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A tensor consists of data, $t$, equipped with a multilinear product $\langle t|u_1,\ldots, u_{\ell}\rangle$, called a tensor contraction. Each vector $u_a$ comes from a space $U_a$ called an axis (or mode), the output $\langle t|u_1,\ldots, u_{\ell}\rangle$ 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 $(i_0,i_1,\ldots, i_{\ell})$, $1\leq i_a\leq k_a$, where \[ (i_0,i_1,\ldots, i_{\ell})\not\in \Delta \qquad \Longrightarrow \qquad \langle t|X_{1,i_1},\ldots, X_{\ell,i_{\ell}}\rangle \leq \sum_{j\neq i_0} X_{0,j}. \] 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., Comment: 17 pages, 8 figures

The use of measured vibration data from structures has a long history of enabling the development of methods for inference and monitoring. In particular, applications based on system identification and structural health monitoring have risen to prominence over recent decades and promise significant benefits when implemented in practice. However, significant challenges remain in the development of these methods. The introduction of realistic, full-scale datasets will be an important contribution to overcoming these challenges. This paper presents a new benchmark dataset capturing the dynamic response of a decommissioned BAE Systems Hawk T1A. The dataset reflects the behaviour of a complex structure with a history of service that can still be tested in controlled laboratory conditions, using a variety of known loading and damage simulation conditions. As such, it provides a key stepping stone between simple laboratory test structures and in-service structures. In this paper, the Hawk structure is described in detail, alongside a comprehensive summary of the experimental work undertaken. Following this, key descriptive highlights of the dataset are presented, before a discussion of the research challenges that the data present. Using the dataset, non-linearity in the structure is demonstrated, as well as the sensitivity of the structure to damage of different types. The dataset is highly applicable to many academic enquiries and additional analysis techniques which will enable further advancement of vibration-based engineering techniques.