206 research outputs found
Classification and Geometry of General Perceptual Manifolds
Perceptual manifolds arise when a neural population responds to an ensemble
of sensory signals associated with different physical features (e.g.,
orientation, pose, scale, location, and intensity) of the same perceptual
object. Object recognition and discrimination requires classifying the
manifolds in a manner that is insensitive to variability within a manifold. How
neuronal systems give rise to invariant object classification and recognition
is a fundamental problem in brain theory as well as in machine learning. Here
we study the ability of a readout network to classify objects from their
perceptual manifold representations. We develop a statistical mechanical theory
for the linear classification of manifolds with arbitrary geometry revealing a
remarkable relation to the mathematics of conic decomposition. Novel
geometrical measures of manifold radius and manifold dimension are introduced
which can explain the classification capacity for manifolds of various
geometries. The general theory is demonstrated on a number of representative
manifolds, including L2 ellipsoids prototypical of strictly convex manifolds,
L1 balls representing polytopes consisting of finite sample points, and
orientation manifolds which arise from neurons tuned to respond to a continuous
angle variable, such as object orientation. The effects of label sparsity on
the classification capacity of manifolds are elucidated, revealing a scaling
relation between label sparsity and manifold radius. Theoretical predictions
are corroborated by numerical simulations using recently developed algorithms
to compute maximum margin solutions for manifold dichotomies. Our theory and
its extensions provide a powerful and rich framework for applying statistical
mechanics of linear classification to data arising from neuronal responses to
object stimuli, as well as to artificial deep networks trained for object
recognition tasks.Comment: 24 pages, 12 figures, Supplementary Material
Matching Image Sets via Adaptive Multi Convex Hull
Traditional nearest points methods use all the samples in an image set to
construct a single convex or affine hull model for classification. However,
strong artificial features and noisy data may be generated from combinations of
training samples when significant intra-class variations and/or noise occur in
the image set. Existing multi-model approaches extract local models by
clustering each image set individually only once, with fixed clusters used for
matching with various image sets. This may not be optimal for discrimination,
as undesirable environmental conditions (eg. illumination and pose variations)
may result in the two closest clusters representing different characteristics
of an object (eg. frontal face being compared to non-frontal face). To address
the above problem, we propose a novel approach to enhance nearest points based
methods by integrating affine/convex hull classification with an adapted
multi-model approach. We first extract multiple local convex hulls from a query
image set via maximum margin clustering to diminish the artificial variations
and constrain the noise in local convex hulls. We then propose adaptive
reference clustering (ARC) to constrain the clustering of each gallery image
set by forcing the clusters to have resemblance to the clusters in the query
image set. By applying ARC, noisy clusters in the query set can be discarded.
Experiments on Honda, MoBo and ETH-80 datasets show that the proposed method
outperforms single model approaches and other recent techniques, such as Sparse
Approximated Nearest Points, Mutual Subspace Method and Manifold Discriminant
Analysis.Comment: IEEE Winter Conference on Applications of Computer Vision (WACV),
201
Support Vector Machines and Radon's Theorem
A support vector machine (SVM) is an algorithm which finds a hyperplane that
optimally separates labeled data points in into positive and
negative classes. The data points on the margin of this separating hyperplane
are called support vectors. We study the possible configurations of support
vectors for points in general position. In particular, we connect the possible
configurations to Radon's theorem, which provides guarantees for when a set of
points can be divided into two classes (positive and negative) whose convex
hulls intersect. If the positive and negative support vectors in a generic SVM
configuration are projected to the separating hyperplane, then these projected
points will form a Radon configuration. Further, with a particular type of
general position, we show there are at most support vectors. This can be
used to test the level of machine precision needed in a support vector machine
implementation. We also show the projections of the convex hulls of the support
vectors intersect in a single Radon point, and under a small enough
perturbation, the points labeled as support vectors remain labeled as support
vectors. We furthermore consider computations studying the expected number of
support vectors for randomly generated data
PROBABILISTIC AND GEOMETRIC APPROACHES TO THE ANALYSIS OF NON-STANDARD DATA
This dissertation explores topics in machine learning, network analysis, and the foundations of statistics using tools from geometry, probability and optimization. The rise of machine learning has brought powerful new (and old) algorithms for data analysis. Much of classical statistics research is about understanding how statistical algorithms behave depending on various aspects of the data. The first part of this dissertation examines the support vector machine classifier (SVM). Leveraging Karush-Kuhn-Tucker conditions we find surprising connections between SVM and several other simple classifiers. We use these connections to explain SVM’s behavior in a variety of data scenarios and demonstrate how these insights are directly relevant to the data analyst. The next part of this dissertation studies networks which evolve over time. We first develop a method to empirically evaluate vertex centrality metrics in an evolving network. We then apply this methodology to investigate the role of precedent in the US legal system. Next, we shift to a probabilistic perspective on temporally evolving networks. We study a general probabilistic model of an evolving network that undergoes an abrupt change in its evolution dynamics. In particular, we examine the effect of such a change on the network’s structural properties. We develop mathematical techniques using continuous time branching processes to derive quantitative error bounds for functionals of a major class of these models about their large network limits. Using these results, we develop general theory to understand the role of abrupt changes in the evolution dynamics of these models. Based on this theory we derive a consistent, non-parametric change point detection estimator. We conclude with a discussion on foundational topics in statistics, commenting on debates both old and new. First, we examine the false confidence theorem which raises questions for data practitioners making inferences based on epistemic uncertainty measures such as Bayesian posterior distributions. Second, we give an overview of the rise of “data science" and what it means for statistics (and vice versa), touching on topics such as reproducibility, computation, education, communication and statistical theory.Doctor of Philosoph
A framework of face recognition with set of testing images
We propose a novel framework to solve the face recognition problem base on set of testing images. Our framework can handle the case that no pose overlap between training set and query set. The main techniques used in this framework are manifold alignment, face normalization and discriminant learning. Experiments on different databases show our system outperforms some state of the art methods
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