Stacked SVD or SVD stacked? A Random Matrix Theory perspective on data integration
Authors: Tavor Z. Baharav, Phillip B. Nicol, Rafael A. Irizarry, Rong Ma
Organizations: Eric and Wendy Schmidt Center, Broad Institute, Cambridge, MA, 02142, USA · Department of Data Science, Dana Farber Cancer Institute, Boston, MA, 02115, USA · Department of Biostatistics, Harvard T.H. Chan School of Public Health, Boston, MA, 02115, USA
Modern data analysis increasingly requires identifying shared latent structure across multiple high-dimensional datasets. A commonly used model assumes that the data matrices are noisy observations of low-rank matrices with a shared singular subspace. In this case, two primary methods have emerged for estimating this shared structure, which vary in how they integrate information across datasets. The first approach, termed Stack-SVD, concatenates all the datasets, and then performs a singular value decomposition (SVD). The second approach, termed SVD-Stack, first performs an SVD separately for each dataset, then aggregates the top singular vectors across these datasets, and finally computes a consensus amongst them. While these methods are widely used, they have not been rigorously studied in the proportional asymptotic regime, which is of great practical relevance in today's world of increasing data size and dimensionality. Consequently, it remains unclear when one method should be preferred over another. In this work, we derive exact expressions for the asymptotic performance and phase transitions of these two methods and develop optimal weighting schemes to further improve both methods. Our analysis reveals that while neither method uniformly dominates the other in the unweighted case, optimally weighted Stack-SVD dominates optimally weighted SVD-Stack when the low rank signal is fully shared across the datasets. We then analyze multiple, partially shared components per dataset and demonstrate that SVD-Stack can yield improved performance without requiring estimation of subspace alignment. Finally, we provide practical algorithms for estimating optimal weights from data, offering theoretical guidance for method selection in practical data integration problems. Extensive numerical simulations and semi-synthetic experiments on genomic data corroborate our theoretical findings.
We study null-space estimation from a noisy matrix. For a simple left null space, we first derive an exact compact expression for the error of the smallest left singular vector. We then give an all-order series for the SVD vector and projector, followed by compact and consistently truncated series forms for the fixed-realization empirical risk and conditional population generalization risk. The recursion extends to a multiple-dimensional null space by following the complete invariant subspace. The convergence radius is not inferred from an error plot: it is computed independently from the nearest complex exceptional point that joins a retained eigenvalue branch to its complement. A reduced-nullity experiment shows that moving this spectral boundary can increase the radius, although the improvement is not monotone in the retained nullity. For individually ordered null directions under Gaussian training with τ≥m, we prove that the Wishart splitting matrix W gives a strict second-order empirical ranking. Gaussian averaging equalizes the leading generalization risks at both small and very large noise, while a column-swap theorem proves strict expected generalization ranking for an isotropic signal subspace. For unequal spikes, an exact population-overlap criterion and a simultaneous 99% Monte Carlo confidence certificate explain the observed intermediate ranking. A sixth-order risk correction improves the lower-crossover estimate in the reported experiment. This equal--ranked--equal phenomenon is a finite-sample diagnostic related to spectral mixing, but its tolerance crossings, the exceptional-point radius, and the asymptotic BBP threshold are three distinct quantities.
Given a matrix X, and two ranks r1 and r2, the Hadamard decomposition (HD) looks for two low-rank matrices, X1 of rank r1 and X2 of rank r2, both of the same size as X, such that X≈X1∘X2, where ∘ is the Hadamard (element-wise) product. In most cases, HD is more expressive than standard low-rank approximations such as the truncated singular value decomposition (TSVD), as it can represent higher-rank matrices with the same number of parameters; this is because the rank of X1∘X2 is generically equal to r1r2. In this paper, we first present some theoretical insights for HD, in particular a useful reformulation X≈WH⊤ where W and H have r1r2 columns and belong to certain manifolds. These allow us to develop three new algorithms for computing HD. The first one uses the representation X≈X1∘X2 and relies on the Manopt toolbox. The other two rely on the reformulation X≈WH⊤: one is a block projected gradient method, and the other is a manifold-based gradient descent algorithm that does not require projection onto the feasible set. The last two algorithms are particularly effective for handling large sparse data. We also propose new initializations that allow us to improve the accuracy of the HD. We compare our algorithms and initialization strategies with the TSVD and with the state of the art. Numerical results show that the new methods are efficient and competitive on both synthetic and real data.
Spectral co-clustering is a useful tool for discovering latent structure in word-document matrices, but its reliance on singular value decomposition (SVD) can make standard formulations expensive on high-dimensional data. This paper presents two randomized approximations for normalized spectral co-clustering of bipartite text data when the numbers of document and word clusters may differ. The first method uses randomized SVD through random projection, while the second combines partial SVD with element-wise random sampling. Across real-world and synthetic datasets, both methods reduce runtime relative to the full-SVD baseline, but their behavior depends on matrix sparsity. The random projection method is the more reliable approximation across the tested settings, whereas the sampling-based method is most useful on denser matrices and provides limited benefit on already sparse text data. These results show that randomized approximations for spectral co-clustering should be selected according to the underlying structure of the data.