Effiziente Algorithmen

[no abstract available

Compressive Data Gathering in Wireless Sensor Networks

The thesis focuses on collecting data from wireless sensors which are deployed randomly in a region. These sensors are widely used in applications ranging from tracking to the monitoring of environment, traffic and health among others. These energy constrained sensors, once deployed may receive little or no maintenance. Hence gathering data in the most energy efficient manner becomes critical for the longevity of wireless sensor networks (WSNs). Recently, Compressive data gathering (CDG) has emerged as a useful method for collecting sensory data in WSN; this technique is able to reduce global scale communication cost without introducing intensive computation, and is capable of extending the lifetime of the entire sensor network by balancing the forwarding load across the network. This is particularly true due to the benefits obtained from in-network data compression. With CDG, the central unit, instead of receiving data from all sensors in the network, it may receive very few compressed or weighted sums from sensors, and eventually recovers the original data. To prolong the lifetime of WSN, in this thesis, we present data gathering methods based on CDG. More specifically, we propose data gathering schemes using CDG by building up data aggregation trees from sensor nodes to a central unit (sink). Our problem aims at minimizing the number of links in the forwarding trees to minimize the number of overall transmissions. First, we mathematically formulate the problem and solve it using optimization program. Owing to its complexity, we present real-time algorithmic (centralized and decentralized) methods to efficiently solve the problem. We also explore the benefits one may obtain when jointly applying compressive data gathering with network coding in a wireless sensor network. Finally, and in the context of compressive data gathering, we study the problem of joint forwarding tree construction and scheduling under a realistic interference model, and propose some efficient distributed methods for solving it. We also present a primal dual decomposition method, using the theory of column generation, to solve this complex problem

Approximating Continuous Functions on Persistence Diagrams Using Template Functions

The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map persistence diagrams into $\mathbb{R}^n$, in a way which maximizes the structure preserved. This process is commonly referred to as featurization. In this paper, we describe a mathematical framework for featurization using template functions. These functions are general as they are only required to be continuous and compactly supported. We discuss two realizations: tent functions, which emphasize the local contributions of points in a persistence diagram, and interpolating polynomials, which capture global pairwise interactions. We combine the resulting features with classification and regression algorithms on several examples including shape data and the Rossler system. Our results show that using template functions yields high accuracy rates that match and often exceed those of existing featurization methods. One counter-intuitive observation is that in most cases using interpolating polynomials, where each point contributes globally to the feature vector, yields significantly better results than using tent functions, where the contribution of each point is localized. Along the way, we provide a complete characterization of compactness in the space of persistence diagrams

Associative Memories in the Feature Space

An autoassociative memory model is a function that, given a set of data points, takes as input an arbitrary vector and outputs the most similar data point from the memorized set. However, popular memory models fail to retrieve images even when the corruption is mild and easy to detect for a human evaluator. This is because similarities are evaluated in the raw pixel space, which does not contain any semantic information about the images. This problem can be easily solved by computing \emph{similarities} in an embedding space instead of the pixel space. We show that an effective way of computing such embeddings is via a network pretrained with a contrastive loss. As the dimension of embedding spaces is often significantly smaller than the pixel space, we also have a faster computation of similarity scores. We test this method on complex datasets such as CIFAR10 and STL10. An additional drawback of current models is the need of storing the whole dataset in the pixel space, which is often extremely large. We relax this condition and propose a class of memory models that only stores low-dimensional semantic embeddings, and uses them to retrieve similar, but not identical, memories. We demonstrate a proof of concept of this method on a simple task on the MNIST dataset.Comment: 8 Pages, 4 Figures, accepted for publication at ECAI 202