Learning representations that capture rich semantic relationships and accommodate propositional calculus poses a significant challenge. Existing approaches are either contrastive, lacking theoretical guarantees, or fall short in effectively representing the partial orders inherent to rich visual-semantic hierarchies. In this paper, we propose a novel approach for learning visual representations that not only conform to a specified semantic structure but also facilitate probabilistic propositional reasoning. Our approach is based on a new nuclear norm-based loss. We show that its minimum encodes the spectral geometry of the semantics in a subspace lattice, where logical propositions can be represented by projection operators.

The precise estimation of camera poses within large camera networks is a foundational problem in computer vision and robotics, with broad applications spanning autonomous navigation, surveillance, and augmented reality. In this paper, we introduce a novel methodology that extends state-of-the-art Pose Graph Optimization (PGO) techniques. Departing from the conventional PGO paradigm, which primarily relies on camera-camera edges, our approach centers on the introduction of a dynamic element - any rigid object free to move in the scene - whose pose can be reliably inferred from a single image. Specifically, we consider the bipartite graph encompassing cameras, object poses evolving dynamically, and camera-object relative transformations at each time step. This shift not only offers a solution to the challenges encountered in directly estimating relative poses between cameras, particularly in adverse environments, but also leverages the inclusion of numerous object poses to ameliorate and integrate errors, resulting in accurate camera pose estimates. Though our framework retains compatibility with traditional PGO solvers, its efficacy benefits from a custom-tailored optimization scheme. To this end, we introduce an iterative primal-dual algorithm, capable of handling large graphs. Empirical benchmarks, conducted on a new dataset of simulated indoor environments, substantiate the efficacy and efficiency of our approach.

Recent research in representation learning has shown that hierarchical data lends itself to low-dimensional and highly informative representations in hyperbolic space. However, even if hyperbolic embeddings have gathered attention in image recognition, their optimization is prone to numerical hurdles. Further, it remains unclear which applications stand to benefit the most from the implicit bias imposed by hyperbolicity, when compared to traditional Euclidean features. In this paper, we focus on prototypical hyperbolic neural networks. In particular, the tendency of hyperbolic embeddings to converge to the boundary of the Poincar\'e ball in high dimensions and the effect this has on few-shot classification. We show that the best few-shot results are attained for hyperbolic embeddings at a common hyperbolic radius. In contrast to prior benchmark results, we demonstrate that better performance can be achieved by a fixed-radius encoder equipped with the Euclidean metric, regardless of the embedding dimension.