Tensor-Based Chaotic Perception
Abstract
Symbols can provide compact representations of complex sensory information. Mathematically, chaotic attractors can provide a basis for such representations, with object identity encoded in the topological organization of the attractor. In three-dimensional phase space, this organization can be characterized by the linking matrix of unstable periodic orbits, which serves as a topological fingerprint. Because the linking matrix has no closed-form dependence on the learnable parameters, we learn the chaotic series generating the target attractors rather than optimizing the invariant directly; the reconstructed attractor then determines its unstable periodic orbits and linking matrix through its topology. In this paper, we propose a third-order tensor that maps inputs to connection matrices that generate the corresponding chaotic series. Exploiting the duality between a vector and its sequential representation as a series, we lift inputs into a higher-dimensional space and reorder their components, allowing the input to be presented sequentially rather than simultaneously. We first demonstrate that this representation can discriminate between classes in the \href{https://archive.ics.uci.edu/dataset/151/connectionist+bench+sonar+mines+vs+rocks}{Sonar} dataset. We then examine perceptual constancy using the \href{https://www.csc.kth.se/cvap/databases/kth-tips/index.html}{KTH-TIPS2-a} dataset, where variations in color and texture within a class are mapped to distinct chaotic series associated with the same topological attractor class. The results demonstrate that the proposed tensor-based mapping can preserve attractor class while accommodating variations in the sensory realization of an object.
Related
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BibTeX
@misc{majd2026tensorbased,
title = {{Tensor-Based Chaotic Perception}},
author = {Amir M. Majd},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13278},
eprint = {2609.13278},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13278},
}
RIS
TY - UNPB
TI - Tensor-Based Chaotic Perception
AU - Amir M. Majd
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.13278
UR - https://arxiv.org/abs/2609.13278
AB - Symbols can provide compact representations of complex sensory information. Mathematically, chaotic attractors can provide a basis for such representations, with object identity encoded in the topological organization of the attractor. In three-dimensional phase space, this organization can be characterized by the linking matrix of unstable periodic orbits, which serves as a topological fingerprint. Because the linking matrix has no closed-form dependence on the learnable parameters, we learn the chaotic series generating the target attractors rather than optimizing the invariant directly; the reconstructed attractor then determines its unstable periodic orbits and linking matrix through its topology. In this paper, we propose a third-order tensor that maps inputs to connection matrices that generate the corresponding chaotic series. Exploiting the duality between a vector and its sequential representation as a series, we lift inputs into a higher-dimensional space and reorder their components, allowing the input to be presented sequentially rather than simultaneously. We first demonstrate that this representation can discriminate between classes in the \href{https://archive.ics.uci.edu/dataset/151/connectionist+bench+sonar+mines+vs+rocks}{Sonar} dataset. We then examine perceptual constancy using the \href{https://www.csc.kth.se/cvap/databases/kth-tips/index.html}{KTH-TIPS2-a} dataset, where variations in color and texture within a class are mapped to distinct chaotic series associated with the same topological attractor class. The results demonstrate that the proposed tensor-based mapping can preserve attractor class while accommodating variations in the sensory realization of an object.
ER -