Tensor Linear Regression: Degeneracy and Solution

Zhou, Yicheng; Wong, Raymond K. W.; He, Kejun

doi:10.1109/access.2021.3049494

Cited by 6 publications

(12 citation statements)

References 35 publications

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“…Compared with the canonical TLR, Suzuki (2015) shows that the convergence rate of a Bayesian tensor estimator can be as fast as

(}{, R \sum_{d} p_{d} normallog ()(, n \sum_{d} p_{d})) false/ n

, and Zhou et al (2021) shows that the convergence rate of the least squares estimator is upper bounded by

(}{, R \sum_{d} p_{d} + R^{D}) false/ n

. These two upper bounds are very close to

\sum_{d} R p_{d} false/ n

, the minimax lower bound of the canonical TLR (Suzuki, 2015).…”

Section: Theorymentioning

confidence: 98%

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An improved tensor regression model via location smoothing

Zhou

2021

Stat

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Many applications of regression study the predictors with complex forms such as tensors. Besides low dimensional assumption, the effects of predictors with a tensor structure typically show clustered or smooth phenomena in the sense that adjacent elements have a similar effect on the response. To simultaneously incorporate the low‐rank and smoothness in tensor regression, we generalize the CANDECOMP/PARAFAC (CP) decomposition to a smooth version and propose a novel regression model based on the smoothed decomposition. The asymptotic theory of the proposed method is studied, which shows a faster rate of convergence than the one without incorporating the smoothness. The experiments on both synthetic and real data confirm that the proposed method has advantages over the alternative especially when the sample size is small.

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“…Compared with the canonical TLR, Suzuki (2015) shows that the convergence rate of a Bayesian tensor estimator can be as fast as

(}{, R \sum_{d} p_{d} normallog ()(, n \sum_{d} p_{d})) false/ n

, and Zhou et al (2021) shows that the convergence rate of the least squares estimator is upper bounded by

(}{, R \sum_{d} p_{d} + R^{D}) false/ n

. These two upper bounds are very close to

\sum_{d} R p_{d} false/ n

, the minimax lower bound of the canonical TLR (Suzuki, 2015).…”

Section: Theorymentioning

confidence: 98%

“…Thus, (9) may not have a solution. To avoid such an ill-posed situation and improve the numerical stability, we propose to add a penalty term (Zhou et al, 2021) in the objective function ( 9) aiming to solve arg min…”

Section: Least Squares Criterionmentioning

confidence: 99%

An improved tensor regression model via location smoothing

Zhou

2021

Stat

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“…The structure (e.g., geometry and topology) of these higher dimensional manifolds carries more information compared to the original time-series corresponding to cross-sectional kimesurface foliation curves. Modelbased methods, such as tensor-based linear modeling [7], and model-free artificial intelligence (AI) approaches, such as unsupervised clustering and classification [3,8,9], can then be designed and applied to the kimesurface representations of the time-varying processes.…”

Section: Introductionmentioning

confidence: 99%

“…where N is the number of observations [7]. One example of TLM inference is the prediction of several facial attributes from a set of recorded images [24].…”

Section: Introductionmentioning

confidence: 99%

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Kimesurface representation and tensor linear modeling of longitudinal data

Zhang

Liu

et al. 2022

Neural Comput & Applic

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Many modern techniques for analyzing time-varying longitudinal data rely on parametric models to interrogate the timecourses of univariate or multivariate processes. Typical analytic objectives include utilizing retrospective observations to model current trends, predict prospective trajectories, derive categorical traits, or characterize various relations. Among the many mathematical, statistical, and computational strategies for analyzing longitudinal data, tensor-based linear modeling offers a unique algebraic approach that encodes different characterizations of the observed measurements in terms of state indices. This paper introduces a new method of representing, modeling, and analyzing repeated-measurement longitudinal data using a generalization of event order from the positive reals to the complex plane. Using complex time (kime), we transform classical time-varying signals as 2D manifolds called kimesurfaces. This kime characterization extends the classical protocols for analyzing time-series data and offers unique opportunities to design novel inference, prediction, classification, and regression techniques based on the corresponding kimesurface manifolds. We define complex time and illustrate alternative time-series to kimesurface transformations. Using the Laplace transform and its inverse, we demonstrate the bijective mapping between time-series and kimesurfaces. A proposed general tensor regression based linear model is validated using functional Magnetic Resonance Imaging data. This kimesurface representation method can be used with a wide range of machine learning algorithms, artificial intelligence tools, analytical approaches, and inferential techniques to interrogate multivariate, complex-domain, and complex-range longitudinal processes.

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