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We prove a conjecture of Gao and Kitaev on Wilf-equivalence of sets of patterns {12345,12354} and {45123,45213} that extends the list of 10 related conjectures proved in the literature in a series of papers. To achieve our goals, we prove generalized versions of shape-Wilf-equivalence results of Backelin, West, and Xin and use a particular result on shape-Wilf-equivalence of monotone patterns. We also derive general results on shape-Wilf-equivalence of certain classes of partially ordered patterns and use their specialization (also appearing in a paper by Bloom and Elizalde) as an essential piece in proving the conjecture. Our results allow us to show (shape-)Wilf-equivalence of large classes of sets of patterns, including 11 out of 12 classes found by Bean et al. in relation to the conjecture., Comment: 9 pages, 3 figures

This work studies the learning problem of the energy-based prior model and the multi-layer generator model. The multi-layer generator model, which contains multiple layers of latent variables organized in a top-down hierarchical structure, typically assumes the Gaussian prior model. Such a prior model can be limited in modelling expressivity, which results in a gap between the generator posterior and the prior model, known as the prior hole problem. Recent works have explored learning the energy-based (EBM) prior model as a second-stage, complementary model to bridge the gap. However, the EBM defined on a multi-layer latent space can be highly multi-modal, which makes sampling from such marginal EBM prior challenging in practice, resulting in ineffectively learned EBM. To tackle the challenge, we propose to leverage the diffusion probabilistic scheme to mitigate the burden of EBM sampling and thus facilitate EBM learning. Our extensive experiments demonstrate a superior performance of our diffusion-learned EBM prior on various challenging tasks.

Motivated by the ideal peak-to-average-power ratio and radar sensing capability of traditional frequency-coded radar waveforms, this paper considers the frequency shift keying (FSK) based waveform for joint communications and radar (JCR). An analysis of the probability distributions of its ambiguity function (AF) sidelobe levels (SLs) and peak sidelobe level (PSL) is conducted to study the radar sensing capability of random FSK. Numerical results show that the independent frequency modulation introduces uncontrollable AF PSLs. In order to address this problem, the initial phases of waveform sub-pulses are designed by solving a min-max optimisation problem. Numerical results indicate that the optimisation-based phase design can effectively reduce the AF PSL to a level close to well-designed radar waveforms while having no impact on the data rate and the receiver complexity. For large numbers of waveform sub-pulses and modulation orders, the impact on the error probability is also insignificant., Comment: Submitted to IEEE Transactions on Wireless Communications, 13 pages, 6 figures

Crystal seeding enables a deeper understanding of phase behavior, leading to the development of methods for controlling and manipulating phase transitions in various applications such as materials synthesis, crystallization processes, and phase transformation engineering. How to seed a crystalline in time domain is an open question, which is of great significant and may provide an avenue to understand and control time-dependent quantum many-body physics. Here, we utilize a microwave pulse as a seed to induce the formation of a discrete time crystal in Floquet driven Rydberg atoms. In the experiment, the periodic driving on Rydberg states acts as a seeded crystalline order in subspace, which triggers the time-translation symmetry breaking across the entire ensemble. The behavior of the emergent time crystal is elaborately linked to alterations in the seed, such as the relative phase shift and the frequency difference, which result in phase dependent seeding and corresponding shift in periodicity of the time crystal, leading to embryonic synchronization. This result opens up new possibilities for studying and harnessing time-dependent quantum many-body phenomena, offering insights into the behavior of complex many-body systems under seeding.

The identification of tipping points is essential for prediction of collapses or other sudden changes in complex systems. Applications include studies of ecology, thermodynamics, climatology, and epidemiology. However, detecting early signs of proximity to a tipping is made challenging by complexity and non-linearity. Strongly interacting Rydberg atom gases offer a model systems that offer both complexity and non-linearity, including phase transition and critical slowing down. Here, via an external probe we observe prior warning of the proximity of a phase transition of Rydberg thermal gases. This warning signal is manifested as a cessation of the variance growth with increasing probe intensity. We also observed the dynamics of the critical slowing down behavior versus different time scales, driving intensities, and atomic densities, thus providing insights into the study of a Rydberg atom system's critical behavior. Our experiment suggests that the full critical slowing down dynamics of strongly-interacting Rydberg atoms can be probed systematically, thus providing a benchmark with which to identify critical phenomena in quantum many-body systems.

Engineering a Terahertz (THz) frequency comb spectroscopy at atomic level advances the precisely measurement in spectroscopy and sensing. Current progresses on THz frequency comb rely on difference-frequency generation, optical parametric oscillation, and other methods. Generating a THz frequency comb poses challenges in source stability and achieving a narrow bandwidth, which traditional THz devices are difficult to achieve. Furthermore, accurately measuring the generated THz frequency comb necessitates a high-performance THz detector. Rydberg atoms are well-suited for electric field sensing due to their ultra-wide radio frequency transition energy levels, making them especially sensitive to external electric fields in the DC to THz bandwidth. However, there have been no reports about generating THz frequency comb spectroscopy at the atomic level until now. This work presents a THz frequency comb spectroscopy with Rydberg atoms, in which a Floquet comb-like transition is engineered through a time-periodic drive field. Our approach simplifies the setup required for THz frequency comb spectroscopy while extending the working bandwidth for Rydberg atomic sensors. The THz frequency comb spectroscopy at the atomic level reported in this article shows great potential for various applications in astronomy, remote sensing, spectral detection of biological samples, and other related fields.

Developing microwave electric field sensing based on Rydberg atom has received significant attention due to its unique advantages. However, achieving effective coupling between Rydberg atom and the microwave electric field in the sensing process is a challenging problem that greatly impacts the sensitivity. To address this, we propose the use of a microwave resonant cavity to enhance the effective coupling between the Rydberg atoms and the microwave electric field. In our experiment, we use a three-photon excitation scheme to prepare Rydberg atoms, make measurements of electric fields without and with a microwave cavity in which the vapor cell is put inside. Through experimental testing, we achieve an 18 dB enhancement of power sensitivity. The experiment shows an effective enhancement in electric field pulse signal detection. This result provides a promising direction for enhancing the sensitivity of Rydberg atomic electric field sensors and paves the way for their application in precision electric field measurements.

Generative models have shown strong generation ability while efficient likelihood estimation is less explored. Energy-based models~(EBMs) define a flexible energy function to parameterize unnormalized densities efficiently but are notorious for being difficult to train. Adversarial EBMs introduce a generator to form a minimax training game to avoid expensive MCMC sampling used in traditional EBMs, but a noticeable gap between adversarial EBMs and other strong generative models still exists. Inspired by diffusion-based models, we embedded EBMs into each denoising step to split a long-generated process into several smaller steps. Besides, we employ a symmetric Jeffrey divergence and introduce a variational posterior distribution for the generator's training to address the main challenges that exist in adversarial EBMs. Our experiments show significant improvement in generation compared to existing adversarial EBMs, while also providing a useful energy function for efficient density estimation.

Higher-order and fractional discrete time crystals (DTCs) are exotic phases of matter where the discrete time translation symmetry is broken into higher-order and non-integer category. Generation of these unique DTCs has been widely studied theoretically in different systems. However, no current experimental methods can probe these higher-order and fractional DTCs in any quantum many-body systems. We demonstrate an experimental approach to observe higher-order and fractional DTCs in Floquet-driven Rydberg atomic gases. We have discovered multiple $n$-DTCs with integer values of $n$ = 2, 3, and 4, and others ranging up to 14, along with fractional $n$-DTCs with $n$ values beyond the integers. The system response can transition between adjacent integer DTCs, during which the fractional DTCs are investigated. Study of higher-order and fractional DTCs expands fundamental knowledge of non-equilibrium dynamics and is promising for discovery of more complex temporal symmetries beyond the single discrete time translation symmetry., Comment: Added references

