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Improved Spectral-Norm Bounds for Clustering

Pranjal Awasthi, Or Sheffet

arXiv:1206.3204v2cs.LGcs.DS

TL;DR

Kumar and Kannan’s clustering analysis has bounds that depend heavily on the number of clusters and requires multiple Lloyd iterations. This paper simplifies the analysis around center separation and weakens the proximity condition, while retaining clustering guarantees under those weaker conditions.

  • Problem

    Kumar and Kannan’s analysis has separation and classification bounds that grow linearly and quadratically with k, respectively, and its application to Ostrovsky et al.’s datasets holds only for constant k.

    Source: §1 · p. 2

  • Method

    The analysis relies mainly on center separation and basic inequalities, with a simpler proximity condition and a four-dimensional projection plus subclusters in Part II.

    Source: §1.1 · p. 3, §1.4 · p. 6

  • Results

    Centers are k-times closer to the original centers than those retrieved by Kumar and Kannan after numerous Lloyd iterations; the method also achieves exact recovery under the k-weakened proximity condition or at most (ε + O(c^-4))n misclassifications when at most εn points violate it.

    Source: §1.3 · p. 5, §1.4 · p. 7

  • Takeaways & Limitations

    The misclassification bound remains meaningful for constant ε even when k = ω(1).

    Source: §1.1 · p. 3

    The center-separation bound depends on the entire dataset through ||A−C||, rather than only on the two clusters being compared.

    Source: §6 · p. 15

Abstract

from arXiv · show

Aiming to unify known results about clustering mixtures of distributions under separation conditions, Kumar and Kannan[2010] introduced a deterministic condition for clustering datasets. They showed that this single deterministic condition encompasses many previously studied clustering assumptions. More specifically, their proximity condition requires that in the target $k$-clustering, the projection of a point $x$ onto the line joining its cluster center $\mu$ and some other center $\mu'$, is a large additive factor closer to $\mu$ than to $\mu'$. This additive factor can be roughly described as $k$ times the spectral norm of the matrix representing the differences between the given (known) dataset and the means of the (unknown) target clustering. Clearly, the proximity condition implies center separation -- the distance between any two centers must be as large as the above mentioned bound. In this paper we improve upon the work of Kumar and Kannan along several axes. First, we weaken the center separation bound by a factor of $\sqrt{k}$, and secondly we weaken the proximity condition by a factor of $k$. Using these weaker bounds we still achieve the same guarantees when all points satisfy the proximity condition. We also achieve better guarantees when only $(1-\epsilon)$-fraction of the points satisfy the weaker proximity condition. The bulk of our analysis relies only on center separation under which one can produce a clustering which (i) has low error, (ii) has low $k$-means cost, and (iii) has centers very close to the target centers. Our improved separation condition allows us to match the results of the Planted Partition Model of McSherry[2001], improve upon the results of Ostrovsky et al[2006], and improve separation results for mixture of Gaussian models in a particular setting.

1 Introduction

The paper develops deterministic separation conditions for clustering that unify analyses of several distributional settings. It weakens Kumar and Kannan’s bounds while retaining clustering, cost, and center guarantees.

1 Introduction

  • Kumar and Kannan’s deterministic framework unifies clustering results across distributional settings, but its bounds overshoot some prior results by √k and its Ostrovsky application requires constant k.Earlier work typically developed algorithms for particular distribution families after identifying their high-probability properties.Source: §1 · p. 2

1.2 Notations and Preliminaries

  • The analysis centers on the spectral norm of A − C, while the target clustering’s k-means cost is its squared Frobenius norm.The paper uses projection onto top singular vectors, which gives the best rank-t approximation under both Frobenius and spectral norms.Source: §1 · p. 2, §1.2 · p. 4

1.1 Our Contribution

  • The paper weakens center separation by a factor of √k and the proximity condition by a factor of k, while retaining its clustering guarantees.The output k-means cost is also a (1 + O(1/c))-approximation of the target cost.Source: §1.1 · p. 3
  • The authors’ applications subsume Ostrovsky et al. for non-constant k and match Planted Partition separation results while improving Gaussian-mixture separation in a particular setting.The Gaussian-mixture bound is stated as Ω(σmax … k); the supplied passage truncates its expression.Source: §1.1 · p. 3

