A Graph-Based Semi-Supervised k Nearest-Neighbor Method for Nonlinear Manifold Distributed Data Classification

$k$ Nearest Neighbors ($k$NN) is one of the most widely used supervised learning algorithms to classify Gaussian distributed data, but it does not achieve good results when it is applied to nonlinear manifold distributed data, especially when a very limited amount of labeled samples are available. In this paper, we propose a new graph-based $k$NN algorithm which can effectively handle both Gaussian distributed data and nonlinear manifold distributed data. To achieve this goal, we first propose a constrained Tired Random Walk (TRW) by constructing an $R$-level nearest-neighbor strengthened tree over the graph, and then compute a TRW matrix for similarity measurement purposes. After this, the nearest neighbors are identified according to the TRW matrix and the class label of a query point is determined by the sum of all the TRW weights of its nearest neighbors. To deal with online situations, we also propose a new algorithm to handle sequential samples based a local neighborhood reconstruction. Comparison experiments are conducted on both synthetic data sets and real-world data sets to demonstrate the validity of the proposed new $k$NN algorithm and its improvements to other version of $k$NN algorithms. Given the widespread appearance of manifold structures in real-world problems and the popularity of the traditional $k$NN algorithm, the proposed manifold version $k$NN shows promising potential for classifying manifold-distributed data.Comment: 32 pages, 12 figures, 7 table

Measuring relative opinion from location-based social media: A case study of the 2016 U.S. presidential election

Social media has become an emerging alternative to opinion polls for public opinion collection, while it is still posing many challenges as a passive data source, such as structurelessness, quantifiability, and representativeness. Social media data with geotags provide new opportunities to unveil the geographic locations of users expressing their opinions. This paper aims to answer two questions: 1) whether quantifiable measurement of public opinion can be obtained from social media and 2) whether it can produce better or complementary measures compared to opinion polls. This research proposes a novel approach to measure the relative opinion of Twitter users towards public issues in order to accommodate more complex opinion structures and take advantage of the geography pertaining to the public issues. To ensure that this new measure is technically feasible, a modeling framework is developed including building a training dataset by adopting a state-of-the-art approach and devising a new deep learning method called Opinion-Oriented Word Embedding. With a case study of the tweets selected for the 2016 U.S. presidential election, we demonstrate the predictive superiority of our relative opinion approach and we show how it can aid visual analytics and support opinion predictions. Although the relative opinion measure is proved to be more robust compared to polling, our study also suggests that the former can advantageously complement the later in opinion prediction

Graph Construction for Manifold Discovery

Manifold learning is a class of machine learning methods that exploits the observation that high-dimensional data tend to lie on a smooth lower-dimensional manifold. Manifold discovery is the essential first component of manifold learning methods, in which the manifold structure is inferred from available data. This task is typically posed as a graph construction problem: selecting a set of vertices and edges that most closely approximates the true underlying manifold. The quality of this learned graph is critical to the overall accuracy of the manifold learning method. Thus, it is essential to develop accurate, efficient, and reliable algorithms for constructing manifold approximation graphs. To aid in this investigation of graph construction methods, we propose new methods for evaluating graph quality. These quality measures act as a proxy for ground-truth manifold approximation error and are applicable even when prior information about the dataset is limited. We then develop an incremental update scheme for some quality measures, demonstrating their usefulness for efficient parameter tuning. We then propose two novel methods for graph construction, the Manifold Spanning Graph and the Mutual Neighbors Graph algorithms. Each method leverages assumptions about the structure of both the input data and the subsequent manifold learning task. The algorithms are experimentally validated against state of the art graph construction techniques on a multi-disciplinary set of application domains, including image classification, directional audio prediction, and spectroscopic analysis. The final contribution of the thesis is a method for aligning sequential datasets while still respecting each set’s internal manifold structure. The use of high quality manifold approximation graphs enables accurate alignments with few ground-truth correspondences

