<p>In this paper, we explore a specific optimization problem that combines a differentiable nonconvex function with a nondifferentiable function for multi-block variables, which is particularly relevant to tackle the multilinear rank-(<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation>,<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation>,1) block-term tensor decomposition model with a regularization term. While existing algorithms often suffer from high per-iteration complexity and slow convergence, this paper employs a unified multi-step inertial accelerated doubly stochastic gradient descent method tailored for structured rank-<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\left( L_r, L_r, 1\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>L</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>L</mi> <mi>r</mi> </msub> <mo>,</mo> <mn>1</mn> </mfenced> </math></EquationSource> </InlineEquation> tensor decomposition, referred to as Midas-LL1. We also introduce an extended multi-step variance-reduced stochastic estimator framework. Our analysis under this new framework demonstrates the subsequential and sequential convergence of the proposed algorithm under certain conditions and illustrates the sublinear convergence rate of the subsequence, showing that the Midas-LL1 algorithm requires at most <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {O}(\varepsilon ^{-2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> iterations in expectation to reach an <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-stationary point. The proposed algorithm is evaluated on several datasets, and the results indicate that Midas-LL1 outperforms existing state-of-the-art algorithms in terms of both computational speed and solution quality.</p>

Multi-step Inertial Accelerated Doubly Stochastic Gradient Methods for Block Term Tensor Decomposition

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发表日期：

2026年3月6日

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Zehui Liu,
Qingsong Wang,
Chunfeng Cui

摘要

In this paper, we explore a specific optimization problem that combines a differentiable nonconvex function with a nondifferentiable function for multi-block variables, which is particularly relevant to tackle the multilinear rank-( $L_r$ $L_{r}$ , $L_r$ $L_{r}$ ,1) block-term tensor decomposition model with a regularization term. While existing algorithms often suffer from high per-iteration complexity and slow convergence, this paper employs a unified multi-step inertial accelerated doubly stochastic gradient descent method tailored for structured rank- $\left( L_r, L_r, 1\right) $ $(L_{r}, L_{r}, 1)$ tensor decomposition, referred to as Midas-LL1. We also introduce an extended multi-step variance-reduced stochastic estimator framework. Our analysis under this new framework demonstrates the subsequential and sequential convergence of the proposed algorithm under certain conditions and illustrates the sublinear convergence rate of the subsequence, showing that the Midas-LL1 algorithm requires at most $\mathcal {O}(\varepsilon ^{-2})$ $O (ε^{- 2})$ iterations in expectation to reach an $\varepsilon $ $ε$ -stationary point. The proposed algorithm is evaluated on several datasets, and the results indicate that Midas-LL1 outperforms existing state-of-the-art algorithms in terms of both computational speed and solution quality.