Angle Tree: Nearest Neighbor Search in High Dimensions with Low Intrinsic Dimensionality

We propose an extension of tree-based space-partitioning indexing structures for data with low intrinsic dimensionality embedded in a high dimensional space. We call this extension an Angle Tree. Our extension can be applied to both classical kd-trees as well as the more recent rp-trees. The key idea of our approach is to store the angle (the "dihedral angle") between the data region (which is a low dimensional manifold) and the random hyperplane that splits the region (the "splitter"). We show that the dihedral angle can be used to obtain a tight lower bound on the distance between the query point and any point on the opposite side of the splitter. This in turn can be used to efficiently prune the search space. We introduce a novel randomized strategy to efficiently calculate the dihedral angle with a high degree of accuracy. Experiments and analysis on real and synthetic data sets shows that the Angle Tree is the most efficient known indexing structure for nearest neighbor queries in terms of preprocessing and space usage while achieving high accuracy and fast search time.Comment: To be submitted to IEEE Transactions on Pattern Analysis and Machine Intelligenc

Exploring New Forms of Random Projections for Prediction and Dimensionality Reduction in Big-Data Regimes

The story of this work is dimensionality reduction. Dimensionality reduction is a method that takes as input a point-set P of n points in R^d where d is typically large and attempts to find a lower-dimensional representation of that dataset, in order to ease the burden of processing for down-stream algorithms. In today’s landscape of machine learning, researchers and practitioners work with datasets that either have a very large number of samples, and or include high-dimensional samples. Therefore, dimensionality reduction is applied as a pre-processing technique primarily to overcome the curse of dimensionality. Generally, dimensionality reduction improves time and storage space required for processing the point-set, removes multi-collinearity and redundancies in the dataset where different features may depend on one another, and may enable simple visualizations of the dataset in 2-D and 3-D making the relationships in the data easy for humans to comprehend. Dimensionality reduction methods come in many shapes and sizes. Methods such as Principal Component Analysis (PCA), Multi-dimensional Scaling, IsoMaps, and Locally Linear Embeddings are amongst the most commonly used method of this family of algorithms. However, the choice of dimensionality reduction method proves critical in many applications as there is no one-size-fits-all solution, and special care must be considered for different datasets and tasks. Furthermore, the aforementioned popular methods are data-dependent, and commonly rely on computing either the Kernel / Gram matrix or the covariance matrix of the dataset. These matrices scale with increasing number of samples and increasing number of data dimensions, respectively, and are consequently poor choices in today’s landscape of big-data applications. Therefore, it is pertinent to develop new dimensionality reduction methods that can be efficiently applied to large and high-dimensional datasets, by either reducing the dependency on the data, or side-stepping it altogether. Furthermore, such new dimensionality reduction methods should be able to perform on par with, or better than, traditional methods such as PCA. To achieve this goal, we turn to a simple and powerful method called random projections. Random projections are a simple, efficient, and data-independent method for stably embedding a point-set P of n points in R^d to R^k where d is typically large and k is on the order of log n. Random projections have a long history of use in dimensionality reduction literature with great success. In this work, we are inspired to build on the ideas of random projection theory, and extend the framework and build a powerful new setup of random projections for large high-dimensional datasets, with comparable performance to state-of-the-art data-dependent and nonlinear methods. Furthermore, we study the use of random projections in domains other than dimensionality reduction, including prediction, and show the competitive performance of such methods for processing small dataset regimes

Self-Supervised Motion Retargeting with Safety Guarantee

In this paper, we present self-supervised shared latent embedding (S3LE), a data-driven motion retargeting method that enables the generation of natural motions in humanoid robots from motion capture data or RGB videos. While it requires paired data consisting of human poses and their corresponding robot configurations, it significantly alleviates the necessity of time-consuming data-collection via novel paired data generating processes. Our self-supervised learning procedure consists of two steps: automatically generating paired data to bootstrap the motion retargeting, and learning a projection-invariant mapping to handle the different expressivity of humans and humanoid robots. Furthermore, our method guarantees that the generated robot pose is collision-free and satisfies position limits by utilizing nonparametric regression in the shared latent space. We demonstrate that our method can generate expressive robotic motions from both the CMU motion capture database and YouTube videos

