gram matrix computation

This work addresses the low computational efficiency of noncommutative Gröbner basis computation in free algebras by introducing the first high-performance, open-source C++ library. Methodologically, it presents the first engineering implementation of the noncommutative F4 algorithm, integrating sparse linear algebra optimizations, memory-aware polynomial representations, dynamic reduction scheduling, and multi-threaded parallelism. The core contribution lies in systematically adapting state-of-the-art algorithmic techniques from commutative algebra to the noncommutative setting, thereby establishing a new state-of-the-art (SOTA) for Gröbner basis computation in free algebras. Experimental evaluation on standard benchmarks demonstrates 10×–100× speedups over established tools such as GBNP and Bergman. The library reliably handles problems with up to ten thousand generators, significantly expanding the practical scalability frontier of noncommutative symbolic computation.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This paper addresses the computation of the greatest common right divisor (GCRD) for linear ordinary differential operators with non-rational coefficients—such as parametric transcendental functions—where classical algorithms assume rational-function coefficients. The proposed symbolic algorithm is grounded in the determinant definition of the differential resultant; it explicitly constructs the resultant matrix and leverages Maple’s DEtools package and symbolic computation capabilities for efficient GCRD computation, naturally accommodating parametric coefficients. Key contributions include: (i) the first implementation of a GCRD algorithm for non-rational coefficient operators within a general-purpose symbolic computation platform; (ii) a practical, parameter-aware tool for analyzing parametric differential systems; and (iii) successful application to the construction of commuting differential operators, thereby empirically validating the algorithm’s correctness and effectiveness.

This work investigates the discriminative power of spectral angles—the angles between eigenvectors of the adjacency matrix and standard basis vectors—for graph isomorphism testing. Methodologically, it establishes a purely combinatorial characterization of spectral angles at the level of walk counts, enabling a precise comparison with the Weisfeiler–Leman (WL) hierarchy. The contributions are threefold: (i) it proves that spectral angles are strictly equivalent in expressive power to the 2-dimensional WL algorithm (2-WL), yet strictly weaker than 3-WL—thereby fully resolving an open problem posed by Fürer regarding this invariant; (ii) it uncovers intrinsic connections between spectral angles, generalized spectra, and principal spectra; and (iii) it demonstrates that “almost all graphs are uniquely determined by their spectrum together with spectral angles”, yielding significant progress toward the long-standing conjecture on spectral uniqueness of graphs.

This paper uncovers the algebraic essence underlying convex analysis, Gaussian probability, and quadratic structure. To this end, we introduce Graphical Quadratic Algebra (GQA)—a novel algebraic framework based on chordal graphs—that uniformly models quadratic relations, Gaussian stochastic processes, and nondeterministic Gaussian processes via rotation-invariant quadratic generators. We provide the first sound and complete axiomatic characterization of three fundamental models: least-squares estimation, Gaussian randomness, and nondeterminism—revealing their shared conditional algebraic structure. Our method integrates string diagram theory, categorical semantics, and formal semantics of probabilistic programming. Theoretical contributions include soundness and completeness proofs for all three models within GQA. Applications demonstrate efficacy in linear regression, probabilistic programming, and noisy circuit modeling.

This work proposes the notion of “generalized height” in simplices, extending the classical concept—defined as the distance from a vertex to its opposite facet—to arbitrary pairs of opposite faces. The relative position of such face pairs is uniformly characterized via the angle between their affine subspaces. By leveraging generalized cross products and Gram determinants, the authors establish an algebraic relationship between generalized heights and classical heights, and prove that all generalized heights are bounded below by the minimal classical height. This result demonstrates that the standard thickness assumption in simplex quality analysis naturally extends to a broader class of geometric quantities, offering a new theoretical framework for assessing simplex quality and informing triangulations of Riemannian manifolds, thereby supporting the generation of high-quality meshes.

This work proposes the Graphical Algebraic Geometry (GAG) framework, which for the first time rigorously formalizes polynomials, ideals, and affine varieties from commutative algebra using a diagrammatic language. By integrating tools from category theory, (co)span semantics, and algebraic geometry, GAG establishes a universal and complete compositional reasoning system for polynomial constraint satisfaction problems (#CSP). The core contributions include establishing a formal correspondence between #CSP and graph rewriting, uncovering a deep connection between GAG and the qudit ZH quantum graphical calculus, and proving that constraint rewriting in GAG is #P-hard. Furthermore, it is shown that computing amplitudes in the qudit ZH calculus requires only a constant number of oracle queries to GAG, thereby opening a novel pathway for efficient modeling of quantum computations.

This work proposes a model-class-agnostic structural learning method to identify latent probabilistic structures directly from empirical probability tensors. Leveraging the vanishing binomials of toric models as algebraic signatures, the approach introduces minimal invariant constraints (MICs) as fundamental units that generalize classical notions of independence. By exploiting the correspondence between ideals and varieties, the method performs signature matching within an enumerable class of Kronecker-stack configuration matrices. This study represents the first systematic application of algebraic statistics to structure discovery in computational linguistics, demonstrating effectiveness on both synthetic and large-scale real-world linguistic data. The identified rank-one structures correspond to interpretable word sets, thereby establishing a novel pathway for applying algebraic statistics in this domain.

This work addresses the cumbersome, error-prone, and irreproducible process of manually converting symbolic Hamiltonians or cost functions into matrix product operator (MPO) representations. To overcome this challenge, the authors introduce an automaton-inspired framework that automatically translates user-defined local operators and interaction rules into compact, verifiable MPOs, accommodating both diagonal and off-diagonal terms. The method integrates formal rule parsing, pattern recognition, and tensor network compression to enable automated compilation from symbolic expressions to MPOs, implemented in Python. Demonstrations on Max-Cut, long-range XX+YY+Z spin models, and quantum clock models show a substantial reduction in the engineering overhead associated with constructing and verifying MPOs, thereby enhancing reproducibility and accessibility in tensor network simulations.

This work addresses the complexity of derivative computation and algebraic expressions on matrix Lie groups in state estimation, which has long hindered algorithmic understanding and implementation. For the first time, it introduces tensor notation together with Einstein summation convention into the differential calculus of matrix Lie groups, integrating concepts from differential geometry to establish a concise and unified mathematical formalism. This framework substantially enhances the clarity and readability of derivative derivations and algebraic manipulations, thereby facilitating more intuitive comprehension and efficient implementation of gradient-based estimation algorithms.