optimize matrix operations

Recursive matrix multiplication faces a fundamental trade-off between numerical accuracy and computational efficiency. Method: We propose a unified framework for automatically constructing high-accuracy, low-complexity recursive bilinear algorithms. First, we unify and extend norm-based error bounds—covering Frobenius, spectral, and induced norms—to arbitrary recursive bilinear schemes. Second, we introduce a three-stage optimization pipeline: (i) heuristic search minimizing operand count in bilinear formulations; (ii) growth-factor minimization over tensor decomposition orbits; and (iii) sparsification via alternative bases. Contribution/Results: Our framework yields a novel non-commutative 2×2 matrix multiplication algorithm requiring only seven scalar multiplications—matching Strassen’s asymptotic complexity O(n^log₂7) while substantially improving numerical stability. The approach generalizes to established recursive algorithms including those of Smirnov et al., demonstrating broad applicability and enabling scalable, automated generation of high-accuracy fast matrix multiplication schemes.

This work addresses the challenge of efficiently executing floating-point matrix multiplication (GEMM) on integer-tensor-accelerated hardware (e.g., NVIDIA A100/H100). We propose a precision-controllable integer tiling reconstruction method that decomposes floating-point GEMM into multiple exact integer matrix multiplications followed by floating-point accumulation. Our key contributions are: (i) the first lightweight error propagation model enabling precision-driven, adaptive estimation of the required number of integer tiles; and (ii) a systematic analysis revealing the dual impact of row/column scaling imbalance on both numerical accuracy and computational throughput. Experiments validate our theoretical error bounds and demonstrate accurate identification of precision failure boundaries under realistic imbalance scenarios. The method achieves floating-point-level accuracy while significantly improving integer hardware utilization, enabling explicit, tunable trade-offs between performance and precision.

This work addresses the limitations of traditional memory-bandwidth-based performance models—such as Roofline and ECM—in accurately predicting the performance of tensor n-mode product kernels under high arithmetic intensity, particularly on processors with high SIMD instruction latency like the Fujitsu A64FX. To overcome this challenge, the authors propose a learning-augmented performance model that integrates dependency chain analysis with XGBoost regression, introducing instruction-level dependency features for the first time into performance prediction for high-order finite element computations. The approach effectively evaluates the execution efficiency of various loop tiling strategies, achieving mean absolute percentage errors of only 1%–24% on both the A64FX and Intel Xeon Gold 6230 platforms, substantially outperforming the Roofline (42%–256%) and ECM (5%–117%) models.

This work addresses efficiency bottlenecks in solving large-scale linear systems and approximating matrix norms. We propose a multilevel randomized sketching preconditioned iterative method, integrating Nyström low-rank approximation, sparse random sketching, and multilevel preconditioning. It establishes the first multilevel sketched preconditioning framework grounded in the natural average condition number. Theoretical contributions include: (1) optimal complexity $ ilde{O}(n^2 + d_lambda^omega)$ for solving regularized linear systems; (2) accelerated complexity $ ilde{O}(n^{2.065} + k^omega)$ for systems with $k$ outlying singular values; and (3) Schatten-$p$ norm approximation—particularly the nuclear norm—at $ ilde{O}(n^{2.11})$, improving upon the prior best $ ilde{O}(n^{2.18})$. These advances significantly enhance computational efficiency for key subproblems in applications such as Gaussian process regression.

Existing iterative solvers for large-scale linear systems suffer from strong dependence on the global condition number and coarse-grained complexity analyses. Method: We introduce the *spectral tail condition number* $kappa_ell$, a new fine-grained spectral measure, and develop a refined time-complexity framework. Our approach formally defines $kappa_ell$, integrates it with the Sketch-and-Project paradigm, Nesterov acceleration, determinant point process sampling, and universality theory for Gaussian matrices, thereby exposing an intrinsic connection between iteration complexity and the matrix multiplication exponent $omega$. Contribution/Results: Our analysis achieves a sharper separation between deterministic and randomized algorithms, yielding an $ ilde{O}(kappa_ell n^2 log(1/varepsilon))$ bound for computing an $varepsilon$-accurate solution—valid for $ell$ up to $O(n^{0.729})$. This significantly improves the fine-grained analysis of the conjugate gradient method and establishes a novel theoretical benchmark for iterative algorithm design.

This work addresses the significant performance degradation of high-order finite element stiffness operators in end-to-end scenarios due to inefficient utilization of modern processor matrix engines. Focusing on the stiffness operator in SPECFEM3D running on Arm LX2 CPUs, the study proposes a holistic operator-level co-optimization methodology that transcends conventional approaches limited to optimizing tensor contraction kernels alone. By systematically co-designing pointwise computations, field data layouts, and coefficient memory access patterns—through explicit SIMD optimization, data relayout, and vectorized blocked streaming loads—the approach achieves a matrix engine speedup of 1.6× for the full stiffness operator, up from 1.1×, approaching the theoretical upper bound. Factorization-based diagnostics and contraction-free ablation experiments validate the efficacy and necessity of this full-path co-optimization strategy.

This work addresses the gap between algorithmic prototypes and efficient implementations in scientific research by proposing a lightweight approach to translate statistical and machine learning algorithms—such as kernel ridge regression and stochastic gradient descent matrix factorization—from mathematical formulations into readable, high-performance C++ code. Leveraging the Eigen template library for core linear algebra operations—including kernel matrix construction, regularized solvers, and vectorized updates—the implementation seamlessly integrates into the Python ecosystem via pybind11, enabling efficient interoperability with NumPy arrays. The project provides concise, reproducible code examples that encapsulate common computational patterns in research, significantly lowering the barrier for researchers to adopt C++ for high-performance development while balancing performance, readability, and usability.

This work presents the first systematic evaluation of Rust’s suitability for high-performance sparse linear algebra, addressing the longstanding trade-off between performance and memory safety in traditional scientific computing that relies on C/C++ and Fortran. The authors natively implement core operations—including sparse matrix-vector multiplication, the Lanczos Krylov method, and matrix exponential computation—leveraging compile-time monomorphization, SIMD vectorization, and careful FFI boundary analysis. Comprehensive benchmarks against established libraries such as Intel oneMKL, Eigen, PETSc, and PSBLAS demonstrate that Rust achieves performance on par with Eigen and PSBLAS in CSC format, approaching state-of-the-art levels while preserving memory safety. However, it still lags behind PETSc in block CSR optimizations, highlighting both the promise and current limitations of Rust for building efficient, safe numerical software stacks.