SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning

• URL: http://arxiv.org/abs/2510.24170v1
• Date: Tue, 28 Oct 2025 08:25:03 GMT
• Title: SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning
• Authors: Hong Wang, Jie Wang, Minghao Ma, Haoran Shao, Haoyang Liu,
• Abstract summary: Symbolic Matrix Preconditioning (SymMaP) learns efficient symbolic expressions for preconditioning parameters.<n>We employ a neural network to search the high-dimensional discrete space for expressions that can accurately predict the optimal parameters.<n> Experimental results show that SymMaP consistently outperforms traditional strategies across various benchmarks.
• Score: 5.546260420622416
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Matrix preconditioning is a critical technique to accelerate the solution of linear systems, where performance heavily depends on the selection of preconditioning parameters. Traditional parameter selection approaches often define fixed constants for specific scenarios. However, they rely on domain expertise and fail to consider the instance-wise features for individual problems, limiting their performance. In contrast, machine learning (ML) approaches, though promising, are hindered by high inference costs and limited interpretability. To combine the strengths of both approaches, we propose a symbolic discovery framework-namely, Symbolic Matrix Preconditioning (SymMaP)-to learn efficient symbolic expressions for preconditioning parameters. Specifically, we employ a neural network to search the high-dimensional discrete space for expressions that can accurately predict the optimal parameters. The learned expression allows for high inference efficiency and excellent interpretability (expressed in concise symbolic formulas), making it simple and reliable for deployment. Experimental results show that SymMaP consistently outperforms traditional strategies across various benchmarks.

Related papers

• Variables Ordering Optimization in Boolean Characteristic Set Method Using Simulated Annealing and Machine Learning-based Time Prediction [1.654967376694554]
  This paper introduces a novel framework that integrates machine learning (ML)-based time prediction with simulated annealing (SA)<n>We train an accurate ML predictor ft(X) to estimate solving time for any given variables ordering.<n>Experiments demonstrate that our method substantially outperforms the standard BCS algorithm.
  arXiv  Detail & Related papers  (2025-09-18T09:02:32Z)
• Generalized Tensor-based Parameter-Efficient Fine-Tuning via Lie Group Transformations [50.010924231754856]
  Adapting pre-trained foundation models for diverse downstream tasks is a core practice in artificial intelligence.<n>To overcome this, parameter-efficient fine-tuning (PEFT) methods like LoRA have emerged and are becoming a growing research focus.<n>We propose a generalization that extends matrix-based PEFT methods to higher-dimensional parameter spaces without compromising their structural properties.
  arXiv  Detail & Related papers  (2025-04-01T14:36:45Z)
• PROMISE: Preconditioned Stochastic Optimization Methods by Incorporating Scalable Curvature Estimates [17.777466668123886]
  We introduce PROMISE ($textbfPr$econditioned $textbfO$ptimization $textbfM$ethods by $textbfI$ncorporating $textbfS$calable Curvature $textbfE$stimates), a suite of sketching-based preconditioned gradient algorithms. PROMISE includes preconditioned versions of SVRG, SAGA, and Katyusha.
  arXiv  Detail & Related papers  (2023-09-05T07:49:10Z)
• Numerical Optimizations for Weighted Low-rank Estimation on Language Model [73.12941276331316]
  Singular value decomposition (SVD) is one of the most popular compression methods that approximates a target matrix with smaller matrices. Standard SVD treats the parameters within the matrix with equal importance, which is a simple but unrealistic assumption. We show that our method can perform better than current SOTA methods in neural-based language models.
  arXiv  Detail & Related papers  (2022-11-02T00:58:02Z)
• MARS: Meta-Learning as Score Matching in the Function Space [79.73213540203389]
  We present a novel approach to extracting inductive biases from a set of related datasets. We use functional Bayesian neural network inference, which views the prior as a process and performs inference in the function space. Our approach can seamlessly acquire and represent complex prior knowledge by metalearning the score function of the data-generating process.
  arXiv  Detail & Related papers  (2022-10-24T15:14:26Z)
• Sparse high-dimensional linear regression with a partitioned empirical Bayes ECM algorithm [62.997667081978825]
  We propose a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are used through the use of plug-in empirical Bayes estimates. The proposed approach is implemented in the R package probe.
  arXiv  Detail & Related papers  (2022-09-16T19:15:50Z)
• Making Linear MDPs Practical via Contrastive Representation Learning [101.75885788118131]
  It is common to address the curse of dimensionality in Markov decision processes (MDPs) by exploiting low-rank representations. We consider an alternative definition of linear MDPs that automatically ensures normalization while allowing efficient representation learning. We demonstrate superior performance over existing state-of-the-art model-based and model-free algorithms on several benchmarks.
  arXiv  Detail & Related papers  (2022-07-14T18:18:02Z)
• HyperImpute: Generalized Iterative Imputation with Automatic Model Selection [77.86861638371926]
  We propose a generalized iterative imputation framework for adaptively and automatically configuring column-wise models. We provide a concrete implementation with out-of-the-box learners, simulators, and interfaces.
  arXiv  Detail & Related papers  (2022-06-15T19:10:35Z)
• The Parametric Cost Function Approximation: A new approach for multistage stochastic programming [4.847980206213335]
  We show that a parameterized version of a deterministic optimization model can be an effective way of handling uncertainty without the complexity of either programming or dynamic programming. This approach can handle complex, high-dimensional state variables, and avoids the usual approximations associated with scenario trees or value function approximations.
  arXiv  Detail & Related papers  (2022-01-01T23:25:09Z)
• Symbolic Regression by Exhaustive Search: Reducing the Search Space Using Syntactical Constraints and Efficient Semantic Structure Deduplication [2.055204980188575]
  Symbolic regression is a powerful system identification technique in industrial scenarios where no prior knowledge on model structure is available. In this chapter we introduce a deterministic symbolic regression algorithm specifically designed to address these issues. A finite enumeration of all possible models is guaranteed by structural restrictions as well as a caching mechanism for detecting semantically equivalent solutions.
  arXiv  Detail & Related papers  (2021-09-28T17:47:51Z)
• Meta-Learning for Symbolic Hyperparameter Defaults [2.928016570228877]
  Hyperparameter optimization in machine learning (ML) deals with the problem of empirically learning an optimal algorithm configuration from data. We propose a zero-shot method to meta-learn symbolic default hyperparameter configurations that are expressed in terms of the properties of the dataset. This enables a much faster, but still data-dependent, configuration of the ML algorithm.
  arXiv  Detail & Related papers  (2021-06-10T14:20:28Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.