Optimization on the Extended Tensor-Train Manifold with Shared Factors

Alexander Molozhavenko; M. Rakhuba

doi:10.1007/s40314-025-03605-0

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Optimization on the Extended Tensor-Train Manifold with Shared Factors

Computational and Applied Mathematics. 2026. Vol. 45. No. 6. Article 221.

Alexander Molozhavenko, Rakhuba M.

This paper studies tensors that admit decomposition in the Extended Tensor Train (ETT) format, with a key focus on the case where some decomposition factors are constrained to be equal. This factor sharing introduces additional challenges, as it breaks the multilinear structure of the decomposition. Nevertheless, we show that Riemannian optimization methods can naturally handle such constraints and prove that the underlying manifold is indeed smooth. We develop efficient algorithms for key Riemannian optimization components, including a retraction operation based on quasi-optimal approximation in the new format, as well as tangent space projection using automatic differentiation. Finally, we demonstrate the practical effectiveness of our approach through tensor approximation tasks and multidimensional eigenvalue problem.

Research target: Mathematics Computer Science

Language: English

DOI

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Keywords: Riemannian optimization Риманова оптимизация tensor decompositions тензорные разложения тензорный поезд Tensor Train decomposition

A Multi-Feature Fusion Deep Neural Network with Attention for Multi-Task Crop Monitoring

Obsie E. Y., Zhan Y., Liu Y., Engineering Research Express 2026 Article 7

Maintaining plant health is crucial for crop quality, food security, and agricultural productivity. Although deep learning methods have shown strong potential for species, disease, and severity classification, most existing approaches tackle these tasks separately, failing to exploit shared representations and interdependencies among them. Furthermore, the variability of disease symptoms, such as subtle differences in size, ...

Added: October 22, 2026

A Neural Network Approach for Generating Libration Point Orbits

Lev Pyatko, Sergey Aksenov, Regular and Chaotic Dynamics 2026 Vol. 31 No. 5 P. 794–813

In the circular restricted three-body problem, families of periodic and quasi-periodic orbits exist in the vicinity of the libration points and are of significant interest for practical applications. This work presents a neural network-based algorithm for orbit construction and station-keeping, applicable to both periodic and quasi-periodic orbits near collinear libration points. The method employs a ...

Added: October 19, 2026

Основы теории информации

Vorontsova T., М.: Академия, 2025.

The textbook has been prepared in accordance with the requirements of the federal state educational standards for secondary vocational education and is intended for studying the professional modules “Information Security in Information and Telecommunication Systems and Networks Using Software and Hardware‑Software Tools, Including Cryptographic Protection Tools” and “Information Security in Automated Systems Using Software and ...

Added: October 19, 2026

Моментные оценки для стохастической аппроксимации

Naumov A., Рубцов А. С., Samsonov S. et al., Успехи математических наук 2026 Т. 81 № 4 (490) С. 161–162

Краткое сообщение рассматривает Моментные оценки для стохастической аппроксимации ...

Added: October 17, 2026

An infinite series of Gorenstein local algebras failing the affine homogeneity property

Roman Avdeev, Yulia Zaitseva, Mediterranean Journal of Mathematics 2026 Vol. 23 No. 7 Article 224

We provide an infinite series of commutative finite-dimensional Gorenstein local algebras A_n for n ≥ 2. We give an elementary proof that the maximal ideal of every algebra A_n possesses a one-dimensional subspace that is different from the socle and invariant under the automorphism group of A_n. The latter implies that the algebras A_n fail ...

Added: October 17, 2026

Turbulent Exchange Tensor in the Atmospheric Boundary Layer: An Inverse-Problem Approach Using Radiosonde Wind Observations with High Vertical Resolution

Gordin V. A., Efremenko D., Boundary-Layer Meteorology 2026 Vol. 192 No. 51

An inverse problem of reconstructing the turbulent exchange tensor from large archives of observed wind profiles within the Åkerblom–Ekman (A-E) model framework is investigated using high-vertical-resolution radiosonde observations. The reconstructed tensor provides improved agreement between modeled and observed wind profiles compared to classical formulations based on real turbulent exchange coefficient. Instead of the classical A-E ...

Added: October 16, 2026

Diversity of chaos with additional zero Lyapunov exponents in heterogeneous ensemble of quasi-periodic and chaotic systems

Kuznetsov A. P., Sedova Y. V., Stankevich N., Chaos, Solitons and Fractals 2026 Vol. 213 No. 12 Article 119309

We analyze the interaction of a quasi-periodic subsystem, in which two-, three-, and four- frequency quasi-periodic oscillations are possible, with a chaotic subsystem. A discrete version is obtained via the Euler scheme from the corresponding flow model. The dynamics of individual subsystems are discussed, including quasi-periodic bifurcations of invariant tori and the occur rence of ...

