Christian Sohler

Bio: Christian Sohler is an academic researcher from University of Cologne. The author has contributed to research in topics: Property testing & Approximation algorithm. The author has an hindex of 36, co-authored 142 publications receiving 5104 citations. Previous affiliations of Christian Sohler include Technical University of Dortmund & University of Paderborn.

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

Abstract: @d can be approximated up to (1 + e)-factor, for an arbitrary small e > 0, using the O(k/e2)-rank approximation of A and a constant. This implies, for example, that the optimal k-means clustering of the rows of A is (1 + e)-approximated by an optimal k-means clustering of their projection on the O(k/e2) first right singular vectors (principle components) of A.A (j, k)-coreset for projective clustering is a small set of points that yields a (1 + e)-approximation to the sum of squared distances from the n rows of A to any set of k affine subspaces, each of dimension at most j. Our embedding yields (0, k)-coresets of size O(k) for handling k-means queries, (j, 1)-coresets of size O(j) for PCA queries, and (j, k)-coresets of size (log n)O(jk) for any j, k ≥ 1 and constant e e (0, 1/2). Previous coresets usually have a size which is linearly or even exponentially dependent of d, which makes them useless when d ~ n.Using our coresets with the merge-and-reduce approach, we obtain embarrassingly parallel streaming algorithms for problems such as k-means, PCA and projective clustering. These algorithms use update time per point and memory that is polynomial in log n and only linear in d.For cost functions other than squared Euclidean distances we suggest a simple recursive coreset construction that produces coresets of size

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

Abstract: 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.

StreamKM++: A clustering algorithm for data streams

Abstract: We develop a new k-means clustering algorithm for data streams of points from a Euclidean space. We call this algorithm StreamKM++. Our algorithm computes a small weighted sample of the data stream and solves the problem on the sample using the k-means++ algorithm of Arthur and Vassilvitskii (SODA '07). To compute the small sample, we propose two new techniques. First, we use an adaptive, nonuniform sampling approach similar to the k-means++ seeding procedure to obtain small coresets from the data stream. This construction is rather easy to implement and, unlike other coreset constructions, its running time has only a small dependency on the dimensionality of the data. Second, we propose a new data structure, which we call coreset tree. The use of these coreset trees significantly speeds up the time necessary for the adaptive, nonuniform sampling during our coreset construction.We compare our algorithm experimentally with two well-known streaming implementations: BIRCH [Zhang et al. 1997] and StreamLS [Guha et al. 2003]. In terms of quality (sum of squared errors), our algorithm is comparable with StreamLS and significantly better than BIRCH (up to a factor of 2). Besides, BIRCH requires significant effort to tune its parameters. In terms of running time, our algorithm is slower than BIRCH. Comparing the running time with StreamLS, it turns out that our algorithm scalesmuch better with increasing number of centers. We conclude that, if the first priority is the quality of the clustering, then our algorithm provides a good alternative to BIRCH and StreamLS, in particular, if the number of cluster centers is large. We also give a theoretical justification of our approach by proving that our sample set is a small coreset in low-dimensional spaces.

Counting triangles in data streams

Abstract: We present two space bounded random sampling algorithms that compute an approximation of the number of triangles in an undirected graph given as a stream of edges. Our first algorithm does not make any assumptions on the order of edges in the stream. It uses space that is inversely related to the ratio between the number of triangles and the number of triples with at least one edge in the induced subgraph, and constant expected update time per edge. Our second algorithm is designed for incidence streams (all edges incident to the same vertex appear consecutively). It uses space that is inversely related to the ratio between the number of triangles and length 2 paths in the graph and expected update time O(log|V|⋅(1+s⋅|V|/|E|)), where s is the space requirement of the algorithm. These results significantly improve over previous work [20, 8]. Since the space complexity depends only on the structure of the input graph and not on the number of nodes, our algorithms scale very well with increasing graph size and so they provide a basic tool to analyze the structure of large graphs. They have many applications, for example, in the discovery of Web communities, the computation of clustering and transitivity coefficient, and discovery of frequent patterns in large graphs.We have implemented both algorithms and evaluated their performance on networks from different application domains. The sizes of the considered graphs varied from about 8,000 nodes and 40,000 edges to 135 million nodes and more than 1 billion edges. For both algorithms we run experiments with parameter s=1,000, 10,000, 100,000, 1,000,000 to evaluate running time and approximation guarantee. Both algorithms appear to be time efficient for these sample sizes. The approximation quality of the first algorithm was varying significantly and even for s=1,000,000 we had more than 10% deviation for more than half of the instances. The second algorithm performed much better and even for s=10,000 we had an average deviation of less than 6% (taken over all but the largest instance for which we could not compute the number of triangles exactly).