Single View Reconstruction for Human Face and Motion with Priors

Single view reconstruction is fundamentally an under-constrained problem. We aim to develop new approaches to model human face and motion with model priors that restrict the space of possible solutions. First, we develop a novel approach to recover the 3D shape from a single view image under challenging conditions, such as large variations in illumination and pose. The problem is addressed by employing the techniques of non-linear manifold embedding and alignment. Specifically, the local image models for each patch of facial images and the local surface models for each patch of 3D shape are learned using a non-linear dimensionality reduction technique, and the correspondences between these local models are then learned by a manifold alignment method. Local models successfully remove the dependency of large training databases for human face modeling. By combining the local shapes, the global shape of a face can be reconstructed directly from a single linear system of equations via least square. Unfortunately, this learning-based approach cannot be successfully applied to the problem of human motion modeling due to the internal and external variations in single view video-based marker-less motion capture. Therefore, we introduce a new model-based approach for capturing human motion using a stream of depth images from a single depth sensor. While a depth sensor provides metric 3D information, using a single sensor, instead of a camera array, results in a view-dependent and incomplete measurement of object motion. We develop a novel two-stage template fitting algorithm that is invariant to subject size and view-point variations, and robust to occlusions. Starting from a known pose, our algorithm first estimates a body configuration through temporal registration, which is used to search the template motion database for a best match. The best match body configuration as well as its corresponding surface mesh model are deformed to fit the input depth map, filling in the part that is occluded from the input and compensating for differences in pose and body-size between the input image and the template. Our approach does not require any makers, user-interaction, or appearance-based tracking. Experiments show that our approaches can achieve good modeling results for human face and motion, and are capable of dealing with variety of challenges in single view reconstruction, e.g., occlusion

Manifold Learning for Natural Image Sets, Doctoral Dissertation August 2006

The field of manifold learning provides powerful tools for parameterizing high-dimensional data points with a small number of parameters when this data lies on or near some manifold. Images can be thought of as points in some high-dimensional image space where each coordinate represents the intensity value of a single pixel. These manifold learning techniques have been successfully applied to simple image sets, such as handwriting data and a statue in a tightly controlled environment. However, they fail in the case of natural image sets, even those that only vary due to a single degree of freedom, such as a person walking or a heart beating. Parameterizing data sets such as these will allow for additional constraints on traditional computer vision problems such as segmentation and tracking. This dissertation explores the reasons why classical manifold learning algorithms fail on natural image sets and proposes new algorithms for parameterizing this type of data

High-dimensional polytopes defined by oracles: algorithms, computations and applications

