arXiv subjects

Alaa El Ichi

Publications and source records attributed to Alaa El Ichi.

6 recordsLinked to original sources

arXiv2026

The tubal Arnoldi method for tensor function approximation

This paper describes Krylov subspace methods based on the tensor t-product for approximating quantities associated with third-order tensor functions. Specifically, it introduces the tensor-tubal Arnoldi method, which projects highdimensional tensor problems onto lower-dimensional tensor Krylov subspaces. The derived method exploits novel algebraic properties of the t-product to address large-scale tensor computations efficiently. Applications include the evaluation of parameter-dependent tensor functions and the solution of multidimensional ordinary differential equations. Numerical experiments illustrate the effectiveness and accuracy of the proposed methods compared with other approaches.

math.NA↗

arXiv2026

Multidimensional Task Learning: A Unified Tensor Framework for Computer Vision Tasks

This paper introduces Multidimensional Task Learning (MTL), a unified mathematical framework based on Generalized Einstein MLPs (GE-MLPs) that operate directly on tensors via the Einstein product. We argue that current computer vision task formulations are inherently constrained by matrix-based thinking: standard architectures rely on matrix-valued weights and vectorvalued biases, requiring structural flattening that restricts the space of naturally expressible tasks. GE-MLPs lift this constraint by operating with tensor-valued parameters, enabling explicit control over which dimensions are preserved or contracted without information loss. Through rigorous mathematical derivations, we demonstrate that classification, segmentation, and detection are special cases of MTL, differing only in their dimensional configuration within a formally defined task space. We further prove that this task space is strictly larger than what matrix-based formulations can natively express, enabling principled task configurations such as spatiotemporal or cross modal predictions that require destructive flattening under conventional approaches. This work provides a mathematical foundation for understanding, comparing, and designing computer vision tasks through the lens of tensor algebra.

cs.CV↗

arXiv2026

Structured Multidimensional Representation Learning for Large Language Models

Transformer architectures achieve state-of-the-art performance across a wide range of pattern recognition and natural language processing tasks, but their scaling is accompanied by substantial parameter growth and redundancy in the embedding dimension. In this work, we introduce a structured spectral factorization of the embedding space based on the L-product for third-order tensors. By reshaping token representations into spectral tensor slices and performing attention and feed-forward operations in the transform domain, we obtain a Tensor Transformer architecture that decomposes the encoder into p independent spectral sub-transformers while preserving standard Transformer semantics. We prove that the proposed L-Transformer is spectrally equivalent to p parallel Transformers operating on reduceddimensional embeddings, which yields approximately 1/p reduction (up to lower-order terms such as biases and normalization parameters) in encoder parameters under fixed total embedding size. When instantiated with a real-valued Discrete Cosine Transform (DCT), the method remains fully differentiable and compatible with existing training pipelines. Beyond compression, the spectral decomposition introduces an inductive bias over embedding frequencies, enabling slice-dependent frequency scaling that improves generalization. Experiments on IMDB and AG~News show that the proposed model can substantially reduce encoder parameters (up to 75\% for p=4) while maintaining competitive accuracy. On IMDB, the tensorized encoder matches or improves upon the standard baseline under compression, whereas on AG~News at moderate width we observe a small accuracy decrease in exchange for a 4 times encoder reduction; at BERT-base width (d=768), performance returns to parity.

cs.CL↗

arXiv2026

A Computationally Efficient Multidimensional Vision Transformer

Vision Transformers have achieved state-of-the-art performance in a wide range of computer vision tasks, but their practical deployment is limited by high computational and memory costs. In this paper, we introduce a novel tensor-based framework for Vision Transformers built upon the Tensor Cosine Product (Cproduct). By exploiting multilinear structures inherent in image data and the orthogonality of cosine transforms, the proposed approach enables efficient attention mechanisms and structured feature representations. We develop the theoretical foundations of the tensor cosine product, analyze its algebraic properties, and integrate it into a new Cproduct-based Vision Transformer architecture (TCP-ViT). Numerical experiments on standard classification and segmentation benchmarks demonstrate that the proposed method achieves a uniform 1/C parameter reduction (where C is the number of channels) while maintaining competitive accuracy.

cs.LG↗

arXiv2026

Dimensionality Reduction on Riemannian Manifolds in Data Analysis

In this work, we investigate Riemannian geometry based dimensionality reduction methods that respect the underlying manifold structure of the data. In particular, we focus on Principal Geodesic Analysis (PGA) as a nonlinear generalization of PCA for manifold valued data, and extend discriminant analysis through Riemannian adaptations of other known dimensionality reduction methods. These approaches exploit geodesic distances, tangent space representations, and intrinsic statistical measures to achieve more faithful low dimensional embeddings. We also discuss related manifold learning techniques and highlight their theoretical foundations and practical advantages. Experimental results on representative datasets demonstrate that Riemannian methods provide improved representation quality and classification performance compared to their Euclidean counterparts, especially for data constrained to curved spaces such as hyperspheres and symmetric positive definite manifolds. This study underscores the importance of geometry aware dimensionality reduction in modern machine learning and data science applications.

cs.LG↗

arXiv2026

A Unified Trace-Optimization Framework for Multidimensionality Reduction

This paper presents a comprehensive overview of several multidimensional reduction methods focusing on Multidimensional Principal Component Analysis (MPCA), Multilinear Orthogonal Neighborhood Preserving Projection (MONPP), Multidimensional Locally Linear Embedding (MLLE), and Multidimensional Laplacian Eigenmaps (MLE). These techniques are formulated within a unified framework based on trace optimization, where the dimensionality reduction problem is expressed as maximization or minimization problems. In addition to the linear MPCA and MONPP approaches, kernel-based extensions of these methods also are presented. The latter methods make it possible to capture nonlinear relations between high-dimensional data. A comparative analysis highlights the theoretical foundations, assumptions, and computational efficiency of each method, as well as their practical applicability. The study provides insights and guidelines for selecting an appropriate dimensionality reduction technique suited to the application at hand.

math.NA↗