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Hyperspectral Anomaly Detection via Group Sparse Low-Rank Tensor Factorization With Automatic Anomaly Grouping

Quan Yu, Yu-Hong Dai, Xiongjun Zhang

arXiv:2609.08121v1cs.CVmath.OC

TL;DR

Existing tensor-based anomaly detection faces costly low-rank modeling and limited flexibility when anomaly groups are fixed at the pixel level. GSAA uses group-sparse tensor factors and data-learned anomaly grouping, with GSAA-SS fusing spectral- and spatial-domain maps. Across five real hyperspectral scenes, the authors report accurate detection and competitive computational efficiency relative to representative methods.

Abstract

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Low-rank tensor modeling has become an effective tool for hyperspectral anomaly detection. However, existing methods still suffer from high computational cost and limited flexibility in characterizing spatially structured anomalies. To address these issues, this paper proposes a hyperspectral anomaly detection method based on group sparse low-rank tensor factorization with automatic anomaly grouping (GSAA). Specifically, the low tubal rank background is characterized by imposing group sparsity on tensor factors, which provides an efficient alternative to direct tensor rank regularization. For anomaly modeling, a latent grouping map is introduced to build an automatic anomaly grouping penalty, allowing anomaly groups to be adaptively inferred from the data rather than predefined at the pixel level. To further exploit complementary spectral and spatial information, GSAA is applied in both domains, and the resulting detection maps are fused to form a spectral--spatial version of GSAA, termed GSAA-SS. An efficient linearized alternating direction method of multipliers algorithm with convergence guarantee is developed to solve the resulting model. Experimental results on five real hyperspectral datasets demonstrate that the proposed method achieves superior detection performance and competitive computational efficiency compared with several state-of-the-art methods.

I. Introduction

Hyperspectral anomaly detection must separate weak, spatially structured anomalies from complex backgrounds, while existing tensor methods face costly optimization and inflexible pixel-level anomaly groups. The paper introduces GSAA and its spectral–spatial extension, GSAA-SS, and provides tensor preliminaries for the model.

I. Introduction

  • Hyperspectral backgrounds can be modeled as low-rank structures and anomalies as sparse deviations; tensor formulations preserve the data’s spectral–spatial structure.The paper situates tensor methods alongside matrix-based low-rank decomposition and representation approaches.Source: §I · pp. 1–2
  • Tensor low-rank methods face repeated-SVD costs, while pixel-level anomaly groups cannot flexibly represent regions with unknown shapes and extents.The background may be complex and heterogeneous, and anomalies may be weak and spatially structured.Source: §I · pp. 1–2
  • GSAA uses group sparsity on tensor factors for low-tubal-rank background modeling and a latent grouping map for adaptive anomaly modeling; GSAA-SS fuses spectral- and spatial-domain detection maps.The paper also develops an LADMM solver with convergence analysis.Source: §I · p. 2

II. Preliminaries

  • The preliminaries establish tensor notation and finite-difference operators, alongside t-SVD and tubal-rank concepts used by the model.Tubal rank is defined as the number of nonzero singular tubes in the t-SVD.Source: §II · pp. 2–3

III. A GSAA-SS model for HAD

The paper frames hyperspectral anomaly detection as separating a structured background from sparse anomalies. Its GSAA framework targets the computational cost of low-rank regularization and the rigidity of predefined pixel-level anomaly groups.

A. General Tensor Based Detection Framework

  • The observed hyperspectral tensor is modeled as a low-rank background plus a sparse anomaly component, reflecting background spectral–spatial correlation.The practical model depends on how both components are regularized.Source: A. General Tensor Based Detection… · p. 3
  • Direct rank minimization is NP-hard, and tractable tensor surrogates typically require repeated SVDs that burden large-scale hyperspectral images.The cited approaches include weighted nuclear norms, ε-shrinkage tensor nuclear norms, and nonconvex penalties.Source: A. General Tensor Based Detection… · p. 3
  • Mixed ℓ2–ℓ1 anomaly penalties preserve spectral grouping but fix each pixel as a group, limiting representation of spatial regions with unknown shapes and extents.Such regions are described as typical anomalous targets in HAD.Source: A. General Tensor Based Detection… · p. 3

III. A GSAA-SS model for HAD

  • GSAA addresses these issues with group-sparse factorization for efficient background modeling and data-driven learning of anomaly grouping.The broader model applies the components in spectral and spatial domains and fuses the resulting maps.Source: A. General Tensor Based Detection… · p. 3, §III · p. 3

B. Group Sparse Low-Rank Tensor Factorization

GSAA models low-rank backgrounds through group-sparse tensor factors and adapts anomaly groups through a latent spatial grouping map. The map’s structure controls grouping flexibility while connecting the penalty to both coarse and fine spatial partitions.