StreamKM++: a clustering algorithm for data streams

Abstract: We develop a new k-means clustering algorithm for data streams, which we call StreamKM++. Our algorithm computes a small weighted sample of the data stream and solves the problem on the sample using the k-means++ algorithm [1]. To compute the small sample, we propose two new techniques. First, we use a non-uniform sampling approach similar to the k-means++ seeding procedure to obtain small coresets from the data stream. This construction is rather easy to implement and, unlike other coreset constructions, its running time has only a low dependency on the dimensionality of the data. Second, we propose a new data structure which we call a coreset tree. The use of these coreset trees significantly speeds up the time necessary for the non-uniform sampling during our coreset construction. We compare our algorithm experimentally with two well-known streaming implementations (BIRCH [16] and StreamLS [4, 9]). In terms of quality (sum of squared errors), our algorithm is comparable with StreamLS and significantly better than BIRCH (up to a factor of 2). In terms of running time, our algorithm is slower than BIRCH. Comparing the running time with StreamLS, it turns out that our algorithm scales much better with increasing number of centers. We conclude that, if the first priority is the quality of the clustering, then our algorithm provides a good alternative to BIRCH and StreamLS, in particular, if the number of cluster centers is large. We also give a theoretical justification of our approach by proving that our sample set is a small coreset in low dimensional spaces.

Data streams: algorithms and applications

Abstract: In the data stream scenario, input arrives very rapidly and there is limited memory to store the input. Algorithms have to work with one or few passes over the data, space less than linear in the input size or time significantly less than the input size. In the past few years, a new theory has emerged for reasoning about algorithms that work within these constraints on space, time, and number of passes. Some of the methods rely on metric embeddings, pseudo-random computations, sparse approximation theory and communication complexity. The applications for this scenario include IP network traffic analysis, mining text message streams and processing massive data sets in general. Researchers in Theoretical Computer Science, Databases, IP Networking and Computer Systems are working on the data stream challenges. This article is an overview and survey of data stream algorithmics and is an updated version of [1].

Data Streams: Algorithms and Applications

Abstract: 1 Introduction 2 Map 3 The Data Stream Phenomenon 4 Data Streaming: Formal Aspects 5 Foundations: Basic Mathematical Ideas 6 Foundations: Basic Algorithmic Techniques 7 Foundations: Summary 8 Streaming Systems 9 New Directions 10 Historic Notes 11 Concluding Remarks Acknowledgements References

Visual Place Recognition: A Survey

Abstract: Visual place recognition is a challenging problem due to the vast range of ways in which the appearance of real-world places can vary. In recent years, improvements in visual sensing capabilities, an ever-increasing focus on long-term mobile robot autonomy, and the ability to draw on state-of-the-art research in other disciplines—particularly recognition in computer vision and animal navigation in neuroscience—have all contributed to significant advances in visual place recognition systems. This paper presents a survey of the visual place recognition research landscape. We start by introducing the concepts behind place recognition—the role of place recognition in the animal kingdom, how a “place” is defined in a robotics context, and the major components of a place recognition system. Long-term robot operations have revealed that changing appearance can be a significant factor in visual place recognition failure; therefore, we discuss how place recognition solutions can implicitly or explicitly account for appearance change within the environment. Finally, we close with a discussion on the future of visual place recognition, in particular with respect to the rapid advances being made in the related fields of deep learning, semantic scene understanding, and video description.

Property Testing and its connection to Learning and Approximation

Abstract: In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.