Low-Rank Matrix Completion Theory via Plücker Coordinates

In the competitive landscape of online learning, developing robust and effective learning resource recommendation systems is paramount, yet the field faces challenges due to high-dimensional, sparse matrices and intricate user–resource interactions. Our study focuses on geometric matrix completion (GMC) and introduces a novel approach, graph-based truncated norm regularization (GBTNR) for problem solving. GBTNR innovatively incorporates truncated Dirichlet norms for both user and item graphs, enhancing the model’s ability to handle complex data structures. This method synergistically combines the benefits of truncated norm regularization with the insightful analysis of user–user and resource–resource graph relationships, leading to a significant improvement in recommendation performance. Our model’s unique application of truncated Dirichlet norms distinctively positions it to address the inherent complexities in user and item data structures more effectively than existing methods. By bridging the gap between theoretical robustness and practical applicability, the GBTNR approach offers a substantial leap forward in the field of learning resource recommendations. This advancement is particularly critical in the realm of online education, where understanding and adapting to diverse and intricate user–resource interactions is key to developing truly personalized learning experiences. Moreover, our work includes a thorough theoretical analysis, complete with proofs, to establish the convergence property of the GMC-GBTNR model, thus reinforcing its reliability and effectiveness in practical applications. Empirical validation through extensive experiments on diverse real-world datasets affirms the model’s superior performance over existing methods, marking a groundbreaking advancement in personalized education and deepening our understanding of the dynamics in learner–resource interactions.

Section: Geometric Matrix Completionmentioning

confidence: 99%

Geometric Matrix Completion via Graph-Based Truncated Norm Regularization for Learning Resource Recommendation

Yang,

Shi,

Zhou

et al. 2024

“…An open issue is how to identify matrix patterns with unique or a finite number of completions. In [421], three families of matrix patterns are presented for lowrank matrix completion (in terms of Plücker coordinates). In [422], a deterministic sampling method for matrix completion using an asymmetric Ramanujan graph and its sufficient conditions for the matrix completion are derived.…”

Section: A Few Topics For Future Researchmentioning

confidence: 99%

Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Swamy

Wang

et al. 2023

Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are topics similar to sparse coding. Matrix completion is the process of recovering a data matrix from a subset of its entries, and it extends the principles of compressed sensing and sparse approximation. The nonnegative matrix factorization is a low-rank matrix factorization technique for nonnegative data. All of these low-rank matrix factorization techniques are unsupervised learning techniques, and can be used for data analysis tasks, such as dimension reduction, feature extraction, blind source separation, data compression, and knowledge discovery. In this paper, we survey a few emerging matrix factorization techniques that are receiving wide attention in machine learning, signal processing, and statistics. The treated topics are compressed sensing, dictionary learning, sparse representation, matrix completion and matrix recovery, nonnegative matrix factorization, the Nyström method, and CUR matrix decomposition in the machine learning framework. Some related topics, such as matrix factorization using metaheuristics or neurodynamics, are also introduced. A few topics are suggested for future investigation in this article.

“…In recent years, the problem of recovering an unknown low-rank matrix from its limited number of observed entries has been actively studied in many scientific applications such as Netflix problem [1], image processing [2], system identification [3], video denoising [4], signal processing [5], subspace learning [6], and so on. In mathematics, this problem can be modeled by the following low-rank matrix completion problem: min X∈R m×n rank(X) s.t.…”

Section: Introductionmentioning

confidence: 99%

A Designed Thresholding Operator for Low-Rank Matrix Completion

Cui,

He,

Yang

2024

In this paper, a new thresholding operator, namely, designed thresholding operator, is designed to recover the low-rank matrices. With the change of parameter in designed thresholding operator, the designed thresholding operator can apply less bias to the larger singular values of a matrix compared with the classical soft thresholding operator. Based on the designed thresholding operator, an iterative thresholding algorithm for recovering the low-rank matrices is proposed. Numerical experiments on some image inpainting problems show that the proposed thresholding algorithm performs effectively in recovering the low-rank matrices.