1,616 research outputs found
Regression on fixed-rank positive semidefinite matrices: a Riemannian approach
The paper addresses the problem of learning a regression model parameterized
by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear
nature of the search space and on scalability to high-dimensional problems. The
mathematical developments rely on the theory of gradient descent algorithms
adapted to the Riemannian geometry that underlies the set of fixed-rank
positive semidefinite matrices. In contrast with previous contributions in the
literature, no restrictions are imposed on the range space of the learned
matrix. The resulting algorithms maintain a linear complexity in the problem
size and enjoy important invariance properties. We apply the proposed
algorithms to the problem of learning a distance function parameterized by a
positive semidefinite matrix. Good performance is observed on classical
benchmarks
Bounded-Distortion Metric Learning
Metric learning aims to embed one metric space into another to benefit tasks
like classification and clustering. Although a greatly distorted metric space
has a high degree of freedom to fit training data, it is prone to overfitting
and numerical inaccuracy. This paper presents {\it bounded-distortion metric
learning} (BDML), a new metric learning framework which amounts to finding an
optimal Mahalanobis metric space with a bounded-distortion constraint. An
efficient solver based on the multiplicative weights update method is proposed.
Moreover, we generalize BDML to pseudo-metric learning and devise the
semidefinite relaxation and a randomized algorithm to approximately solve it.
We further provide theoretical analysis to show that distortion is a key
ingredient for stability and generalization ability of our BDML algorithm.
Extensive experiments on several benchmark datasets yield promising results
Model-based learning of local image features for unsupervised texture segmentation
Features that capture well the textural patterns of a certain class of images
are crucial for the performance of texture segmentation methods. The manual
selection of features or designing new ones can be a tedious task. Therefore,
it is desirable to automatically adapt the features to a certain image or class
of images. Typically, this requires a large set of training images with similar
textures and ground truth segmentation. In this work, we propose a framework to
learn features for texture segmentation when no such training data is
available. The cost function for our learning process is constructed to match a
commonly used segmentation model, the piecewise constant Mumford-Shah model.
This means that the features are learned such that they provide an
approximately piecewise constant feature image with a small jump set. Based on
this idea, we develop a two-stage algorithm which first learns suitable
convolutional features and then performs a segmentation. We note that the
features can be learned from a small set of images, from a single image, or
even from image patches. The proposed method achieves a competitive rank in the
Prague texture segmentation benchmark, and it is effective for segmenting
histological images
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