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Scalable Second-order Riemannian Optimization for KK-means Clustering

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
K-means clusteringmanifold optimizationNewton's methodnonconvex
Abstract:

Clustering is a hard discrete optimization problem. Nonconvex approaches such as low-rank semidefinite programming (SDP) have recently demonstrated promising statistical and local algorithmic guarantees for cluster recovery. Due to the combinatorial structure of the KK-means clustering problem, current relaxation algorithms struggle to balance their constraint feasibility and objective optimality, presenting tremendous challenges in computing the second-order critical points with rigorous guarantees. In this paper, we provide a new formulation of the KK-means problem as a smooth unconstrained optimization over a submanifold and characterize its Riemannian structures to allow it to be solved using a second-order cubic-regularized Riemannian Newton algorithm. By factorizing the KK-means manifold into a product manifold, we show how each Newton subproblem can be solved in linear time. Our numerical experiments show that the proposed method converges significantly faster than the state-of-the-art first-order nonnegative low-rank factorization method, while achieving similarly optimal statistical accuracy.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper proposes a smooth unconstrained Riemannian manifold formulation of K-means clustering, solved via a cubic-regularized Newton algorithm with linear-time subproblem solvers. It resides in the 'Second-order and Cubic-Regularized Riemannian Methods' leaf, which contains only one sibling paper among the 47 papers surveyed. This indicates a relatively sparse research direction within the broader field of Riemannian K-means optimization, where most work focuses on first-order gradient methods or intrinsic formulations on homogeneous manifolds.

The taxonomy reveals that neighboring leaves include 'Manifold Relaxations and Gradient-Based Methods' (three papers using first-order approaches) and 'Intrinsic K-means on Homogeneous Manifolds' (two papers defining geodesic-based clustering). The paper diverges from these by emphasizing second-order curvature exploitation rather than gradient descent or intrinsic geodesic computations. Its focus on computational scalability through manifold factorization also distinguishes it from theoretical branches like 'Consistency and Asymptotic Theory' and application-oriented subtopics such as 'Wireless Communications and Signal Processing'.

Among 30 candidates examined, the Riemannian manifold formulation (Contribution 1) shows one refutable candidate out of 10 examined, suggesting some prior work on manifold relaxations exists. The linear-time Newton subproblem solver (Contribution 2) found no refutations among 10 candidates, indicating potential novelty in the computational approach. The scalable cubic-regularized Newton algorithm (Contribution 3) identified one refutable candidate among 10, likely reflecting existing second-order Riemannian methods. The limited search scope means these findings reflect top-30 semantic matches rather than exhaustive coverage.

Given the sparse population of the second-order methods leaf and the limited search scope, the work appears to occupy a less-explored niche within Riemannian K-means. The manifold factorization approach for linear-time subproblems shows the strongest novelty signal, while the core formulation and second-order framework have some precedent among the examined candidates. The analysis is constrained by the top-30 search window and does not capture the full landscape of optimization literature beyond this specific clustering context.

Taxonomy

Core-task Taxonomy Papers
47
3
Claimed Contributions
30
Contribution Candidate Papers Compared
2
Refutable Paper

Research Landscape Overview

Core task: Riemannian optimization for K-means clustering. This field extends classical K-means to data residing on Riemannian manifolds, where Euclidean distance is replaced by geodesic metrics. The taxonomy reveals several major branches: Core Riemannian K-means Formulations and Algorithms develop fundamental methods such as first-order gradient-based schemes and second-order approaches that exploit manifold curvature; Theoretical Foundations and Convergence Analysis establish guarantees for these algorithms; Extensions and Variants explore probabilistic models, robust formulations, and multi-view clustering; Manifold Learning and Graph-Based Clustering address scenarios where the manifold structure itself must be inferred; Applications to Specific Domains demonstrate the utility of these methods in areas like wireless communications and medical imaging; and specialized branches examine Learning Manifolds and Approximation Theory as well as Curvature Regularization in autoencoder latent spaces. Representative works include Manifold optimization for k-means[1], which laid early groundwork, and Statistical initialization of intrinsic[3] and Intrinsic K-means clustering over[4], which refine initialization and algorithmic strategies. A particularly active line of research focuses on advancing optimization efficiency through higher-order methods. While many studies rely on first-order Riemannian gradient descent, a smaller cluster investigates second-order and cubic-regularized techniques that promise faster convergence by incorporating curvature information. Scalable Second-order Riemannian Optimization[0] sits squarely within this branch, emphasizing computational scalability for large-scale problems—a critical concern given the expense of Hessian computations on manifolds. This contrasts with simpler first-order schemes like those in Simple algorithms for optimization[5], which trade off convergence speed for per-iteration simplicity, and with works such as Rethinking k-means from manifold[17] that revisit foundational formulations. The interplay between algorithmic sophistication and practical scalability remains an open question, with Scalable Second-order Riemannian Optimization[0] contributing methods that aim to harness second-order information without prohibitive cost, thereby bridging theoretical power and real-world applicability.

Claimed Contributions

Riemannian manifold formulation of K-means clustering

The authors reformulate the constrained K-means optimization problem as smooth unconstrained optimization over a Riemannian manifold by establishing a submersion from a product manifold to the constraint set. This enables applying Riemannian optimization algorithms with global convergence guarantees to first- and second-order critical points.

10 retrieved papers
Can Refute
Linear-time Newton subproblem solver via manifold factorization

The authors demonstrate that by factorizing the K-means manifold into a product manifold structure, the Newton subproblems in second-order Riemannian optimization can be solved in O(n) time per iteration, exploiting block-diagonal-plus-low-rank structure in the Riemannian Hessian.

10 retrieved papers
Scalable second-order Riemannian cubic-regularized Newton algorithm

The authors develop a practical implementation of Riemannian cubic-regularized Newton method for K-means that achieves O(n·ε^(-3/2)·poly(r,d)) total time complexity to compute ε-second-order critical points, combining fast per-iteration cost with rapid convergence.

10 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Riemannian manifold formulation of K-means clustering

The authors reformulate the constrained K-means optimization problem as smooth unconstrained optimization over a Riemannian manifold by establishing a submersion from a product manifold to the constraint set. This enables applying Riemannian optimization algorithms with global convergence guarantees to first- and second-order critical points.

Contribution

Linear-time Newton subproblem solver via manifold factorization

The authors demonstrate that by factorizing the K-means manifold into a product manifold structure, the Newton subproblems in second-order Riemannian optimization can be solved in O(n) time per iteration, exploiting block-diagonal-plus-low-rank structure in the Riemannian Hessian.

Contribution

Scalable second-order Riemannian cubic-regularized Newton algorithm

The authors develop a practical implementation of Riemannian cubic-regularized Newton method for K-means that achieves O(n·ε^(-3/2)·poly(r,d)) total time complexity to compute ε-second-order critical points, combining fast per-iteration cost with rapid convergence.

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