Low-degree Lower bounds for clustering in moderate dimension

• URL: http://arxiv.org/abs/2602.23023v1
• Date: Thu, 26 Feb 2026 14:03:55 GMT
• Title: Low-degree Lower bounds for clustering in moderate dimension
• Authors: Alexandra Carpentier, Nicolas Verzelen,
• Abstract summary: We study the fundamental problem of clustering $n$ points into $K$ groups drawn from a mixture of isotropic Gaussians in $mathbbRd$.<n>We show that while the difficulty of clustering for $n leq dK$ is driven by dimension reduction and spectral methods, the moderate-dimensional regime involves more delicate phenomena leading to a "non-optimal rate"<n>We provide a novel non-spectral algorithm matching this rate, shedding new light on the computational limits of the clustering problem in moderate dimension.
• Score: 53.03724383992195
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the fundamental problem of clustering $n$ points into $K$ groups drawn from a mixture of isotropic Gaussians in $\mathbb{R}^d$. Specifically, we investigate the requisite minimal distance $Δ$ between mean vectors to partially recover the underlying partition. While the minimax-optimal threshold for $Δ$ is well-established, a significant gap exists between this information-theoretic limit and the performance of known polynomial-time procedures. Although this gap was recently characterized in the high-dimensional regime ($n \leq dK$), it remains largely unexplored in the moderate-dimensional regime ($n \geq dK$). In this manuscript, we address this regime by establishing a new low-degree polynomial lower bound for the moderate-dimensional case when $d \geq K$. We show that while the difficulty of clustering for $n \leq dK$ is primarily driven by dimension reduction and spectral methods, the moderate-dimensional regime involves more delicate phenomena leading to a "non-parametric rate". We provide a novel non-spectral algorithm matching this rate, shedding new light on the computational limits of the clustering problem in moderate dimension.

Related papers

• Dimension-Independent Convergence of Underdamped Langevin Monte Carlo in KL Divergence [50.719298242863744]
  Underdamped Langevin dynamics (ULD) is a widely-used sampler for Gibbs distributions $propto e-V$.<n>We prove the first dimension-free KL divergence bounds for discretized ULD.
  arXiv  Detail & Related papers  (2026-03-02T22:14:38Z)
• Near-Optimal Clustering in Mixture of Markov Chains [74.3828414695655]
  We study the problem of clustering $T$ trajectories of length $H$, each generated by one of $K$ unknown ergodic Markov chains over a finite state space of size $S$.<n>We derive an instance-dependent, high-probability lower bound on the clustering error rate, governed by the weighted KL divergence between the transition kernels of the chains.<n>We then present a novel two-stage clustering algorithm.
  arXiv  Detail & Related papers  (2025-06-02T05:10:40Z)
• Generalization Bounds for Stochastic Gradient Descent via Localized $\varepsilon$-Covers [16.618918548497223]
  We propose a new covering technique localized for the trajectories of SGD. This localization provides an algorithm-specific clustering measured by the bounds number. We derive these results in various contexts and improve the known state-of-the-art label rates.
  arXiv  Detail & Related papers  (2022-09-19T12:11:07Z)
• Subspace clustering in high-dimensions: Phase transitions \& Statistical-to-Computational gap [24.073221004661427]
  A simple model to study subspace clustering is the high-dimensional $k$-Gaussian mixture model. We provide an exact characterization of the statistically optimal reconstruction error in this model in the high-dimensional regime with extensive sparsity.
  arXiv  Detail & Related papers  (2022-05-26T17:47:35Z)
• Minimax Supervised Clustering in the Anisotropic Gaussian Mixture Model: A new take on Robust Interpolation [5.98367009147573]
  We study the supervised clustering problem under the two-component anisotropic Gaussian mixture model. We show that in the high-dimensional regime, the linear discriminant analysis (LDA) classifier turns out to be sub-optimal in the minimax sense.
  arXiv  Detail & Related papers  (2021-11-13T05:19:37Z)
• Lattice partition recovery with dyadic CART [79.96359947166592]
  We study piece-wise constant signals corrupted by additive Gaussian noise over a $d$-dimensional lattice. Data of this form naturally arise in a host of applications, and the tasks of signal detection or testing, de-noising and estimation have been studied extensively in the statistical and signal processing literature. In this paper we consider instead the problem of partition recovery, i.e.of estimating the partition of the lattice induced by the constancy regions of the unknown signal. We prove that a DCART-based procedure consistently estimates the underlying partition at a rate of order $sigma2 k*
  arXiv  Detail & Related papers  (2021-05-27T23:41:01Z)
• Multiscale regression on unknown manifolds [13.752772802705978]
  We construct low-dimensional coordinates on $mathcalM$ at multiple scales and perform multiscale regression by local fitting. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our algorithm has quasilinear complexity in the sample size, with constants linear in $D$ and exponential in $d$.
  arXiv  Detail & Related papers  (2021-01-13T15:14:31Z)
• High-Dimensional Sparse Linear Bandits [67.9378546011416]
  We derive a novel $Omega(n2/3)$ dimension-free minimax regret lower bound for sparse linear bandits in the data-poor regime. We also prove a dimension-free $O(sqrtn)$ regret upper bound under an additional assumption on the magnitude of the signal for relevant features.
  arXiv  Detail & Related papers  (2020-11-08T16:48:11Z)
• Computationally efficient sparse clustering [67.95910835079825]
  We provide a finite sample analysis of a new clustering algorithm based on PCA. We show that it achieves the minimax optimal misclustering rate in the regime $|theta infty$.
  arXiv  Detail & Related papers  (2020-05-21T17:51:30Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.