NeuMatC: A General Neural Framework for Fast Parametric Matrix Operation

TL;DR

NeuMatC introduces a general neural framework for fast parametric matrix operations by learning a low-rank, continuous mapping from a continuous parameter to matrix operation results. It encodes each output component via a mode-3 product with a latent tensor and a parameter-embedding MLP, enabling efficient inference after unsupervised training with an algebraic-structure loss and failure-informed adaptive sampling. Theoretical results support existence and Lipschitz continuity of the compact mapping, while experiments on synthetic and real data show up to ~10x speedups for parametric SVD and ~3x for parametric inversion with acceptable accuracy. The approach scales to large matrices, benefits from GPU acceleration, and generalizes to multi-dimensional parameter domains and additional matrix operations, suggesting broad practical impact in high-throughput scientific computing and wireless applications.

Abstract

Matrix operations (e.g., inversion and singular value decomposition (SVD)) are fundamental in science and engineering. In many emerging real-world applications (such as wireless communication and signal processing), these operations must be performed repeatedly over matrices with parameters varying continuously. However, conventional methods tackle each matrix operation independently, underexploring the inherent low-rankness and continuity along the parameter dimension, resulting in significantly redundant computation. To address this challenge, we propose \textbf{\textit{Neural Matrix Computation Framework} (NeuMatC)}, which elegantly tackles general parametric matrix operation tasks by leveraging the underlying low-rankness and continuity along the parameter dimension. Specifically, NeuMatC unsupervisedly learns a low-rank and continuous mapping from parameters to their corresponding matrix operation results. Once trained, NeuMatC enables efficient computations at arbitrary parameters using only a few basic operations (e.g., matrix multiplications and nonlinear activations), significantly reducing redundant computations. Experimental results on both synthetic and real-world datasets demonstrate the promising performance of NeuMatC, exemplified by over $3\times$ speedup in parametric inversion and $10\times$ speedup in parametric SVD compared to the widely used NumPy baseline in wireless communication, while maintaining acceptable accuracy.

Figures (8)

• Figure 1: Comparison between conventional numerical methods and the proposed NeuMatC. Conventional methods tackle each matrix operation independently, leading to redundant computation. In contrast, NeuMatC learns a low-rank and continuous mapping from parameters to the corresponding operation results using only a few basic operations (i.e., matrix multiplications and nonlinear activations). Validated on real wireless communication scenarios, NeuMatC achieves over $3\times$ speedup in parametric inversion and $10\times$ speedup in parametric SVD compared to the widely used NumPy baseline, while maintaining acceptable accuracy (see Section \ref{['sec:experiments']}).
• Figure 2: Singular value decay of matrices constructed by stacking vectorized forms of $\mathbf{A}(p)$, $\mathbf{A}^{-1}(p)$, and their SVD components $\mathbf{U}(p)$, $\mathbf{S}(p)$, and $\mathbf{V}(p)$ across $m$ parameter instances. The rapid decay of singular values indicates low-rank structure along the parameter dimension.
• Figure 3: Representative sampled entries of a real-world channel matrix $\mathbf{A}(p)$, its inverse $\mathbf{A}(p)^{-1}$, and its SVD outputs (singular values $\mathbf{S}(p)$ and left and right singular vectors $\mathbf{U}(p)$ and $\mathbf{V}(p)$) along the parameter dimension $p$, demonstrating the continuity of these matrices. The green and orange curves correspond to two different sampled entries
• Figure 4: Time cost across (a) matrix size and (b) inferred parameter number on the inversion task. In subfigure (a), we report the average time cost for processing a single matrix, whereas in subfigure (b) we fix the matrix size at $512 \times 512$.
• Figure 5: Influence of different sampling strategies on the real-world SVD task.

Theorems & Definitions (15)

• Theorem 1: Existence of the Compact Mapping
• Remark 1
• Theorem 2: Lipschitz Continuity in NeuMatC
• Remark 2
• Remark 3: Connection to Physics-Informed Neural Networks (PINNs)
• Definition 1: Mode-3 Tensor Folding and Unfolding kernfeld2015tproducts
• Definition 2: Mode-$3$ Matrix-Tensor Product kernfeld2015tproducts
• Lemma 1: Continuity of Matrix Product
• proof
• Theorem 3: Continuity of Matrix Inversion