Research On Riemannian Manifold Learning Based Visual Recognition

With the continuous development of society and technology,in the computer vision community,the collected visual data,compared to the past,has two obvious characteristics: 1.larger amount of data; 2.richer nonlinear information encoded in the data.First,the increasing amount of data makes it possible for set-based methods.Different from a single image,image sets contain more spatiotemporal appearance information about the content of interest.Besides,since traditional machine learning methods are based on Euclidean space,it is difficult for them to effectively mine the nonlinear structural information contained in the visual data.However,methods based on geometry theories,like differential geometry,Riemannian geometry,etc.,can effectively cope with nonlinear structure.They have therefore begun to receive growing attention from researchers.This paper focuses mainly on Riemannian manifold learning and its applications in image set classification.Starting from the mathematical theories beneath manifold learning,this paper reviews the classical manifold learning algorithms and proposes improved algorithms.The main contributions of this paper are as follows.1.From the aspects of analysis,algebra,geometry,optimization and so on,the principal mathematical theories related to manifold learning are systematically introduced.Some intuitive explanations are also offered for a straightforward understanding of some abstract mathematical theories.Furthermore,four frequently studied Riemannian manifolds in computer vision are introduced.At the same time,according to the depth of the mathematical theories involved in the algorithms,this paper classifies the classical manifold learning algorithms from a novel perspective.2.In Riemannian manifold learning methods,complementary statistical information is encoded in heterogeneous Riemannian manifolds.It is hence difficult for single-manifold models to take full advantage of this complementary information,undermining its effectiveness when facing complex scenarios.To tackle this issue,some researchers began to develop learning algorithms on hybrid manifolds.However,the computational burden caused by multiple manifolds limits the application of these methods.This paper therefore proposes a hybrid Riemannian graph-embedding metric learning framework.With the proposed sparse graph embedding,the computational burden is greatly alleviated.This approach has achieved superior performance on different visual tasks.3.With the vibrant progress of deep learning,some researchers begin to build deep networks on Riemannian manifolds.However,most of the existing Riemannian deep networks focus on global geometric information and fail to pay attention to local geometric information.To deal with this problem,based on category theory,this paper transfers the local mechanism prevailing in traditional machine learning methods to deep manifold networks,and obtains promising results on multiple datasets.For the best of our knowledge,this work is the first attempt to mine local information in manifold.4.In the existing deep Riemannian networks,there are few studies on dimensionality reduction layer and Euclidean embedding layer.However,this kind of fundamental research is very important.Based on the Riemannian geometry,this paper conducts an in-depth analysis of the matrix logarithm,and studies several kinds of mappings induced by the matrix logarithm,including two Euclidean embedding mappings,Lie group homomorphism and Riemannian submersion(immersion).For the proposed two adaptive Euclidean embedding layers,we verify their effectiveness on two datasets.For the proposed theoretical framework of Lie group homomorphism and submersion(immersion),their specific applications to deep Riemannian networks are also briefly discussed.