arXiv subjects

Abhijeet Vyas

Publications and source records attributed to Abhijeet Vyas.

7 recordsLinked to original sources

arXiv2026

Programmable Quantum-Like bits from Signed Regular Graphs

Extending upon observations of the emergence of quantum-like (QL) states from classical complex synchronized networks, this work adds mathematical rigor to the analysis of single QL bits constructed from adjacency-matrix eigenvectors. First, we rigorously show that symmetric construction of such networks (regular undirected/symmetric bipartite graph $G_C$ connecting two regular undirected subgraphs $G_A,\,G_B$) leads to an equal superposition of the $|+\rangle, |-\rangle$ Hadamard states (with basis $|0\rangle,\,|1\rangle$ set from eigenvectors of the subgraphs), and provide an analysis of sufficient conditions on the network for construction of such states. Second, we prove two methods to construct arbitrary single qubit states $|ψ\rangle = a|0\rangle + b|1\rangle,\, |a|^2+|b|^2=1$, and give switching lemmas for their boundaries: (i) by detuning the two subgraphs regularities and (ii) by asymmetrically allowing the bipartite connection matrix $C$ to be directed and detuning those regularities. Although motivated by using complex synchronized networks for quantum information storage and computation, the proofs for these methods rely only on the structure of the graph embedded in the adjacency matrix. Thus, synchronization is unnecessary; QL bits arise when edge weights are unit (or near-unit) and subgraphs are regular. Results on combinations of random k-regular graphs (more precisely Erdős-Rényi graphs) may be independently interesting.

quant-ph↗

arXiv2026

Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, and more. While this has led to solving many complex phenomena, there are some limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Difference Methods (FDMs) require considerable time and are computationally expensive. In contrast, data-driven machine learning-based methods, such as neural networks, provide a faster, fairly accurate alternative, and, in particular, focus on neural operators, which have certain advantages such as discretization invariance and resolution invariance. This article aims to provide a comprehensive insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems, while also noting some of the open problems of machine learning-based approaches. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics.

cs.AI↗

arXiv2026

Mirror-Free Proximal Methods

We present a \emph{mirror-free} mirror prox (MFMP) algorithm, which extends the classic approach of Nemirovski (2004) to allow for proximal-like updates without the explicit need for a mirror map. We further analyze the convergence of our method under suitable notions of relative smoothness and relative Lipschitzness, for which we introduce a relaxation of the standard Bregman divergence in terms of more general potential operators. Finally, we show how a strongly monotone variant of our method allows us to solve regularized Taylor-expansion subproblems that appear in both second- and third-order smooth min-max optimization.

math.OC↗

arXiv2026

Beyond First-Order Methods for $\ell_p$-Structured Non-Monotone Variational Inequalities

We propose novel high-order algorithms for a class of $\ell_p$-structured non-monotone variational inequalities. In particular, work by Diakonikolas et al. (2021), which introduced the weak Minty variational inequality (weak-MVI) setting, showed how to find an approximate first-order Euclidean stationary point for a strictly positive range of the weak-MVI parameter $ρ$. However, for the $\ell_p$-norm stationary point setting ($p \neq 2$), their guarantees are limited to $ρ=0$, which recovers the standard MVI setting. In this work, we address this gap by presenting a suite of high-order methods that converge to $\ell_p$-norm stationary points for a suitable range of $ρ> 0$, thereby circumventing previous fundamental challenges in $\ell_p$ settings. We further show convergence for high-order smooth \textit{monotone} operators, generalizing Adil et al. (2022) to the case where $p \geq 2$, and we extend our Euclidean techniques to continuous-time settings.

math.OC↗

arXiv2026

Online Min-Max Optimization: From Individual Regrets to Cumulative Saddle Points

We propose and study an online version of min-max optimization based on cumulative saddle points under a variety of performance measures beyond convex-concave settings. After first observing the incompatibility of (static) Nash equilibrium (SNE-Reg$_T$) with individual regrets even for strongly convex-strongly concave functions, we propose an alternate \emph{static} duality gap (SDual-Gap$_T$) inspired by the online convex optimization (OCO) framework. We provide algorithms that, using a reduction to classic OCO problems, achieve bounds for SDual-Gap$_T$~and a novel \emph{dynamic} saddle point regret (DSP-Reg$_T$), which we suggest naturally represents a min-max version of the dynamic regret in OCO. We derive our bounds for SDual-Gap$_T$~and DSP-Reg$_T$~under strong convexity-strong concavity and a min-max notion of exponential concavity (min-max EC), and in addition we establish a class of functions satisfying min-max EC~that captures a two-player variant of the classic portfolio selection problem. Finally, for a dynamic notion of regret compatible with individual regrets, we derive bounds under a two-sided Polyak-Łojasiewicz (PL) condition.

cs.LG↗

arXiv2023

Beyond first-order methods for non-convex non-concave min-max optimization

We propose a study of structured non-convex non-concave min-max problems which goes beyond standard first-order approaches. Inspired by the tight understanding established in recent works [Adil et al., 2022, Lin and Jordan, 2022b], we develop a suite of higher-order methods which show the improvements attainable beyond the monotone and Minty condition settings. Specifically, we provide a new understanding of the use of discrete-time $p^{th}$-order methods for operator norm minimization in the min-max setting, establishing an $O(1/ε^\frac{2}{p})$ rate to achieve $ε$-approximate stationarity, under the weakened Minty variational inequality condition of Diakonikolas et al. [2021]. We further present a continuous-time analysis alongside rates which match those for the discrete-time setting, and our empirical results highlight the practical benefits of our approach over first-order methods.

math.OC↗

arXiv2022

Competitive Gradient Optimization

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of $α$-coherent function for which we provide convergence analysis. We show that for strictly $α$-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in $α$-coherent class of functions.

math.OC↗