Using Centroidal Voronoi Tessellations to Scale Up the Multi-dimensional Archive of Phenotypic Elites Algorithm

The recently introduced Multi-dimensional Archive of Phenotypic Elites (MAP-Elites) is an evolutionary algorithm capable of producing a large archive of diverse, high-performing solutions in a single run. It works by discretizing a continuous feature space into unique regions according to the desired discretization per dimension. While simple, this algorithm has a main drawback: it cannot scale to high-dimensional feature spaces since the number of regions increase exponentially with the number of dimensions. In this paper, we address this limitation by introducing a simple extension of MAP-Elites that has a constant, pre-defined number of regions irrespective of the dimensionality of the feature space. Our main insight is that methods from computational geometry could partition a high-dimensional space into well-spread geometric regions. In particular, our algorithm uses a centroidal Voronoi tessellation (CVT) to divide the feature space into a desired number of regions; it then places every generated individual in its closest region, replacing a less fit one if the region is already occupied. We demonstrate the effectiveness of the new "CVT-MAP-Elites" algorithm in high-dimensional feature spaces through comparisons against MAP-Elites in maze navigation and hexapod locomotion tasks

Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors

[See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the $c$-Approximate Near Neighbor Search problem. For the $d$-dimensional Euclidean space and $n$-point datasets, we develop a data structure with space $n^{1 + \rho_u + o(1)} + O(dn)$ and query time $n^{\rho_q + o(1)} + d n^{o(1)}$ for every $\rho_u, \rho_q \geq 0$ such that: \begin{equation} c^2 \sqrt{\rho_q} + (c^2 - 1) \sqrt{\rho_u} = \sqrt{2c^2 - 1}. \end{equation} This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor $c > 1$, improving upon [Kapralov, PODS 2015]. The data structure is a culmination of a long line of work on the problem for all space regimes; it builds on Spherical Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni, Razenshteyn, STOC 2015]. Our matching lower bounds are of two types: conditional and unconditional. First, we prove tightness of the whole above trade-off in a restricted model of computation, which captures all known hashing-based approaches. We then show unconditional cell-probe lower bounds for one and two probes that match the above trade-off for $\rho_q = 0$, improving upon the best known lower bounds from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than the one-probe bound. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.Comment: 62 pages, 5 figures; a merger of arXiv:1511.07527 [cs.DS] and arXiv:1605.02701 [cs.DS], which subsumes both of the preprints. New version contains more elaborated proofs and fixed some typo

Lower Bounds on Time-Space Trade-Offs for Approximate Near Neighbors

We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In articular, our lower bound matches the upper bound recently shown in [Laarhoven 2015] for the random instance on a Euclidean sphere (which we show in fact extends to the entire space $\mathbb{R}^d$ using the techniques from [Andoni, Razenshteyn 2015]). We also show tight, unconditional cell-probe lower bounds for one and two probes, improving upon the best known bounds from [Panigrahy, Talwar, Wieder 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than for one probe. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.Comment: 47 pages, 2 figures; v2: substantially revised introduction, lots of small corrections; subsumed by arXiv:1608.03580 [cs.DS] (along with arXiv:1511.07527 [cs.DS]

