Beyond the Flat-Spike: Adaptive Sparse CCA for Decaying and Unbalanced Signals
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908975043182592 |
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| author | Xu, Mengchu Wang, Jian Eldar, Yonina C. |
| author_facet | Xu, Mengchu Wang, Jian Eldar, Yonina C. |
| contents | Sparse Canonical Correlation Analysis (SCCA) is a fundamental statistical tool for identifying linear relationships in high-dimensional, multi-view data. While minimax theory establishes an optimal sample complexity scaling additively with the sparsity levels of the canonical vectors, computationally efficient algorithms typically suffer from a suboptimal multiplicative dependence. This computational-statistical gap is intrinsically tied to worst-case ``flat'' signal assumptions. In practice, however, multi-view signals frequently exhibit structured energy concentration, such as a power-law decay. To exploit this structural concentration and bypass the worst-case bottleneck, we propose Bilateral Spectral Energy Pursuit (Bi-SEP). Operating directly on the cross-covariance matrix, Bi-SEP is a stagewise adaptive algorithm that utilizes a proxy refinement step to dynamically track and capture cross-view signal energy. Theoretically, we establish a profile-adaptive sample complexity bound governed by the coupled energy profiles of the two views. Notably, under power-law decay models, we reveal a synergistic phase transition: the optimal linear sample complexity is attainable provided that the aggregate decay rate of the two views is sufficiently large. This result demonstrates that a highly concentrated signal in one view allows the model to accommodate a completely flat signal in its partner. Numerical experiments validate our theoretical findings, illustrating the advantages of Bi-SEP in structured, non-flat signal regimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16435 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Beyond the Flat-Spike: Adaptive Sparse CCA for Decaying and Unbalanced Signals Xu, Mengchu Wang, Jian Eldar, Yonina C. Signal Processing Information Theory Statistics Theory Sparse Canonical Correlation Analysis (SCCA) is a fundamental statistical tool for identifying linear relationships in high-dimensional, multi-view data. While minimax theory establishes an optimal sample complexity scaling additively with the sparsity levels of the canonical vectors, computationally efficient algorithms typically suffer from a suboptimal multiplicative dependence. This computational-statistical gap is intrinsically tied to worst-case ``flat'' signal assumptions. In practice, however, multi-view signals frequently exhibit structured energy concentration, such as a power-law decay. To exploit this structural concentration and bypass the worst-case bottleneck, we propose Bilateral Spectral Energy Pursuit (Bi-SEP). Operating directly on the cross-covariance matrix, Bi-SEP is a stagewise adaptive algorithm that utilizes a proxy refinement step to dynamically track and capture cross-view signal energy. Theoretically, we establish a profile-adaptive sample complexity bound governed by the coupled energy profiles of the two views. Notably, under power-law decay models, we reveal a synergistic phase transition: the optimal linear sample complexity is attainable provided that the aggregate decay rate of the two views is sufficiently large. This result demonstrates that a highly concentrated signal in one view allows the model to accommodate a completely flat signal in its partner. Numerical experiments validate our theoretical findings, illustrating the advantages of Bi-SEP in structured, non-flat signal regimes. |
| title | Beyond the Flat-Spike: Adaptive Sparse CCA for Decaying and Unbalanced Signals |
| topic | Signal Processing Information Theory Statistics Theory |
| url | https://arxiv.org/abs/2604.16435 |