Added: October 16, 2026

A Reproducible Microscopy Acquisition Pipeline and Microscopy-Aware Attention for Classification, Detection, and Segmentation of Microorganisms

Aleksei Samarin, Nazarenko A., Nazarenko D. et al., Machine Learning and Knowledge Extraction 2026 Vol. 8 No. 10 Article 330

Automated microscopic image analysis is essential for quantitative microbiology, yet manual acquisition and generic recognition models struggle with morphological variability and scene artifacts. We address both ends of this pipeline. First, we present a low-cost, open-source acquisition system, based on an Arduino UNO R3 microcontroller and NEMA 17 stepper motors, that automates focusing, magnification, and ...

Added: October 14, 2026

On Calculation of Robustness and Stability of Formal Concepts Based on Delta Measure

Parkhomenko V., Kulikova S., Buzmakov A. V., Lobachevskii Journal of Mathematics 2026 Vol. 47 No. 6 P. 2979–2992

The stability of formal concepts and its generalization as robustness are considered in the paper. We use an approximation of stability in the form of Δ-measure for fast calculations. A computationally efficient method for calculating robustness based on subcontexts is presented. We also present a new method to analytically evaluate the number of preimages for ...

Added: October 14, 2026

Shrimps as a fundamental object of nonlinear dynamics. Overview, properties and quasi-periodic generalization

Kuznetsov A. P., Sedova Y. V., Sataev I. R. et al., Mathematics and Computers in Simulation 2027 Vol. 2 P. 574–592

Universal bifurcation structures on the parameter plane of nonlinear systems, known as shrimps, are discussed. An overview of numerous examples from various fields of science is given. Shrimps are typical for both maps of different dimensions and flows. The varieties of shrimp are discussed and relevant examples are given. The similarity property on the parameter ...

Added: October 14, 2026

Studies in Computational Intelligence

Springer, 2026.

This book highlights cutting-edge research in the field of network science, offering scientists, researchers, students and practitioners a unique update on the latest advances in theory and a multitude of applications. It presents the peer-reviewed proceedings of the XIV International Conference on Complex Networks and their Applications (COMPLEX NETWORKS 2025). The carefully selected papers cover ...

Added: October 13, 2026

Proceedings of the 3rd International Conference on Sustainability: Developments and Innovations. Proceedings of ICSDI 2026, Volume 4. Lecture Notes in Civil Engineering (LNCE, volume 886)

Springer, 2026.

This book offers a comprehensive overview of the latest research, innovations, and practical applications in the field of sustainability, with a focus on construction materials and their use in building and civil engineering structures. As an outcome of the 3rd International Conference on Sustainability: Developments and Innovations (ICSDI 2026), this book presents a selection of ...

Added: October 13, 2026

Sandpile Cellular Automata with Manna Rule on the Chung-Lu Graph: Application to X Social Network

Dmitriev A., Kornilov V., Batienkov R., Lecture Notes in Civil Engineering 2026 Vol. 886 P. 249–255

Numerous studies have provided compelling evidence that the probability distribution of avalanche sizes follows a power law, and that 1/f noise is present in the stochastic dynamics of landslides. We present a theoretical explanation of these phenomena within the framework of self-organized criticality theory, proposing a discrete-time model in the form of a sandpile cellular ...

Added: October 13, 2026

Sandpile Cellular Automata with Manna Rule on the Chung-Lu Graph: Application to X Social Network

Dmitriev A., Shulman I., Lebedev A., Studies in Computational Intelligence 2026 Vol. 1265 P. 266–273

A discrete-time model is presented for the self-organization of the X/Twitter social network to the edge of a phase transition, characterized by avalanche-like propagation of information in the network. The model is based on the analogy between the critical dynamics of the X social network and sandpile cellular automata with Manna rule on the Chung-Lu ...

Added: October 13, 2026

When universal machine learning models fail in capacitance prediction for carbon-based supercapacitors: Leakage-aware validation and electrolyte screening

Makarov D. M., Kalikin N., Fashchevsky K. A. et al., Electrochimica Acta 2026 Vol. 579 Article 150159

Machine learning (ML) has become a powerful tool for accelerating the development of carbon-based supercapacitors; however, most published models assume that a single predictive framework can describe both conventional porous carbons and hybrid carbon electrodes. Here, we test this assumption using a comprehensive benchmark dataset and an extended literature-curated dataset enriched with synthesis- and electrolyte-related descriptors. Three ML algorithms ...

