singular-value truncation

This work investigates the impact of matrix concatenation operations on singular value spectra, aiming to bridge a theoretical gap in the structural stability of SVDs under concatenation. Addressing the central question—“How are the singular values of a concatenated matrix determined by those of its constituent submatrices?”—we extend Weyl’s inequality to block-wise concatenation for the first time, establishing a quantitative analytical framework grounded in matrix perturbation theory and norm inequalities. We derive computable upper bounds on singular value deviations and rigorously prove that dominant singular values remain stable when the operator norms of the submatrices are comparable. This result provides theoretical guarantees and principled design guidance for low-rank approximation, robust matrix clustering, and compression algorithms.

This work addresses the issue of uncontrolled reconstruction errors in SVD-based compression of large collections of matrices when heuristic grouping is employed prior to concatenation. To overcome this limitation, the authors propose a theory-driven compressive clustering framework grounded in spectral analysis of horizontally concatenated matrices. They establish, for the first time, a globally provable upper bound on SVD reconstruction error and derive two novel spectral bounds based on a lower bound for singular value growth. Building upon these theoretical guarantees, they design three clustering algorithms with explicit error control, integrated with incremental approximate SVD to efficiently estimate compression error without explicitly forming the full concatenated matrix. The resulting approach achieves a favorable balance among speed, accuracy, and scalability, significantly enhancing the reliability and practicality of SVD compression in applications such as multi-view learning, signal processing, and neural network compression.

This work addresses the spectral-norm robustness of low-rank pseudoinverse approximations to matrix inversion under observational noise. To handle noisy perturbations, we introduce, for the first time, a compact non-asymptotic perturbation bound based on contour integral techniques applied to the non-holomorphic function $f(z) = 1/z$, overcoming the asymptotic and loose nature of classical bounds—achieving up to $sqrt{n}$-fold improvement in theoretical accuracy. The bound explicitly characterizes error dependence on eigenvalue gaps, spectral decay, and alignment between noise and low-curvature directions. By integrating matrix perturbation theory with spectral analysis, our framework enables quantifiable robustness modeling. Experiments demonstrate that the derived bound tightly tracks empirical errors and significantly outperforms existing estimates across diverse synthetic and real-world datasets. This provides a new theoretical guarantee for efficient, spectrum-aware matrix computation in noisy environments.

This paper investigates the statistical stability of randomized singular value decomposition (RSVD) under a signal-plus-noise model, focusing on ℓ₂ and ℓ_{2,∞} errors between approximate and true left singular vectors, as well as entrywise errors after projection. Methodologically, it integrates Gaussian random sketching, power iteration acceleration, and refined perturbation analysis. The contributions are threefold: (i) it establishes the first error bounds explicitly dependent on the signal-to-noise ratio (SNR); (ii) it characterizes a sharp phase-transition threshold for the number of power iterations (g) governing estimation accuracy; and (iii) it proves row-wise and entrywise asymptotic normality of the RSVD estimators. These results provide near-optimal theoretical guarantees for community detection, PCA with missing data, and matrix completion. Crucially, the derived error bounds quantitatively reveal the synergistic interplay between SNR and iteration count—highlighting how increased iterations mitigate low-SNR degradation but exhibit diminishing returns beyond the phase transition.

This paper investigates the detectability and reconstruction of low multilinear-rank signals in high-dimensional spiked tensor models near computational phase transition thresholds. To address key bottlenecks—degraded performance of conventional methods in the critical signal-to-noise ratio (SNR) regime and lack of theoretical convergence guarantees for the Higher-Order Orthogonal Iteration (HOOI) algorithm—the authors systematically apply random matrix theory to analyze the spectral properties of tensor unfoldings. They derive a novel SNR criterion characterizing statistical detectability, precisely quantify the reconstruction error of truncated multilinear singular value decomposition (MLSVD) in the nontrivial regime, and rigorously prove that, in the large-dimensional limit, HOOI converges to the global optimum in a single iteration. These results establish tight theoretical bounds for low-rank tensor estimation and provide provably efficient algorithmic guarantees.

This work addresses the limitations of classical perturbation analyses for CUR decomposition, which rely solely on global noise levels and fail to capture how sampling structures influence local reconstruction errors. By employing a local tangent space expansion, the study establishes, for the first time, a precise connection between the first-order perturbation error of rank-truncated CUR mappings with fixed index sets and the oblique projection operators induced by sampling. This reveals a mechanism whereby invisible perturbations are automatically eliminated under first-order approximation. Leveraging Fréchet derivatives and local Taylor expansions, the authors theoretically derive first- and second-order local convergence rates. Numerical experiments further validate the perturbation-removal effect across distinct subspaces, highlighting CUR’s unique advantage over truncated SVD in structural sensitivity.

This study addresses the numerical instability in extreme learning machine (ELM) training caused by ill-conditioned hidden-layer matrices during pseudoinverse computation. From a spectral perspective, it reveals how perturbations in output weights are amplified by the smallest singular value and quantifies instability via the condition number. The work establishes, for the first time, a systematic theoretical link between ELM numerical stability and the singular value structure of the hidden-layer matrix, proposing a spectral-based stability criterion. Leveraging singular value decomposition (SVD) and iterative hyperpower methods to compute the pseudoinverse, combined with random feature theory, it analyzes how network width influences the condition number. Experiments demonstrate that SVD is the most robust under ill-conditioned scenarios, whereas iterative methods exhibit greater sensitivity to spectral properties, confirming that stability is predominantly governed by the singular value spectrum.

This work addresses the issue of excessively large dictionaries and high computational costs in sparse approximation of high-dimensional functions, which arises from overly conservative $L^\infty$ truncation error bounds. By introducing i.i.d. sampling randomness into the truncation error analysis for the first time, the authors derive a sharp discrete $L^2$ error bound that accurately captures the decay behavior of the continuous $L^2$ norm. This approach substantially reduces the size of the truncated index set and yields improved sparse recovery guarantees in both weighted Wiener spaces and anisotropic Sobolev spaces compared to existing results. Additionally, the measurement conditions under bounded Riesz systems are refined to exhibit weaker dependence on the Riesz constants and to possess scale invariance. The theoretical framework integrates tools from compressed sensing, random sampling, function space analysis, and Riesz system theory.

This study addresses a fundamental challenge in system identification: distinguishing spurious eigenvalues arising from limited data from those genuinely reflecting the underlying system dynamics. To this end, the paper introduces—for the first time—the probabilistic sampling pseudospectrum \( P(\lambda) \) and its computationally efficient estimator \( \hat{P}(\lambda) \). By leveraging resampling and statistical inference, this framework quantifies the uncertainty of eigenvalues across the complex plane. The proposed approach provides a general and rigorous statistical criterion for data-driven methods such as Dynamic Mode Decomposition and subspace identification, substantially enhancing the reliability of identifying true dynamical modes from noisy, finite-length observations.

This study investigates whether stochastic rounding (SR) retains its regularizing effect in matrices with constant aspect ratios and examines its impact on the singular value spectrum. By integrating singular value analysis, a stochastic rounding quantization model, and spectral theory, the work demonstrates for the first time that SR not only enhances the smallest singular value but also collectively elevates multiple singular values in the tail of the spectrum. This finding reveals that the regularizing influence of SR extends beyond extreme aspect ratio regimes. The results establish SR as a universal spectral regularization mechanism, thereby broadening its theoretical foundation and application potential in numerical computation and low-precision machine learning.