1.3 Formal Description of the Algorithm and Our Theorems

  • The algorithm has three parts and aims to identify most points correctly, keep k-means cost comparable to the target, and find centers close to the true centers.Part I projects onto the top k singular vectors and runs a 10-approximation k-means algorithm on the projected data.Source: §1.3 · p. 5
  • Under center separation, Part I misclassifies at most an O(c^-2) fraction per cluster and achieves k-means cost at most (1 + O(1/c)) times the target cost.When (1−ε)n points satisfy the k-weaker proximity condition, misclassification is at most (ε + O(c^-4))n; if all points satisfy it, Part III recovers the target partition exactly.Source: §1.3 · p. 5

1.4 Organization and Proofs Overview

  • Part II simplifies a Kumar–Kannan relation by projecting into four dimensions and using subclusters; it applies the main lemma once rather than relying on iterative Lloyd steps.The analysis relies on center separation rather than requiring many points to satisfy the proximity condition.Source: §1.4 · p. 6

2 Related Work

Prior work established separation-based clustering guarantees for several distribution families, while later work addressed other mixture models and deterministic structure. The literature also includes general k-means approximation algorithms and separation-based analyses.

2 Related Work

  • For Gaussian mixtures, separation requirements were reduced from ˜Ω(√n(σ_r+σ_s)) to ˜Ω((σ_r+σ_s)k^1/4) for spherical Gaussians, with extensions to general Gaussians.[DS07] improved the earlier bound to ˜Ω((σ_r+σ_s)n^1/4).Source: §2 · p. 7
  • [KSV08] require separation of ˜Ω(k^3/2) in the cited mixture setting.The passage also notes that other distributions were addressed, although with larger separation requirements.Source: §2 · p. 7
  • Other work covered product and heavy-tailed mixtures, hyperplane-separated Gaussian mixtures, and deterministic clustering based on low-rank structure [BV08].Some algorithms for k>2 Gaussians run in time exponential in k [MV10].Source: §2 · p. 7
  • General k-means approximation work includes a (9 + ǫ)-approximation with polynomial dependence on k and dimension [KMN+02], while Lloyd’s heuristic has separation-based guarantees.For constant k, give a PTAS.Source: §2 · p. 7

3 Part I of the Algorithm

Part I assigns projected points to the nearest estimated center and guarantees low per-cluster misclassification and near-target k-means cost under the stated separation setting. For ORSS-separated datasets, it yields a (1 + O(√ǫ))-approximation for any k.

3 Part I of the Algorithm

  • O(1/c^2)|T_s| points are misclassified from each target cluster, with at most O(1/c^2)|T_r| foreign points added to each output cluster.The theorem also bounds overall misclassification by O(1/c^2)|T_r2|, where T_r2 is the second-largest target cluster.Source: §3 · p. 8
  • The clustering Z has k-means cost at most (1 + O(1/c)) times the target clustering cost.The analysis shows that assigning points in Z to their corresponding target centers costs each point at most (1 + O(1/c)) times its target-center cost.Source: §3 · p. 9

3.1 Application: The ORSS-Separation

  • For ORSS-separated datasets, Part I yields a (1 + O(√ǫ))-approximation for any k.The passage contrasts this with Kumar and Kannan’s (1 + O(kǫ))-approximation.Source: §3.1 · p. 9

4 Part II of the Algorithm

Part II uses projected points to form sets whose centers are close to the target centers, while bounding misassignments through distance comparisons. Part III weakens the proximity threshold and gives guarantees for clustering and Lloyd’s method under stated conditions.

