Questions tagged [clustering]

Clustering is an unsupervised learning problem. It deals with finding "clusters" or groups in a collection of unlabeled data. A cluster is therefore a collection of objects which are “similar” and are “dissimilar” to the objects belonging to other clusters.

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43 views

k-Median Problem With Restricted Centers

The $k$-median problem is defined as follows: Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ of $k$ facility in $L$ such that the ...
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1answer
112 views

Divide and Conquer Algorithm for 1-Median Problem

Let $P_1$ and $P_2$ be two disjoint point sets in $\mathbb{R}^d$ and $n = \vert P_1\vert = \vert P_2\vert$ and $P = P_1\cup P_2$. Let $c^\star$ be the optimal 1-median for $P$ and $opt^\star$ is the ...
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59 views

Exact FPT Algorithm for Continuous Euclidean $k$-Means

The continuous Euclidean $k$-means problem is defined as follows: Given a set $X$ of $n$ points in $d$ dimensional Euclidean space $\mathbb{R}^{d}$. Given a parameter $k>0$, find a partitioning $P$ ...
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1answer
41 views

partitioning points in the plane into two clusters to minimize maximum cluster diameter

What is a fast algorithm for the following problem? input: a set of $n$ pairs of points in the Euclidean plane output: a partition of the points into two clusters so that, for each given pair, the ...
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49 views

Failing to understand a lemma regarding Robust Low Rank Approximation

I am reading Low Rank Approximation in the Presence of Outliers by Bhaskara and Kumar and kind of stuck at the proof of Lemma 9. The paper studies robust (to outliers) low rank approximation problem. ...
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92 views

How general cost function for $p = \log n$ is the $k$-center cost function?

The $k$-clustering problem is defined as follows: Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ of $k$ facility in $L$ such that the ...
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44 views

Low Rank Approximation of a hidden subset

Let $P$ be a set of $n$ points in $\mathbb{R}^d$ and $Q\subseteq P$ with $\vert Q\vert \geq \alpha n$ for some constant $\alpha\in(0,1]$. Given a $j$-dimensional affine subspace(flat) $F$ consider the ...
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38 views

A variant of k-median clustering

Suppose $\mathcal{P} =\{P_1,\cdots,P_n\}$ is a family of $n$ finite sets in $\mathbb{R}^d$. Given set $C=\{c_1,\cdots,c_k\}$ of $k$ points, consider the follwoing objective funtion $cost(\mathcal{P},C)...
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1answer
51 views

How to build the tree with the "most different" solutions of a clustering?

Illustrate the question with an example : we have a similarity matrix for 1000 people, and the similarity represents how much their hobbies are the same (it does not really matter how it's built). Let'...
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248 views

Is there any Bi-criteria PTAS for Metric $k$-Median?

The $k$-median problem is defined as follows: Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ of $k$ facility in $L$ such that the ...
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115 views

Incorrect Lower Bound of k-Means++ Algorithm

The $k$-means++ algorithm is composed of two parts: Initialization part: the initial $k$ centers are chosen based on $D^2$ sampling. Expectation maximization part: the standard $k$-means algorithm (...
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82 views

Extension of a sampling Lemma of Indyk for $k$-median cost

A famous sampling lemma of Indyk (Theorem 31 here) states that if the $k$-median cost of a finite point set(in a metric space) is large with respect to the optimum, then this is true for the case of a ...
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172 views

Is the Fermat-Weber problem $\mathsf{NP}$-hard?

Given a set of $n$ points in a Euclidean space, the Fermat-Weber problem asks to find a center that minimizes the sum of distances of points to that center. There are iterative algorithms known for ...
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171 views

Does k-Median problem become any easier when L = C?

