Tensor Envelopes in the Nonconvex Setting
Abstract
We introduce a tensor envelope (TEE) for composite objectives φ=f+g, built from a qth-order Taylor model of the smooth term f regularized by a pth power with p>q. To construct it, we first establish a tensor descent lemma and characterize its behavior in the three regimes q<p<q+1, p=q+1, and p>q+1. Under suitable prox-boundedness and parameter conditions, the associated tensor operator (TOP) has nonempty compact values, and TEE is finite-valued and continuous, preserves global minimizers, and inherits the level-boundedness of φ. We further show that TEE is locally Lipschitz continuous and directionally differentiable, characterize its Fréchet differentiability, and prove that it is C1 whenever TOP is single-valued. As such, TEE preserves the essential optimization landscape properties of φ while enjoying a richer analytical structure, which facilitates the design and analysis of tensor-based optimization methods. Finally, TEE serves as a Lyapunov function for the exact tensor iteration: every cluster point is critical, and, for bounded trajectories with q<p≤ q+1, an ε-stationary point is reached within 𝒪(ε−p/(p−1)) iterations. To our knowledge, both the tensor descent lemma and this complexity bound are the first of their kind obtained without assuming global Lipschitz or Hölder continuity of the qth-order derivative of f.
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BibTeX
@misc{kabgani2026tensor,
title = {{Tensor Envelopes in the Nonconvex Setting}},
author = {Alireza Kabgani and Masoud Ahookhosh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04610},
eprint = {2610.04610},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04610},
}
RIS
TY - UNPB
TI - Tensor Envelopes in the Nonconvex Setting
AU - Alireza Kabgani
AU - Masoud Ahookhosh
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04610
UR - https://arxiv.org/abs/2610.04610
AB - We introduce a tensor envelope (TEE) for composite objectives φ=f+g, built from a qth-order Taylor model of the smooth term f regularized by a pth power with p>q. To construct it, we first establish a tensor descent lemma and characterize its behavior in the three regimes q<p<q+1, p=q+1, and p>q+1. Under suitable prox-boundedness and parameter conditions, the associated tensor operator (TOP) has nonempty compact values, and TEE is finite-valued and continuous, preserves global minimizers, and inherits the level-boundedness of φ. We further show that TEE is locally Lipschitz continuous and directionally differentiable, characterize its Fréchet differentiability, and prove that it is C^1 whenever TOP is single-valued. As such, TEE preserves the essential optimization landscape properties of φ while enjoying a richer analytical structure, which facilitates the design and analysis of tensor-based optimization methods. Finally, TEE serves as a Lyapunov function for the exact tensor iteration: every cluster point is critical, and, for bounded trajectories with q<p≤ q+1, an ε-stationary point is reached within 𝒪(ε^−p/(p−1)) iterations. To our knowledge, both the tensor descent lemma and this complexity bound are the first of their kind obtained without assuming global Lipschitz or Hölder continuity of the qth-order derivative of f.
ER -