arXiv subjects
Publications and source records attributed to Mahdi Dehghani.
We introduced and study a family of generalized norm on $C^*$-algebras and the notion of Birkhoff-James orthogonality and norm-parallelism with respect to these family of generalized norms are investigated.
Large Language Models (LLMs) suffer from hallucinations and outdated knowledge due to their reliance on static training data. Retrieval-Augmented Generation (RAG) mitigates these issues by integrating external dynamic information for improved factual grounding. With advances in multimodal learning, Multimodal RAG extends this approach by incorporating multiple modalities such as text, images, audio, and video to enhance the generated outputs. However, cross-modal alignment and reasoning introduce unique challenges beyond those in unimodal RAG. This survey offers a structured and comprehensive analysis of Multimodal RAG systems, covering datasets, benchmarks, metrics, evaluation, methodologies, and innovations in retrieval, fusion, augmentation, and generation. We review training strategies, robustness enhancements, loss functions, and agent-based approaches, while also exploring the diverse Multimodal RAG scenarios. In addition, we outline open challenges and future directions to guide research in this evolving field. This survey lays the foundation for developing more capable and reliable AI systems that effectively leverage multimodal dynamic external knowledge bases. All resources are publicly available at https://github.com/llm-lab-org/Multimodal-RAG-Survey.
It has been shown that if $T$ is a complex matrix, then {\small\begin{align*} ω(T)&=\frac{1}{n}\sup\left\{|\mathrm{Tr}\ X|;\ X\in W^n(T)\right\}\\ &=\frac{1}{n}\sup\left\{\|X\|_1;\ X\in W^n(T)\right\}\\ &= \sup\left\{ ω(X);\ X\in W^n(T)\right\} \end{align*} } where $n$ is a positive integer, $ω(T)$ is the numerical radius and $W^n(T)$ is the $n$'th matricial range of $T$.
The aim of this paper is to present some results concerning the $ρ_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T\,: X \longrightarrow Y$ is a nonzero linear $(I, ρ_*)$-orthogonality preserving mapping between real normed spaces, then $$\frac{1}{3}\|T\|\|x\|\leq\|Tx\|\leq 3[T]\|x\|, \qquad (x\in X)$$ where $[T]:=\inf\{\|Tx\|: \,x\in X, \|x\|=1\}$. We also show that the pair $(X,\perp_{ρ_*})$ is an orthogonality space in the sense of Rätz. Some characterizations of smooth spaces are given based on the $ρ_*$-orthogonality.
Utilizing the notion of positive multilinear mappings, we give some matrix inequalities. In particular, Choi--Davis--Jensen and Kantorovich type inequalities including positive multilinear mappings are presented.