Improvements on the k-center problem for uncertain data

In real applications, there are situations where we need to model some problems based on uncertain data. This leads us to define an uncertain model for some classical geometric optimization problems and propose algorithms to solve them. In this paper, we study the $k$-center problem, for uncertain input. In our setting, each uncertain point $P_i$ is located independently from other points in one of several possible locations $\{P_{i,1},\dots, P_{i,z_i}\}$ in a metric space with metric $d$, with specified probabilities and the goal is to compute $k$-centers $\{c_1,\dots, c_k\}$ that minimize the following expected cost $$Ecost(c_1,\dots, c_k)=\sum_{R\in \Omega} prob(R)\max_{i=1,\dots, n}\min_{j=1,\dots k} d(\hat{P}_i,c_j)$$ here $\Omega$ is the probability space of all realizations $$R=\{\hat{P}_1,\dots, \hat{P}_n\}$$ of given uncertain points and $$prob(R)=\prod_{i=1}^n prob(\hat{P}_i).$$ In restricted assigned version of this problem, an assignment $A:\{P_1,\dots, P_n\}\rightarrow \{c_1,\dots, c_k\}$ is given for any choice of centers and the goal is to minimize $$Ecost_A(c_1,\dots, c_k)=\sum_{R\in \Omega} prob(R)\max_{i=1,\dots, n} d(\hat{P}_i,A(P_i)).$$ In unrestricted version, the assignment is not specified and the goal is to compute $k$ centers $\{c_1,\dots, c_k\}$ and an assignment $A$ that minimize the above expected cost. We give several improved constant approximation factor algorithms for the assigned versions of this problem in a Euclidean space and in a general metric space. Our results significantly improve the results of \cite{guh} and generalize the results of \cite{wang} to any dimension. Our approach is to replace a certain center point for each uncertain point and study the properties of these certain points. The proposed algorithms are efficient and simple to implement

Approximation and Streaming Algorithms for Projective Clustering via Random Projections

Let $P$ be a set of $n$ points in $\mathbb{R}^d$. In the projective clustering problem, given $k, q$ and norm $\rho \in [1,\infty]$, we have to compute a set $\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\sum_{p\in P}d(p, \mathcal{F})^\rho)^{1/\rho}$ is minimized; here $d(p, \mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\mathcal{F}$. We let $f_k^q(P,\rho)$ denote the minimal value and interpret $f_k^q(P,\infty)$ to be $\max_{r\in P}d(r, \mathcal{F})$. When $\rho=1,2$ and $\infty$ and $q=0$, the problem corresponds to the $k$-median, $k$-mean and the $k$-center clustering problems respectively. For every $0 < \epsilon < 1$, $S\subset P$ and $\rho \ge 1$, we show that the orthogonal projection of $P$ onto a randomly chosen flat of dimension $O(((q+1)^2\log(1/\epsilon)/\epsilon^3) \log n)$ will $\epsilon$-approximate $f_1^q(S,\rho)$. This result combines the concepts of geometric coresets and subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence, an orthogonal projection of $P$ to an $O(((q+1)^2 \log ((q+1)/\epsilon)/\epsilon^3) \log n)$ dimensional randomly chosen subspace $\epsilon$-approximates projective clusterings for every $k$ and $\rho$ simultaneously. Note that the dimension of this subspace is independent of the number of clusters~$k$. Using this dimension reduction result, we obtain new approximation and streaming algorithms for projective clustering problems. For example, given a stream of $n$ points, we show how to compute an $\epsilon$-approximate projective clustering for every $k$ and $\rho$ simultaneously using only $O((n+d)((q+1)^2\log ((q+1)/\epsilon))/\epsilon^3 \log n)$ space. Compared to standard streaming algorithms with $\Omega(kd)$ space requirement, our approach is a significant improvement when the number of input points and their dimensions are of the same order of magnitude.Comment: Canadian Conference on Computational Geometry (CCCG 2015

Streaming Coreset Constructions for M-Estimators

We introduce a new method of maintaining a (k,epsilon)-coreset for clustering M-estimators over insertion-only streams. Let (P,w) be a weighted set (where w : P - > [0,infty) is the weight function) of points in a rho-metric space (meaning a set X equipped with a positive-semidefinite symmetric function D such that D(x,z) <=rho(D(x,y) + D(y,z)) for all x,y,z in X). For any set of points C, we define COST(P,w,C) = sum_{p in P} w(p) min_{c in C} D(p,c). A (k,epsilon)-coreset for (P,w) is a weighted set (Q,v) such that for every set C of k points, (1-epsilon)COST(P,w,C) <= COST(Q,v,C) <= (1+epsilon)COST(P,w,C). Essentially, the coreset (Q,v) can be used in place of (P,w) for all operations concerning the COST function. Coresets, as a method of data reduction, are used to solve fundamental problems in machine learning of streaming and distributed data. M-estimators are functions D(x,y) that can be written as psi(d(x,y)) where ({X}, d) is a true metric (i.e. 1-metric) space. Special cases of M-estimators include the well-known k-median (psi(x) =x) and k-means (psi(x) = x^2) functions. Our technique takes an existing offline construction for an M-estimator coreset and converts it into the streaming setting, where n data points arrive sequentially. To our knowledge, this is the first streaming construction for any M-estimator that does not rely on the merge-and-reduce tree. For example, our coreset for streaming metric k-means uses O(epsilon^{-2} k log k log n) points of storage. The previous state-of-the-art required storing at least O(epsilon^{-2} k log k log^{4} n) points

BigFCM: Fast, Precise and Scalable FCM on Hadoop

Clustering plays an important role in mining big data both as a modeling technique and a preprocessing step in many data mining process implementations. Fuzzy clustering provides more flexibility than non-fuzzy methods by allowing each data record to belong to more than one cluster to some degree. However, a serious challenge in fuzzy clustering is the lack of scalability. Massive datasets in emerging fields such as geosciences, biology and networking do require parallel and distributed computations with high performance to solve real-world problems. Although some clustering methods are already improved to execute on big data platforms, but their execution time is highly increased for large datasets. In this paper, a scalable Fuzzy C-Means (FCM) clustering named BigFCM is proposed and designed for the Hadoop distributed data platform. Based on the map-reduce programming model, it exploits several mechanisms including an efficient caching design to achieve several orders of magnitude reduction in execution time. Extensive evaluation over multi-gigabyte datasets shows that BigFCM is scalable while it preserves the quality of clustering