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Three-dimensional (3D) topological states resemble truly localised, particle-like objects in physical space. Among the richest such structures are 3D skyrmions and hopfions, that realise integer topological numbers in their configuration via homotopic mappings from real space to the hypersphere (sphere in 4D space) or the 2D sphere. They have received tremendous attention as exotic textures in particle physics, cosmology, superfluids, and many other systems. Here we experimentally create and measure a topological 3D skyrmionic hopfion in fully structured light. By simultaneously tailoring the polarisation and phase profile, our beam establishes the skyrmionic mapping by realising every possible optical state in the propagation volume. The resulting light field’s Stokes parameters and phase are synthesised into a Hopf fibration texture. We perform volumetric full-field reconstruction of the Π 3 mapping, measuring a quantised topological charge, or Skyrme number, of 0.945. Such topological state control opens avenues for 3D optical data encoding and metrology. The Hopf characterisation of the optical hypersphere endows a fresh perspective to topological optics, offering experimentally-accessible photonic analogues to the gamut of particle-like 3D topological textures, from condensed matter to high-energy physics. One way to describe a particle is as a localised, 3-dimensional topological state, such as a skyrmion or hopfion. Here, the authors demonstrate and characterise particle-like skyrmionic hopfions in a free-space structured light beam.

Structured light fields embody strong spatial variations of polarization, phase, and amplitude. Understanding, characterization, and exploitation of such fields can be achieved through their topological properties. Three-dimensional (3D) topological solitons, such as hopfions, are 3D localized continuous field configurations with nontrivial particle-like structures that exhibit a host of important topologically protected properties. Here, we propose and demonstrate photonic counterparts of hopfions with exact characteristics of Hopf fibration, Hopf index, and Hopf mapping from real-space vector beams to homotopic hyperspheres representing polarization states. We experimentally generate photonic hopfions with on-demand high-order Hopf indices and independently controlled topological textures, including Néel-, Bloch-, and antiskyrmionic types. We also demonstrate a robust free-space transport of photonic hopfions, thus showing the potential of hopfions for developing optical topological informatics and communications.

Loss functions is one of the main challenges in face recognition problems. Recent works focus on designing loss functions that make learned features more discriminative by a larger angular or cosine distance. In this paper, in addition to the method based on additional angle margins, we propose a Multi-Sphere Radius Loss (MsrFace) to add radius constraints. MsrFace pushes learned features to hyperspheres with different spherical radii and the classes can be separated more strictly. We present experiments on several widely used benchmarks to show that MsrFace has a better performance in comparison with some recent state-of-the-art face recognition methods.

Computer experiments with qualitative and quantitative factors occur frequently in various applications in science and engineering. Analysis of such experiments is not yet completely resolved. In this work, we propose an additive Gaussian process model for computer experiments with qualitative and quantitative factors. The proposed method considers an additive correlation structure for qualitative factors, and assumes that the correlation function for each qualitative factor and the correlation function of quantitative factors are multiplicative. It inherits the flexibility of unrestrictive correlation structure for qualitative factors by using the hypersphere decomposition, embracing more flexibility in modeling the complex systems of computer experiments. The merits of the proposed method are illustrated by several numerical examples and a real data application. Supplementary materials for this article are available online.

A step towards the next generation of high-capacity, noise-resilient communication and computing technologies is a substantial increase in the dimensionality of information space and the synthesis of superposition states on an N -dimensional ( N  > 2) Hilbert space featuring exotic group symmetries. Despite the rapid development of photonic devices and systems, on-chip information technologies are mostly limited to two-level systems owing to the lack of sufficient reconfigurability to satisfy the stringent requirement for 2( N  − 1) degrees of freedom, intrinsically associated with the increase of synthetic dimensionalities. Even with extensive efforts dedicated to recently emerged vector lasers and microcavities for the expansion of dimensionalities 1 – 10 , it still remains a challenge to actively tune the diversified, high-dimensional superposition states of light on demand. Here we demonstrate a hyperdimensional, spin–orbit microlaser for chip-scale flexible generation and manipulation of arbitrary four-level states. Two microcavities coupled through a non-Hermitian synthetic gauge field are designed to emit spin–orbit-coupled states of light with six degrees of freedom. The vectorial state of the emitted laser beam in free space can be mapped on a Bloch hypersphere defining an SU(4) symmetry, demonstrating dynamical generation and reconfiguration of high-dimensional superposition states with high fidelity. A fully integrated semiconductor microlaser that exploits spin–orbit coupling of light emits in a four-dimensional Hilbert space, with flexible control of up to six degrees of freedom.