In the $k$-median problem, $L$ defines as set of feasible facility locations and $C$ defines a set of client locations in a metric space. The current best approximation guarantee for the problem is $2....
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88 views

Hardness of Approximation of Continuous Metric k-Median

First let me describe the metric $k$-median problem. Definition (Metric $k$-Median): Given a set $C$ of clients and a set $L$ of facility locations defined over a distance metric $d$. Open a set $F$ ...
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36 views

A continuum version of the 1D k-means clustering problem: constant factor approximations

Modify the k-means clustering problem in 1D by assuming that, instead of a finite number of observations, we must classify into $K$ clusters a continuum of observations, distributed on the unit ...
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106 views

Exact algorithms for $k$-means

Lets recall the definition of $k$-means clustering for euclidean spaces. Let $X$ be a set of $n$ points in $R^d$ and $k$ a given natural number. Let $C$ any $k$ clustering of $X$. Define the cost of $...
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1answer
82 views

Al-Mubaid's Similarity Measure for Ontological Concepts

Al-Mubaid et al. proposed a semantic similarity measure in their research paper [1]. They see ontologies as connected graphs but refer to clusters within ontology graphs without ever defining what ...
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1answer
213 views

Good Survey paper for k-means/k-median/k-center/facility-location

I have stated 4 problems in the Question title. All these problems are closely related and are studied in various variations. For example: Space: Euclidean/metric/discrete/continuous/non-metric/2-...
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2answers
173 views

kmeans++ for arbitrary metric spaces and general potential function

I was reading this popular paper "k-means++: The Advantages of Careful Seeding". It appeared in SODA 2007. Since this technique is the most popular clustering technique, I am hoping that my question ...
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1answer
153 views

Approximation Ratio of Local search for $k-$center problem

In the $k-$center problem, you're given $V$ points in Eucledian space, and you're asked to get a subset $C\subset V, |C|=k$ such that $\max _{v\in V}d(v, Closest-Center(C,v))$ is minimized. Now I am ...
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43 views

What is the meaning of an Oracle in data clustering?

I am not sure whether this is the best place to ask this question. I am in the process of researching the area in data clustering as well as the algorithms that are associated with it and the term ...
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32 views

Does optimal fitting flat must pass through the mean of the point set?

I am confused about a statement made in the paper Linear Time Algorithm for Projective Clustering, section 5.1, second paragraph, second line. Project clustering is a natural generalization of k-...
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80 views

k-center 2.0: A stronger k-center condition

Given an unweighted, undirected graph, we can use the classical 2-appx for $k$-center to select a set $S$ of centers such that every vertex is within a distance of 2 of some center in $S$. Note that ...
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1answer
87 views

Centroid in $\ell_2$ distance

Given points $x_1, x_2, \cdots, x_n \in \mathbb{R}^d$. What is the complexity of computing $$ argmin_{x}\left(\sum_{i=1}^n ||x_i-x||_2\right) $$
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1answer
186 views

Kleinberg-consistency of spectral clustering

Spectral clustering refers to a family of graph-based algorithms, which usually rely on a similarity function rather than a metric, though a metric $\rho(x,y)$ can always be converted to a similarity ...
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104 views

algorithms for a large submatrix / general factor / quasi-biclique problem?

Given a sparse 0/1 matrix $X$, too large to fit in memory, with $m$ rows and $n$ columns, I'm looking for an algorithm for finding a submatrix (when one exists) with maximum number of rows such that ...
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1answer
244 views

Cluster Edge Deletion on 2-trees

Definitions: Cluster Edge Deletion problem is a graph modification problem, in which we want to remove the minimum number of edges such that the resulting graph does not contain a $P_3$ as an induced ...
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1answer
921 views

Max-sum graph-partition for maximizing intra-edge weights?

I would like to know if the following problem has already been studied, and if so how is it called. In particular I'm interested in approximability results. Input: A graph G with negative or non-...
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55 views

Clustering algorithm for image metric

Im working on image clustering (finding duplicates). I have a metric for images, it uses histogram features (mean, dispersion, skewnewss) for each color channel. So there are 9 dimensions. It is quite ...
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433 views

a geometric variant of k-medians. NP-hard or in P?

