Unsupervised Representation Learning with Minimax Distance Measures

We investigate the use of Minimax distances to extract in a nonparametric way the features that capture the unknown underlying patterns and structures in the data. We develop a general-purpose and computationally efficient framework to employ Minimax distances with many machine learning methods that perform on numerical data. We study both computing the pairwise Minimax distances for all pairs of objects and as well as computing the Minimax distances of all the objects to/from a fixed (test) object. We first efficiently compute the pairwise Minimax distances between the objects, using the equivalence of Minimax distances over a graph and over a minimum spanning tree constructed on that. Then, we perform an embedding of the pairwise Minimax distances into a new vector space, such that their squared Euclidean distances in the new space equal to the pairwise Minimax distances in the original space. We also study the case of having multiple pairwise Minimax matrices, instead of a single one. Thereby, we propose an embedding via first summing up the centered matrices and then performing an eigenvalue decomposition to obtain the relevant features. In the following, we study computing Minimax distances from a fixed (test) object which can be used for instance in K-nearest neighbor search. Similar to the case of all-pair pairwise Minimax distances, we develop an efficient and general-purpose algorithm that is applicable with any arbitrary base distance measure. Moreover, we investigate in detail the edges selected by the Minimax distances and thereby explore the ability of Minimax distances in detecting outlier objects. Finally, for each setting, we perform several experiments to demonstrate the effectiveness of our framework.Comment: 32 page

Nonparametric Feature Extraction from Dendrograms

We propose feature extraction from dendrograms in a nonparametric way. The Minimax distance measures correspond to building a dendrogram with single linkage criterion, with defining specific forms of a level function and a distance function over that. Therefore, we extend this method to arbitrary dendrograms. We develop a generalized framework wherein different distance measures can be inferred from different types of dendrograms, level functions and distance functions. Via an appropriate embedding, we compute a vector-based representation of the inferred distances, in order to enable many numerical machine learning algorithms to employ such distances. Then, to address the model selection problem, we study the aggregation of different dendrogram-based distances respectively in solution space and in representation space in the spirit of deep representations. In the first approach, for example for the clustering problem, we build a graph with positive and negative edge weights according to the consistency of the clustering labels of different objects among different solutions, in the context of ensemble methods. Then, we use an efficient variant of correlation clustering to produce the final clusters. In the second approach, we investigate the sequential combination of different distances and features sequentially in the spirit of multi-layered architectures to obtain the final features. Finally, we demonstrate the effectiveness of our approach via several numerical studies

Inference of Effective Pairwise Relations for Data Processing

In various data science and artificial intelligence areas, representation learning is a performance-critical step. While different representation learning methods can detect different descriptive and latent features, many representation learning methods reflect on pairwise relations. The thesis consists of two parts, studying pairwise relations from two points of view: i) Pairwise relations between the states of a Markov chain. ii) Pairwise relations between objects in a dataset based on a desired (dis)similarity measure. In the first part of the thesis, we consider Markov chains, noting that pairwise relations between its states are naturally modeled by the state-transition matrix. We propose a method for modeling the performance of a synchronization method for a multi-processor architecture. Our model introduces and builds upon a cache line bouncing process that models the interaction of threads accessing the shared cache lines. In the second part of the thesis, we consider representation learning using the transitive-aware Minimax distance, which enables the extraction of elongated manifolds and structures in the data. While recent work has made Minimax distances computationally feasible, little attention has been put to its memory footprint, which is naturally O(N^2), the cost of storing all pairwise distances. We do, however, compute a novel hierarchical representation of the data, requiring O(N) memory, from which pairwise Minimax distances can then be efficiently inferred, in total requiring O(N) memory, at the cost of higher computational cost. An alternative sampling-based approach is also derived, which computes approximate Minimax distances, also in O(N) memory but with a significantly reduced computational cost, while still yielding a good approximation, as verified by impressive results on clustering benchmarks. Finally, we develop an unsupervised learning framework for clustering vehicle trajectories based on Minimax distances. The performance of the framework is validated on real-world datasets collected from real driving scenarios, on which satisfactory performance is demonstrated

Offline and Online Models for Learning Pairwise Relations in Data

Pairwise relations between data points are essential for numerous machine learning algorithms. Many representation learning methods consider pairwise relations to identify the latent features and patterns in the data. This thesis, investigates learning of pairwise relations from two different perspectives: offline learning and online learning.The first part of the thesis focuses on offline learning by starting with an investigation of the performance modeling of a synchronization method in concurrent programming using a Markov chain whose state transition matrix models pairwise relations between involved cores in a computer process.Then the thesis focuses on a particular pairwise distance measure, the minimax distance, and explores memory-efficient approaches to computing this distance by proposing a hierarchical representation of the data with a linear memory requirement with respect to the number of data points, from which the exact pairwise minimax distances can be derived in a memory-efficient manner. Then, a memory-efficient sampling method is proposed that follows the aforementioned hierarchical representation of the data and samples the data points in a way that the minimax distances between all data points are maximally preserved. Finally, the thesis proposes a practical non-parametric clustering of vehicle motion trajectories to annotate traffic scenarios based on transitive relations between trajectories in an embedded space.The second part of the thesis takes an online learning perspective, and starts by presenting an online learning method for identifying bottlenecks in a road network by extracting the minimax path, where bottlenecks are considered as road segments with the highest cost, e.g., in the sense of travel time. Inspired by real-world road networks, the thesis assumes a stochastic traffic environment in which the road-specific probability distribution of travel time is unknown. Therefore, it needs to learn the parameters of the probability distribution through observations by modeling the bottleneck identification task as a combinatorial semi-bandit problem. The proposed approach takes into account the prior knowledge and follows a Bayesian approach to update the parameters. Moreover, it develops a combinatorial variant of Thompson Sampling and derives an upper bound for the corresponding Bayesian regret. Furthermore, the thesis proposes an approximate algorithm to address the respective computational intractability issue.Finally, the thesis considers contextual information of road network segments by extending the proposed model to a contextual combinatorial semi-bandit framework and investigates and develops various algorithms for this contextual combinatorial setting

