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Coreset selection, a technique for compressing large datasets while preserving performance, is crucial for modern machine learning. This paper presents a novel method for generating high-quality Wasserstein coresets using the Sinkhorn loss, a powerful tool with computational advantages. However, existing approaches suffer from numerical instability in Sinkhorn’s algorithm. We address this by proposing stable algorithms for the computation and differentiation of the Sinkhorn optimization problem, including an analytical formula for the derivative of the Sinkhorn loss and a rigorous stability analysis of our method. Extensive experiments demonstrate that our approach significantly outperforms existing methods in terms of sample selection quality, computational efficiency, and achieving a smaller Wasserstein distance. 

Generative models based on latent variables, such as generative adversarial networks (GANs) and variationalauto-encoders (VAEs), have gained lots of interests due to their impressive performance in many fields.However, many data such as natural images usually do not populate the ambient Euclidean space but insteadreside in a lower-dimensional manifold. Thus an inappropriate choice of the latent dimension fails to uncoverthe structure of the data, possibly resulting in mismatch of latent representations and poor generativequalities. Toward addressing these problems, we propose a novel framework called the latent WassersteinGAN (LWGAN) that fuses the Wasserstein auto-encoder and the Wasserstein GAN so that the intrinsicdimension of the data manifold can be adaptively learned by a modified informative latent distribution. Weprove that there exist an encoder network and a generator network in such a way that the intrinsic dimensionof the learned encoding distribution is equal to the dimension of the data manifold. We theoreticallyestablish that our estimated intrinsic dimension is a consistent estimate of the true dimension of the datamanifold. Meanwhile, we provide an upper bound on the generalization error of LWGAN, implying that weforce the synthetic data distribution to be similar to the real data distribution from a population perspective.Comprehensive empirical experiments verify our framework and show that LWGAN is able to identify thecorrect intrinsic dimension under several scenarios, and simultaneously generate high-quality syntheticdata by sampling from the learned latent distribution. Supplementary materials for this article are availableonline, including a standardized description of the materials available for reproducing the work. 

Organic trisradicals featuring three-fold symmetry have attracted significant interest because of their unique magnetic properties associated with spin frustration. Herein, we describe the synthesis and characterization of a triangular prism-shaped organic cage for which we have coined the name PrismCage6+ and its trisradical trication—TR3(•+). PrismCage6+ is composed of three 4,4'-bipyridinium dications and two 1,3,5-phenylene units bridged by six methylene groups. In the solid state, PrismCage6+ adopts a highly twisted conformation with close to C3 symmetry as a result of encapsulating one PF6− anion as a guest. PrismCage6+ undergoes stepwise reduction to its mono-, di- and trisradical cations in MeCN on account of strong electronic communication between its 4,4'-bipyridinium units. TR3(•+), which is obtained by reduction of PrismCage6+ employing CoCp2, adopts a triangular prism-shaped conformation with close to C2v symmetry in the solid state. Temperature-dependent continuous-wave and nutation frequency-selective EPR spectra of TR3(•+) in frozen N,N-dimethylformamide indicate its doublet ground state. The doublet-quartet energy gap of TR3(•+) is estimated to be −0.06 kcal mol−1 and the critical temperature of spin-state conversion is found to be ca. 50 K, suggesting that it displays pronounced spin-frustration at the molecular level. To the best of our knowledge, this example is the first organic radical cage to exhibit spin frustration. The trisradical trication of PrismCage6+ opens up new possibilities for fundamental investigations and potential applications in the fields of both organic cages and spin chemistry. 

Abstract With the motivation to study how non-magnetic ion site disorder affects the quantum magnetism of Ba 3 CoSb 2 O 9 , a spin-1/2 equilateral triangular lattice antiferromagnet, we performed DC and AC susceptibility, specific heat, elastic and inelastic neutron scattering measurements on single crystalline samples of Ba 2.87 Sr 0.13 CoSb 2 O 9 with Sr doping on non-magnetic Ba 2+ ion sites. The results show that Ba 2.87 Sr 0.13 CoSb 2 O 9 exhibits (i) a two-step magnetic transition at 2.7 K and 3.3 K, respectively; (ii) a possible canted 120 degree spin structure at zero field with reduced ordered moment as 1.24 μ B /Co; (iii) a series of spin state transitions for both H ∥ ab -plane and H ∥ c -axis. For H ∥ ab -plane, the magnetization plateau feature related to the up–up–down phase is significantly suppressed; (iv) an inelastic neutron scattering spectrum with only one gapped mode at zero field, which splits to one gapless and one gapped mode at 9 T. All these features are distinctly different from those observed for the parent compound Ba 3 CoSb 2 O 9 , which demonstrates that the non-magnetic ion site disorder (the Sr doping) plays a complex role on the magnetic properties beyond the conventionally expected randomization of the exchange interactions. We propose the additional effects including the enhancement of quantum spin fluctuations and introduction of a possible spatial anisotropy through the local structural distortions. 

Abstract The magnetic ground state of the pyrochlore Yb 2 GaSbO 7 has not been established. The persistent spin fluctuations observed by muon spin-relaxation measurements at low temperatures have not been adequately explained for this material using existing theories for quantum magnetism. Here we report on the synthesis and characterisation of Yb 2 GaSbO 7 to revisit the nature of the magnetic ground state. Through DC and AC magnetic susceptibility, heat capacity, and neutron scattering experiments, we observe evidence for a dynamical ground state that makes Yb 2 GaSbO 7 a promising candidate for disorder-induced spin-liquid or spin-singlet behaviour. This state is quite fragile, being tuned to a splayed ferromagnet in a modest magnetic field μ 0 H c  ~ 1.5 T.