Η επεξεργασία και ανάλυση γεωμετρικών δεδομένων σε υψηλές διαστάσεις διαδραματίζει ένα θεμελιώδη ρόλο σε διάφορους κλάδους της επιστήμης και της μηχανικής. Τις τελευταίες δεκαετίες έχουν αναπτυχθεί πολλοί επιτυχημένοι γεωμετρικοί αλγόριθμοι σε 2 και 3 διαστάσεις. Ωστόσο, στις περισσότερες περιπτώσεις, οι επιδόσεις τους σε υψηλότερες διαστάσεις δεν είναι ικανοποιητικές. Αυτή η συμπεριφορά είναι ευρέως γνωστή ως κατάρα των μεγάλων διαστάσεων (curse of dimensionality). Δυο πλαίσια λύσης που έχουν υιοθετηθεί για να ξεπεραστεί αυτή η δυσκολία είναι η εκμετάλλευση της ειδικής δομής των δεδομένων, όπως σε περιπτώσεις αραιών (sparse) δεδομένων ή στην περίπτωση που τα δεδομένα βρίσκονται σε χώρο χαμηλότερης διάστασης, και ο σχεδιασμός προσεγγιστικών αλγορίθμων. Στη διατριβή αυτή μελετάμε προβλήματα μέσα σε αυτά τα πλαίσια. Το κύριο ερευνητικό πεδίο της παρούσας εργασίας είναι η διακριτή και υπολογιστικής γεωμετρία και οι σχέσεις της με τους κλάδους της επιστήμης των υπολογιστών και τα εφαρμοσμένα μαθηματικά, όπως είναι η θεωρία πολυτόπων, οι υλοποιήσεις αλγορίθμων, οι πιθανοθεωρητικοί γεωμετρικοί αλγόριθμοι, η υπολογιστική αλγεβρική γεωμετρία και η βελτιστοποίηση. Τα θεμελιώδη γεωμετρικά αντικείμενα της μελέτης μας είναι τα πολύτοπα, και οι βασικές τους ιδιότητες είναι η κυρτότητα και ότι ορίζονται από ένα μαντείο (oracle) σε ένα χώρο υψηλής διάστασης. Η επεξεργασία και ανάλυση γεωμετρικών δεδομένων σε υψηλές διαστάσεις διαδραματίζει ένα θεμελιώδη ρόλο σε διάφορους κλάδους της επιστήμης και της μηχανικής. Τις τελευταίες δεκαετίες έχουν αναπτυχθεί πολλοί επιτυχημένοι γεωμετρικοί αλγόριθμοι σε 2 και 3 διαστάσεις. Ωστόσο, στις περισσότερες περιπτώσεις, οι επιδόσεις τους σε υψηλότερες διαστάσεις δεν είναι ικανοποιητικές. Δυο πλαίσια λύσης που έχουν υιοθετηθεί για να ξεπεραστεί αυτή η δυσκολία είναι η εκμετάλλευση της ειδικής δομής των δεδομένων, όπως σε περιπτώσεις αραιών (sparse) δεδομένων ή στην περίπτωση που τα δεδομένα βρίσκονται σε χώρο χαμηλότερης διάστασης, και ο σχεδιασμός προσεγγιστικών αλγορίθμων. Το κύριο ερευνητικό πεδίο της παρούσας εργασίας είναι η διακριτή και υπολογιστικής γεωμετρία και οι σχέσεις της με τους κλάδους της επιστήμης των υπολογιστών και τα εφαρμοσμένα μαθηματικά. Η συμβολή αυτής της διατριβής είναι τριπλή. Πρώτον, στο σχεδιασμό και την ανάλυση των γεωμετρικών αλγορίθμων για προβλήματα σε μεγάλες διαστάσεις. Δεύτερον, θεωρητικά αποτελέσματα σχετικά με το συνδυαστικό χαρακτηρισμό βασικών οικογενειών πολυτόπων. Τρίτον, η εφαρμογή και πειραματική ανάλυση των προτεινόμενων αλγορίθμων και μεθόδων. Η ανάπτυξη λογισμικού ανοιχτού κώδικα, που είναι διαθέσιμο στο κοινό και βασίζεται και επεκτείνει διαδεδομένες γεωμετρικές και αλγεβρικές βιβλιοθήκες λογισμικού, όπως η CGAL και το polymake.The processing and analysis of high dimensional geometric data plays a fundamental role in disciplines of science and engineering. The last decades many successful geometric algorithms has been developed in 2 and 3 dimensions. However, in most cases their performance in higher dimensions is poor. This behavior is commonly called the curse of dimensionality. A solution framework adopted for the healing of the curse of dimensionality is the exploitation of the special structure of the data, such as sparsity or low intrinsic dimension and the design of approximation algorithms. The main research area of this thesis is discrete and computational geometry and its connections to branches of computer science and applied mathematics. The contribution of this thesis is threefold. First, the design and analysis of geometric algorithms for problems concerning high-dimensional, convex polytopes, such as convex hull and volume computation and their applications to computational algebraic geometry and optimization. Second, the establishment of combinatorial characterization results for essential polytope families. Third, the implementation and experimental analysis of the proposed algorithms and methods. The developed software is opensource, publicly available and builds on and extends state-of-the-art geometric and algebraic software libraries such as CGAL and polymake

Hybrid architecture for metric space searches

Every day, new technologies are developed to combine the facilities arranged for shared memory systems with the facilities that provide distributed memory systems. This paper proposes a hybrid system that enables communication between threads running in a shared memory environment and a cluster of computers. To do this we use specific directives provided by MPI to solve a problem of similarity search on metric spaces .This work is part of a larger project that deals with improving query searches over high dimensional spaces, managing large volumes of data, reducing the number of distance evaluations and query response times. While the proposal of this work may be generalized and used for other problems, the results show that the proposed hybrid algorithm allows a significant improvement. This work is part of a larger project that deals with improving the execution of parallel algorithms using a hybrid architecture. The goal is to take advantage of the features and facilities provided by the new parallel architectures that combine distributed and shared memory systems. The former allows to solve large scale problems while the second allows better use of resources.Presentado en el XI Workshop Procesamiento Distribuido y Paralelo (WPDP)Red de Universidades con Carreras en Informática (RedUNCI

