arXiv subjects

Christian Sohler

Publications and source records attributed to Christian Sohler.

At least 37 records · Page 2Linked to original sources

arXiv2020

Fast and Accurate $k$-means++ via Rejection Sampling

$k$-means++ \cite{arthur2007k} is a widely used clustering algorithm that is easy to implement, has nice theoretical guarantees and strong empirical performance. Despite its wide adoption, $k$-means++ sometimes suffers from being slow on large data-sets so a natural question has been to obtain more efficient algorithms with similar guarantees. In this paper, we present a near linear time algorithm for $k$-means++ seeding. Interestingly our algorithm obtains the same theoretical guarantees as $k$-means++ and significantly improves earlier results on fast $k$-means++ seeding. Moreover, we show empirically that our algorithm is significantly faster than $k$-means++ and obtains solutions of equivalent quality.

cs.LG↗

arXiv2019

A characterization of graph properties testable for general planar graphs with one-sided error (It is all about forbidden subgraphs)

The problem of characterizing testable graph properties (properties that can be tested with a number of queries independent of the input size) is a fundamental problem in the area of property testing. While there has been some extensive prior research characterizing testable graph properties in the dense graphs model and we have good understanding of the bounded degree graphs model, no similar characterization has been known for general graphs, with no degree bounds. In this paper we take on this major challenge and consider the problem of characterizing all testable graph properties in general planar graphs. We consider the model in which a general planar graph can be accessed by the random neighbor oracle that allows access to any given vertex and access to a random neighbor of a given vertex. We show that, informally, a graph property $P$ is testable with one-sided error for general planar graphs if and only if testing $P$ can be reduced to testing for a finite family of finite forbidden subgraphs. While our presentation focuses on planar graphs, our approach extends easily to general minor-free graphs. Our analysis of the necessary condition relies on a recent construction of canonical testers in the random neighbor oracle model that is applied here to the one-sided error model for testing in planar graphs. The sufficient condition in the characterization reduces the problem to the task of testing $H$-freeness in planar graphs, and is the main and most challenging technical contribution of the paper: we show that for planar graphs (with arbitrary degrees), the property of being $H$-free is testable with one-sided error for every finite graph $H$, in the random neighbor oracle model.

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arXiv2019

Fully dynamic hierarchical diameter k-clustering and k-center

We develop dynamic data structures for maintaining a hierarchical k-center clustering when the points come from a discrete space $\{1,\ldots,Δ\}^d$. Our first data structure is for the low dimensional setting, i.e., d is a constant, and processes insertions, deletions and cluster representative queries in $\log^{O(1)} (Δn)$ time, where $n$ is the current size of the point set. For the high dimensional case and an integer parameter $\ell > 1$, we provide a randomized data structure that maintains an $O(d \ell)$-approximation. The amortized expected insertion time is $O(d^2 \ell \log n \log Δ)$. The amortized expected deletion time is $O(d^2 n^{1/\ell} \log^2 n \log Δ)$. At any point of time, with probability at least $1-1/n$, the data structure can correctly answer all queries for cluster representatives in $O(d \ell \log n \log Δ)$ time per query.

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arXiv2019

Testable Properties in General Graphs and Random Order Streaming

We present a novel framework closely linking the areas of property testing and data streaming algorithms in the setting of general graphs. It has been recently shown (Monemizadeh et al. 2017) that for bounded-degree graphs, any constant-query tester can be emulated in the random order streaming model by a streaming algorithm that uses only space required to store a constant number of words. However, in a more natural setting of general graphs, with no restriction on the maximum degree, no such results were known because of our lack of understanding of constant-query testers in general graphs and lack of techniques to appropriately emulate in the streaming setting off-line algorithms allowing many high-degree vertices. In this work we advance our understanding on both of these challenges. First, we provide canonical testers for all constant-query testers for general graphs, both, for one-sided and two-sided errors. Such canonizations were only known before (in the adjacency matrix model) for dense graphs (Goldreich and Trevisan 2003) and (in the adjacency list model) for bounded degree (di-)graphs (Goldreich and Ron 2011, Czumaj et al. 2016). Using the concept of canonical testers, we then prove that every property of general graphs that is constant-query testable with one-sided error can also be tested in constant-space with one-sided error in the random order streaming model. Our results imply, among others, that properties like $(s,t)$ disconnectivity, $k$-path-freeness, etc. are constant-space testable in random order streams.

