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Clustering with Spectral Norm and the k-means Algorithm
Amit Kumar, Ravindran Kannan
TL;DR
Existing clustering guarantees often rely on generative models and distribution-specific properties. This paper gives a deterministic proximity-condition framework using k-means, recovers results for studied generative models, and establishes additional guarantees under variance bounds.
Problem
Prior efficient clustering results for well-separated mixtures generally rely heavily on the generative model and particular distributional properties.
Method
The paper uses a proximity condition based on projected distances and spectral-norm standard deviations, with a clustering algorithm relying on k-means.
Results
The proximity condition yields known generative-model results as corollaries and supports clustering all but an ε fraction of points under variance bounds.
Takeaways & Limitations
The framework provides polynomial-time clustering under deterministic assumptions and shows k-means can converge to true centers despite spurious points under suitable initialization and proximity conditions.
Takeaways & Limitations
Results for mixtures with unbounded variance cited by the paper apply only to product distributions.
Abstract
from arXiv · showhide
There has been much progress on efficient algorithms for clustering data points generated by a mixture of $k$ probability distributions under the assumption that the means of the distributions are well-separated, i.e., the distance between the means of any two distributions is at least $Ω(k)$ standard deviations. These results generally make heavy use of the generative model and particular properties of the distributions. In this paper, we show that a simple clustering algorithm works without assuming any generative (probabilistic) model. Our only assumption is what we call a "proximity condition": the projection of any data point onto the line joining its cluster center to any other cluster center is $Ω(k)$ standard deviations closer to its own center than the other center. Here the notion of standard deviations is based on the spectral norm of the matrix whose rows represent the difference between a point and the mean of the cluster to which it belongs. We show that in the generative models studied, our proximity condition is satisfied and so we are able to derive most known results for generative models as corollaries of our main result. We also prove some new results for generative models - e.g., we can cluster all but a small fraction of points only assuming a bound on the variance. Our algorithm relies on the well known $k$-means algorithm, and along the way, we prove a result of independent interest -- that the $k$-means algorithm converges to the "true centers" even in the presence of spurious points provided the initial (estimated) centers are close enough to the corresponding actual centers and all but a small fraction of the points satisfy the proximity condition. Finally, we present a new technique for boosting the ratio of inter-center separation to standard deviation.
1 Introduction
The paper replaces generative-model assumptions with a deterministic proximity condition measured using spectral norm, enabling polynomial-time clustering via SVD initialization and k-means. It also derives generative-model results, handles variance-bounded mixtures with limited errors, and introduces a boosting construction.
- Motivation: Generative-model clustering results typically require well-separated means and rely heavily on distribution-specific properties.The paper targets settings where the data arise from mixtures of probability distributions.
- Deterministic framework: The proximity condition requires each point’s projection toward another center to be Ω(k) standard deviations closer to its own center.Here, the standard-deviation scale is based on the spectral norm of the centered data matrix.
- Main result: The paper gives a polynomial-time deterministic clustering result and describes it as the first such result under only deterministic assumptions.The result also settles an open question raised in.
- Applications: The proximity condition holds in the generative models studied, yielding known generative-model results as corollaries with a qualification for Gaussian variance dependence.The paper’s separation uses whole-data variance rather than, in some Gaussian cases, only individual-density variances.
- Applications: For mixtures of general densities with bounded variance, the paper classifies all but an ε fraction of points without assuming further concentration.Earlier results for distributions that may lack variance apply only to product densities with additional constraints.
- k-means analysis: A technical theorem shows k-means converges to true centers despite spurious points when initialization is sufficiently close and all but an ε fraction satisfy proximity.The proof uses a geometric argument showing misclassified points are far from true centers, limiting their number.
- Algorithm and boosting: The algorithm first finds centers using SVD and then runs standard k-means; the paper also presents a construction that boosts inter-center separation relative to spectral norm.The boosting construction uses two independent sample sets and reduces dependence on the minimum mixture-component weight.
2 Preliminaries and the Main Theorem
The main theorem formalizes proximity for points assigned to cluster means and guarantees polynomial-time classification with an error proportional to the fraction violating the condition. The framework represents clusters and centers in matrix form and applies to sampled mixtures with variance and mixing-weight parameters.
- Setup: The data consist of n points in ℜd divided into k clusters, represented by matrices A and C whose rows are points and corresponding cluster means.For point Ai in Tr, the corresponding row Ci equals the cluster mean µr.
- Parameterization: A sufficiently large constant c determines the theorem’s separation scale through the stated setup.The supplied passage identifies c as a large enough constant.
- Proximity condition: The proximity condition requires each point in Tr to project at least ∆rs closer to µr than to every other center µs.The set G contains the points satisfying this condition.
- Main theorem: If |G| ≥ (1−ε)·n, the algorithm correctly classifies all but O(k^2ε·n) points in polynomial time.When ε = 0, all points are classified correctly.
- Mixture applications: In mixture applications, σr denotes maximum directional variance for distribution Fr, σ is the maximum σr, and wmin is the minimum mixing weight.These parameters connect the deterministic framework to samples drawn from mixture distributions.
3 Previous Work
Previous work developed clustering methods for separated mixtures, often under distributional assumptions such as Gaussian structure, independence, bounded variance, or slope conditions. Related k-means methods can achieve polynomial-time approximation under sufficiently large separation, while some approaches require exponential time.
- Mixture-learning research seeks to recover clusters when component means are sufficiently separated.
