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Consistent Low-Rank Approximation

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.7Download Report PDF
low-rank approximationonline algorithmsconsistencyrecourse
Abstract:

We introduce and study the problem of consistent low-rank approximation, in which rows of an input matrix A∈Rn×d\mathbf{A}\in\mathbb{R}^{n\times d} arrive sequentially and the goal is to provide a sequence of subspaces that well-approximate the optimal rank-kk approximation to the submatrix A(t)\mathbf{A}^{(t)} that has arrived at each time tt, while minimizing the recourse, i.e., the overall change in the sequence of solutions. We first show that when the goal is to achieve a low-rank cost within an additive ε⋅∣∣A(t)∣∣F2\varepsilon\cdot||\mathbf{A}^{(t)}||_F^2 factor of the optimal cost, roughly O(kεlog⁡(nd))\mathcal{O}\left(\frac{k}{\varepsilon}\log(nd)\right) recourse is feasible. For the more challenging goal of achieving a relative (1+ε)(1+\varepsilon)-multiplicative approximation of the optimal rank-kk cost, we show that a simple upper bound in this setting is k2ε2⋅polylog⁡(nd)\frac{k^2}{\varepsilon^2}\cdot\text{poly}\log(nd) recourse, which we further improve to k3/2ε2⋅polylog⁡(nd)\frac{k^{3/2}}{\varepsilon^2}\cdot\text{poly}\log(nd) for integer-bounded matrices and kε2⋅polylog⁡(nd)\frac{k}{\varepsilon^2}\cdot\text{poly}\log(nd) for data streams with polynomial online condition number. We also show that Ω(kεlog⁡nk)\Omega\left(\frac{k}{\varepsilon}\log\frac{n}{k}\right) recourse is necessary for any algorithm that maintains a multiplicative (1+ε)(1+\varepsilon)-approximation to the optimal low-rank cost, even if the full input is known in advance. Finally, we perform a number of empirical evaluations to complement our theoretical guarantees, demonstrating the efficacy of our algorithms in practice.

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Overview

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
25
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: consistent low-rank approximation in streaming data. The field addresses how to maintain compact, low-dimensional representations of large-scale data that arrive sequentially, while ensuring that successive approximations remain stable and interpretable over time. The taxonomy organizes research into three main branches: Matrix Low-Rank Approximation Methods focus on classical streaming SVD, sketching, and robust factorizations for evolving matrices; Tensor Low-Rank Approximation Methods extend these ideas to higher-order data structures using Tucker and tensor-train decompositions; and Application-Driven Low-Rank Methods tailor approximation strategies to specific domains such as recommendation systems, visual tracking, and online learning. Representative works like Streaming Matrix Approximation[4] and StreamSVD[17] illustrate efficient matrix sketching, while Streaming Tensor Factorization[5] and Tucker Streaming[28] demonstrate how tensor methods handle multi-way streaming data. Across these branches, a recurring theme is the trade-off between computational efficiency, memory footprint, and approximation accuracy. Several active lines of work explore contrasting priorities: some emphasize speed and scalability through randomized sketching (e.g., Randomized Low-Rank Techniques[12], FlashSVD[31]), while others prioritize robustness to outliers and missing entries (e.g., Online Robust NMF[6], Outlier-Robust Streaming Tensor[36]). A smaller cluster of studies addresses consistency and recourse minimization—ensuring that updates to the approximation do not drastically alter previously computed results. Consistent Low-Rank[0] sits squarely within this latter cluster, emphasizing stability across streaming updates in a way that contrasts with purely accuracy-driven approaches like Fixed-Rank Streaming[7] or speed-focused methods such as Frequent Directions Tubal[23]. By prioritizing consistency, Consistent Low-Rank[0] addresses an emerging concern in deployment scenarios where abrupt changes in learned representations can disrupt downstream tasks, distinguishing it from neighbors that optimize traditional error metrics without explicit recourse constraints.

Claimed Contributions

Formalization of consistent low-rank approximation problem

The authors formalize a new problem variant of low-rank approximation in the streaming setting that explicitly accounts for consistency by minimizing recourse (the cumulative change in output solutions over time) while maintaining approximation quality at each time step. This problem captures the practical need for stable feature representations in dynamic data environments.

5 retrieved papers
Algorithm achieving additive error with O(k/ε log(nd)) recourse

The authors develop an algorithm that maintains an additive approximation to the optimal low-rank cost at all times while achieving recourse that is sublinear in the number of rows and linear in the rank parameter k. This demonstrates that consistency is achievable even with strong approximation guarantees.

10 retrieved papers
Algorithm achieving relative (1+ε)-approximation with k^(3/2)/ε² · polylog(ndM) recourse

The authors present an algorithm that achieves multiplicative (1+ε)-approximation to the optimal low-rank cost while improving recourse from quadratic to sub-quadratic in k for integer-bounded matrices. This is accomplished through careful case analysis on singular value contributions and online ridge leverage score sampling.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Formalization of consistent low-rank approximation problem

The authors formalize a new problem variant of low-rank approximation in the streaming setting that explicitly accounts for consistency by minimizing recourse (the cumulative change in output solutions over time) while maintaining approximation quality at each time step. This problem captures the practical need for stable feature representations in dynamic data environments.

Contribution

Algorithm achieving additive error with O(k/ε log(nd)) recourse

The authors develop an algorithm that maintains an additive approximation to the optimal low-rank cost at all times while achieving recourse that is sublinear in the number of rows and linear in the rank parameter k. This demonstrates that consistency is achievable even with strong approximation guarantees.

Contribution

Algorithm achieving relative (1+ε)-approximation with k^(3/2)/ε² · polylog(ndM) recourse

The authors present an algorithm that achieves multiplicative (1+ε)-approximation to the optimal low-rank cost while improving recourse from quadratic to sub-quadratic in k for integer-bounded matrices. This is accomplished through careful case analysis on singular value contributions and online ridge leverage score sampling.

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