cs.DS↗

arXiv2018

Planar Graphs: Random Walks and Bipartiteness Testing

We initiate the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of $\tilde{O}(\sqrt{n})$ for graphs on $n$ vertices. The constant-time testability was only known for planar graphs with bounded degree. Our algorithm is based on random walks. Since planar graphs have good separators, i.e., bad expansion, our analysis diverges from standard techniques that involve the fast convergence of random walks on expanders. We reduce the problem to the task of detecting an odd-parity cycle in a multigraph induced by constant-length cycles. We iteratively reduce the length of cycles while preserving the detection probability, until the multigraph collapses to a collection of easily discoverable self-loops. Our approach extends to arbitrary minor-free graphs. We also believe that our techniques will find applications to testing other properties in arbitrary minor-free graphs.

cs.DS↗

arXiv2018

Every Testable (Infinite) Property of Bounded-Degree Graphs Contains an Infinite Hyperfinite Subproperty

One of the most fundamental questions in graph property testing is to characterize the combinatorial structure of properties that are testable with a constant number of queries. We work towards an answer to this question for the bounded-degree graph model introduced in [Goldreich, Ron, 2002], where the input graphs have maximum degree bounded by a constant $d$. In this model, it is known (among other results) that every \emph{hyperfinite} property is constant-query testable [Newman, Sohler, 2013], where, informally, a graph property is hyperfinite, if for every $δ>0$ every graph in the property can be partitioned into small connected components by removing $δn$ edges. In this paper we show that hyperfiniteness plays a role in \emph{every} testable property, i.e. we show that every testable property is either finite (which trivially implies hyperfiniteness and testability) or contains an infinite hyperfinite subproperty. A simple consequence of our result is that no infinite graph property that only consists of expander graphs is constant-query testable. Based on the above findings, one could ask if every infinite testable non-hyperfinite property might contain an infinite family of expander (or near-expander) graphs. We show that this is not true. Motivated by our counter-example we develop a theorem that shows that we can partition the set of vertices of every bounded degree graph into a constant number of subsets and a separator set, such that the separator set is small and the distribution of $k$-disks on every subset of a partition class, is roughly the same as that of the partition class if the subset has small expansion.

cs.DS↗

arXiv2018

Turning Big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering

We develop and analyze a method to reduce the size of a very large set of data points in a high dimensional Euclidean space R d to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set is approximately the same as that for the original point set. For example, computing the first k principal components of the reduced set will return approximately the first k principal components of the original set or computing the centers of a k-means clustering on the reduced set will return an approximation for the original set. Such a reduced set is also known as a coreset. The main new feature of our construction is that the cardinality of the reduced set is independent of the dimension d of the input space and that the sets are mergable. The latter property means that the union of two reduced sets is a reduced set for the union of the two original sets (this property has recently also been called composability, see Indyk et. al., PODS 2014). It allows us to turn our methods into streaming or distributed algorithms using standard approaches. For problems such as k-means and subspace approximation the coreset sizes are also independent of the number of input points. Our method is based on projecting the points on a low dimensional subspace and reducing the cardinality of the points inside this subspace using known methods. The proposed approach works for a wide range of data analysis techniques including k-means clustering, principal component analysis and subspace clustering. The main conceptual contribution is a new coreset definition that allows to charge costs that appear for every solution to an additive constant.

cs.DS↗

arXiv2017

Approximating the Spectrum of a Graph

The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of approximating the spectrum $λ= (λ_1,\dots,λ_{|V|})$, $0 \le λ_1,\le \dots, \le λ_{|V|}\le 2$ of $G$ in the regime where the graph is too large to explicitly calculate the spectrum. We present a sublinear time algorithm that, given the ability to query a random node in the graph and select a random neighbor of a given node, computes a succinct representation of an approximation $\widetilde λ= (\widetilde λ_1,\dots,\widetilde λ_{|V|})$, $0 \le \widetilde λ_1,\le \dots, \le \widetilde λ_{|V|}\le 2$ such that $\|\widetilde λ- λ\|_1 \le ε|V|$. Our algorithm has query complexity and running time $exp(O(1/ε))$, independent of the size of the graph, $|V|$. We demonstrate the practical viability of our algorithm on 15 different real-world graphs from the Stanford Large Network Dataset Collection, including social networks, academic collaboration graphs, and road networks. For the smallest of these graphs, we are able to validate the accuracy of our algorithm by explicitly calculating the true spectrum; for the larger graphs, such a calculation is computationally prohibitive. In addition we study the implications of our algorithm to property testing in the bounded degree graph model.