- Gaussian mixtures: Gaussian-mixture algorithms use separation conditions whose strength depends on the model and algorithm, including random projection, EM, and spectral techniques.
- Heavy-tailed distributions: Heavy-tailed mixture results commonly assume product distributions, coordinate independence, or slope conditions on directions joining means.
- Bounded-variance mixtures: For arbitrary distributions with bounded variance in every direction, prior work obtained correct classification under mean separation and a suitable slope condition.
- Planted partition: The planted partition model can be viewed as a mixture-learning problem over binary vectors whose means encode group memberships.
- k-means: Lloyd’s k-means algorithm alternates nearest-center assignment with cluster-mean updates, but can take superpolynomial time and requires good initialization for provable guarantees.
4 Our Contributions
The paper combines an SVD-based initialization with standard k-means to cluster deterministically under a spectral-norm proximity condition. It derives results across mixture settings, gives a bounded-variance classification guarantee and a k-means PTAS, and introduces a separation-boosting construction.
- Core algorithm: The main algorithm first obtains candidate centers using SVD, then runs standard k-means initialized with those centers.
- Core theorem: The paper proves correct classification for point sets satisfying a deterministic proximity condition based on spectral norm.
- Mixture applications: The method matches prior arbitrary-Gaussian and planted-partition results, except that separation depends on maximum standard deviation rather than individual distribution variances.
- Mixture applications: For bounded-variance distributions, the algorithm classifies all but an ε fraction of points under a suitable mean-separation condition.
- PTAS: The paper obtains a PTAS because the separation condition of Ostrovsky et al. is stronger than the proximity condition.
- Separation boosting: The algorithm applies across the settings discussed and includes a construction that increases spectral norm more slowly than inter-mean distance as sample size grows.
- Separation boosting: For arbitrary Gaussians, the boosting technique yields separation depending only logarithmically on the minimum mixture weight.
- Mixture applications: For power-law distributions with sufficiently large constant exponent γ, the paper learns all but an ε fraction of samples and reduces ε-dependence for large γ.
5 Proof of Theorem 2.2
The proof analyzes an SVD-initialized, Lloyd-style k-means procedure and shows that sufficiently accurate centers improve geometrically despite a small fraction of spurious points. The resulting iteration bounds establish correct classification under the proximity condition.
- Algorithm: The algorithm begins with a near-optimal k-means solution after projecting points onto the best k-dimensional SVD subspace.These centers form the base case for subsequent Lloyd-like iterations.
- Center refinement: The k-means update replaces assigned points by their cluster means, producing new centers that are closer to the true centers.The proof controls means after removing misclassified or spurious points using bounds on how much deleting points can shift a cluster mean.
- Misclassification bound: If every pair of current centers is sufficiently close to its corresponding true centers, misclassified points must lie far from their true centers.Theorem 5.4 uses this geometric fact to bound their number through the total squared deviation from true centers.
- Convergence: In the absence of spurious points, each iteration halves the upper bound on center error once the initialization is sufficiently accurate.With an ε fraction of exceptional points, the proof propagates corresponding error bounds through the iterations.
- Conclusion: After sufficiently many iterations, the accumulated misclassification bound yields the main theorem’s classification guarantee.The proof substitutes the final center-accuracy bound into the misclassification estimate and sums over cluster pairs.
6 Applications
The paper applies spectral-norm clustering to planted models, Gaussian mixtures, and general bounded-variance distributions. These applications recover exact classification in structured settings and classify all but a small fraction under weaker variance assumptions.
- General tool: A spectral-norm bound for independent-row random matrices supplies the main technical tool for the applications.The bound uses the matrix E[Y^T Y] together with row-norm control.
- Planted models: For planted partition models, concentration and a union bound show that the proximity condition holds with high probability.The resulting theorem derives McSherry’s clustering result as a corollary of the paper’s deterministic theorem.
- Gaussian mixtures: For Gaussian mixtures, separation relative to the maximum directional variance implies correct classification of all sampled points.Gaussian tail bounds control projected deviations, while spectral-norm estimates control the sample matrix.
- Bounded variance: For general distributions with bounded directional variance, the algorithm correctly classifies at least 1−ε fraction of points under the stated separation conditions.This result does not require the stronger concentration properties used for thin-tailed distributions.
- Error control: The analysis bounds the number of bad, non-nice, and removed points to control the centers retained during the algorithm.These bounds support the approximate classification guarantee for the bounded-variance setting.
7 Boosting
The boosting construction transforms independently sampled mixture data into higher-dimensional points whose inter-center separation is amplified relative to spectral norm. The resulting procedure supports clustering guarantees under the proximity condition and yields a PTAS application.
- Construction: The construction samples two independent datasets and combines them through inner products after translating by the overall mean.The transformed points encode relationships between samples from the two sets.
- Amplification: The transformed cluster means separate by a larger factor relative to the transformed data’s spectral norm.The proof separately lower-bounds inter-center distance and upper-bounds ||X−Z|| using matrix concentration.
- Clustering analysis: The first clustering step produces few errors, after which the center-refinement argument controls the induced clustering of the original samples.The analysis transfers the transformed-space guarantees back to the original clusters.
- Guarantee: Under the stated mixture conditions, the final theorem correctly classifies all but O(k^2ε) fraction of sampled points.This theorem is then used in applications including Gaussian mixtures and bounded-variance distributions.
- Graph modification: A graph-based modification handles components whose cluster relationships are not known in advance by assigning a large value across different components.Within components, the original inner-product construction is retained; across components, the large value preserves separation.