Nachiketa Mishra

Snigdhashree Nayak, Hemant Sharma, Nachiketa Mishra

This article introduces an algebraic framework for establishing eigenvalue bounds for symmetric positive definite tensors by leveraging intrinsic invariants, specifically the trace and determinant (resultant). We derive a hierarchy of inequalities via the Arithmetic Mean-Geometric Mean (AM-GM) inequality that yields progressively tighter upper and lower bounds for the tensor spectral radius and smallest eigenvalue. A comprehensive comparative analysis demonstrates that our invariant-based approach significantly outperforms classical coordinate-dependent methods such as the Gershgorin circle theorem. We explicitly show that our bounds remain robust and informative in scenarios where Gershgorin bounds fail, particularly for tensors with negative off-diagonal entries, where algebraic cancellations occur, and higher-order tensors, where combinatorial growth leads to loose estimates. Furthermore, we validate the practical utility of these bounds by applying them to certify the positive definiteness of Lyapunov functions in the stability analysis of nonlinear autonomous systems.

Hemant Sharma, Nachiketa Mishra

This article introduces a trace-based metric on the space of positive semi-definite (PSD) tensors, offering a geometric perspective that connects their algebraic structure to their intrinsic geometric properties. It defines the Bures-Wasserstein distance on tensor spaces, establishing clear measurements between tensors. Moreover, the study derives trace-based eigenvalue bounds for PSD tensors and analyzes how these bounds depend on the PSD condition. The behavior of these bounds is further explored when the PSD requirement is relaxed, with illustrative examples provided to support the theoretical findings. In addition, a detailed complexity analysis is carried out for the methods proposed in this study.

Hemant Sharma, Nachiketa Mishra

We develop and analyze iterative methods for computing the principal square root of third-order tensors under the T-product framework. Tensor extensions of the Newton iteration (quadratic convergence) and the Denman--Beavers iteration (geometric convergence with simultaneous computation of the inverse square root) are proposed, with rigorous convergence guarantees established via the Fourier-domain block-diagonalization of the T-product. We apply these methods to image processing, introducing Tensor Decorrelated Grayscale conversion, T-Whitening, and optimal color transfer under the T-product geometry. We also formulate the Tensor Bures--Wasserstein distance and prove it defines a valid metric on the space of T-positive definite tensors. Numerical experiments confirm rapid convergence and demonstrate that the proposed tensor-based techniques offer improved structural preservation and cross-channel decorrelation compared to classical methods.

Hemant Sharma, Nachiketa Mishra

In earlier work \cite{sharma2025} we developed deterministic analytical bounds on the T-eigenvalues of symmetric positive definite (SPD) third-order tensors under the Kilmer--Martin T-product: the trace--determinant (TDet) bounds via the AM--GM inequality, and the trace-dependent (TDep) bounds generalizing Samuelson's inequality. While these bounds are cheap and guaranteed-valid, their relative gap grows as $\sqrt{d-1}$ in the tensor dimension $d = np$, limiting their usefulness for large tensors. This paper develops randomized estimators for the extreme T-eigenvalues of T-SPD tensors that complement the deterministic bounds. We adapt the Halko--Martinsson--Tropp framework \cite{halko2011} to the T-product setting and introduce four methods: (i) a randomized power method that produces a lower bound on $λ_1$ with exponential convergence; (ii) a randomized subspace iteration with a tensor-analogue HMT error bound; (iii) a two-sided rigorous bracket combining the randomized lower bound with the deterministic TDep upper bound; and (iv) a Hutchinson-based fully randomized TDep bound for matvec-only settings.

Madhab Barman, Madhurima Panja, Nachiketa Mishra et al.

Tuberculosis (TB) remains a formidable global health challenge, driven by complex spatiotemporal transmission dynamics and influenced by factors such as population mobility and behavioral changes. We propose an Epidemic-Guided Deep Learning (EGDL) approach that fuses mechanistic epidemiological principles with advanced deep learning techniques to enhance early warning systems and intervention strategies for TB outbreaks. Our framework is built upon a modified networked Susceptible-Infectious-Recovered (MN-SIR) model augmented with a saturated incidence rate and graph Laplacian diffusion, capturing both long-term transmission dynamics and region-specific population mobility patterns. Compartmental model parameters are rigorously estimated using Bayesian inference via the Markov Chain Monte Carlo approach. Theoretical analysis leveraging the comparison principle and Green's formula establishes global stability properties of the disease-free and endemic equilibria. Building on these epidemiological insights, we design two forecasting architectures, EGDL-Parallel and EGDL-Series, that integrate the mechanistic outputs of the MN-SIR model within deep neural networks. This integration mitigates the overfitting risks commonly encountered in data-driven methods and filters out noise inherent in surveillance data, resulting in reliable forecasts of real-world epidemic trends. Experiments conducted on TB incidence data from 47 prefectures in Japan and 31 provinces in mainland China demonstrate that our approach delivers robust and accurate predictions across multiple time horizons (short to medium-term forecasts), supporting its generalizability across regions with different population dynamics.