191 research outputs found
Manifold-valued Image Generation with Wasserstein Generative Adversarial Nets
Generative modeling over natural images is one of the most fundamental
machine learning problems. However, few modern generative models, including
Wasserstein Generative Adversarial Nets (WGANs), are studied on manifold-valued
images that are frequently encountered in real-world applications. To fill the
gap, this paper first formulates the problem of generating manifold-valued
images and exploits three typical instances: hue-saturation-value (HSV) color
image generation, chromaticity-brightness (CB) color image generation, and
diffusion-tensor (DT) image generation. For the proposed generative modeling
problem, we then introduce a theorem of optimal transport to derive a new
Wasserstein distance of data distributions on complete manifolds, enabling us
to achieve a tractable objective under the WGAN framework. In addition, we
recommend three benchmark datasets that are CIFAR-10 HSV/CB color images,
ImageNet HSV/CB color images, UCL DT image datasets. On the three datasets, we
experimentally demonstrate the proposed manifold-aware WGAN model can generate
more plausible manifold-valued images than its competitors.Comment: Accepted by AAAI 201
DeepKSPD: Learning Kernel-matrix-based SPD Representation for Fine-grained Image Recognition
Being symmetric positive-definite (SPD), covariance matrix has traditionally
been used to represent a set of local descriptors in visual recognition. Recent
study shows that kernel matrix can give considerably better representation by
modelling the nonlinearity in the local descriptor set. Nevertheless, neither
the descriptors nor the kernel matrix is deeply learned. Worse, they are
considered separately, hindering the pursuit of an optimal SPD representation.
This work proposes a deep network that jointly learns local descriptors,
kernel-matrix-based SPD representation, and the classifier via an end-to-end
training process. We derive the derivatives for the mapping from a local
descriptor set to the SPD representation to carry out backpropagation. Also, we
exploit the Daleckii-Krein formula in operator theory to give a concise and
unified result on differentiating SPD matrix functions, including the matrix
logarithm to handle the Riemannian geometry of kernel matrix. Experiments not
only show the superiority of kernel-matrix-based SPD representation with deep
local descriptors, but also verify the advantage of the proposed deep network
in pursuing better SPD representations for fine-grained image recognition
tasks
Efficient Continuous Manifold Learning for Time Series Modeling
Modeling non-Euclidean data is drawing attention along with the unprecedented
successes of deep neural networks in diverse fields. In particular, symmetric
positive definite (SPD) matrix is being actively studied in computer vision,
signal processing, and medical image analysis, thanks to its ability to learn
appropriate statistical representations. However, due to its strong
constraints, it remains challenging for optimization problems or inefficient
computation costs, especially, within a deep learning framework. In this paper,
we propose to exploit a diffeomorphism mapping between Riemannian manifolds and
a Cholesky space, by which it becomes feasible not only to efficiently solve
optimization problems but also to reduce computation costs greatly. Further, in
order for dynamics modeling in time series data, we devise a continuous
manifold learning method by integrating a manifold ordinary differential
equation and a gated recurrent neural network in a systematic manner. It is
noteworthy that because of the nice parameterization of matrices in a Cholesky
space, it is straightforward to train our proposed network with Riemannian
geometric metrics equipped. We demonstrate through experiments that the
proposed model can be efficiently and reliably trained as well as outperform
existing manifold methods and state-of-the-art methods in two classification
tasks: action recognition and sleep staging classification
Graph Neural Networks on SPD Manifolds for Motor Imagery Classification: A Perspective from the Time-Frequency Analysis
Motor imagery (MI) classification is one of the most widely-concern research
topics in Electroencephalography (EEG)-based brain-computer interfaces (BCIs)
with extensive industry value. The MI-EEG classifiers' tendency has changed
fundamentally over the past twenty years, while classifiers' performance is
gradually increasing. In particular, owing to the need for characterizing
signals' non-Euclidean inherence, the first geometric deep learning (GDL)
framework, Tensor-CSPNet, has recently emerged in the BCI study. In essence,
Tensor-CSPNet is a deep learning-based classifier on the second-order
statistics of EEGs. In contrast to the first-order statistics, using these
second-order statistics is the classical treatment of EEG signals, and the
discriminative information contained in these second-order statistics is
adequate for MI-EEG classification. In this study, we present another GDL
classifier for MI-EEG classification called Graph-CSPNet, using graph-based
techniques to simultaneously characterize the EEG signals in both the time and
frequency domains. It is realized from the perspective of the time-frequency
analysis that profoundly influences signal processing and BCI studies. Contrary
to Tensor-CSPNet, the architecture of Graph-CSPNet is further simplified with
more flexibility to cope with variable time-frequency resolution for signal
segmentation to capture the localized fluctuations. In the experiments,
Graph-CSPNet is evaluated on subject-specific scenarios from two well-used
MI-EEG datasets and produces near-optimal classification accuracies.Comment: 16 pages, 5 figures, 9 Tables; This work has been submitted to the
IEEE for possible publication. Copyright may be transferred without notice,
after which this version may no longer be accessibl
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