The following problem is a special case of k-medians. Is it NP-hard? Is it in P? Input: $n$ points $(x_1,y_1), (x_2,y_2), \ldots, (x_n, y_n)$ with each $y_i \ge 0$, and an integer $k$. Output: a set ...
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40 views

Concept of 'shape' in clustering

Is there any abstract definition for 'shapes' of a cluster? I am currently working on providing for a set of axioms to study clustering. In my work, I have found a need for an abstract definition for ...
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1answer
213 views

Determining the number of clusters using property testing algorithm

We say a set of $n$ points in $R^d$ are $k$-clusterable, if all points are covered by k unit balls. We have a property testing algorithm (see section 5 of paper) which consider a promise version of ...
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374 views

Finding similar vectors in subquadratic time

Let $d:\{0,1\}^k\times \{0,1\}^k \to \mathbb{R}$ be a function which we refer to as the similarity function. Examples of similarity function are cosine distance, $l_2$ norm, Hamming distance, Jaccard ...
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1answer
100 views

Approximating the value of k in $k$-mean clustering problem

Consider a set of $n$ points in $R^d$ which are covered by some finitely many (say $k$) unit balls. Can we approximate the value of $k$ by querying only sublinear many points. More precisely, by ...
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211 views

Clustering in sublinear time/query

Given a set of $n$ points in $R^d$, the goal is to cover them with (finitely many) unit balls such that following conditions satisfy: 1) Minimizing the number of balls that are required to cover all ...
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1answer
585 views

What is a minimum vertex separator as in this definition?

In a research paper the following definition appears that I'm not able to understand completely. Let $G=(V,E)$ be an undirected unweighted graph with vertex set $V$ and edge set $E$, no self-loops, ...
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1answer
183 views

an axiomatic framework for clustering by jon kleinberg may have a problem?

In the paper An Impossibility Theorem for Clustering, Jon Kleinberg introduced an axiomatic framework for clustering and showed that his set of axioms are inconsistent. One of the axioms is the ...
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1answer
251 views

Bisecting a set of points into two optimal subsets

I want to divide a set of points into two equally-sized subsets such that the within-cluster sum of squares is minimized. We can assume that the points are in two-dimensional Euclidian space. I'm ...
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610 views

Splitting a graph into size constrained clusters

I ran across a problem while working on an algorithm for a game I'm making on the side. It's basically a clustering problem where we have a graph G and want to split it into clusters of equal size ...
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136 views

Quality measure for clusters of a metric space embedding of a graph?

When evaluating clustering algorithms for networks, we have well-established metrics like Modularity and Surprise for evaluating the quality of the resulting partition. If we then embed our graph (...
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1answer
2k views

Time complexity analysis of random forest and k-means?

I am working with random forest for a supervised classification problem, and I am using the k-means clustering algorithm to split the data at each node, where $n$ is the number of points, $K$ is ...
2
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1answer
489 views

The non-metric k-median problem

It is well-known that the non-metric $k$-median problem cannot be approximated better than $O(\log(n))$ (by a gap preserving reduction from the set cover problem). Is there any logarithmic ...
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526 views

Time complexity of clustering based on random walk

What is the time complexity of the following algorithm (from this paper suggested by Zhou) to partition directed graph? Can I use the complexity of eigen vector computation for this purpose? The ...
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0answers
227 views

Grouping a set of rectangles in larger rectangular regions

I have a set of rectangles, which I want to cluster (group) as shown here(I can not post images yet, so please bear with me). The approach I took was to consider central points of each rectangle as a ...
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2answers
2k views

Hyperspherical nature of K-means and similar clustering methods

Jain, Murty, and Flynn state in their article Data Clustering: A Review all squared error based clustering methods like K-means tend to generate hyperspherical clusters. However, they do not give a ...
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1answer
424 views

Canopy clustering: what should we do with samples in overlapping canopies?

In canopy clustering http://www.kamalnigam.com/papers/canopy-kdd00.pdf, if a sample falls in an overlap of 2 canopies, how do we choose its cluster?
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1answer
207 views

Clustering massive data sets in practice

If you have a very large data set of $n$ vectors and you want to cluster them according to some metric measure, what is the current state of the art when you can not afford to do more than $\Theta(n)$ ...
5
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1answer
147 views

PTAS Algorithm for K-Clustering when Distance Computation is Costly

Can anyone throw any light on any PTAS algorithm that I can apply for K-Clustering algorithm when the distance computation between the clustering points is costly. In details, I have a set of N ...
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2answers
183 views

Classic parallel clustering algorithms

I'm starting a research about parallel clustering. I see a ton of articles on this topic, so that I don't know where to start. I'd like to get familiar with classic methods of parallelizing clustering....