B. Group Sparse Low-Rank Tensor Factorization

  • Group sparsity on tensor factors encourages low-rank structure because active factor groups determine the reconstructed background’s effective tubal rank.The factor penalties are also connected to tensor Schatten-p regularization.Source: B. Group Sparse Low-Rank Tensor… · p. 3
  • Factor regularization avoids direct tensor rank regularization and repeated SVDs; the auxiliary dimension adapts and approaches the usually smaller tubal rank in practice.This is presented as a computational advantage for large hyperspectral images.Source: B. Group Sparse Low-Rank Tensor… · p. 3

C. Automatic Anomaly Grouping

  • Unlike structured sparsity models that assume known groups or fixed pixel-level partitions, AAG learns the spatial partition while keeping the spectral dimension within each group.It is designed for anomalous regions whose spatial extent is not known in advance.Source: C. Automatic Anomaly Grouping · p. 4
  • A latent map Θ assigns shared values to pixels in each group, with connected constant regions encoding the partition and its boundaries appearing as nonzero finite differences.Parameters α1 and α2 constrain vertical and horizontal changes, respectively, controlling grouping flexibility.Source: C. Automatic Anomaly Grouping · p. 4
  • The AAG penalty bridges the coarsest and finest spatial partitions, with the limiting finest-partition case reducing to the conventional ℓ2,1 penalty.At α1 = α2 = 0, the theorem gives Ψ(0,0)(E) = √n1n2∥E∥.Source: C. Automatic Anomaly Grouping · p. 4
  • AAG estimates both the anomaly tensor and grouping map, whose piecewise-constant structure provides a complementary spatially coherent anomaly response.This contrasts with detectors based only on the anomaly tensor’s magnitude map.Source: C. Automatic Anomaly Grouping · p. 4

D. Proposed GSAA-SS Model

The unified GSAA model combines group-sparse low-rank background factorization with adaptive anomaly grouping. GSAA is applied in spectral and spatial domains, and the resulting maps are fused for GSAA-SS.

D. Proposed GSAA-SS Model

  • The unified model uses group-sparse tensor factors for the low-rank background and a latent grouping map to adaptively structure the anomaly component.The factors X and Y model the background, while E and Θ represent anomalies and their learned grouping.Source: D. Proposed GSAA-SS Model · p. 4
  • In the spectral branch, matrix GSAA on the mode-3 unfolding identifies pixels with large residuals from the common background spectral subspace as spectral anomalies.The mode-3 unfolding is the spectral-domain input to this branch.Source: D. Proposed GSAA-SS Model · pp. 4–5
  • In the spatial branch, PCA reduces spectral dimension before t-product GSAA models the spatial background, and the two branches’ AAG maps are fused element-wise.The PCA tensor retains the main spatial structure; the branch maps are fused by element-wise multiplication.Source: D. Proposed GSAA-SS Model · pp. 4–5

IV. Optimization by Linearized Alternating Direction

The optimization section develops LADMM to solve the GSAA model. Its updates linearize the factorization subproblem and use closed-form or proximal steps for the resulting subproblems.

IV. Optimization by Linearized Alternating Direction

  • LADMM solves the GSAA formulation after converting its constrained model into a penalized one to account for Gaussian observation noise.The method introduces auxiliary variables to rewrite the problem and alternates variable updates.Source: §IV · p. 5, A. LADMM Algorithm · p. 5

A. LADMM Algorithm

  • The X and Y updates linearize the factorization term at the current iterate, while the resulting subproblems are handled with closed-form or proximal solutions.The algorithm also updates anomaly variables, auxiliary variables, and projections in alternating steps.Source: A. LADMM Algorithm · pp. 5–6
  • The Υ update is solved in the Fourier domain after diagonalizing the block-circulant operators with a two-dimensional FFT.The convolution theorem yields a closed-form solution.Source: A. LADMM Algorithm · p. 6

B. Computation Complexity

The per-iteration cost is dominated by t-products and FFT operations, with separate costs for factor, auxiliary, and projection updates.

B. Computation Complexity

  • The overall per-iteration complexity is O((d(n1 + n2) + n1n2)n3 log n3 + dn1n2n3 + n1n2 log(n1n2)).The dominant costs come from t-products and FFT operations.Source: B. Computation Complexity · p. 7

C. Convergence Analysis

The convergence analysis establishes subsequential convergence of Algorithm 4.1 for problem (6), with accumulation points satisfying the stated KKT-point conclusion.

C. Convergence Analysis

  • The analysis establishes subsequential convergence of Algorithm 4.1 for problem (6), with its accumulation points satisfying the stated KKT-point conclusion.The result is stated for the sequence produced by Algorithm 4.1.Source: C. Convergence Analysis · p. 7

V. Numerical Experiments

The experiments evaluate GSAA-SS on five real hyperspectral scenes against ten representative detectors, using AUC as the primary accuracy metric.

V. Numerical Experiments

  • The evaluation uses five real scenes and compares GSAA-SS with ten detectors spanning statistical, matrix and tensor low-rank, and deep-learning approaches.The comparison includes RX, RPCA, LRASR, Turbo-GoDec, PTA, TPCA, PCA-TLRSR, RGAE, GAED, and LCRS.Source: §V · p. 7
  • All methods use raw hyperspectral data without additional preprocessing and were implemented in MATLAB R2022a on an Intel Core i5-12500H computer with 16 GB RAM.The CPU runs at 2.50 GHz.Source: §V · p. 7
  • AUC of the ROC curve is the primary metric; the ROC curve plots probability of detection against false-alarm rate, and larger AUC indicates better accuracy.The experiments also report running time in seconds.Source: §V · p. 7

A. Dataset Description

The study tests five scenes drawn from two public sources, covering airport, beach, urban, and San Diego imagery.