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arXiv2017

Estimating Graph Parameters from Random Order Streams

We develop a new algorithmic technique that allows to transfer some constant time approximation algorithms for general graphs into random order streaming algorithms. We illustrate our technique by proving that in random order streams with probability at least $2/3$, $\bullet$ the number of connected components of $G$ can be approximated up to an additive error of $\varepsilon n$ using $(\frac{1}{\varepsilon})^{O(1/\varepsilon^3)}$ space, $\bullet$ the weight of a minimum spanning tree of a connected input graph with integer edges weights from $\{1,\dots,W\}$ can be approximated within a multiplicative factor of $1+\varepsilon$ using $\big(\frac{1}{\varepsilon}\big)^{\tilde O(W^3/\varepsilon^3)}$ space, $\bullet$ the size of a maximum independent set in planar graphs can be approximated within a multiplicative factor of $1+\varepsilon$ using space $2^{(1/\varepsilon)^{(1/\varepsilon)^{\log^{O(1)} (1/\varepsilon)}}}$.

cs.DS↗

arXiv2017

Testable Bounded Degree Graph Properties Are Random Order Streamable

We study which property testing and sublinear time algorithms can be transformed into graph streaming algorithms for random order streams. Our main result is that for bounded degree graphs, any property that is constant-query testable in the adjacency list model can be tested with constant space in a single-pass in random order streams. Our result is obtained by estimating the distribution of local neighborhoods of the vertices on a random order graph stream using constant space. We then show that our approach can also be applied to constant time approximation algorithms for bounded degree graphs in the adjacency list model: As an example, we obtain a constant-space single-pass random order streaming algorithms for approximating the size of a maximum matching with additive error $εn$ ($n$ is the number of nodes). Our result establishes for the first time that a large class of sublinear algorithms can be simulated in random order streams, while $Ω(n)$ space is needed for many graph streaming problems for adversarial orders.

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arXiv2017

Clustering High Dimensional Dynamic Data Streams

We present data streaming algorithms for the $k$-median problem in high-dimensional dynamic geometric data streams, i.e. streams allowing both insertions and deletions of points from a discrete Euclidean space $\{1, 2, \ldots Δ\}^d$. Our algorithms use $k ε^{-2} poly(d \log Δ)$ space/time and maintain with high probability a small weighted set of points (a coreset) such that for every set of $k$ centers the cost of the coreset $(1+ε)$-approximates the cost of the streamed point set. We also provide algorithms that guarantee only positive weights in the coreset with additional logarithmic factors in the space and time complexities. We can use this positively-weighted coreset to compute a $(1+ε)$-approximation for the $k$-median problem by any efficient offline $k$-median algorithm. All previous algorithms for computing a $(1+ε)$-approximation for the $k$-median problem over dynamic data streams required space and time exponential in $d$. Our algorithms can be generalized to metric spaces of bounded doubling dimension.

cs.DS↗

arXiv2016

Theoretical Analysis of the $k$-Means Algorithm - A Survey

The $k$-means algorithm is one of the most widely used clustering heuristics. Despite its simplicity, analyzing its running time and quality of approximation is surprisingly difficult and can lead to deep insights that can be used to improve the algorithm. In this paper we survey the recent results in this direction as well as several extension of the basic $k$-means method.

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arXiv2015

Clustering time series under the Fréchet distance

The Fréchet distance is a popular distance measure for curves. We study the problem of clustering time series under the Fréchet distance. In particular, we give $(1+\varepsilon)$-approximation algorithms for variations of the following problem with parameters $k$ and $\ell$. Given $n$ univariate time series $P$, each of complexity at most $m$, we find $k$ time series, not necessarily from $P$, which we call \emph{cluster centers} and which each have complexity at most $\ell$, such that (a) the maximum distance of an element of $P$ to its nearest cluster center or (b) the sum of these distances is minimized. Our algorithms have running time near-linear in the input size for constant $\varepsilon$, $k$ and $\ell$. To the best of our knowledge, our algorithms are the first clustering algorithms for the Fréchet distance which achieve an approximation factor of $(1+\varepsilon)$ or better. Keywords: time series, longitudinal data, functional data, clustering, Fréchet distance, dynamic time warping, approximation algorithms.

cs.CG↗

arXiv2015

Random projections for Bayesian regression

This article deals with random projections applied as a data reduction technique for Bayesian regression analysis. We show sufficient conditions under which the entire $d$-dimensional distribution is approximately preserved under random projections by reducing the number of data points from $n$ to $k\in O(\operatorname{poly}(d/\varepsilon))$ in the case $n\gg d$. Under mild assumptions, we prove that evaluating a Gaussian likelihood function based on the projected data instead of the original data yields a $(1+O(\varepsilon))$-approximation in terms of the $\ell_2$ Wasserstein distance. Our main result shows that the posterior distribution of Bayesian linear regression is approximated up to a small error depending on only an $\varepsilon$-fraction of its defining parameters. This holds when using arbitrary Gaussian priors or the degenerate case of uniform distributions over $\mathbb{R}^d$ for $β$. Our empirical evaluations involve different simulated settings of Bayesian linear regression. Our experiments underline that the proposed method is able to recover the regression model up to small error while considerably reducing the total running time.

