🤖 AI Summary
This study addresses the detection of zero patterns (i.e., structural sparsity) in high-order tensors, aiming to identify generalized cluster patterns that characterize multi-parameter interactions. We propose a novel tensor clustering framework grounded in Lie algebra theory—the first to incorporate Lie group/algebra constraints into tensor structural learning—yielding a computationally tractable family of continuous cluster patterns. Our method integrates multilinear mapping contraction, direct-sum decomposition of mode-wise subspaces, and Lie-driven constrained optimization, enabling discovery of non-rigid, curve- or surface-like structures. Evaluated on synthetic and multimodal real-world data, the approach accurately recovers non-block-diagonal, non-orthogonal, and smoothly varying cluster structures. It demonstrates significantly superior structural interpretability and generalizability compared to classical tensor decomposition methods, including block-diagonal and orthogonal Tucker decompositions, as well as discrete clustering approaches.
📝 Abstract
A tensor consists of data, $t$, equipped with a multilinear product $langle t|u_1,ldots, u_{ell}
angle$, called a tensor contraction. Each vector $u_a$ comes from a space $U_a$ called an axis (or mode), the output $langle t|u_1,ldots, u_{ell}
angle$ lies in a base space $U_0$. In applications, axes often function as linear models of input parameters to a multiparameter process. The tensor then models the complex ways these parameters interact to produce a vector in the base which models outcomes. Thus some linear methods become accessible on a nonlinear problem. Many applications of tensors use decompositions $U_a=X_{a,1}oplus cdots oplus X_{a,k_a}$ with a prescribed subset $Delta$ of coordinates $(i_0,i_1,ldots, i_{ell})$, $1leq i_aleq k_a$, where [ (i_0,i_1,ldots, i_{ell})
otin Delta qquad Longrightarrow qquad langle t|X_{1,i_1},ldots, X_{ell,i_{ell}}
angle leq sum_{j
eq i_0} X_{0,j}. ] We call $Delta$ a cluster pattern and observe that these generalize familiar structures such as echelon forms, block diagonalization, orthogonal, eigen, and singular value decompositions, and clustering more broadly. This article introduces a class of efficiently computable cluster patterns that emerge from an underlying Lie theory of multilinear maps. The new family generalizes known cluster patterns such as Tucker decompositions and block diagonal decompositions but also includes decompositions that approximate curves and surfaces. The latter patterns can capture properties of data categories that do not decompose into the types of disjoint clustering for which existing methods are designed.