提出非负张量分解的可识别性条件，确保源信号唯一还原。
Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory
- 基于稀疏性假设，通过最小化核心张量切片体积实现唯一分解
- 只需核心张量部分展开具满列秩，不要求非负
- 适用于信号分离、数据溯源等需唯一还原场景
张量分解已成为数据科学的核心工具，广泛应用于数据分析、信号处理和机器学习。许多张量分解（如CP分解）具有可识别性：因子在平凡缩放与排列模糊下唯一，从而可恢复生成数据的真实源。张量分解（TD）是核心且广泛应用的模型，但通常不可识别。本文研究非负张量分解（nTD）的可识别性。通过扩展非负矩阵分解（NMF）的可识别性结果，我们给出nTD的唯一性条件：要求非负矩阵因子满足稀疏性（即满足可分性或充分分散条件），而核心张量仅需某些切片（或其线性组合）或展开具有满列秩（无需非负）。在此条件下，我们提出若干方法，通过最小化输入张量展开或切片的体积，获得可识别的nTD。
原文摘要 · Abstract (English)
Tensor decompositions have become a central tool in data science, with applications in areas such as data analysis, signal processing, and machine learning. A key property of many tensor decompositions, such as the canonical polyadic decomposition, is identifiability: the factors are unique, up to trivial scaling and permutation ambiguities. This allows one to recover the groundtruth sources that generated the data. The Tucker decomposition (TD) is a central and widely used tensor decomposition model. However, it is in general not identifiable. In this paper, we study the identifiability of the nonnegative TD (nTD). By adapting and extending identifiability results of nonnegative matrix factorization (NMF), we provide uniqueness results for nTD. Our results require the nonnegative matrix factors to have some degree of sparsity (namely, satisfy the separability condition, or the sufficiently scattered condition), while the core tensor only needs to have some slices (or linear combinations of them) or unfoldings with full column rank (but does not need to be nonnegative). Under such conditions, we derive several procedures, using either unfoldings or slices of the input tensor, to obtain identifiable nTDs by minimizing the volume of unfoldings or slices of the core tensor.
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