Three-dimensional Laser-based Classification in Outdoor Environments

Robotics research strives for deploying autonomous systems in populated environments, such as inner city traffic. Autonomous cars need a reliable collision avoidance, but also an object recognition to distinguish different classes of traffic participants. For both tasks, fast three-dimensional laser range sensors generating multiple accurate laser range scans per second, each consisting of a vast number of laser points, are often employed. In this thesis, we investigate and develop classification algorithms that allow us to automatically assign semantic labels to laser scans. We mainly face two challenges: (1) we have to ensure consistent and correct classification results and (2) we must efficiently process a vast number of laser points per scan. In consideration of these challenges, we cover both stages of classification -- the feature extraction from laser range scans and the classification model that maps from the features to semantic labels. As for the feature extraction, we contribute by thoroughly evaluating important state-of-the-art histogram descriptors. We investigate critical parameters of the descriptors and experimentally show for the first time that the classification performance can be significantly improved using a large support radius and a global reference frame. As for learning the classification model, we contribute with new algorithms that improve the classification efficiency and accuracy. Our first approach aims at deriving a consistent point-wise interpretation of the whole laser range scan. By combining efficient similarity-preserving hashing and multiple linear classifiers, we considerably improve the consistency of label assignments, requiring only minimal computational overhead compared to a single linear classifier. In the last part of the thesis, we aim at classifying objects represented by segments. We propose a novel hierarchical segmentation approach comprising multiple stages and a novel mixture classification model of multiple bag-of-words vocabularies. We demonstrate superior performance of both approaches compared to their single component counterparts using challenging real world datasets.Ziel des Forschungsbereichs Robotik ist der Einsatz autonomer Systeme in natürlichen Umgebungen, wie zum Beispiel innerstädtischem Verkehr. Autonome Fahrzeuge benötigen einerseits eine zuverlässige Kollisionsvermeidung und andererseits auch eine Objekterkennung zur Unterscheidung verschiedener Klassen von Verkehrsteilnehmern. Verwendung finden vorallem drei-dimensionale Laserentfernungssensoren, die mehrere präzise Laserentfernungsscans pro Sekunde erzeugen und jeder Scan besteht hierbei aus einer hohen Anzahl an Laserpunkten. In dieser Dissertation widmen wir uns der Untersuchung und Entwicklung neuartiger Klassifikationsverfahren zur automatischen Zuweisung von semantischen Objektklassen zu Laserpunkten. Hierbei begegnen wir hauptsächlich zwei Herausforderungen: (1) wir möchten konsistente und korrekte Klassifikationsergebnisse erreichen und (2) die immense Menge an Laserdaten effizient verarbeiten. Unter Berücksichtigung dieser Herausforderungen untersuchen wir beide Verarbeitungsschritte eines Klassifikationsverfahrens -- die Merkmalsextraktion unter Nutzung von Laserdaten und das eigentliche Klassifikationsmodell, welches die Merkmale auf semantische Objektklassen abbildet. Bezüglich der Merkmalsextraktion leisten wir ein Beitrag durch eine ausführliche Evaluation wichtiger Histogrammdeskriptoren. Wir untersuchen kritische Deskriptorparameter und zeigen zum ersten Mal, dass die Klassifikationsgüte unter Nutzung von großen Merkmalsradien und eines globalen Referenzrahmens signifikant gesteigert wird. Bezüglich des Lernens des Klassifikationsmodells, leisten wir Beiträge durch neue Algorithmen, welche die Effizienz und Genauigkeit der Klassifikation verbessern. In unserem ersten Ansatz möchten wir eine konsistente punktweise Interpretation des gesamten Laserscans erreichen. Zu diesem Zweck kombinieren wir eine ähnlichkeitserhaltende Hashfunktion und mehrere lineare Klassifikatoren und erreichen hierdurch eine erhebliche Verbesserung der Konsistenz der Klassenzuweisung bei minimalen zusätzlichen Aufwand im Vergleich zu einem einzelnen linearen Klassifikator. Im letzten Teil der Dissertation möchten wir Objekte, die als Segmente repräsentiert sind, klassifizieren. Wir stellen eine neuartiges hierarchisches Segmentierungsverfahren und ein neuartiges Klassifikationsmodell auf Basis einer Mixtur mehrerer bag-of-words Vokabulare vor. Wir demonstrieren unter Nutzung von praxisrelevanten Datensätzen, dass beide Ansätze im Vergleich zu ihren Entsprechungen aus einer einzelnen Komponente zu erheblichen Verbesserungen führen