Congratulation to our faculty member Kelum Gajamannage! Kelum has been awarded an NSF grant:

**DMS-2418826 LEAPS-MPS: Hadamard Deep Autoencoders and Alternating Directional Methods of Multipliers for Manifold Learning Enabled Distance Preserving Matrix Completion**.

Dates: September 1, 2024 — August 31, 2026 (Estimated). Amount: $249,926.00.

https://www.nsf.gov/awardsearch/showAward?AWD_ID=2418826&HistoricalAwards=false

ABSTRACT:

Data sampled from real-world applications such as image inpainting (e.g. reconstruction of the area covered by the mask of a criminal) and recommender systems (e.g. suggesting products based on a user’s Amazon purchase history) often consists of partially observed or unobserved entries; thus, estimating those entries before any analysis is vital. The technique of recovering missing entries of a data set, specifically, a data matrix, is known as Matrix Completion (MC) in which the data matrix is decomposed into a low-rank component representing features of the data, and a sparse component representing anomalies and noise. However, conventional MC frameworks have limited transferability and robustness when applied in diverse domains since such methods do not consider the natural correlation of the data. Thus, the principal investigator (PI) develops a highly transferable and robust MC framework by harnessing the natural correlation of the data. The optimization scheme of the MC framework is numerically implemented using both an efficient algebraic approach and a high-precision Deep Neural Network (DNN) approach. The performance of this MC framework is validated using both a theoretical analysis, as well as synthetic and real-world benchmark datasets.

Real-world data with natural correlation underlies low-rank nonlinear manifold representations; thus, robust MC methods should guarantee the manifold’s primary characteristic of distance-preserving ability within the low-rank component of the data matrix, which results in meticulous sparse information retention ability within the sparse component. The PI will develop the MC model with a new mathematical foundation to assure the aforementioned characteristics within decomposed low-rank and sparse components of the data matrix. Especially, the method intakes training data as bounds in any distance of interest (e.g., geodesic, hamming, hop), which helps incorporate observed, unobserved, and fully observed data instances so that the method produces the recovered matrix in the same type of distance. The PI adopts the truncated nuclear norm convex relaxation on the low-rank component of the data matrix as a surrogate to the non-convex and discontinuous truncated rank minimization. The distance-preserving ability of the nonlinear manifold is attained by the adaptation of a special constraint into the optimization scheme that emphasizes the Gramian matrix of low-rank component is positive semi-definite. Anomalies in the real-world data are structured; thus, the PI extracts the sparse component by utilizing both square integrable and intergable norm minimization in contrast to the use of only integrable norm minimization. This MC model is numerically implemented by the algebraic approach Alternating Directional Methods of Multipliers and the DNN approach Hadamard Deep Autoencoders. This project is jointly funded by the Launching Early-Career Academic Pathways in the Mathematical and Physical Sciences Program and the Established Program to Stimulate Competitive Research (EPSCoR).