decomposition analysis

This paper addresses the weak theoretical foundations of matrix decomposition in machine learning by systematically constructing a self-consistent, comprehensive, and modern-application-oriented pedagogical framework. Methodologically, it grounds the exposition in numerical linear algebra and matrix analysis, unifying classical decompositions—including LU, QR, SVD, and block triangular factorizations—while integrating numerical stability analysis and Hermitian/Hilbert space theory. Crucially, it bridges traditional numerical analysis with deep learning’s backpropagation setting, emphasizing differentiability and computational robustness of decompositions in algorithm design and model optimization. The primary contribution is a compact, dual-purpose (teaching and research) knowledge system that fills critical gaps in both theoretical coherence and machine-learning relevance present in existing literature, thereby providing rigorous mathematical foundations for high-dimensional data modeling and efficient training.

Traditional neural networks suffer from limited interpretability and weak theoretical foundations. Method: This paper proposes a novel machine learning paradigm grounded in infinite-dimensional Hilbert spaces, centering on linear operators. It integrates reproducing kernel Hilbert spaces (RKHS), spectral operator learning, wavelet representations, scattering transforms, and Koopman operator theory to formulate learning tasks as sampling, approximation, and dynamical inference in infinite-dimensional function spaces. Contribution/Results: We establish the first unified Hilbert-space-theoretic framework bridging spectral learning and symbolic reasoning. The approach significantly enhances mathematical rigor and model interpretability by grounding learning in well-defined functional-analytic principles. Moreover, it provides a rigorous mathematical foundation and new methodological pathways for deep interdisciplinary integration between signal processing and machine learning—enabling principled analysis of structured data, hierarchical feature extraction, and nonlinear dynamical system modeling.

This paper addresses the fragmentation and lack of interoperability among structural complexity measures across graph theory, geometric group theory, and dynamical systems. Methodologically, it introduces a unified “structured decomposition” framework grounded in category theory: (i) it is the first to formalize diverse domain-specific decomposition paradigms categorically; (ii) it establishes a general duality theory linking decompositions to object completions; and (iii) it defines composable width functors enabling cross-model quantification, comparison, and translation of structural complexity. Key contributions include: a unified categorical characterization of over ten complexity parameters—including treewidth, layered treewidth, and hypergraph treewidth—revealing their intrinsic structural relationships; and a novel parameterized tractability paradigm for NP-hard problems, grounded in decomposition width. The framework achieves both theoretical unification and algorithmic realizability.

Bradley–Terry experimental design for large-scale pairwise comparisons (n > 150) becomes computationally intractable due to spectral decomposition of the high-dimensional pairwise covariance matrix. Method: We propose a novel dynamic experimental design framework based on dimensionality-reduction basis decomposition, which avoids explicit construction of the full covariance matrix. Leveraging matrix approximation theory and spectral analysis, we characterize the low-rank structure of the design matrix and derive tight eigenvalue bounds to ensure approximation fidelity. Contribution/Results: Theoretically and empirically, our method accelerates computation by over 100× for n ≥ 64; reduces design time for a 452-region spatial study to under 7 minutes; and cuts update latency in classroom peer assessment from 15 minutes to 15 seconds—while maintaining negligible estimation error. This work is the first to systematically integrate dimensionality-reduction basis decomposition into optimal pairwise comparison design, establishing a scalable, high-accuracy, and real-time paradigm for large-scale preference learning.

This work addresses the multi-level low-rank (MLR) matrix approximation problem under the Frobenius norm, tackling three core challenges: hierarchical structural partitioning (row/column stratification), rank allocation (optimizing individual block ranks under a total storage budget), and joint factor fitting. We propose the first end-to-end joint optimization framework for MLR matrices, unifying structural design, rank assignment, and factor learning within a single model. Our approach employs hierarchical block-diagonal parameterization, alternating optimization, and a constrained rank allocation algorithm to achieve coordinated optimization. The resulting approximation preserves matrix-vector multiplication complexity at O(n). Empirical evaluation on multiple benchmark datasets shows that our method reduces approximation error by 35% on average compared to single-level low-rank baselines, significantly improving both accuracy and storage efficiency. The implementation is publicly available.

This work proposes a highly scalable parallel recursive spectral bisection (parRSB) method tailored for Exascale computing to address the challenge of efficient, high-quality graph partitioning of large-scale spectral element meshes. The approach computes the Fiedler vector in parallel on the mesh dual graph by integrating the Lanczos algorithm with conjugate gradient-based inverse iteration, augmented with several numerical optimization strategies to enhance computational efficiency. Experimental results on the Summit and Frontier supercomputing platforms demonstrate that parRSB significantly reduces communication overhead while preserving partition quality, exhibiting excellent strong and weak scalability as well as superior partitioning speedup.

This work addresses a key limitation in traditional decomposition-based program synthesis: reliance on ground-truth subgoals that ignore the solver’s actual capabilities, often yielding logically correct but practically unsolvable subtasks. To overcome this, the authors propose Solver-Aware Decomposition (SAD), a framework that retains supervision from ground-truth subgoals while incorporating feedback signals from a frozen synthesizer to refine the decomposer. This approach reveals, for the first time, that decomposition quality depends more on the solver than on the task itself, demonstrating that ground-truth subgoals are not universally optimal. By combining supervised learning with reinforcement learning—using synthesizer loss as a reward signal—the method guides decomposition toward subgoals better aligned with the solver’s strengths. Experiments in two programming domains show substantial gains in synthesis success rate and end-to-end accuracy, even solving tasks previously intractable to existing methods.