An Undirected Weighted Network (UWN) is frequently encountered in a big-data-related application concerning the complex interactions among numerous nodes, e.g., a protein interaction network from a bioinformatics application. A Symmetric High-Dimensional and Incomplete (SHDI) matrix can smoothly illustrate such an UWN, which contains rich knowledge like node interaction behaviors and local complexes. To extract desired knowledge from an SHDI matrix, an analysis model should carefully consider its symmetric-topology for describing an UWN's intrinsic symmetry. Representation learning to an UWN borrows the success of a pyramid of symmetry-aware models like a Symmetric Nonnegative Matrix Factorization (SNMF) model whose objective function utilizes a sole Latent Factor (LF) matrix for representing SHDI's symmetry rigorously. However, they suffer from the following drawbacks: 1) their computational complexity is high; and 2) their modeling strategy narrows their representation features, making them suffer from low learning ability. Aiming at addressing above critical issues, this paper proposes a Multi-constrained Symmetric Nonnegative Latent-factor-analysis (MSNL) model with two-fold ideas: 1) introducing multi-constraints composed of multiple LF matrices, i.e., inequality and equality ones into a data-density-oriented objective function for precisely representing the intrinsic symmetry of an SHDI matrix with broadened feature space; and 2) implementing an Alternating Direction Method of Multipliers (ADMM)-incorporated learning scheme for precisely solving such a multi-constrained model. Empirical studies on three SHDI matrices from a real bioinformatics or industrial application demonstrate that the proposed MSNL model achieves stronger representation learning ability to an SHDI matrix than state-of-the-art models do.

An Undirected Weighted Network (UWN) is commonly found in big data-related applications. Note that such a network's information connected with its nodes, and edges can be expressed as a Symmetric, High-Dimensional and Incomplete (SHDI) matrix. However, existing models fail in either modeling its intrinsic symmetry or low-data density, resulting in low model scalability or representation learning ability. For addressing this issue, a Proximal Symmetric Nonnegative Latent-factor-analysis (PSNL) model is proposed. It incorporates a proximal term into symmetry-aware and data density-oriented objective function for high representation accuracy. Then an adaptive Alternating Direction Method of Multipliers (ADMM)-based learning scheme is implemented through a Tree-structured of Parzen Estimators (TPE) method for high computational efficiency. Empirical studies on four UWNs demonstrate that PSNL achieves higher accuracy gain than state-of-the-art models, as well as highly competitive computational efficiency.

Large-scale undirected weighted networks are usually found in big data-related research fields. It can naturally be quantified as a symmetric high-dimensional and incomplete (SHDI) matrix for implementing big data analysis tasks. A symmetric non-negative latent-factor-analysis (SNL) model is able to efficiently extract latent factors (LFs) from an SHDI matrix. Yet it relies on a constraint-combination training scheme, which makes it lack flexibility. To address this issue, this paper proposes an unconstrained symmetric nonnegative latent-factor-analysis (USNL) model. Its main idea is two-fold: 1) The output LFs are separated from the decision parameters via integrating a nonnegative mapping function into an SNL model; and 2) Stochastic gradient descent (SGD) is adopted for implementing unconstrained model training along with ensuring the output LFs nonnegativity. Empirical studies on four SHDI matrices generated from real big data applications demonstrate that an USNL model achieves higher prediction accuracy of missing data than an SNL model, as well as highly competitive computational efficiency.