SVD-inv Framework for Scalable SVD

• SVD-inv framework is a robust method that uses multiple independent Gaussian random sketches and integrates them via orthonormal subspace averaging for accurate low-rank matrix approximations.
• It leverages constrained optimization on the Stiefel manifold with Kolmogorov–Nagumo-type averaging to optimally combine subspaces while ensuring statistical consistency and geometric fidelity.
• Empirical and theoretical analyses confirm that the framework reduces approximation errors and variance, outperforming standard randomized SVD in large-scale, parallelizable settings.

The SVD-inv framework encompasses a class of methodologies that leverage singular value decomposition (SVD) and inversion principles for robust, scalable, and accurate matrix computations in high-dimensional and large-scale settings. It incorporates advances in random sketching, manifold optimization, statistical analyses, and practical algorithmic design to overcome limitations of traditional SVD approaches, especially in the context of data analytics, scientific computing, and inverse problems.

1. Multiple Random Sketches and Subspace Integration

The SVD-inv framework departs from conventional randomized SVD—which typically generates a single random sketch of the data matrix—by instead constructing $N$ independent Gaussian random sketches using projection matrices $\Omega[i] \in \mathbb{R}^{n \times \ell}$, where $\ell = k + p$ (target rank $k$ and oversampling parameter $p$). For each sketch,

$Y[i] = A \Omega[i].$

An orthonormal basis $Q[i] \in \mathbb{R}^{m \times \ell}$ is extracted via QR or SVD such that $Q[i] Q[i]^T$ approximates the projection onto the dominantly sampled subspace in that sketch. The ensemble $\{Q[1],\ldots,Q[N]\}$ serves as a basis for integrating subspace information. This Monte Carlo–type approach enables parallelism and statistical averaging to reduce stochastic variation inherent to one-time sketching procedures.

2. Optimization over the Stiefel Manifold

Integration of multiple subspaces is formalized as a constrained optimization over the Stiefel manifold $S_{m,\ell}$ (matrices with orthonormal columns). The goal is to find an optimal subspace representative $\overline{Q}$ solving: $$\overline{Q} = \arg\min_{Q \in S_{m,\ell}} \sum_{i=1}^N \| Q[i] Q[i]^T - Q Q^T \|_F^2$$ where $\|\cdot\|_F$ denotes the Frobenius norm. This is equivalent to minimizing the squared error between the mean projection $\overline{P} = \frac{1}{N}\sum Q[i]Q[i]^T$ and the candidate $Q Q^T$, up to orthogonal transformation. Iterative optimization methods are employed, taking into account the manifold constraint $Q^T Q = I_\ell$. This ensures geometric optimality and consistency in representing the aggregate sampled subspace.

3. Kolmogorov–Nagumo-type Averaging on Manifolds

Arithmetic averages are inadequate for points on the Stiefel manifold. The framework therefore implements Kolmogorov–Nagumo-type (KN) averaging, wherein each basis $Q[i]$ is mapped via lifting $\varphi_{Q_c}(Q[i])$ to the tangent space at the current iterate $Q_c$. The arithmetic mean in this Euclidean tangent space is calculated, and the result is retracted back to the manifold using $\varphi_{Q_c}^{-1}$. A specific lifting is given by: $$\varphi_{Q_c}(Q[i]) = (I - Q_c Q_c^T) Q[i] Q[i]^T Q_c$$ and the retraction by: $$Q_+ = Q_c C + \left( \frac{1}{N} \sum \varphi_{Q_c}(Q[i]) \right) C^{-1}$$ where

$$C = \left( \frac{I}{2} + \left[ \frac{I}{4} - \left( \frac{1}{N} \sum \varphi_{Q_c}(Q[i]) \right)^T \left( \frac{1}{N} \sum \varphi_{Q_c}(Q[i]) \right) \right]^{1/2} \right)^{1/2}$$

This iterative KN-average yields the optimal integrated subspace representative, respecting both geometry and statistical properties.

4. Theoretical Guarantees: Population Averaging, Consistency, and Rates of Convergence

A theoretical analysis confirms that, as the number of random sketches $N \to \infty$, the population average projection $E[Q[i] Q[i]^T]$ approaches $U \Lambda U^T$, where $U$ contains true left singular vectors of $A$, and $\Lambda$ is a diagonal matrix of strictly decreasing scalars in $(0,1)$. Thus, population averaging robustly recovers ordering and orientation information intrinsic to $A$'s SVD. Consistency results (Theorem 5.2) further establish that $\overline{Q} \overline{Q}^T$ converges almost surely to the rank-$\ell$ principal subspace, and singular vectors computed from $\overline{Q}$ converge pointwise (up to sign) to the true singular vectors.

5. Statistical Analysis: Strong Law and Central Limit Theorem

Statistical analysis elucidates the probabilistic guarantees of the integrated approach. The Strong Law of Large Numbers ensures that

$$\lim_{N\to\infty} \frac{1}{N}\sum_{i=1}^N Q[i] Q[i]^T = U \Lambda U^T$$

almost surely. The Central Limit Theorem shows that for large $N$,

$$\frac{1}{\sqrt{N}} \sum_{i=1}^N (Q[i] Q[i]^T - U \Lambda U^T)$$

is asymptotically Gaussian. The delta-method yields error rates in singular vector estimation of $O_p(N^{-1/2})$, with pointwise convergence distributions for each computed singular vector.

6. Numerical Validation and Empirical Outcomes

Empirical results confirm the theoretical properties:

• As $N$ increases, cosine similarity $|\hat{u}_j^T u_j|$ between estimated and true singular vectors increases, with inter-run variance decreasing.
• Integrated SVD (iSVD) typically outperforms standard randomized SVD (rSVD) in Frobenius-norm error, for comparable total sampled subspace dimension.
• The parallel nature of random sketching allows for scalable computation and pooled statistical accuracy on large matrices.

7. Practical Integration and Deployment within SVD-inv Framework

The SVD-inv framework is ideally suited for large-scale, parallelizable SVD applications:

• Use many small independent sketches instead of a single large random projection to minimize memory and computational cost.
• Integrate and optimize subspace estimates using manifold KN-averaging, yielding a statistically optimal global subspace.
• Scalability is inherent, as sketch generation is embarrassingly parallel; integration steps are mathematically tractable and efficiently implementable.
• The computed subspace $\overline{Q}$ enables low-rank SVD via smaller projected problems ($\overline{Q}^T A$), ensuring robust subspace estimation for downstream inverse tasks—such as preconditioning, regularization, and solving ill-posed problems.
• Statistical rates guarantee reduced error and variance with increasing $N$, and empirical evidence suggests improved rank-$k$ approximation performance compared to single-sketch methods.

This framework generalizes to other matrix approximation tasks and underpinning algorithms, making KN-averaged iSVD a foundational model for principled, high-precision, large-scale SVD computations.