Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks

This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter, induced by the given dissimilarity structures. Our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissimilarities are reduced, the network may become more clustered but not less. Several admissible methods are constructed and two particular methods, termed reciprocal and nonreciprocal clustering, are shown to provide upper and lower bounds in the space of admissible methods. Alternative clustering methodologies and axioms are further considered. Allowing the outcome of hierarchical clustering to be asymmetric, so that it matches the asymmetry of the original data, leads to the inception of quasi-clustering methods. The existence of a unique quasi-clustering method is shown. Allowing clustering in a two-node network to proceed at the minimum of the two dissimilarities generates an alternative axiomatic construction. There is a unique clustering method in this case too. The paper also develops algorithms for the computation of hierarchical clusters using matrix powers on a min-max dioid algebra and studies the stability of the methods proposed. We proved that most of the methods introduced in this paper are such that similar networks yield similar hierarchical clustering results. Algorithms are exemplified through their application to networks describing internal migration within states of the United States (U.S.) and the interrelation between sectors of the U.S. economy.Comment: This is a largely extended version of the previous conference submission under the same title. The current version contains the material in the previous version (published in ICASSP 2013) as well as material presented at the Asilomar Conference on Signal, Systems, and Computers 2013, GlobalSIP 2013, and ICML 2014. Also, unpublished material is included in the current versio

What are the true clusters?

Constructivist philosophy and Hasok Chang's active scientific realism are used to argue that the idea of "truth" in cluster analysis depends on the context and the clustering aims. Different characteristics of clusterings are required in different situations. Researchers should be explicit about on what requirements and what idea of "true clusters" their research is based, because clustering becomes scientific not through uniqueness but through transparent and open communication. The idea of "natural kinds" is a human construct, but it highlights the human experience that the reality outside the observer's control seems to make certain distinctions between categories inevitable. Various desirable characteristics of clusterings and various approaches to define a context-dependent truth are listed, and I discuss what impact these ideas can have on the comparison of clustering methods, and the choice of a clustering methods and related decisions in practice

Learning image segmentation and hierarchies by learning ultrametric distances

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Brain and Cognitive Sciences, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 100-105).In this thesis I present new contributions to the fields of neuroscience and computer science. The neuroscientific contribution is a new technique for automatically reconstructing complete neural networks from densely stained 3d electron micrographs of brain tissue. The computer science contribution is a new machine learning method for image segmentation and the development of a new theory for supervised hierarchy learning based on ultrametric distance functions. It is well-known that the connectivity of neural networks in the brain can have a dramatic influence on their computational function . However, our understanding of the complete connectivity of neural circuits has been quite impoverished due to our inability to image all the connections between all the neurons in biological network. Connectomics is an emerging field in neuroscience that aims to revolutionize our understanding of the function of neural circuits by imaging and reconstructing entire neural circuits. In this thesis, I present an automated method for reconstructing neural circuitry from 3d electron micrographs of brain tissue. The cortical column, a basic unit of cortical microcircuitry, will produce a single 3d electron micrograph measuring many 100s terabytes once imaged and contain neurites from well over 100,000 different neurons. It is estimated that tracing the neurites in such a volume by hand would take several thousand human years. Automated circuit tracing methods are thus crucial to the success of connectomics. In computer vision, the circuit reconstruction problem of tracing neurites is known as image segmentation. Segmentation is a grouping problem where image pixels belonging to the same neurite are clustered together. While many algorithms for image segmentation exist, few have parameters that can be optimized using groundtruth data to extract maximum performance on a specialized dataset. In this thesis, I present the first machine learning method to directly minimize an image segmentation error. It is based the theory of ultrametric distances and hierarchical clustering. Image segmentation is posed as the problem of learning and classifying ultrametric distances between image pixels. Ultrametric distances on point set have the special property that(cont.) they correspond exactly to hierarchical clustering of the set. This special property implies hierarchical clustering can be learned by directly learning ultrametric distances. In this thesis, I develop convolutional networks as a machine learning architecture for image processing. I use this powerful pattern recognition architecture with many tens of thousands of free parameters for predicting affinity graphs and detecting object boundaries in images. When trained using ultrametric learning, the convolutional network based algorithm yields an extremely efficient linear-time segmentation algorithm. In this thesis, I develop methods for assessing the quality of image segmentations produced by manual human efforts or by automated computer algorithms. These methods are crucial for comparing the performance of different segmentation methods and is used through out the thesis to demonstrate the quality of the reconstructions generated by the methods in this thesis.by Srinivas C. Turaga.Ph.D

A Cybernetics Update for Competitive Deep Learning System

A number of recent reports in the peer-reviewed literature have discussed irreproducibility of results in biomedical research. Some of these articles suggest that the inability of independent research laboratories to replicate published results has a negative impact on the development of, and confidence in, the biomedical research enterprise. To get more resilient data and to achieve higher reproducible result, we present an adaptive and learning system reference architecture for smart learning system interface. To get deeper inspiration, we focus our attention on mammalian brain neurophysiology. In fact, from a neurophysiological point of view, neuroscientist LeDoux finds two preferential amygdala pathways in the brain of the laboratory mouse. The low road is a pathway which is able to transmit a signal from a stimulus to the thalamus, and then to the amygdala, which then activates a fast-response in the body. The high road is activated simultaneously. This is a slower road which also includes the cortical parts of the brain, thus creating a conscious impression of what the stimulus is (to develop a rational mechanism of defense for instance). To mimic this biological reality, our main idea is to use a new input node able to bind known information to the unknown one coherently. Then, unknown "environmental noise" or/and local "signal input" information can be aggregated to known "system internal control status" information, to provide a landscape of attractor points, which either fast or slow and deeper system response can computed from. In this way, ideal cybernetics system interaction levels can be matched exactly to practical system modeling interaction styles, with no paradigmatic operational ambiguity and minimal information loss. The present paper is a relevant contribute to classic cybernetics updating towards a new General Theory of Systems, a post-Bertalanffy Systemics

On Folding and Twisting (and whatknot): towards a characterization of workspaces in syntax

Syntactic theory has traditionally adopted a constructivist approach, in which a set of atomic elements are manipulated by combinatory operations to yield derived, complex elements. Syntactic structure is thus seen as the result or discrete recursive combinatorics over lexical items which get assembled into phrases, which are themselves combined to form sentences. This view is common to European and American structuralism (e.g., Benveniste, 1971; Hockett, 1958) and different incarnations of generative grammar, transformational and non-transformational (Chomsky, 1956, 1995; and Kaplan & Bresnan, 1982; Gazdar, 1982). Since at least Uriagereka (2002), there has been some attention paid to the fact that syntactic operations must apply somewhere, particularly when copying and movement operations are considered. Contemporary syntactic theory has thus somewhat acknowledged the importance of formalizing aspects of the spaces in which elements are manipulated, but it is still a vastly underexplored area. In this paper we explore the consequences of conceptualizing syntax as a set of topological operations applying over spaces rather than over discrete elements. We argue that there are empirical advantages in such a view for the treatment of long-distance dependencies and cross-derivational dependencies: constraints on possible configurations emerge from the dynamics of the system.Comment: Manuscript. Do not cite without permission. Comments welcom