Robust Schatten-p Norm Based Approach for Tensor Completion

Gao, Shangqi; Fan, Qing‐Hua

doi:10.1007/s10915-019-01108-9

Cited by 16 publications

(8 citation statements)

References 30 publications

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“…Definition 2. (Tensor Schatten p-norm (Gao and Fan, 2020)). For a N -th mode tensor X ∈ R I1×I2×•••×I N , its Schatten p-norm is defined as follow :…”

Section: Model Formulationmentioning

confidence: 99%

Truncated tensor Schatten p-norm based approach for spatiotemporal traffic data imputation with complicated missing patterns

Nie,

Qin,

Sun

2022

Preprint

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Rapid advances in sensor, wireless communication, cloud computing and data science have brought unprecedented amount of data to assist transportation engineers and researchers in making better decisions. However, traffic data in reality often has corrupted or incomplete values due to detector and communication malfunctions. Data imputation is thus required to ensure the effectiveness of downstream data-driven applications. To this end, numerous tensor-based methods treating the imputation problem as the low-rank tensor completion (LRTC) have been attempted in previous works. To tackle rank minimization, which is at the core of the LRTC, most of aforementioned methods utilize the tensor nuclear norm (NN) as a convex surrogate for the minimization. However, the over-relaxation issue in NN refrains it from desirable performance in practice. In this paper, we define an innovative nonconvex truncated Schatten p-norm for tensors (TSpN) to approximate tensor rank and impute missing spatiotemporal traffic data under the LRTC framework. We model traffic data into a thirdorder tensor structure of time intervals×locations (sensors)×days and introduce four complicated missing patterns, including random missing and three fiber-like missing cases according to the tensor mode-n fibers. Despite nonconvexity of the objective function in our model, we derive the global optimal solutions by integrating the alternating direction method of multipliers (ADMM) with generalized soft-thresholding (GST). In addition, we design a truncation rate decay strategy to deal with varying missing rate scenarios. Comprehensive experiments are finally conducted using four different types of real-world spatiotemporal traffic datasets, which demonstrate that the proposed LRTC-TSpN method performs well under various missing cases even with high rates of data loss, meanwhile outperforming other state-of-the-art tensor-based imputation models in almost all scenarios. In general, the performance of the proposed method achieves up to a 52% improvement in mean absolute error (MAE) compared to baseline methods even when the missing ratio is as high as 90%. Moreover, both theoretical and empirical evidences indicate that our method has robust and efficient convergence performances.

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“…Definition 2. (Tensor Schatten p-norm (Gao and Fan, 2020)). For a N -th mode tensor X ∈ R I1×I2×•••×I N , its Schatten p-norm is defined as follow :…”

Section: Model Formulationmentioning

confidence: 99%

Truncated tensor Schatten p-norm based approach for spatiotemporal traffic data imputation with complicated missing patterns

Nie,

Qin,

Sun

2022

Preprint

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show abstract

“…where c ¼ jjBjj Sq jjAjj Sp and J p ðÁÞ and J q ðÁÞ are the duality mappings for the ' p and ' q norms, respectively (see (5)). If p ¼ 1, then a necessary condition for the saturation of the H€ older inequality is that rankðAÞ r 1 rankðBÞ, (…”

Section: Duality Mapping In Schatten Spacesmentioning

confidence: 99%

“…In linear algebra and matrix analysis, Schatten norms are a family of spectral matrix norms that are defined via the singular-value decomposition [1]. They have appeared in many applications such as image reconstruction [2,3], image denoising [4], and tensor decomposition [5], to name a few.…”

Section: Introductionmentioning

confidence: 99%

Duality Mapping for Schatten Matrix Norms

Aziznejad

Unser

2021

Numerical Functional Analysis and Optimization

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In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the H€ older inequality for Schatten norms. We prove in our main result that, for p 2 ð1, 1Þ, the duality mapping over the space of real-valued matrices with Schatten-p norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case p ¼ 1, the mapping is setvalued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable single-valued function for which we also provide a closed-form expression.

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“…where c = B Sq A Sp and J p (•) and J q (•) are the duality mappings for the ℓ p and ℓ q norms, respectively (see (5)).…”

Section: Duality Mapping In Schatten Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Duality Mapping for Schatten Matrix Norms

Aziznejad¹,

Unser²

2020

Preprint

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In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Hölder inequality for Schatten norms. We prove in our main result that, for p ∈ (1, ∞), the duality mapping over the space of real-valued matrices with Schatten-p norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case p = 1, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borelmeasurable single-valued function for which we also provide a closed-form expression.

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Robust Schatten-p Norm Based Approach for Tensor Completion

Cited by 16 publications

References 30 publications

Truncated tensor Schatten p-norm based approach for spatiotemporal traffic data imputation with complicated missing patterns

Truncated tensor Schatten p-norm based approach for spatiotemporal traffic data imputation with complicated missing patterns

Duality Mapping for Schatten Matrix Norms

Duality Mapping for Schatten Matrix Norms

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