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Shape manifolds, Procrustean metrics, and complex projective spaces
 Bulletin of the London Mathematical Society
, 1984
"... 2. Shapespaces and shapemanifolds 82 3. Procrustes analysis, and the invariant (quotient) metric on I j.... 87 4. Shapemeasures and shapedensities 93 5. The manifold carrying the shapes of triangles 96 ..."
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Cited by 186 (0 self)
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2. Shapespaces and shapemanifolds 82 3. Procrustes analysis, and the invariant (quotient) metric on I j.... 87 4. Shapemeasures and shapedensities 93 5. The manifold carrying the shapes of triangles 96
Latent Space Approaches to Social Network Analysis
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
"... Network models are widely used to represent relational information among interacting units. In studies of social networks, recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation be ..."
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Cited by 174 (16 self)
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Network models are widely used to represent relational information among interacting units. In studies of social networks, recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors. We develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved "social space." Inference for the social space is developed within a maximum likelihood and Bayesian framework, and Markov chain Monte Carlo procedures are proposed for making inference on latent positions and the effects of observed covariates. We present analyses of three standard datasets from the social networks literature, and compare the method to an alternative stochastic blockmodeling approach. In addition to improving upon model fit, our method provides a visual and interpretable modelbased spatial representation of social relationships, and improves upon existing methods by allowing the statistical uncertainty in the social space to be quantified and graphically represented.
Transformation Invariance in Pattern Recognition  Tangent Distance and Tangent Propagation
 Lecture Notes in Computer Science
, 1998
"... . In pattern recognition, statistical modeling, or regression, the amount of data is a critical factor affecting the performance. If the amount of data and computational resources are unlimited, even trivial algorithms will converge to the optimal solution. However, in the practical case, given ..."
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Cited by 136 (2 self)
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. In pattern recognition, statistical modeling, or regression, the amount of data is a critical factor affecting the performance. If the amount of data and computational resources are unlimited, even trivial algorithms will converge to the optimal solution. However, in the practical case, given limited data and other resources, satisfactory performance requires sophisticated methods to regularize the problem by introducing a priori knowledge. Invariance of the output with respect to certain transformations of the input is a typical example of such a priori knowledge. In this chapter, we introduce the concept of tangent vectors, which compactly represent the essence of these transformation invariances, and two classes of algorithms, "tangent distance" and "tangent propagation", which make use of these invariances to improve performance. 1 Introduction Pattern Recognition is one of the main tasks of biological information processing systems, and a major challenge of compute...
Dynamic social network analysis using latent space models
 SIGKDD Explorations, Special Issue on Link Mining
"... This paper explores two aspects of social network modeling. First, we generalize a successful static model of relationships into a dynamic model that accounts for friendships drifting over time. Second, we show how to make it tractable to learn such models from data, even as the number of entities n ..."
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Cited by 72 (3 self)
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This paper explores two aspects of social network modeling. First, we generalize a successful static model of relationships into a dynamic model that accounts for friendships drifting over time. Second, we show how to make it tractable to learn such models from data, even as the number of entities n gets large. The generalized model associates each entity with a point in pdimensional Euclidean latent space. The points can move as time progresses but large moves in latent space are improbable. Observed links between entities are more likely if the entities are close in latent space. We show how to make such a model tractable (subquadratic in the number of entities) by the use of appropriate kernel functions for similarity in latent space; the use of low dimensional KDtrees; a new efficient dynamic adaptation of multidimensional scaling for a first pass of approximate projection of entities into latent space; and an efficient conjugate gradient update rule for nonlinear local optimization in which amortized time per entity during an update is O(log n). We use both synthetic and realworld data on up to 11,000 entities which indicate nearlinear scaling in computation time and improved performance over four alternative approaches. We also illustrate the system operating on twelve years of NIPS coauthorship data. 1.
Principal Component Analysis based on Robust Estimators of the Covariance or Correlation Matrix: Influence Functions and Efficiencies
 BIOMETRIKA
, 2000
"... A robust principal component analysis can be easily performed by computing the eigenvalues and eigenvectors of a robust estimator of the covariance or correlation matrix. In this paper we derive the influence functions and the corresponding asymptotic variances for these robust estimators of eige ..."
