Abstract Zero-shot recognition is centered around learning representations to transfer knowledge from seen to unseen classes. Where foundational approaches perform the transfer with semantic embedding spaces, e.g., from attributes or word vectors, the current state-of-the-art relies on prompting pre-trained vision-language models to obtain class embeddings. Whether zero-shot learning is performed with attributes, CLIP, or something else, current approaches de facto assume that there is a pre-defined embedding space in which seen and unseen classes can be positioned. Our work is concerned with real-world zero-shot settings where a pre-defined embedding space can no longer be assumed. This is natural in domains such as biology and medicine, where class names are not common English words, rendering vision-language models useless; or neuroscience, where class relations are only given with non-semantic human comparison scores. We find that there is one data structure enabling zero-shot learning in both standard and non-standard settings: a similarity matrix spanning the seen and unseen classes. We introduce four similarity-based zero-shot learning challenges, tackling open-ended scenarios such as learning with uncommon class names, learning from multiple partial sources, and learning with missing knowledge. As the first step for zero-shot learning beyond a pre-defined semantic embedding space, we propose $$\kappa $$ κ -MDS, a general approach that obtains a prototype for each class on any manifold from similarities alone, even when part of the similarities are missing. Our approach can be plugged into any standard, hyperspherical, or hyperbolic zero-shot learner. Experiments on existing datasets and the new benchmarks show the promise and challenges of similarity-based zero-shot learning.

It has repeatedly been observed that loss minimization by stochastic gradient descent (SGD) leads to heavy-tailed distributions of neural network parameters. Here, we analyze a continuous diffusion approximation of SGD, called homogenized stochastic gradient descent (hSGD), and show in a regularized linear regression framework that it leads to an asymptotically heavy-tailed parameter distribution, even though local gradient noise is Gaussian. We give explicit upper and lower bounds on the tail-index of the resulting parameter distribution and validate these bounds in numerical experiments. Moreover, the explicit form of these bounds enables us to quantify the interplay between optimization hyperparameters and the tail-index. Doing so, we contribute to the ongoing discussion on links between heavy tails and the generalization performance of neural networks as well as the ability of SGD to avoid suboptimal local minima.

Abstract Deep representation learning is a ubiquitous part of modern computer vision. While Euclidean space has been the de facto standard manifold for learning visual representations, hyperbolic space has recently gained rapid traction for learning in computer vision. Specifically, hyperbolic learning has shown a strong potential to embed hierarchical structures, learn from limited samples, quantify uncertainty, add robustness, limit error severity, and more. In this paper, we provide a categorization and in-depth overview of current literature on hyperbolic learning for computer vision. We research both supervised and unsupervised literature and identify three main research themes in each direction. We outline how hyperbolic learning is performed in all themes and discuss the main research problems that benefit from current advances in hyperbolic learning for computer vision. Moreover, we provide a high-level intuition behind hyperbolic geometry and outline open research questions to further advance research in this direction.

Hyperbolic space has become a popular choice of manifold for representation learning of various datatypes from tree-like structures and text to graphs. Building on the success of deep learning with prototypes in Euclidean and hyperspherical spaces, a few recent works have proposed hyperbolic prototypes for classification. Such approaches enable effective learning in low-dimensional output spaces and can exploit hierarchical relations amongst classes, but require privileged information about class labels to position the hyperbolic prototypes. In this work, we propose Hyperbolic Busemann Learning. The main idea behind our approach is to position prototypes on the ideal boundary of the Poincar\'{e} ball, which does not require prior label knowledge. To be able to compute proximities to ideal prototypes, we introduce the penalised Busemann loss. We provide theory supporting the use of ideal prototypes and the proposed loss by proving its equivalence to logistic regression in the one-dimensional case. Empirically, we show that our approach provides a natural interpretation of classification confidence, while outperforming recent hyperspherical and hyperbolic prototype approaches.