Complex queries and complex data

With the widespread availability of wearable computers, equipped with sensors such as GPS or cameras, and with the ubiquitous presence of micro-blogging platforms, social media sites and digital marketplaces, data can be collected and shared on a massive scale. A necessary building block for taking advantage from this vast amount of information are efficient and effective similarity search algorithms that are able to find objects in a database which are similar to a query object. Due to the general applicability of similarity search over different data types and applications, the formalization of this concept and the development of strategies for evaluating similarity queries has evolved to an important field of research in the database community, spatio-temporal database community, and others, such as information retrieval and computer vision. This thesis concentrates on a special instance of similarity queries, namely k-Nearest Neighbor (kNN) Queries and their close relative, Reverse k-Nearest Neighbor (RkNN) Queries. As a first contribution we provide an in-depth analysis of the RkNN join. While the problem of reverse nearest neighbor queries has received a vast amount of research interest, the problem of performing such queries in a bulk has not seen an in-depth analysis so far. We first formalize the RkNN join, identifying its monochromatic and bichromatic versions and their self-join variants. After pinpointing the monochromatic RkNN join as an important and interesting instance, we develop solutions for this class, including a self-pruning and a mutual pruning algorithm. We then evaluate these algorithms extensively on a variety of synthetic and real datasets. From this starting point of similarity queries on certain data we shift our focus to uncertain data, addressing nearest neighbor queries in uncertain spatio-temporal databases. Starting from the traditional definition of nearest neighbor queries and a data model for uncertain spatio-temporal data, we develop efficient query mechanisms that consider temporal dependencies during query evaluation. We define intuitive query semantics, aiming not only at returning the objects closest to the query but also their probability of being a nearest neighbor. After theoretically evaluating these query predicates we develop efficient querying algorithms for the proposed query predicates. Given the findings of this research on nearest neighbor queries, we extend these results to reverse nearest neighbor queries. Finally we address the problem of querying large datasets containing set-based objects, namely image databases, where images are represented by (multi-)sets of vectors and additional metadata describing the position of features in the image. We aim at reducing the number of kNN queries performed during query processing and evaluate a modified pipeline that aims at optimizing the query accuracy at a small number of kNN queries. Additionally, as feature representations in object recognition are moving more and more from the real-valued domain to the binary domain, we evaluate efficient indexing techniques for binary feature vectors.Nicht nur durch die Verbreitung von tragbaren Computern, die mit einer Vielzahl von Sensoren wie GPS oder Kameras ausgestattet sind, sondern auch durch die breite Nutzung von Microblogging-Plattformen, Social-Media Websites und digitale Marktplätze wie Amazon und Ebay wird durch die User eine gigantische Menge an Daten veröffentlicht. Um aus diesen Daten einen Mehrwert erzeugen zu können bedarf es effizienter und effektiver Algorithmen zur Ähnlichkeitssuche, die zu einem gegebenen Anfrageobjekt ähnliche Objekte in einer Datenbank identifiziert. Durch die Allgemeinheit dieses Konzeptes der Ähnlichkeit über unterschiedliche Datentypen und Anwendungen hinweg hat sich die Ähnlichkeitssuche zu einem wichtigen Forschungsfeld, nicht nur im Datenbankumfeld oder im Bereich raum-zeitlicher Datenbanken, sondern auch in anderen Forschungsgebieten wie dem Information Retrieval oder dem Maschinellen Sehen entwickelt. In der vorliegenden Arbeit beschäftigen wir uns mit einem speziellen Anfrageprädikat im Bereich der Ähnlichkeitsanfragen, mit k-nächste Nachbarn (kNN) Anfragen und ihrem Verwandten, den Revers k-nächsten Nachbarn (RkNN) Anfragen. In einem ersten Beitrag analysieren wir den RkNN Join. Obwohl das Problem von reverse nächsten Nachbar Anfragen in den letzten Jahren eine breite Aufmerksamkeit in der Forschungsgemeinschaft erfahren hat, wurde das Problem eine Menge von RkNN Anfragen gleichzeitig auszuführen nicht ausreichend analysiert. Aus diesem Grund formalisieren wir das Problem des RkNN Joins mit seinen monochromatischen und bichromatischen Varianten. Wir identifizieren den monochromatischen RkNN Join als einen wichtigen und interessanten Fall und entwickeln entsprechende Anfragealgorithmen. In einer detaillierten Evaluation vergleichen wir die ausgearbeiteten Verfahren auf einer Vielzahl von synthetischen und realen Datensätzen. Nach diesem Kapitel über Ähnlichkeitssuche auf sicheren Daten konzentrieren wir uns auf unsichere Daten, speziell im Bereich raum-zeitlicher Datenbanken. Ausgehend von der traditionellen Definition von Nachbarschaftsanfragen und einem Datenmodell für unsichere raum-zeitliche Daten entwickeln wir effiziente Anfrageverfahren, die zeitliche Abhängigkeiten bei der Anfragebearbeitung beachten. Zu diesem Zweck definieren wir Anfrageprädikate die nicht nur die Objekte zurückzugeben, die dem Anfrageobjekt am nächsten sind, sondern auch die Wahrscheinlichkeit mit der sie ein nächster Nachbar sind. Wir evaluieren die definierten Anfrageprädikate theoretisch und entwickeln effiziente Anfragestrategien, die eine Anfragebearbeitung zu vertretbaren Laufzeiten gewährleisten. Ausgehend von den Ergebnissen für Nachbarschaftsanfragen erweitern wir unsere Ergebnisse auf Reverse Nachbarschaftsanfragen. Zuletzt behandeln wir das Problem der Anfragebearbeitung bei Mengen-basierten Objekten, die zum Beispiel in Bilddatenbanken Verwendung finden: Oft werden Bilder durch eine Menge von Merkmalsvektoren und zusätzliche Metadaten (zum Beispiel die Position der Merkmale im Bild) dargestellt. Wir evaluieren eine modifizierte Pipeline, die darauf abzielt, die Anfragegenauigkeit bei einer kleinen Anzahl an kNN-Anfragen zu maximieren. Da reellwertige Merkmalsvektoren im Bereich der Objekterkennung immer öfter durch Bitvektoren ersetzt werden, die sich durch einen geringeren Speicherplatzbedarf und höhere Laufzeiteffizienz auszeichnen, evaluieren wir außerdem Indexierungsverfahren für Binärvektoren

On the expected diameter, width, and complexity of a stochastic convex-hull

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both $n$ and $d$. For width, two approximation algorithms are provided: a deterministic $O(1)$-approximation running in $O(n^{d+1} \log n)$ time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact $O(n^d)$-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest