Learning Discriminative αβ-Divergences for Positive Definite Matrices

Cherian, Anoop; Stanitsas, Panagiotis; Harandi, Mehrtash; Morellas, Vassilios; Papanikolopoulos, N.

doi:10.1109/iccv.2017.458

Cited by 9 publications

(6 citation statements)

References 56 publications

(165 reference statements)

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“…In particular, visual representations often rely on SPD manifolds, such as the kernel matrix, the covariance descriptor [26], and the diffusion tensor image [8]. Optimization problems on SPD manifolds are especially important in the medical imaging field [4,8]. Furthermore, optimization problems in hyperbolic spaces are important in natural language processing.…”

Section: Introductionmentioning

confidence: 99%

Global convergence of Hager–Zhang type Riemannian conjugate gradient method

Sakai

Sato

Iiduka

2023

Applied Mathematics and Computation

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Section: Introductionmentioning

confidence: 99%

Global convergence of Hager–Zhang type Riemannian conjugate gradient method

Sakai

Sato

Iiduka

2023

Applied Mathematics and Computation

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“…The input of a standard Euclidean-space-learning classifier (e.g., the support vector machine) is a feature vector that lies in the Euclidean space, while the SPD manifold is clearly not a Euclidean space. Most existing SPD manifold classification methods convert an SPD representation to a requisite vector by the tangent approximation [7], [8], the kernel method [9], [10], [11], [12], or the coding technique [13], [14], [15]. These methods are not superior because the vector operation on an SPD representation inevitably distorts the matrix structure.…”

mentioning

confidence: 99%

A Robust Distance Measure for Similarity-Based Classification on the SPD Manifold

Gao

Harandi

et al. 2020

IEEE Trans. Neural Netw. Learning Syst.

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The symmetric positive definite (SPD) matrices, forming a Riemannian manifold, are commonly used as visual representations. The non-Euclidean geometry of the manifold often makes developing learning algorithms (e.g., classifiers) difficult and complicated. The concept of similarity-based learning has been shown to be effective to address various problems on SPD manifolds. This is mainly because the similaritybased algorithms are agnostic to the geometry and purely work based on the notion of similarities/distances. However, existing similarity-based models on SPD manifolds opt for holistic representations, ignoring characteristics of information captured by SPD matrices. To circumvent this limitation, we propose a novel SPD distance measure for the similarity-based algorithm. Specifically, we introduce the concept of point-to-set transformation, which enables us to learn multiple lower-dimensional and discriminative SPD manifolds from a higher-dimensional one. For lower-dimensional SPD manifolds obtained by the point-to-set transformation, we propose a tailored set-to-set distance measure by making use of the family of alpha-beta divergences. We further propose to learn the point-to-set transformation and the setto-set distance measure jointly, yielding a powerful similaritybased algorithm on SPD manifolds. Our thorough evaluations on several visual recognition tasks (e.g., action classification, face recognition) suggest that our algorithm comfortably outperforms various state-of-the-art algorithms.

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“…The training data used for JHMDB is considerably larger than the other tests and is included to see how our algorithms perform on higher dimensional data. GBWML, Log-Euclidean Metric Learning (LEML) and and Information-Divergence Dictionary Learning [132] (IDDL -Problem 2.21) are also included in the k-NN performance tests for relevant comparison. GBMWL is the most similar to our algorithms and provides the best comparison for their performance.…”

Section: Discussionmentioning

confidence: 99%

“…This approach learns a set of SPD atoms B i ∈ S n ++ that can be used to map from a test point X i ∈ S n ++ to a classification vector. Information Divergence Dictionary Learning (IDDL) [132,133] is a fairly recent dictionary learning algorithm that uses an information-theoretic formulation. The optimisation problem takes the form:…”

Section: Spd Metric Learningmentioning

confidence: 99%

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Applications of the Bures-Wasserstein Distance in Linear Metric Learning

Cooper¹

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Metric Learning has proved valuable in information retrieval and classification problems, with many algorithms using statistical formulations to their advantage. Optimal Transport, a powerful framework for computing distances between probability distributions, has seen significant uptake in Machine Learning and some applications in Metric Learning. The Bures-Wasserstein distance, a particular formulation of Optimal Transport, has seen limited application in Metric Learning despite being well suited to the field. In this thesis Linear Metric Learning algorithms incorporating the Bures-Wasserstein distance are developed, utilising its novel properties. The first set of algorithms we contribute use the Bures-Wasserstein distance for learning linear metrics using Cyclic Projections, Riemannian barycenters and gradient-based algorithms. The convergence properties and classification performance of these algorithms are then compared to benchmark Linear Metric Learning algorithms. Our algorithms achieve competitive classification performance on feature and image datasets and converge reliability. The second set of contributions extend Linear Metric Learning with the Bures-Wasserstein distance to a low-rank context to address issues related to scalability. An extension of the previous Cyclic Projections algorithm and a Riemannian optimisation method are developed. Convergence and classification experiments analogous to the full-rank case are conducted to examine how the reduced rank affects both of these performance metrics. Both algorithms converge reliably under a variety of synthetic conditions. Reduction in rank generally decreased the training time for algorithms until effects caused by the size of the training data dominated. Reducing rank either left classification error unchanged or increased it. With respect to other low-rank Linear Metric Learning algorithms our algorithms again performed competitively with respect to classification error. The last set of contributions consider Linear Metric Learning on Symmetric Positive Definite (SPD) matrices. Some datasets are expressed as or are suited to a SPD representation. As such, Metric Learning algorithms that respect the structure of these matrices have found success when adapted to this kind of data. Another extension of the Cyclic Projections algorithm is developed to use the Generalised Bures-Wasserstein distance on SPD matrices. Two other methods based on a ground metric formulation of the Bures-Wasserstein distance for learning linear metrics on SPD data are also introduced. One ground metric formulation uses an approximation (inspired by the Sinkhorn distance) to increase its scalability. Convergence experiments on synthetic problems adapted to SPD matrices are conducted on the ground metric algorithms to establish their convergence properties. Classification experiments are also conducted with respect to several types of SPD data. While the Cyclic Projections method is competitive, the ground metric algorithms have poor performance. Despite their poor classification performance, the approximation method does evaluate faster than the Generalised Bures-Wasserstein distance, potentially offering some scalability improvements.

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Learning Discriminative αβ-Divergences for Positive Definite Matrices

Cited by 9 publications

References 56 publications

Global convergence of Hager–Zhang type Riemannian conjugate gradient method

Global convergence of Hager–Zhang type Riemannian conjugate gradient method

A Robust Distance Measure for Similarity-Based Classification on the SPD Manifold

Applications of the Bures-Wasserstein Distance in Linear Metric Learning

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