A. Dataset Description

  • Four selected Airport-Beach-Urban scenes comprise two airport scenes, one beach scene, and one urban scene.Images are either 100 × 100 or 150 × 150 pixels, with approximately 100 or 200 spectral bands depending on the sensor.Source: A. Dataset Description · p. 7
  • The five-scene set also includes San Diego; Fig. 2 pairs pseudo-color images with ground-truth maps for Airport1, Airport2, Beach, Urban, and San Diego.The four Airport-Beach-Urban scenes appear in Fig. 2(a)–(d), and San Diego appears in Fig. 2(e).Source: Figure 2 · p. 7, A. Dataset Description · p. 7

B. Detection Performance

GSAA-SS leads the five-scene detection comparison in AUC, with favorable runtime and strong qualitative separability. Parameter analyses report stable ranges and scene-dependent grouping-map settings.

B. Detection Performance

  • GSAA-SS produces compact, prominent responses with fewer background artifacts on Urban and better target-background separation on Airport2; ROC advantages appear across nearly all Airport2 false-alarm rates and at stated thresholds elsewhere.On Beach and San Diego, its ROC advantage is particularly pronounced when PF > 10^-3; on Urban when PF > 10^-2; and on Airport1 when PF > 10^-1.Source: B. Detection Performance · p. 8
  • GSAA-SS achieves the highest AUC on all five datasets, exceeding second-best LCRS, GAED, Turbo-GoDec, PCA-TLRSR, and PCA-TLRSR by 3.17%, 2.41%, 1.71%, 1.02%, and 0.09%, respectively.Among methods other than RX, GSAA-SS has the lowest average runtime; it is about twice as fast as PCA-TLRSR and at least twenty-five times faster than RGAE and GAED on average.Source: B. Detection Performance · p. 8
  • GSAA-SS shows clearer anomaly-background separation than competing methods on Airport1, Airport2, and Beach, with comparable separation to selected methods on Urban and San Diego.On San Diego, PCA-TLRSR and GSAA-SS have similar separability, while PCA-TLRSR is less stable on Airport1, Airport2, and Beach.Source: B. Detection Performance · p. 8

1) Parameter analysis:

  • AUC is stable for γ ∈ {5 × 10^-2, 10^-1} and p ∈ [0.4, 0.7]; the following experiments set γ = 5 × 10^-2 and p = 0.6.The tested γ values range from 10^-2 to 1, and p ranges from 0.2 to 1 in increments of 0.1.Source: 1) Parameter analysis · p. 8
  • AUC varies by less than 0.5% across datasets except Beach, where the largest variation is about 2%; α1 tends to increase with more anomaly blocks.The experiments fix α2 = 10 and set α1 = 1 for scenes with few anomaly blocks or 50 for scenes with more blocks; α1 and α2 control vertical and horizontal changes in the learned grouping map.Source: 1) Parameter analysis · p. 8

2) Adaptive dimensional reduction during iterations:

The model rapidly reduces its factorization dimension, then evaluates automatic anomaly grouping and spectral–spatial fusion. The results report gains from coherent grouping and fusion, while the conclusion emphasizes efficient factorization and competitive detection.

2) Adaptive dimensional reduction during iterations:

  • The factorization dimension d collapses from the full tensor size to a small stable value within a few iterations, reducing subsequent optimization cost.After the early drop, d remains fixed for the rest of optimization.Source: 2) Adaptive dimensional reduction… · p. 8

3) Effects of automatic anomaly grouping and spectral– spatial fusion:

  • The spatial and spectral anomaly magnitude maps and grouping maps are fused by element-wise multiplication.The fused maps are Efus = Espa ⊙ Espe and Θfus = Θspa ⊙ Θspe.Source: 3) Effects of automatic anomaly… · p. 8
  • Finite α1 and α2 improve anomaly modeling over the limiting conventional ∥·∥2,1 penalty, and Θ maps outperform E maps in both domains.The learned grouping map provides a more spatially coherent anomaly response than the anomaly magnitude map alone.Source: 3) Effects of automatic anomaly… · p. 10
  • Θfus achieves the highest AUC on four of five datasets and is only 0.0029 below the best single-domain result on Airport1.Fused maps generally outperform their single-domain counterparts, with minor exceptions.Source: 3) Effects of automatic anomaly… · p. 10, Table II · p. 10

VI. Conclusions

  • GSAA uses group sparsity on tensor factors for efficient background modeling and a latent grouping map to learn spatially coherent anomalies from data.GSAA-SS applies the method across spectral and spatial domains; the paper reports an efficient LADMM solver with theoretical convergence analysis.Source: §VI · pp. 10–11

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