Matrix factorization is a fundamental characterization model in machine learning and is usually solved using mathematical decomposition reconstruction loss. However, matrix factorization is a data-driven model whose results depend on data quality, making it susceptible to noise. Inspired by physics, the law of conservation of energy is used to introduce physical laws into matrix factorization, which is called Physics-informed Matrix Factorization operator (PiMF). The PiMF operator uses the heat conduction equation to construct the energy objective function for matrix factorization, thereby retaining the mathematical model's decomposition meaning and satisfying the interpretability of physics. The PiMF follows the physical laws, thereby suppressing irregular or sudden noise signals that violate these physical principles. The solutions of the PiMF operator include more comprehensive knowledge of mathematics and physics, which improves the ability to generalize complex data, especially for noisy data. We demonstrate the consistency of the energy objective function and the mathematical model, which verifies the feasibility of matrix factorization using physical energy laws. In addition, the physical interpretability of the PiMF operator is proved from the perspective of energy decline. This study proposes two practical algorithms for PiMF in classification and clustering tasks, enhancing the practicability of matrix factorization by incorporating task-specific prior information constraints. The experimental results of PiMF for classification and clustering demonstrate the advantages of the proposed operator. The importance of physics-informed matrix factorization is verified, especially for noisy data.

In feature learning (FL), structural information shows advantages in retaining information and maintaining stability. Graph diffusion, a graph learning method that can focus on neighborhood structure and transmit information, has great research potential. In this study, a novel dynamic diffusion graph nonnegative matrix factorization (DGNMF) method is proposed, which uses a diffusion graph to improve the performance of FL and further enhances the effectiveness and stability of downstream classification tasks. DGNMF aims to mine and retain structural information more deeply in FL to build a more powerful and stable FL method. First, the model embeds graph learning into FL to obtain features containing structural information. Second, dynamic diffusion graph learning is used to mine deeper and more global structural information. Finally, we construct an updateable indicator matrix to enhance the discriminability of features. The classification experimental results of DGNMF on six databases demonstrate its advantages, verify its effectiveness and stability, and prove the importance of diffusion graph in improving FL.

Matrix factorization (MF) is a fundamental problem in machine learning, which is usually used as a feature learning method in various fields. For complex data involving spatiotemporal interactions, MF that only handles 2-D data will disrupt spatial dependence or temporal dynamics, failing to effectively couple spatial information with temporal factors. According to Markov chain principle, the spatial information of the present time is related to the spatial state of the previous time. We propose a spatial-temporal diffusion model for MF (STDMF), which uses graph diffusion to couple spatial-temporal information. Then, MF is used to learn the joint feature of data and spatial-temporal diffusion graph. Specifically, STDMF utilizes the graph diffusion with physical laws to generate spatial-temporal structure information. It obtains the underlying core structure of complex systems from a global perspective, which enhances the generalization ability of MF in noisy time-series data. To learn the lowest rank subspace of MF in time-series data, STDMF uses structural learning to constrain the rank of the learned features. Finally, STDMF is applied to clustering and anomaly detection of dynamic graph. The effectiveness of this method is verified by sufficient experiments, especially for noisy data.

For complex data, high dimension and high noise are challenging problems, and deep matrix factorization shows great potential in data dimensionality reduction. In this article, a novel robust and effective deep matrix factorization framework is proposed. This method constructs a dual-angle feature for single-modal gene data to improve the effectiveness and robustness, which can solve the problem of high-dimensional tumor classification. The proposed framework consists of three parts, deep matrix factorization, double-angle decomposition, and feature purification. First, a robust deep matrix factorization (RDMF) model is proposed in the feature learning, to enhance the classification stability and obtain better feature when faced with noisy data. Second, a double-angle feature (RDMF-DA) is designed by cascading the RDMF features with sparse features, which contains the more comprehensive information in gene data. Third, to avoid the influence of redundant genes on the representation ability, a gene selection method is proposed to purify the features by RDMF-DA, based on the principle of sparse representation (SR) and gene coexpression. Finally, the proposed algorithm is applied to the gene expression profiling datasets, and the performance of the algorithm is fully verified.