construct measure-preserving maps

This study addresses the limitation that universal approximation of distribution-to-distribution mapping operators readily fails under atomic inputs, exposing deficiencies in existing theories. To overcome this, we propose the uniform level set condition, integrating measure theory, optimal transport theory, and Transformer architecture analysis to construct a continuous measure-dependent pushforward model based on the Wasserstein distance. We prove that, on compact sets satisfying this condition, the pushforward model can uniformly approximate arbitrary continuous operators, thereby effectively overcoming the constraints imposed by atomic inputs. This work establishes a comprehensive theoretical framework for universal approximation, providing a rigorous foundation for measure-theoretic Transformers and cross-attention mechanisms.

When machine learning models are employed as measurement instruments, it remains unclear whether their outputs genuinely reflect stable and consistent latent constructs beyond merely achieving predictive performance. This work formally introduces the concept of “learned measurement functions” and proposes “measurement stability” as a distinct evaluation criterion. Through theoretical analysis and empirical case studies, we demonstrate that conventional metrics—such as generalization error, calibration, and robustness—do not guarantee measurement consistency. Our findings reveal that models with comparable predictive accuracy can implement systematically inequivalent measurement functions, and that these discrepancies become pronounced under distributional shifts, thereby exposing critical limitations in current evaluation frameworks.

This paper addresses the lack of an information-theoretic characterization and a point-to-set principle for finite-state dimension (FS dimension). It introduces a precise quantification of the information content of real numbers under finite precision, integrating effective dimension theory, finite-state automaton complexity, and relativized algorithmic information theory. The work establishes, for the first time, a point-to-set principle for FS dimension and thereby defines a robust notion of relative normality. It then rigorously proves the equivalence between relative normality and FS dimension. The main contributions are: (1) an information-theoretic, exact definition of FS dimension; (2) a necessary and sufficient characterization linking relative normality to FS dimension; and (3) a foundational framework enabling future investigation of its equidistribution properties.

Traditional Takens embedding fails under realistic sparse and noisy data due to its deterministic and noise-free assumptions. To address this, this work establishes, for the first time, a time-delay embedding framework grounded in probability measure spaces—thereby relaxing these restrictive assumptions. Methodologically, we generalize time-delay embedding to the measure level, model dynamics from an Eulerian perspective, and define a probabilistic pushforward map via optimal transport, enabling robust state reconstruction under stochasticity and observational degradation. Our key theoretical contribution is a falsifiable embedding theorem in the measure domain, accompanied by a corresponding algorithm. Extensive validation on diverse datasets—including the Lorenz-63 system, NOAA sea surface temperature, and ERA5 wind fields—demonstrates substantial improvements in reconstruction accuracy and stability under sparse-noisy conditions. This work provides a new paradigm for state perception in complex dynamical systems.

This paper investigates the expressive power of Transformers as arbitrary input-to-output measure mappings. Method: We reformulate Transformers from a measure-theoretic perspective, modeling them as differentiable maps on continuous measure spaces—departing from conventional discrete token-based interpretations. Leveraging continuity equations to describe particle dynamics, we design an attention mechanism incorporating spherical geometry constraints and optimal transport theory. Contribution/Results: We propose the first Transformer architecture provably capable of exact matching between arbitrary input–target measure pairs. Under the minimal assumption that a transport map exists between each pair, a single model achieves precise matching for N arbitrary measure pairs. We establish theoretical completeness by proving that Transformers serve as universal interpolators between measures and provide explicit parameter constructions. This work fundamentally characterizes the expressive capacity of Transformers for measure transformation tasks.

本文提出了一种通过有限维映射ψ来近似函数空间中测度的方法，该方法保留了参考测度的条件分布，并且适用于低维结构。

该研究通过Copula理论解决了iDCI与原始联合DCI解之间的关系问题，提出了一种基于Copula变换的方法来改进iDCI算法的准确性。

This work addresses the limitation of classical test theory, which relies on commutative algebra and thus fails to capture non-commutative measurement phenomena prevalent in the social sciences. By introducing measurement state vectors and a matrix Lie group representation, the study constructs a faithful matrix representation of the Heisenberg group and reveals, for the first time, a symmetry between classical measurement transformations and the Heisenberg group through the latter’s conjugation action under general measurement transformations. Employing non-commutative algebraic and geometric tools—such as Lie groups, matrix conjugation, and automorphisms—the paper demonstrates that this conjugation action preserves the Heisenberg group’s commutator structure. Moreover, when components of the measurement state vector are scaled proportionally, the Heisenberg geometric structure remains strictly invariant, thereby establishing a rigorous mathematical foundation for extending classical test theory into a non-commutative framework.

本文提出了一种名为机制断层扫描的方法，通过设计测量来恢复内部机制和干预效果，从而提高模型的可解释性。

This study investigates the preservation of several classical combinatorial properties—such as recurrence, morphicity, and factor frequencies—under the action of deterministic finite-state transducers on infinite words. To address this problem, the authors introduce a unified analytical framework by combining Krohn–Rhodes decomposition theory with ergodic methods from symbolic dynamics for the first time. Within this framework, they systematically characterize the capacity of transducers to preserve these combinatorial properties and establish a comprehensive set of preservation theorems encompassing all the aforementioned features. This work provides a theoretical foundation for understanding the structural stability of sequences under automaton-based transformations.