Averaging data on the unit sphere S d (also called a unit hypersphere) is a common problem in computer vision, robotics and other fields, with applications ranging from motion planning to DNA modelling. In this paper, we introduce a new method for averaging data represented as points on the unit sphere S d −1 using the d-dimensional generalized Kuramoto model. Our method is verified on a range of benchmark data sets and compared with common data averaging algorithms. Also, we showcase the applicability of this method for solving rotation averaging problem.

Computer graphics, robotics, and physics are one of the many domains where interpolation on the unit sphere S n (often called a unit hypersphere or unit n-sphere) plays a crucial role. In this paper, we introduce a novel approach for achieving smooth and precise interpolation on the unit sphere S n −1 using the n-dimensional generalized Kuramoto model. The proposed algorithm finds the shortest and most direct path between two points on that non-Euclidean manifold. Our simulation results demonstrate that it achieves performance comparable to that of a Spherical Linear Interpolation algorithm. Also, the paper proposes the application of our algorithm in the interpolation of rotations that are presented in the form of four-dimensional data.

Single-cell RNA-Seq (scRNA-seq) is invaluable for studying biological systems. Dimensionality reduction is a crucial step in interpreting the relation between cells in scRNA-seq data. However, current dimensionality reduction methods are often confounded by multiple simultaneous technical and biological variability, result in “crowding” of cells in the center of the latent space, or inadequately capture temporal relationships. Here, we introduce scPhere, a scalable deep generative model to embed cells into low-dimensional hyperspherical or hyperbolic spaces to accurately represent scRNA-seq data. ScPhere addresses multi-level, complex batch factors, facilitates the interactive visualization of large datasets, resolves cell crowding, and uncovers temporal trajectories. We demonstrate scPhere on nine large datasets in complex tissue from human patients or animal development. Our results show how scPhere facilitates the interpretation of scRNA-seq data by generating batch-invariant embeddings to map data from new individuals, identifies cell types affected by biological variables, infers cells’ spatial positions in pre-defined biological specimens, and highlights complex cellular relations. Single-cell RNA-seq allows the study of tissues at cellular resolution. Here, the authors demonstrate how deep learning can be used to gain biological insight from such data by accounting for biological and technical variability. Data exploration is improved by accurately visualizing cells on an interactive 3D surface.

We prove that the four-dimensional round sphere contains a minimally embedded hypertorus, as well as infinitely many, pairwise non-isometric, immersed ones. Our analysis also yields infinitely many, pairwise non-isometric, minimally embedded hyperspheres and thus provides a self-contained solution to Chern's spherical Bernstein conjecture in dimensions four and six.

In the field of face recognition, improving recognition accuracy is an important research direction. Due to the fact that face images may be disturbed during acquisition, transmission and storage, the recognition algorithms are not robust enough to meet the requirements of practical applications. Therefore, the introduction of an effective loss function is particularly critical. In this paper, the advantages and disadvantages of different loss functions will be derived by analysing the softmax, ArcFace and AdaFace loss functions and comparing the differences between them. Through the analysis, this paper concludes that the loss functions still have different defects in the application of real scenarios and cannot effectively identify data with fewer samples. For example, Arcace loss is not able to perfectly make the samples evenly distributed on the hypersphere in real life. Therefore, in the future design and optimisation of the self-supervised pre-training framework should be strengthened to make a greater improvement in the recognition accuracy of face recognition.