Indexing and Retrieval of 3D Articulated Geometry Models

In this PhD research study, we focus on building a content-based search engine for 3D articulated geometry models. 3D models are essential components in nowadays graphic applications, and are widely used in the game, animation and movies production industry. With the increasing number of these models, a search engine not only provides an entrance to explore such a huge dataset, it also facilitates sharing and reusing among different users. In general, it reduces production costs and time to develop these 3D models. Though a lot of retrieval systems have been proposed in recent years, search engines for 3D articulated geometry models are still in their infancies. Among all the works that we have surveyed, reliability and efficiency are the two main issues that hinder the popularity of such systems. In this research, we have focused our attention mainly to address these two issues. We have discovered that most existing works design features and matching algorithms in order to reflect the intrinsic properties of these 3D models. For instance, to handle 3D articulated geometry models, it is common to extract skeletons and use graph matching algorithms to compute the similarity. However, since this kind of feature representation is complex, it leads to high complexity of the matching algorithms. As an example, sub-graph isomorphism can be NP-hard for model graph matching. Our solution is based on the understanding that skeletal matching seeks correspondences between the two comparing models. If we can define descriptive features, the correspondence problem can be solved by bag-based matching where fast algorithms are available. In the first part of the research, we propose a feature extraction algorithm to extract such descriptive features. We then convert the skeletal matching problems into bag-based matching. We further define metric similarity measure so as to support fast search. We demonstrate the advantages of this idea in our experiments. The improvement on precision is 12\% better at high recall. The indexing search of 3D model is 24 times faster than the state of the art if only the first relevant result is returned. However, improving the quality of descriptive features pays the price of high dimensionality. Curse of dimensionality is a notorious problem on large multimedia databases. The computation time scales exponentially as the dimension increases, and indexing techniques may not be useful in such situation. In the second part of the research, we focus ourselves on developing an embedding retrieval framework to solve the high dimensionality problem. We first argue that our proposed matching method projects 3D models on manifolds. We then use manifold learning technique to reduce dimensionality and maximize intra-class distances. We further propose a numerical method to sub-sample and fast search databases. To preserve retrieval accuracy using fewer landmark objects, we propose an alignment method which is also beneficial to existing works for fast search. The advantages of the retrieval framework are demonstrated in our experiments that it alleviates the problem of curse of dimensionality. It also improves the efficiency (3.4 times faster) and accuracy (30\% more accurate) of our matching algorithm proposed above. In the third part of the research, we also study a closely related area, 3D motions. 3D motions are captured by sticking sensor on human beings. These captured data are real human motions that are used to animate 3D articulated geometry models. Creating realistic 3D motions is an expensive and tedious task. Although 3D motions are very different from 3D articulated geometry models, we observe that existing works also suffer from the problem of temporal structure matching. This also leads to low efficiency in the matching algorithms. We apply the same idea of bag-based matching into the work of 3D motions. From our experiments, the proposed method has a 13\% improvement on precision at high recall and is 12 times faster than existing works. As a summary, we have developed algorithms for 3D articulated geometry models and 3D motions, covering feature extraction, feature matching, indexing and fast search methods. Through various experiments, our idea of converting restricted matching to bag-based matching improves matching efficiency and reliability. These have been shown in both 3D articulated geometry models and 3D motions. We have also connected 3D matching to the area of manifold learning. The embedding retrieval framework not only improves efficiency and accuracy, but has also opened a new area of research

Density-based Clustering by Means of Bridge Point Identification

Density-based clustering focuses on defining clusters consisting of contiguous regions characterized by similar densities of points. Traditional approaches identify core points first, whereas more recent ones initially identify the cluster borders and then propagate cluster labels within the delimited regions. Both strategies encounter issues in presence of multi-density regions or when clusters are characterized by noisy borders. To overcome the above issues, we present a new clustering algorithm that relies on the concept of bridge point. A bridge point is a point whose neighborhood includes points of different clusters. The key idea is to use bridge points, rather than border points, to partition points into clusters. We have proved that a correct bridge point identification yields a cluster separation consistent with the expectation. To correctly identify bridge points in absence of a priori cluster information we leverage an established unsupervised outlier detection algorithm. Specifically, we empirically show that, in most cases, the detected outliers are actually a superset of the bridge point set. Therefore, to define clusters we spread cluster labels like a wildfire until an outlier, acting as a candidate bridge point, is reached. The proposed algorithm performs statistically better than state-of-the-art methods on a large set of benchmark datasets and is particularly robust to the presence of intra-cluster multiple densities and noisy borders

Hyperspectral image representation and processing with binary partition trees

The optimal exploitation of the information provided by hyperspectral images requires the development of advanced image processing tools. Therefore, under the title Hyperspectral image representation and Processing with Binary Partition Trees, this PhD thesis proposes the construction and the processing of a new region-based hierarchical hyperspectral image representation: the Binary Partition Tree (BPT). This hierarchical region-based representation can be interpreted as a set of hierarchical regions stored in a tree structure. Hence, the Binary Partition Tree succeeds in presenting: (i) the decomposition of the image in terms of coherent regions and (ii) the inclusion relations of the regions in the scene. Based on region-merging techniques, the construction of BPT is investigated in this work by studying hyperspectral region models and the associated similarity metrics. As a matter of fact, the very high dimensionality and the complexity of the data require the definition of specific region models and similarity measures. Once the BPT is constructed, the fixed tree structure allows implementing efficient and advanced application-dependent techniques on it. The application-dependent processing of BPT is generally implemented through a specific pruning of the tree. Accordingly, some pruning techniques are proposed and discussed according to different applications. This Ph.D is focused in particular on segmentation, object detection and classification of hyperspectral imagery. Experimental results on various hyperspectral data sets demonstrate the interest and the good performances of the BPT representatio

