In this paper, we develop novel perturbation bounds for the high-order orthogonal iteration (HOOI) [DLDMV00b]. Under mild regularity conditions, we establish blockwise tensor perturbation bounds for HOOI with guarantees for both tensor reconstruction in Hilbert-Schmidt norm $\|\widehat{\bcT} - \bcT \|_{\tHS}$ and mode-$k$ singular subspace estimation in Schatten-$q$ norm $\| \sin \Theta (\widehat{\U}_k, \U_k) \|_q$ for any $q \geq 1$. We show the upper bounds of mode-$k$ singular subspace estimation are unilateral and converge linearly to a quantity characterized by blockwise errors of the perturbation and signal strength. For the tensor reconstruction error bound, we express the bound through a simple quantity $\xi$, which depends only on perturbation and the multilinear rank of the underlying signal. Rate matching deterministic lower bound for tensor reconstruction, which demonstrates the optimality of HOOI, is also provided. Furthermore, we prove that one-step HOOI (i.e., HOOI with only a single iteration) is also optimal in terms of tensor reconstruction and can be used to lower the computational cost. The perturbation results are also extended to the case that only partial modes of $\bcT$ have low-rank structure. We support our theoretical results by extensive numerical studies. Finally, we apply the novel perturbation bounds of HOOI on two applications, tensor denoising and tensor co-clustering, from machine learning and statistics, which demonstrates the superiority of the new perturbation results.

翻译：在本文中, 我们开发了高阶或远端迭代( HOOI ) [DLDMV00b] 的新扰动界限。 在轻微的常规条件下, 我们为 HOOI 建立块状的振动界限, 保证Hilbert- Schmidt 规范的快速重建 $\\ 百拉特=bcT} -\ bcT\\ tHS} -\\ k$ 和 模式- k$ 标准 的单方位子空间估计值 。 在全局 或高端变代( 全局 $UQk,\ U_ k) 。 在任何 $\ geg 1 的常规条件下, 我们为 HOI 设定了块状的超振动调调调调 。 我们通过一个简单数量的 $xxxxion 来表达 。 仅取决于 roturb 和 基本信号的多线级值 $x@c, rob rodeal rodeal rudeal rodudeal rodudeal rodudeal rodudeal rodudeal rodudeal ral lax. lax hal lax lax laut lax lax lax lax laut laut laut laut laut laut laut unt.