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Cited by 52 (11 self)
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A robust principal component analysis can be easily performed by computing the eigenvalues and eigenvectors of a robust estimator of the covariance or correlation matrix. In this paper we derive the influence functions and the corresponding asymptotic variances for these robust estimators of eigenvalues and eigenvectors. The behavior of several of these estimators is investigated by a simulation study. Finally, the use of empirical influence functions is illustrated by a real data example.
A Review of Medical Image Registration
 Interactive imageguided neurosurgery
, 1993
"... Introduction The ever expanding gamut of medical imaging techniques provides the clinician an increasingly multifaceted view of brain function and anatomy. The information provided by the various imaging modalities is often complementary (i.e. provides separate but useful information) and synergist ..."
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Cited by 29 (0 self)
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Introduction The ever expanding gamut of medical imaging techniques provides the clinician an increasingly multifaceted view of brain function and anatomy. The information provided by the various imaging modalities is often complementary (i.e. provides separate but useful information) and synergistic (i.e. the combination of information provides useful extra information). For example, Xray computed tomography (CT) and magnetic resonance (MR) imaging exquisitely demonstrate brain anatomy but provide little functional information. Positron emission tomography (PET) and single photon emission computed tomography (SPECT) scans display aspects of brain function and allow metabolic measurements but poorly delineate anatomy. Furthermore, CT and MR images describe complementary morphologic features. For example, bone and calcifications are best seen on CT images, while softtissue structures are better differentiated by MR imaging. Clinical diagnosis and therapy planning and evaluatio
The Distribution of Target Registration Error in Rigidbody, Pointbased Registration
, 1999
"... Introduction The pointbased registration problem is as follows: given a set of homologous points in two spaces, nd a transformation that brings the points into approximate alignment. In many cases the appropriate transformations are rigid, consisting of translations and rotations. Medical applicat ..."
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Cited by 24 (0 self)
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Introduction The pointbased registration problem is as follows: given a set of homologous points in two spaces, nd a transformation that brings the points into approximate alignment. In many cases the appropriate transformations are rigid, consisting of translations and rotations. Medical applications abound in neurosurgery, for example, where the head can be treated as a rigid body [1], [2], [3], [4], [5], [6], [7]. The points, which we will call ducial points, may be anatomical landmarks or may be produced articially by means of attached markers. In the case that we address here, the spaces are three dimensional and may consist, for example, of two MR volumes, a CT volume and an MR volume or PET volume, or, in the case of imageguided neurosurgical applications, an image volume and the physical space of the operating room itself. The rigidbody, pointbased image registration problem is typically dened to be the problem of nding the translation vector an
Determining the dimensionality of multidimensional scaling representations for cognitive modeling
 Journal of Mathematical Psychology
, 2001
"... Multidimensional scaling models of stimulus domains are widely used as a representational basis for cognitive modeling. These representations associate stimuli with points in a coordinate space that has some predetermined number of dimensions. Although the choice of dimensionality can significantly ..."
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Cited by 21 (7 self)
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Multidimensional scaling models of stimulus domains are widely used as a representational basis for cognitive modeling. These representations associate stimuli with points in a coordinate space that has some predetermined number of dimensions. Although the choice of dimensionality can significantly influence cognitive modeling, it is often made on the basis of unsatisfactory heuristics. To address this problem, a Bayesian approach to dimensionality determination, based on the Bayesian Information Criterion (BIC), is developed using a probabilistic formulation of multidimensional scaling. The BIC approach formalizes the tradeoff between datafit and model complexity implicit in the problem of dimensionality determination and allows for the explicit introduction of information regarding data precision. Monte Carlo simulations are presented that indicate, by using this approach, the determined dimensionality is likely to be accurate if either a significant number of stimuli are considered or a reasonable estimate of precision is available. The approach is demonstrated using an established data set involving the judged pairwise similarities between a set of geometric stimuli. 2001 Academic Press COGNITIVE MODELING AND MULTIDIMENSIONAL SCALING