stat.CO↗

arXiv2015

Testing Cluster Structure of Graphs

We study the problem of recognizing the cluster structure of a graph in the framework of property testing in the bounded degree model. Given a parameter $\varepsilon$, a $d$-bounded degree graph is defined to be $(k, ϕ)$-clusterable, if it can be partitioned into no more than $k$ parts, such that the (inner) conductance of the induced subgraph on each part is at least $ϕ$ and the (outer) conductance of each part is at most $c_{d,k}\varepsilon^4ϕ^2$, where $c_{d,k}$ depends only on $d,k$. Our main result is a sublinear algorithm with the running time $\widetilde{O}(\sqrt{n}\cdot\mathrm{poly}(ϕ,k,1/\varepsilon))$ that takes as input a graph with maximum degree bounded by $d$, parameters $k$, $ϕ$, $\varepsilon$, and with probability at least $\frac23$, accepts the graph if it is $(k,ϕ)$-clusterable and rejects the graph if it is $\varepsilon$-far from $(k, ϕ^*)$-clusterable for $ϕ^* = c'_{d,k}\frac{ϕ^2 \varepsilon^4}{\log n}$, where $c'_{d,k}$ depends only on $d,k$. By the lower bound of $Ω(\sqrt{n})$ on the number of queries needed for testing graph expansion, which corresponds to $k=1$ in our problem, our algorithm is asymptotically optimal up to polylogarithmic factors.

cs.DS↗

arXiv2014

Asymptotically exact streaming algorithms

We introduce a new computational model for data streams: asymptotically exact streaming algorithms. These algorithms have an approximation ratio that tends to one as the length of the stream goes to infinity while the memory used by the algorithm is restricted to polylog(n) size. Thus, the output of the algorithm is optimal in the limit. We show positive results in our model for a series of important problems that have been discussed in the streaming literature. These include computing the frequency moments, clustering problems and least squares regression. Our results also include lower bounds for problems, which have streaming algorithms in the ordinary setting but do not allow for sublinear space algorithms in our model.

cs.DS↗

arXiv2014

Analysis of Agglomerative Clustering

The diameter $k$-clustering problem is the problem of partitioning a finite subset of $\mathbb{R}^d$ into $k$ subsets called clusters such that the maximum diameter of the clusters is minimized. One early clustering algorithm that computes a hierarchy of approximate solutions to this problem (for all values of $k$) is the agglomerative clustering algorithm with the complete linkage strategy. For decades, this algorithm has been widely used by practitioners. However, it is not well studied theoretically. In this paper, we analyze the agglomerative complete linkage clustering algorithm. Assuming that the dimension $d$ is a constant, we show that for any $k$ the solution computed by this algorithm is an $O(\log k)$-approximation to the diameter $k$-clustering problem. Our analysis does not only hold for the Euclidean distance but for any metric that is based on a norm. Furthermore, we analyze the closely related $k$-center and discrete $k$-center problem. For the corresponding agglomerative algorithms, we deduce an approximation factor of $O(\log k)$ as well.

cs.DS↗

arXiv2013

Property-Testing in Sparse Directed Graphs: 3-Star-Freeness and Connectivity

We study property testing in directed graphs in the bounded degree model, where we assume that an algorithm may only query the outgoing edges of a vertex, a model proposed by Bender and Ron in 2002. As our first main result, we we present a property testing algorithm for strong connectivity in this model, having a query complexity of $\mathcal{O}(n^{1-ε/(3+α)})$ for arbitrary $α>0$; it is based on a reduction to estimating the vertex indegree distribution. For subgraph-freeness we give a property testing algorithm with a query complexity of $\mathcal{O}(n^{1-1/k})$, where $k$ is the number of connected componentes in the queried subgraph which have no incoming edge. We furthermore take a look at the problem of testing whether a weakly connected graph contains vertices with a degree of least $3$, which can be viewed as testing for freeness of all orientations of $3$-stars; as our second main result, we show that this property can be tested with a query complexity of $\mathcal{O}(\sqrt{n})$ instead of, what would be expected, $Ω(n^{2/3})$.

cs.DS↗