Isometry and convexity in dimensionality reduction

The size of data generated every year follows an exponential growth. The number of data points as well as the dimensions have increased dramatically the past 15 years. The gap between the demand from the industry in data processing and the solutions provided by the machine learning community is increasing. Despite the growth in memory and computational power, advanced statistical processing on the order of gigabytes is beyond any possibility. Most sophisticated Machine Learning algorithms require at least quadratic complexity. With the current computer model architecture, algorithms with higher complexity than linear O(N) or O(N logN) are not considered practical. Dimensionality reduction is a challenging problem in machine learning. Often data represented as multidimensional points happen to have high dimensionality. It turns out that the information they carry can be expressed with much less dimensions. Moreover the reduced dimensions of the data can have better interpretability than the original ones. There is a great variety of dimensionality reduction algorithms under the theory of Manifold Learning. Most of the methods such as Isomap, Local Linear Embedding, Local Tangent Space Alignment, Diffusion Maps etc. have been extensively studied under the framework of Kernel Principal Component Analysis (KPCA). In this dissertation we study two current state of the art dimensionality reduction methods, Maximum Variance Unfolding (MVU) and Non-Negative Matrix Factorization (NMF). These two dimensionality reduction methods do not fit under the umbrella of Kernel PCA. MVU is cast as a Semidefinite Program, a modern convex nonlinear optimization algorithm, that offers more flexibility and power compared to iv KPCA. Although MVU and NMF seem to be two disconnected problems, we show that there is a connection between them. Both are special cases of a general nonlinear factorization algorithm that we developed. Two aspects of the algorithms are of particular interest: computational complexity and interpretability. In other words computational complexity answers the question of how fast we can find the best solution of MVU/NMF for large data volumes. Since we are dealing with optimization programs, we need to find the global optimum. Global optimum is strongly connected with the convexity of the problem. Interpretability is strongly connected with local isometry1 that gives meaning in relationships between data points. Another aspect of interpretability is association of data with labeled information. The contributions of this thesis are the following: 1. MVU is modified so that it can scale more efficient. Results are shown on 1 million speech datasets. Limitations of the method are highlighted. 2. An algorithm for fast computations for the furthest neighbors is presented for the first time in the literature. 3. Construction of optimal kernels for Kernel Density Estimation with modern convex programming is presented. For the first time we show that the Leave One Cross Validation (LOOCV) function is quasi-concave. 4. For the first time NMF is formulated as a convex optimization problem 5. An algorithm for the problem of Completely Positive Matrix Factorization is presented. 6. A hybrid algorithm of MVU and NMF the isoNMF is presented combining advantages of both methods. 7. The Isometric Separation Maps (ISM) a variation of MVU that contains classification information is presented. 8. Large scale nonlinear dimensional analysis on the TIMIT speech database is performed. 9. A general nonlinear factorization algorithm is presented based on sequential convex programming. Despite the efforts to scale the proposed methods up to 1 million data points in reasonable time, the gap between the industrial demand and the current state of the art is still orders of magnitude wide.Ph.D.Committee Chair: David Anderson; Committee Co-Chair: Alexander Gray; Committee Member: Anthony Yezzi; Committee Member: Hongyuan Zha; Committee Member: Justin Romberg; Committee Member: Ronald Schafe

Structural Surface Mapping for Shape Analysis

Natural surfaces are usually associated with feature graphs, such as the cortical surface with anatomical atlas structure. Such a feature graph subdivides the whole surface into meaningful sub-regions. Existing brain mapping and registration methods did not integrate anatomical atlas structures. As a result, with existing brain mappings, it is difficult to visualize and compare the atlas structures. And also existing brain registration methods can not guarantee the best possible alignment of the cortical regions which can help computing more accurate shape similarity metrics for neurodegenerative disease analysis, e.g., Alzheimer’s disease (AD) classification. Also, not much attention has been paid to tackle surface parameterization and registration with graph constraints in a rigorous way which have many applications in graphics, e.g., surface and image morphing. This dissertation explores structural mappings for shape analysis of surfaces using the feature graphs as constraints. (1) First, we propose structural brain mapping which maps the brain cortical surface onto a planar convex domain using Tutte embedding of a novel atlas graph and harmonic map with atlas graph constraints to facilitate visualization and comparison between the atlas structures. (2) Next, we propose a novel brain registration technique based on an intrinsic atlas-constrained harmonic map which provides the best possible alignment of the cortical regions. (3) After that, the proposed brain registration technique has been applied to compute shape similarity metrics for AD classification. (4) Finally, we propose techniques to compute intrinsic graph-constrained parameterization and registration for general genus-0 surfaces which have been used in surface and image morphing applications