arXiv subjects

Debraj Banerjee

Publications and source records attributed to Debraj Banerjee.

4 recordsLinked to original sources

arXiv2026

Local-Minima-Preserving Continuous Relaxation of Ising Problems

The generalized Ising problem captures a broad spectrum of hard combinatorial problems, including MAX-CUT, Number Partitioning (NPP), and Maximum Independent Set. In this work, we consider the notion of one-flip local minima for this problem. We construct a polynomial relaxation and prove the landscape equivalence theorem: there exists a one-to-one correspondence between the local minima of the relaxation and the one-flip minima of the original Ising problem. This guarantee reduces the Ising problem to finding the local minima of a smooth function, allowing us to leverage gradient-based optimizers such as ADAM. We demonstrate that our method is scalable and it achieves strong performance across challenging benchmarks, including spin-glass models, MAX-CUT, and NPP.

math.OC↗

arXiv2025

On the Schur Stability of Some Image Reconstruction Operators

We investigate an open problem arising in iterative image reconstruction. In its general form, the problem is to determine the stability of the parametric family of operators $P(t) = W (I-t B)$ and $R(t) = I-W + (I+tB)^{-1} (2W-I)$, where $W$ is a stochastic matrix and $B$ is a nonzero, nonnegative matrix. We prove that if $W$ is primitive, then there exists $T > 0$ such that the spectral radii $\varrho(P(t))$ and $\varrho(R(t))$ are strictly less than $1$ for all $0 < t < T$. The proof combines standard perturbation theory for eigenvalues and an observation about the analyticity of the spectral radius. This argument, however, does not provide an estimate of $T$. To this end, we compute $T$ explicitly for specific classes of $W$ and $B$. Building on these partial results and supporting numerical evidence, we conjecture that if $B$ is positive semidefinite and satisfies certain technical conditions, then $\varrho(P(t)), \, \varrho(R(t))<1$ for all $0 < t < 2/\varrho(B)$. As an application, we show how these results can be applied to establish the convergence of certain iterative imaging algorithms.

math.SP↗

arXiv2025

A Continuous Energy Ising Machine Leveraging Difference-of-Convex Programming

Many combinatorial optimization problems can be reformulated as finding the ground state of the Ising model. Existing Ising solvers are mostly inspired by simulated annealing. Although annealing techniques offer scalability, they lack convergence guarantees and are sensitive to the cooling schedule. We propose solving the Ising problem by relaxing the binary spins to continuous variables and introducing an attraction potential that steers the solution toward binary spin configurations. A key property of this potential is that its combination with the Ising energy produces a Hamiltonian that can be written as a difference of convex polynomials. This enables us to design efficient iterative algorithms that require a single matrix-vector multiplication per iteration and provide convergence guarantees. We implement our Ising solver on a wide range of GPU platforms, from edge devices to high-performance computing clusters, and demonstrate that it consistently outperforms existing solvers across problem sizes ranging from small ($10^3$ spins) to ultra-large ($10^8$ spins).

cs.DC↗

arXiv2025

A James-Stein Estimator based Generalized OMP Algorithm for Robust Signal Recovery using Sparse Representation

In this paper, we introduce a novel algorithm named JS-gOMP, which enhances the generalized Orthogonal Matching Pursuit (gOMP) algorithm for improved noise robustness in sparse signal processing. The JS-gOMP algorithm uniquely incorporates the James-Stein estimator, optimizing the trade-off between signal recovery and noise suppression. This modification addresses the challenges posed by noise in the dictionary, a common issue in sparse representation scenarios. Comparative analyses demonstrate that JS-gOMP outperforms traditional gOMP, especially in noisy environments, offering a more effective solution for signal and image processing applications where noise presence is significant.

eess.SP↗