Added: October 13, 2026

LiteKV: A KV cache compression method for efficient inference on resource-constrained devices

Zhang Z., Ilvovsky D., Information Processing and Management 2026 Vol. 63 No. 7

Transformer-based Large Language Models (LLMs) rely on Key-Value (KV) cache to accelerate autoregressive decoding, but memory and compute requirements still grow with context length, limiting deployment on resource-constrained devices. Existing training-free KV cache compression methods mainly rely on attention-driven retention heuristics, which may under preserve structurally informative tokens under tight cache budgets, especially when such ...

Added: October 13, 2026

Poincaré series of valuations on functions over a subfield of complex numbers

Gusein-Zade S., Campillo A., Delgado F., Mathematische Zeitschrift 2026 Vol. 314 No. 1 P. 1–22

For a subfield $\K$ of the field $\C$ of complex numbers, we consider curve and divisorial valuations on the algebra $\K[[x,y]]$ of formal power series in two variables with the coefficients in $\K$. We compute the semigroup Poincar\'e series and the classical Poincar\'e series of a (discrete, rank one) valuation on it. ...

Added: October 13, 2026

Global optimization of atomic clusters via physically constrained tensor train decomposition

Sozykin K., Rybin N., Chertkov A. et al., Physical Review B: Condensed Matter and Materials Physics 2026 Vol. 113 No. 22 Article 224111

The global optimization of atomic clusters represents a fundamental challenge in computational chemistry and materials science due to the exponential growth of local minima with system size (i.e., the curse of dimensionality). We introduce a framework that overcomes this limitation by exploiting the low-rank structure of potential energy surfaces through tensor train (TT) decomposition. Our ...

Added: July 22, 2026

LoRA meets Riemannion: Muon Optimizer for Parametrization-independent Low-Rank Adapters

Vladimir Bogachev, Aletov V., Alexander Molozhavenko et al., , in: The Fourteenth International Conference on Learning Representations (ICLR 2026).: ICLR, 2026. Ch. 20503 P. 1–26.

This work presents a novel, fully Riemannian framework for Low-Rank Adaptation (LoRA) that geometrically treats low-rank adapters by optimizing them directly on the fixed-rank manifold. This formulation eliminates the parametrization ambiguity present in standard Euclidean optimizers. Our framework integrates three key components to achieve this: (1) we derive Riemannion, a new Riemannian optimizer on the fixed-rank ...

Added: April 29, 2026

Knowledge Graph Completion with Mixed Geometry Tensor Factorization

Yusupov V., Rakhuba M., Frolov E., , in: Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, 3-5 May 2025, Splash Beach Resort in Mai Khao, Thailand, PMLR: vol. 258Vol. 258.: PMLR, 2025. P. 4924–4932.

Added: May 25, 2025

Секция: «Вычислительные технологии и моделирование». Оптимизация на многообразии тензорных поездов с общими факторами.

Molozhavenko A., Rakhuba M., В кн.: Тихоновские чтения : тезисы докладов : научная конференция.: М.: МАКС Пресс, 2024. С. 46–46.

Данная работа посвящена разработке эффективных алгоритмов римановой оптимизации для разложения, представляющего собой тензорный поезд (ТТ) [1, 2] с таккеровским разложением ТТ-ядер. Ключевой особенностью исследования является использование дополнительного условия на равенство некоторых из факторов разложения. Наличие одинаковых факторов ломает полилинейную структуру разложения, а значит делает неприменимым классический подход попеременных наименьших квадратов. Тем не менее, мы показываем, что для такого разложения сохраняется ...

Added: January 17, 2025

Training a Tucker Model With Shared Factors: a Riemannian Optimization Approach

Peshekhonov I., Aleksey Arzhantsev, Rakhuba M., , in: Proceedings of The 27th International Conference on Artificial Intelligence and Statistics (AISTATS 2024), 2-4 May 2024, Palau de Congressos, Valencia, Spain. PMLR: Volume 238Vol. 238.: Valencia: PMLR, 2024. Ch. 238 P. 3304–3312.

Added: November 29, 2024

Tensor methods and recommender systems

Frolov E., Oseledets I., Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 2017 Vol. 7 No. 3

Added: November 16, 2023

Tensor-Based Sequential Learning via Hankel Matrix Representation for Next Item Recommendations

Frolov E., Oseledets I., IEEE Access 2023 Vol. 11 P. 6357–6371

Self-attentive transformer models have recently been shown to solve the next item recommendation task very efficiently. The learned attention weights capture sequential dynamics in user behavior and generalize well. Motivated by the special structure of learned parameter space, we question if it is possible to mimic it with an alternative and more lightweight approach. We ...

Added: November 16, 2023