4 Part II of the Algorithm

  • The sets S_r need not partition the dataset; Part II uses their centers despite leaving some points unassigned.Source: §4 · p. 9
  • Points sufficiently close to μ_r in the projected space must belong to S_r, as shown using separation and triangle inequalities.The proof assumes ∥μ_r − ν_r∥ ≤ 6Δ_r and uses c > 48 in its distance comparison.Source: §4 · p. 10
  • Proposition 4.4 reduces bounding cross-cluster assignments to counting points with a 2:1 projected-distance imbalance, and Lemma 4.5 improves the k-dependence of that bound.Source: §4 · pp. 10–11
  • Part II’s returned centers θ_r satisfy ∥μ_r − θ_r∥ = O(1/c)∥μ_r − μ̂_r∥.The theorem also states that the center of each S_r is close to its target center.Source: §4 · pp. 10, 12

4.1 The Proximity Condition – Part III of the Algorithm

  • For γ-bad points numbering εn, at most O(γ^-2 c^-2 k^-1)n_s good points can be misclassified for each competing cluster.Source: §4.1 · p. 12
  • With γ = 1, the proximity threshold is independent of k; at most O(c^-2) of 1-bad points and O(c^-4) of 1-good points may be misclassified.Source: §4.1 · p. 12
  • When all points are 1-good, Lloyd’s method initialized at the Part II centers converges to the true centers.Source: §4.1 · pp. 12–13

5 Applications

The applications specialize the separation and proximity framework to Gaussian mixtures and McSherry’s planted-partition model. The paper reports high-probability proximity guarantees and matches McSherry’s distance bound, while the relative Gaussian bounds depend on mixture weights and directional variances.

Clustering a mixture of Gaussians

  • For Gaussian mixtures, the paper reports that samples satisfy the proximity condition w.h.p. when every pair of centers is separated by at least cσ_max.Source: §5 · p. 13
  • The Gaussian bound is better in Theorem 5.2 when variances are similar but minimum mixture weight is small; is better when weights are similar but directional variances vary.Source: §5 · p. 13
  • When both Gaussian variances and weights vary substantially, the paper says which bound is better must be determined case by case.Source: §5 · p. 13

McSherry’s Planted Partition Model

  • For the planted-partition model, the authors match McSherry’s distance bound; under the stated condition, every point satisfies the proximity condition with probability at least 1 − δ.Source: §5 · p. 14

6 An Open Problem

The algorithm’s center-separation guarantee depends on the whole dataset, and the authors ask whether it can instead depend only on the two clusters being separated. A local separation condition alone may not make the target centers minimize k-means cost, as a small-cluster/large-bimodal-cluster example illustrates.

6 An Open Problem

  • The current center-separation bound depends on the whole-dataset quantity ∥A−C∥, motivating the open question of a bound based only on the separated clusters.The authors note that much of their analysis uses the ratio between a center’s distance to its estimate and the distance between centers.Source: §6 · p. 15
  • Local separation need not make the target centers k-means-optimal when one cluster is small and low-variance and another is large and bimodal.In the example, placing two centers in the large cluster costs less than placing one at each target center.Source: §6 · p. 15
  • The authors connect this issue to designing a t-approximation for k-means that limits each cluster’s cost increase to a factor of t.Source: §6 · p. 15

A Some Basic Lemmas

These basic lemmas bound the low-rank approximation’s error and match each target center to a nearby algorithmic center. They also formalize bounds for perturbed clusters created by removing and adding points.

A Some Basic Lemmas

  • The rank-2k residual and best rank-k approximation yield ∥Â−C∥F ≤ 2∥A−C∥F.The proof also views Â−Ĉ as projecting A−C to a k-dimensional subspace.Source: A Some Basic Lemmas · pp. 17–18
  • Each target center μr can be matched to a unique center νr within 6∆r.Source: A Some Basic Lemmas · p. 17
  • For a subset X of target cluster Tr, the text uses an equality relating its centroid’s displacement to that of the remaining points.The stated identity follows from the definition of the target center μr.Source: A Some Basic Lemmas · p. 18
  • Fact A.4 considers clusters formed by removing points and adding points from other clusters, with the bound maximized when incoming fractions are evenly split.Source: A Some Basic Lemmas · p. 18
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