Hyperbolic Geometry in Computer Vision: A Survey

Hyperbolic geometry, a Riemannian manifold endowed with constant sectional negative curvature, has been considered an alternative embedding space in many learning scenarios, \eg, natural language processing, graph learning, \etc, as a result of its intriguing property of encoding the data's hierarchical structure (like irregular graph or tree-likeness data). Recent studies prove that such data hierarchy also exists in the visual dataset, and investigate the successful practice of hyperbolic geometry in the computer vision (CV) regime, ranging from the classical image classification to advanced model adaptation learning. This paper presents the first and most up-to-date literature review of hyperbolic spaces for CV applications. To this end, we first introduce the background of hyperbolic geometry, followed by a comprehensive investigation of algorithms, with geometric prior of hyperbolic space, in the context of visual applications. We also conclude this manuscript and identify possible future directions.Comment: First survey paper for the hyperbolic geometry in CV application

Neural Sinkhorn Topic Model

In this paper, we present a new topic modelling approach via the theory of optimal transport (OT). Specifically, we present a document with two distributions: a distribution over the words (doc-word distribution) and a distribution over the topics (doc-topic distribution). For one document, the doc-word distribution is the observed, sparse, low-level representation of the content, while the doc-topic distribution is the latent, dense, high-level one of the same content. Learning a topic model can then be viewed as a process of minimising the transportation of the semantic information from one distribution to the other. This new viewpoint leads to a novel OT-based topic modelling framework, which enjoys appealing simplicity, effectiveness, and efficiency. Extensive experiments show that our framework significantly outperforms several state-of-the-art models in terms of both topic quality and document representations

Approximation vector machines for large-scale online learning

One of the most challenging problems in kernel online learning is to bound the model size and to promote model sparsity. Sparse models not only improve computation and memory usage, but also enhance the generalization capacity -- a principle that concurs with the law of parsimony. However, inappropriate sparsity modeling may also significantly degrade the performance. In this paper, we propose Approximation Vector Machine (AVM), a model that can simultaneously encourage sparsity and safeguard its risk in compromising the performance. In an online setting context, when an incoming instance arrives, we approximate this instance by one of its neighbors whose distance to it is less than a predefined threshold. Our key intuition is that since the newly seen instance is expressed by its nearby neighbor the optimal performance can be analytically formulated and maintained. We develop theoretical foundations to support this intuition and further establish an analysis for the common loss functions including Hinge, smooth Hinge, and Logistic (i.e., for the classification task) and ℓ1, ℓ2, and ε-insensitive (i.e., for the regression task) to characterize the gap between the approximation and optimal solutions. This gap crucially depends on two key factors including the frequency of approximation (i.e., how frequent the approximation operation takes place) and the predefined threshold. We conducted extensive experiments for classification and regression tasks in batch and online modes using several benchmark datasets. The quantitative results show that our proposed AVM obtained comparable predictive performances with current state-of-the-art methods while simultaneously achieving significant computational speed-up due to the ability of the proposed AVM in maintaining the model size