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This paper studies the fundamental problem of multi-layer generator models in learning hierarchical representations. The multi-layer generator model that consists of multiple layers of latent variables organized in a top-down architecture tends to learn multiple levels of data abstraction. However, such multi-layer latent variables are typically parameterized to be Gaussian, which can be less informative in capturing complex abstractions, resulting in limited success in hierarchical representation learning. On the other hand, the energy-based (EBM) prior is known to be expressive in capturing the data regularities, but it often lacks the hierarchical structure to capture different levels of hierarchical representations. In this paper, we propose a joint latent space EBM prior model with multi-layer latent variables for effective hierarchical representation learning. We develop a variational joint learning scheme that seamlessly integrates an inference model for efficient inference. Our experiments demonstrate that the proposed joint EBM prior is effective and expressive in capturing hierarchical representations and modeling data distribution. 

Objective. Dynamic positron emission tomography (PET) imaging, which can provide information on dynamic changes in physiological metabolism, is now widely used in clinical diagnosis and cancer treatment. However, the reconstruction from dynamic data is extremely challenging due to the limited counts received in individual frame, especially in ultra short frames. Recently, the unrolled modelbased deep learning methods have shown inspiring results for low-count PET image reconstruction with good interpretability. Nevertheless, the existing model-based deep learning methods mainly focus on the spatial correlations while ignore the temporal domain. Approach. In this paper, inspired by the learned primal dual (LPD) algorithm, we propose the spatio-temporal primal dual network (STPDnet) for dynamic low-count PET image reconstruction. Both spatial and temporal correlations are encoded by 3D convolution operators. The physical projection of PET is embedded in the iterative learning process of the network, which provides the physical constraints and enhances interpretability. Main results. The experiments of both simulation data and real rat scan data have shown that the proposed method can achieve substantial noise reduction in both temporal and spatial domains and outperform the maximum likelihood expectation maximization, spatio-temporal kernel method, LPD and FBPnet. Significance. Experimental results show STPDnet better reconstruction performance in the low count situation, which makes the proposed method particularly suitable in whole-body dynamic imaging and parametric PET imaging that require extreme short frames and usually suffer from high level of noise. 

This paper studies the fundamental problem of learning multi-layer generator models. The multi-layer generator model builds multiple layers of latent variables as a prior model on top of the generator, which benefits learning complex data distribution and hierarchical representations. However, such a prior model usually focuses on modeling inter-layer relations between latent variables by assuming non-informative (conditional) Gaussian distributions, which can be limited in model expressivity. To tackle this issue and learn more expressive prior models, we propose an energy-based model (EBM) on the joint latent space over all layers of latent variables with the multi-layer generator as its backbone. Such joint latent space EBM prior model captures the intra-layer contextual relations at each layer through layer-wise energy terms, and latent variables across different layers are jointly corrected. We develop a joint training scheme via maximum likelihood estimation (MLE), which involves Markov Chain Monte Carlo (MCMC) sampling for both prior and posterior distributions of the latent variables from different layers. To ensure efficient inference and learning, we further propose a variational training scheme where an inference model is used to amortize the costly posterior MCMC sampling. Our experiments demonstrate that the learned model can be expressive in generating high-quality images and capturing hierarchical features for better outlier detection. 

We present a measurement of the branching fraction and fraction of longitudinal polarization of $B^0\to {\rho }^{+}{\rho }^{-}$decays, which have two ${\pi }^0$’s in the final state. We also measure time-dependent $CP$violation parameters for decays into longitudinally polarized ${\rho }^{+}{\rho }^{-}$pairs. This analysis is based on a data sample containing $(387\pm 6)\times {10}^6$ $\mathrm{\Upsilon }\left(4S\right)$mesons collected with the Belle II detector at the SuperKEKB asymmetric-energy $e^{+}{e}^{-}$collider in 2019–2022. We obtain $\mathcal{B}(B^0\to {\rho }^{+}{\rho }^{-})=\left(2.89_{-0.22}^{+0.23}{}_{-0.27}^{+0.29}\right)\times {10}^{-5}$, $f_L=0.921_{-0.025}^{+0.024}{}_{-0.015}^{+0.017}$, $S=-0.26\pm 0.19\pm 0.08$, and $C=-0.02\pm 0.12_{-0.05}^{+0.06}$, where the first uncertainties are statistical and the second are systematic. We use these results to perform an isospin analysis to constrain the Cabibbo-Kobayashi-Maskawa angle ${\varphi }_2$and obtain two solutions; the result consistent with other Standard Model constraints is ${\varphi }_2=\left({92.6}_{-4.7}^{+4.5}\right)°$. Published by the American Physical Society2025 

A<sc>bstract</sc> We perform the first search forCPviolation in$$ {D}_{(s)}^{+}\to {K}_S^0{K}^{-}{\pi}^{+}{\pi}^{+} $$ $D_{\left(s\right)}^{+}\to K_S^0{K}^{-}{\pi }^{+}{\pi }^{+}$decays. We use a combined data set from the Belle and Belle II experiments, which studye⁺e⁻collisions at center-of-mass energies at or near the Υ(4S) resonance. We use 980 fb⁻¹of data from Belle and 428 fb⁻¹of data from Belle II. We measure sixCP-violating asymmetries that are based on triple products and quadruple products of the momenta of final-state particles, and also the particles’ helicity angles. We obtain a precision at the level of 0.5% for$$ {D}^{+}\to {K}_S^0{K}^{-}{\pi}^{+}{\pi}^{+} $$ $D^{+}\to K_S^0{K}^{-}{\pi }^{+}{\pi }^{+}$decays, and better than 0.3% for$$ {D}_s^{+}\to {K}_S^0{K}^{-}{\pi}^{+}{\pi}^{+} $$ $D_s^{+}\to K_S^0{K}^{-}{\pi }^{+}{\pi }^{+}$decays. No evidence ofCPviolation is found. Our results for the triple-product asymmetries are the most precise to date for singly-Cabibbo-suppressedD⁺decays. Our results for the other asymmetries are the first such measurements performed for charm decays.