and converge to the continuous case as the discretization grid refines; moreover, this estimator is equivalent to standard multivariate regression and inherits its classical statistical properties.

This work addresses the estimation of bilinear forms from noisy, partially observed tensors under a staggered sampling design. The authors propose a spectral algorithm that directly estimates the target functional without requiring full tensor completion, leveraging cross-layer information aggregation. Built upon the Tucker2 model and incorporating an anchored four-block reduction technique, the method accommodates general staggered missingness patterns and reveals a phase transition phenomenon linking the number of layers to estimation performance. Theoretical analysis establishes a non-asymptotic error bound that matches the local minimax lower bound. Experimental results demonstrate that, when the phase transition condition is satisfied, the proposed approach significantly outperforms strategies that forgo information aggregation.

This study addresses the frequent misinterpretation of fluctuations in dominant eigensubspaces and scalar spectral functionals—such as the absorption ratio—in rolling covariance estimation as genuine market structural changes, when they are often artifacts of estimation noise, particularly under shrinkage. By leveraging perturbation analysis and calibrated inference, the work derives, for the first time, the first-order null distribution of eigensubspace variation under overlapping windows and establishes its invariance under rotation-equivariant shrinkage estimators. It further shows that only scale-invariant spectral functionals enjoy first-order immunity to elliptical kurtosis. To correct high-dimensional bias in the absorption ratio, a trace-preserving spiked debiased estimator is proposed. Theoretical results, supported by Davis–Kahan bounds, distribution-free confidence bands, and an estimator-aware bootstrap, are validated through simulations and successfully applied to equity data for reliable detection of true market structural shifts.

This work proposes a unified functional analytic framework that interprets both supervised and unsupervised learning as variational optimization problems within a function space induced by the data distribution. The central insight is that the fundamental distinction between these learning paradigms arises from the choice of the functional being optimized, rather than from differences in the underlying function space itself. Data structure is characterized via operators induced by the distribution, and target functions are estimated in the eigenbasis of these operators. This framework systematically integrates classical algorithms—including kernel methods, spectral clustering, and manifold learning—revealing their intrinsic coherence and underscoring the foundational role of function spaces and associated operators in modern machine learning.

This study addresses the minimax estimation of high-order functionals—such as Rényi and Tsallis entropies—in regimes where the sample size is far smaller than the support size of the distribution or the dimension of the quantum state. It introduces quantum primitives for the first time to construct a unified framework encompassing both classical and quantum estimation, achieving the $L_2$-optimal convergence rate in linear time on a quantum computer. Within the range $\alpha \lesssim n \lesssim \alpha^{3 - o(1)}$, the estimator attains the optimal rate $\alpha/n$, dramatically reducing the required sample complexity from the previous $O(\alpha^2)$ to nearly linear scaling $n \sim \alpha$. Beyond establishing minimax optimality, this work pioneers a new paradigm of quantum-enhanced statistical inference and provides quantum-inspired proofs for classical statistical problems.