Learning representations from dendrograms

We propose unsupervised representation learning and feature extraction from dendrograms. The commonly used Minimax distance measures correspond to building a dendrogram with single linkage criterion, with defining specific forms of a level function and a distance function over that. Therefore, we extend this method to arbitrary dendrograms. We develop a generalized framework wherein different distance measures and representations can be inferred from different types of dendrograms, level functions and distance functions. Via an appropriate embedding, we compute a vector-based representation of the inferred distances, in order to enable many numerical machine learning algorithms to employ such distances. Then, to address the model selection problem, we study the aggregation of different dendrogram-based distances respectively in solution space and in representation space in the spirit of deep representations. In the first approach, for example for the clustering problem, we build a graph with positive and negative edge weights according to the consistency of the clustering labels of different objects among different solutions, in the context of ensemble methods. Then, we use an efficient variant of correlation clustering to produce the final clusters. In the second approach, we investigate the combination of different distances and features sequentially in the spirit of multi-layered architectures to obtain the final features. Finally, we demonstrate the effectiveness of our approach via several numerical studies

Scaling Manifold Ranking Based Image Retrieval

Manifold Ranking is a graph-based ranking algorithm being successfully applied to retrieve images from multimedia databases. Given a query image, Manifold Ranking computes the ranking scores of images in the database by exploiting the relationships among them expressed in the form of a graph. Since Manifold Ranking effectively utilizes the global structure of the graph, it is significantly better at finding intuitive results compared with current approaches. Fundamentally, Manifold Ranking requires an inverse matrix to compute ranking scores and so needs O(n^3) time, where n is the number of images. Manifold Ranking, unfortunately, does not scale to support databases with large numbers of images. Our solution, Mogul, is based on two ideas: (1) It efficiently computes ranking scores by sparse matrices, and (2) It skips unnecessary score computations by estimating upper bounding scores. These two ideas reduce the time complexity of Mogul to O(n) from O(n^3) of the inverse matrix approach. Experiments show that Mogul is much faster and gives significantly better retrieval quality than a state-of-the-art approximation approach

Unsupervised representation learning with Minimax distance measures

We investigate the use of Minimax distances to extract in a nonparametric way the features that capture the unknown underlying patterns and structures in the data. We develop a general-purpose and computationally efficient framework to employ Minimax distances with many machine learning methods that perform on numerical data. We study both computing the pairwise Minimax distances for all pairs of objects and as well as computing the Minimax distances of all the objects to/from a fixed (test) object. We first efficiently compute the pairwise Minimax distances between the objects, using the equivalence of Minimax distances over a graph and over a minimum spanning tree constructed on that. Then, we perform an embedding of the pairwise Minimax distances into a new vector space, such that their squared Euclidean distances in the new space equal to the pairwise Minimax distances in the original space. We also study the case of having multiple pairwise Minimax matrices, instead of a single one. Thereby, we propose an embedding via first summing up the centered matrices and then performing an eigenvalue decomposition to obtain the relevant features. In the following, we study computing Minimax distances from a fixed (test) object which can be used for instance in K-nearest neighbor search. Similar to the case of all-pair pairwise Minimax distances, we develop an efficient and general-purpose algorithm that is applicable with any arbitrary base distance measure. Moreover, we investigate in detail the edges selected by the Minimax distances and thereby explore the ability of Minimax distances in detecting outlier objects. Finally, for each setting, we perform several experiments to demonstrate the effectiveness of our framework

Scaling Manifold Ranking Based Image Retrieval

Manifold Ranking is a graph-based ranking algorithm being successfully applied to retrieve images from multimedia databases. Given a query image, Manifold Ranking computes the ranking scores of images in the database by exploiting the relationships among them expressed in the form of a graph. Since Manifold Ranking effectively utilizes the global structure of the graph, it is significantly better at finding intuitive results compared with current approaches. Fundamentally, Manifold Ranking requires an inverse matrix to compute ranking scores and so needs O(n^3) time, where n is the number of images. Manifold Ranking, unfortunately, does not scale to support databases with large numbers of images. Our solution, Mogul, is based on two ideas: (1) It efficiently computes ranking scores by sparse matrices, and (2) It skips unnecessary score computations by estimating upper bounding scores. These two ideas reduce the time complexity of Mogul to O(n) from O(n^3) of the inverse matrix approach. Experiments show that Mogul is much faster and gives significantly better retrieval quality than a state-of-the-art approximation approach