8 - k-means Clustering
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Questions and Answers

What is a limitation of improved algorithms in clustering?

No theoretical guarantees

What type of distances can k-means be used with?

Bregman divergences

What is the connection between cosine similarity and squared Euclidean distance?

Closely connected

What does Spherical k-means do with input data?

<p>Normalizes it</p> Signup and view all the answers

How does Spherical k-means minimize the average cosine similarity?

<p>By normalizing the new centers</p> Signup and view all the answers

Why can the solution found by the standard k-means algorithm be arbitrarily poor?

<p>Bad starting conditions</p> Signup and view all the answers

What is the main issue with using subtraction in computations for clustering?

<p>Numerically unstable</p> Signup and view all the answers

What is the importance of pairwise squared deviations in clustering?

<p>Minimizing squared deviations from the mean</p> Signup and view all the answers

Why is the standard algorithm for -means considered inefficient?

<p>Little good reason to use it in practice</p> Signup and view all the answers

What is the significance of MacQueen's algorithm in the context of clustering?

<p>First used the name -means for a different algorithm</p> Signup and view all the answers

How does the arithmetic mean relate to the binomial expansion in clustering?

<p>Only holds for the arithmetic mean</p> Signup and view all the answers

What is the main goal of the k-means clustering algorithm?

<p>Divide data into subsets represented by their arithmetic mean to optimize the least squared error.</p> Signup and view all the answers

Why does the k-means algorithm use squared errors instead of other distance metrics?

<p>Squared errors put more weight on larger deviations and the arithmetic mean minimizes the squared Euclidean distance.</p> Signup and view all the answers

What type of problem is k-means clustering considered?

<p>Non-convex problem</p> Signup and view all the answers

Explain why k-means clustering is suitable for signals with normally distributed measurement errors.

<p>K-means is a good choice for signals with normal errors because it optimizes the least squared error.</p> Signup and view all the answers

What theorem is attributed to König, Huygens, and Steiner in the context of clustering?

<p>Steiner Translation Theorem / König-Huygens Theorem</p> Signup and view all the answers

Why is it important to assign every point to its least-squares closest cluster in k-means clustering?

<p>Assigning points to their least-squares closest cluster minimizes the sum of squared errors.</p> Signup and view all the answers

Study Notes

k-Means Clustering

  • k-means clustering is numerically unstable in computations, but useful in proofs.
  • The pairwise sum of squared deviations minimizes squared deviations from the mean.

The Standard Algorithm (Lloyd's Algorithm)

  • The standard algorithm for k-means is not the most efficient algorithm despite being widely taught.
  • There are over 12 variants of the algorithm, including ELKI, which contains multiple variants.
  • Improved algorithms focus on reducing the number of computations for reassignment, but often lack theoretical guarantees.

k-Means for Text Clustering

  • k-means cannot be used with arbitrary distances, only with Bregman divergences.
  • Spherical k-means uses normalized input data and centers, minimizing the average cosine similarity.
  • Spherical k-means uses sparse nearest-centroid computations and can be accelerated using stored bounds.

Limitations of k-Means

  • The solution found by the standard k-means algorithm can be arbitrarily poor.
  • In the worst case, a k-means solution can be arbitrarily worse than the best solution.

Properties of k-Means Clustering

  • k-means divides data into subsets represented by their arithmetic mean.
  • Squared errors put more weight on larger deviations.
  • Arithmetic mean is the maximum likelihood estimator of centrality.
  • Data is split into Voronoi cells.
  • k-means is a non-convex problem.

The Sum of Squares Objective

  • The sum-of-squares objective is minimized by the arithmetic mean.
  • Assigning every point to its least-squares closest cluster reduces the sum of squares.
  • The sum of squares is equivalent to the squared Euclidean distance.

Historical Note

  • The history of least squares estimation is attributed to Legendre and Gauss.

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Related Documents

k-means Clustering PDF

Description

Explore the concept of pairwise sum of squared deviations and the standard algorithm for K-means. Learn about the arithmetic mean and how to minimize squared deviations from the mean. Understand the implications of numerically unstable subtraction in computations.

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