functional tensor decomposition

This work proposes a novel tensor decomposition framework, termed HMD, which addresses a key limitation of traditional low-rank methods such as Tucker and CP decompositions: their inability to capture high-order interactions among modes, as they only model mode-wise independent variations. HMD explicitly incorporates inter-modal high-order couplings within the low-rank approximation by introducing specially designed projection operators that jointly encode both isolated mode-specific effects and cross-modal interaction structures. By transcending the structural constraints inherent in conventional decompositions, the proposed method achieves substantially lower reconstruction errors and demonstrates superior fidelity and robustness across three diverse benchmark datasets, consistently outperforming Tucker and CP decompositions.

This work addresses the challenges in multivariate polynomial models arising from the exponential growth of coefficient tensors with model order, the limited expressivity of existing tensor decomposition methods, and their sensitivity to feature ordering. To overcome these limitations, the authors propose an efficient optimization framework based on matrix product operators (MPOs). The approach leverages learnable MPO-based feature embeddings and compact polynomial weight tensors to achieve permutation-invariant function approximation. Furthermore, structured operators—such as projection, convolution, and masking—are incorporated to explicitly model weight symmetries, thereby transcending the expressivity bottlenecks of conventional tensor decompositions. Experimental results demonstrate that the proposed framework significantly outperforms existing tensor-decomposition-based polynomial models on both regression and classification benchmarks, offering a highly expressive, flexible, and computationally efficient solution for polynomial approximation.

This work addresses the high computational and communication overhead incurred when multiple users compute nonlinearly separable functions in distributed environments. To overcome the limitations of conventional approaches that rely on linear separability assumptions, the paper proposes an efficient computation framework based on sparse tensor representations. By introducing a fixed-support SVD-based sparse tensor decomposition method combined with a multidimensional sub-tensor partitioning strategy, the framework jointly optimizes task allocation and communication patterns. This integrated approach significantly reduces system resource consumption and achieves substantial improvements over state-of-the-art methods in both computational efficiency and communication cost.

本文研究了一类低参数正交矩阵的黎曼结构，并提出有效算法解决其在深度学习中的计算问题。

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 work proposes Functional Tucker Decomposition (FTD) to address the limitation of traditional tensor decomposition, which loses essential continuity structures when discretizing multidimensional data generated by continuous processes. By embedding Reproducing Kernel Hilbert Spaces (RKHS) into the Tucker framework, FTD adaptively learns expressive factors through mode-wise continuity constraints, enabling the modeling of continuous patterns without requiring predefined basis functions while preserving the multilinear subspace structure. The method uniquely unifies structural fidelity with functional flexibility. Experimental results on hyperspectral image classification and multivariate time series analysis demonstrate that FTD significantly outperforms existing approaches, validating the effectiveness of integrating continuous function modeling with tensor decomposition.

本文介绍了一种通过计算交换算子的联合特征向量来解决多项式根和张量分解问题的方法，并分析了其多重结构。

This work proposes a novel approach to overcoming the computational complexity bottleneck in matrix multiplication by explicitly exploiting the intrinsic structural properties of tensor decompositions. By designing tensor decompositions with specialized algebraic structures and integrating techniques from algebraic complexity theory with numerical optimization, the study achieves a reduction in the exponent for 6×6 matrix multiplication from 2.8075 to 2.8016, while maintaining a reasonable leading constant. Notably, this result yields an effective exponent below the theoretical lower bound implied by conventional tensor rank considerations and significantly enhances practical algorithmic efficiency. The findings establish a new structured design paradigm for fast matrix multiplication algorithms, offering both theoretical advancement and practical relevance.

This work proposes a continuous tensor ring (TR) function decomposition method based on implicit neural representations (INRs), addressing the limitations of traditional TR decomposition, which is confined to fixed-grid data and struggles with non-grid high-dimensional signals and high-frequency detail recovery. By integrating INRs into the TR framework for the first time, the approach enables modeling at arbitrary sampling points. A reparameterization strategy expresses each TR factor as a structured combination of learnable latent tensors and fixed basis functions, enhancing both high-frequency modeling capacity and training stability. Theoretical analysis demonstrates improved optimization dynamics and Lipschitz continuity. Experiments show that the method significantly outperforms existing approaches in tasks such as image inpainting, denoising, super-resolution, and point cloud reconstruction, exhibiting superior generalization and multidimensional signal reconstruction capabilities.

This work addresses optimization problems defined over products of simplices, such as low-rank learning of discrete multivariate probability distributions and function data registration based on the Square-Root Velocity Function (SRVF) representation. To tackle the inherent constraints, the authors propose a smooth reparameterization that is strictly convex element-wise, transforming the constrained problem into an unconstrained optimization over a Riemannian manifold. The resulting problem is solved via Riemannian gradient descent (RGD). Theoretical analysis shows that this reparameterization maps second-order KKT points on the manifold to weak second-order KKT points of the original problem, ensuring theoretical soundness while enhancing computational efficiency. Experiments demonstrate that RGD significantly outperforms projected gradient descent (PGD), achieving more accurate shape-preserving registration in functional data and efficiently solving probability tensor decomposition tasks.