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Estimating shared subspace with AJIVE: the power and limitation of multiple data matrices

Yuepeng Yang, Cong Ma

TL;DR

The paper analyzes shared-subspace estimation under the JIVE/AJIVE framework when combining $K$ data matrices. It establishes that AJIVE achieves first-order optimality in high-SNR settings, with estimation error decaying as $1/\, oot 2 extof{K}$, and derives minimax lower bounds that confirm this behavior. In低-SNR regimes, AJIVE exhibits a non-diminishing second-order error that does not vanish with increasing $K$, indicating a fundamental limitation. An oracle-aided spectral estimator is introduced to probe the low-SNR barrier, and its analysis shows even with ideal knowledge of the unique components, the error cannot vanish as $K$ grows. Comprehensive numerical experiments corroborate the theory and illuminate the interplay among SNR, the number of matrices, and subspace misalignment, offering guidance for multi-matrix integration in practice.

Abstract

Integrative data analysis often requires disentangling joint and individual variations across multiple datasets, a challenge commonly addressed by the Joint and Individual Variation Explained (JIVE) model. While numerous methods have been developed to estimate the shared subspace under JIVE, the theoretical understanding of their performance remains limited, particularly in the context of multiple matrices and varying degrees of subspace misalignment. This paper bridges this gap by providing a systematic analysis of shared subspace estimation in multi-matrix settings. We focus on the Angle-based Joint and Individual Variation Explained (AJIVE) method, a two-stage spectral approach, and establish new performance guarantees that uncover its strengths and limitations. Specifically, we show that in high signal-to-noise ratio (SNR) regimes, AJIVE's estimation error decreases with the number of matrices, demonstrating the power of multi-matrix integration. Conversely, in low-SNR settings, AJIVE exhibits a non-diminishing error, highlighting fundamental limitations. To complement these results, we derive minimax lower bounds, showing that AJIVE achieves optimal rates in high-SNR regimes. Furthermore, we analyze an oracle-aided spectral estimator to demonstrate that the non-diminishing error in low-SNR scenarios is a fundamental barrier. Extensive numerical experiments corroborate our theoretical findings, providing insights into the interplay between SNR, the number of matrices, and subspace misalignment.

Estimating shared subspace with AJIVE: the power and limitation of multiple data matrices

TL;DR

The paper analyzes shared-subspace estimation under the JIVE/AJIVE framework when combining data matrices. It establishes that AJIVE achieves first-order optimality in high-SNR settings, with estimation error decaying as , and derives minimax lower bounds that confirm this behavior. In低-SNR regimes, AJIVE exhibits a non-diminishing second-order error that does not vanish with increasing , indicating a fundamental limitation. An oracle-aided spectral estimator is introduced to probe the low-SNR barrier, and its analysis shows even with ideal knowledge of the unique components, the error cannot vanish as grows. Comprehensive numerical experiments corroborate the theory and illuminate the interplay among SNR, the number of matrices, and subspace misalignment, offering guidance for multi-matrix integration in practice.

Abstract

Integrative data analysis often requires disentangling joint and individual variations across multiple datasets, a challenge commonly addressed by the Joint and Individual Variation Explained (JIVE) model. While numerous methods have been developed to estimate the shared subspace under JIVE, the theoretical understanding of their performance remains limited, particularly in the context of multiple matrices and varying degrees of subspace misalignment. This paper bridges this gap by providing a systematic analysis of shared subspace estimation in multi-matrix settings. We focus on the Angle-based Joint and Individual Variation Explained (AJIVE) method, a two-stage spectral approach, and establish new performance guarantees that uncover its strengths and limitations. Specifically, we show that in high signal-to-noise ratio (SNR) regimes, AJIVE's estimation error decreases with the number of matrices, demonstrating the power of multi-matrix integration. Conversely, in low-SNR settings, AJIVE exhibits a non-diminishing error, highlighting fundamental limitations. To complement these results, we derive minimax lower bounds, showing that AJIVE achieves optimal rates in high-SNR regimes. Furthermore, we analyze an oracle-aided spectral estimator to demonstrate that the non-diminishing error in low-SNR scenarios is a fundamental barrier. Extensive numerical experiments corroborate our theoretical findings, providing insights into the interplay between SNR, the number of matrices, and subspace misalignment.
Paper Structure (112 sections, 22 theorems, 353 equations, 13 figures, 2 algorithms)

This paper contains 112 sections, 22 theorems, 353 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Related work
  4. JIVE.
  5. Subspace estimation from a single matrix.
  6. Notation.
  7. Background
  8. Observation models
  9. Identifiability of the shared subspace
  10. Faithfulness: $\mathrm{col}(\bm{U}^{\star}) \subseteq \cap_{1 \leq k \leq K} \mathrm{col}(\bm{A}_{k}^{\star})$.
  11. Exhaustiveness: $\cap_{1 \leq k \leq K} \mathrm{col}(\bm{A}_{k}^\star) \subseteq \mathrm{col}(\bm{U}^\star).$
  12. Angle-based joint and individual variation explained (AJIVE)
  13. A digression: failure of stacked SVD.
  14. Performance guarantees of AJIVE
  15. First-order optimality when SNR is high.
  16. ...and 97 more sections

Key Result

Theorem 1

Assume $n\ge C_{1}\log N$ for some sufficiently large constant $C_{1}>0$. Further assume the following conditions hold for some small enough constants $c_1, c_2 > 0$. Then with probability at least $1-O(KN^{-10})$, the AJIVE estimate $\widehat{\bm{U}}$ output by Algorithm alg:spectral satisfies for some constant $C_{2}>0$. Here $r_{\mathrm{avg}} \coloneqq \frac{1}{K}\sum_{k=1}^{K}r_{k}$.

Figures (13)

  • Figure 1: Estimation error $\|\widehat{\bm{U}}\widehat{\bm{U}}^{\top}-\bm{U}^{\star}\bm{U}^{\star\top}\|$ vs $\sqrt{n}$. The parameter is chosen to be $K=100$ and $n$ ranges from 16 to 400.
  • Figure 2: Estimation error $\|\widehat{\bm{U}}\widehat{\bm{U}}^{\top}-\bm{U}^{\star}\bm{U}^{\star\top}\|$ vs $1/\sqrt{K}$. The parameter is chosen to be $n=20$ and $K$ ranges from 25 to 10000.
  • Figure 3: Estimation error when ${\theta}=1/2$. The parameters are chosen to be $d=20$, $r=2$, $\sigma=0.001$, and $\gamma=0.5$. The loading matrices use the random generation scheme. Each point is an average of 100 trials.
  • Figure 4: Estimation error $\|\widehat{\bm{U}}\widehat{\bm{U}}^{\top}-\bm{U}^{\star}\bm{U}^{\star\top}\|$ vs $\sqrt{n}$.
  • Figure 5: Estimation error $\|\widehat{\bm{U}}\widehat{\bm{U}}^{\top}-\bm{U}^{\star}\bm{U}^{\star\top}\|$ vs $1/\sqrt{K}$.
  • ...and 8 more figures

Theorems & Definitions (25)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Theorem 4
  • Lemma 3
  • ...and 15 more