DEPARTMENT OF COMPUTATIONAL AND DATA SCIENCES
Ph.D. Thesis Colloquium
Speaker: Mr. Biswarup Karmakar
S.R. Number: 06-18-01-10-12-22-1-21055
Title: Tensor generalized inverses and low-rank representation with applications
Research Supervisor: Dr. Ratikanta Behera
Date & Time : January 05 2026 (Monday), 15:00 PM
Venue : #102, CDS Seminar Hall
ABSTRACT
Tensors are mathematical structures that generalize vectors and matrices to higher dimensions, providing efficient computational frameworks for multidimensional data while preserving inter-channel relationships that conventional methods cannot capture. Advanced tensor operation techniques, such as the t-product and M-product, enable Fourier domain computations and mode-product operations, respectively, for flexible factorizations. These establish rigorous foundations for decomposition algorithms, data recovery, and dimensionality reduction while maintaining a multilinear structure. Tensor-based paradigms have advantages in scalable machine learning and the recovery of incomplete data problems.
This dissertation establishes comprehensive analytical and algorithmic frameworks within the t-product and M-product theoretical paradigms for tensor generalized inverses and low-rank recovery methodologies, with applications in high-dimensional signal and image processing. The first contribution introduces a novel M-QDR decomposition under the M-product framework for computing tensor generalized inverses without involving square-root terms, thereby preserving the algebraic structure and enabling the exact computation of the Moore-Penrose and outer inverses of tensors. The second contribution presents efficient iterative algorithms for computing the tensor Moore-Penrose inverse within the t-product framework, utilizing FFT-based diagonalization to decompose tensor operations into independent matrix computations, thereby enhancing tensor least-squares and image restoration applications. The third contribution establishes a robust tensor completion framework employing weighted correlated total variation norms under the M-product, incorporating weighted Schatten-p norms on gradient tensors for low-rank regularization and weighted $l_1$ norms for noise suppression, implemented through an enhanced alternating direction method of multipliers solver for reliable reconstruction in image completion and denoising. Further, iterative methods for quaternion tensor generalized inverses are developed to process multichannel data as unified entities, preserving inter-channel relationships for applications in 3D signal processing, color image inpainting, and video deblurring. This study establishes a unified family of tensor and quaternion computational methods that significantly advance high-dimensional data reconstruction methodologies. Finally, a tensor deep unfolding framework is proposed for hyperspectral and multispectral image reconstruction, which transforms iterative optimization steps into learnable neural modules that integrate interpretability with data-driven adaptability. In conclusion, this dissertation lays the groundwork for future studies on tensors that explore the solution of PDEs, parallel implementations, and hypergraph applications.
ALL ARE WELCOME