A time crystal is an exotic phase of matter where time-translational symmetry is broken; this phase differs from the spatial symmetry breaking induced in crystals in space. Lots of experiments report the transition from a thermal equilibrium phase to time crystal phase. However, there is no experimental method to probe the bifurcation effect of distinct time crystals in quantum many-body systems. Here, in a driven and dissipative many-body Rydberg atom system, we observe multiple continuous dissipative time crystals and emergence of more complex temporal symmetries beyond the single time crystal phase. Bifurcation of time crystals in strongly interacting Rydberg atoms is observed; the process manifests as a transition from a time crystal state of long temporal order to one of short temporal order, or vice versa. By manipulating the driving field parameters, we observe the time crystal's bistability and a hysteresis loop. These investigations indicate new possibilities for control and manipulation of the temporal symmetries of non-equilibrium systems., Comment: Added references

Quantum many-body systems near phase transitions respond collectively to externally applied perturbations. We explore this phenomenon in a laser-driven dissipative Rydberg gas that is tuned to a bistable regime. Here two metastable phases coexist, which feature a low and high density of Rydberg atoms, respectively. The ensuing collective dynamics, which we monitor in situ, is characterized by stochastic collective jumps between these two macroscopically distinct many-body phases. We show that the statistics of these jumps can be controlled using a dual-tone microwave field. In particular, we find that the distribution of jump times develops peaks corresponding to subharmonics of the relative microwave detuning. Our study demonstrates the control of collective statistical properties of dissipative quantum many-body systems without the necessity of fine-tuning or of ultra cold temperatures. Such robust phenomena may find technological applications in quantum sensing and metrology.

This paper studies the fundamental learning problem of the energy-based model (EBM). Learning the EBM can be achieved using the maximum likelihood estimation (MLE), which typically involves the Markov Chain Monte Carlo (MCMC) sampling, such as the Langevin dynamics. However, the noise-initialized Langevin dynamics can be challenging in practice and hard to mix. This motivates the exploration of joint training with the generator model where the generator model serves as a complementary model to bypass MCMC sampling. However, such a method can be less accurate than the MCMC and result in biased EBM learning. While the generator can also serve as an initializer model for better MCMC sampling, its learning can be biased since it only matches the EBM and has no access to empirical training examples. Such biased generator learning may limit the potential of learning the EBM. To address this issue, we present a joint learning framework that interweaves the maximum likelihood learning algorithm for both the EBM and the complementary generator model. In particular, the generator model is learned by MLE to match both the EBM and the empirical data distribution, making it a more informative initializer for MCMC sampling of EBM. Learning generator with observed examples typically requires inference of the generator posterior. To ensure accurate and efficient inference, we adopt the MCMC posterior sampling and introduce a complementary inference model to initialize such latent MCMC sampling. We show that three separate models can be seamlessly integrated into our joint framework through two (dual-) MCMC teaching, enabling effective and efficient EBM learning.

Finding distributions of permutation statistics over pattern-avoiding classes of permutations attracted much attention in the literature. In particular, Bukata et al. found distributions of ascents and descents on permutations avoiding any two patterns of length 3. In this paper, we generalize these results in two different ways: we find explicit formulas for the joint distribution of six statistics (asc, des, lrmax, lrmin, rlmax, rlmin), and also explicit formulas for the joint distribution of four statistics (asc, des, MNA, MND) on these permutations in all cases. The latter result also extends the recent studies by Kitaev and Zhang of the statistics MNA and MND (related to non-overlapping occurrences of ascents and descents) on stack-sortable permutations. All multivariate generating functions in our paper are rational, and we provide combinatorial proofs of five equidistribution results that can be derived from the generating functions., Comment: 25 pages

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 modelling data distribution.

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.

Generation of molecules with desired chemical and biological properties such as high drug-likeness, high binding affinity to target proteins, is critical for drug discovery. In this paper, we propose a probabilistic generative model to capture the joint distribution of molecules and their properties. Our model assumes an energy-based model (EBM) in the latent space. Conditional on the latent vector, the molecule and its properties are modeled by a molecule generation model and a property regression model respectively. To search for molecules with desired properties, we propose a sampling with gradual distribution shifting (SGDS) algorithm, so that after learning the model initially on the training data of existing molecules and their properties, the proposed algorithm gradually shifts the model distribution towards the region supported by molecules with desired values of properties. Our experiments show that our method achieves very strong performances on various molecule design tasks.

Sparsity is a desirable attribute. It can lead to more efficient and more effective representations compared to the dense model. Meanwhile, learning sparse latent representations has been a challenging problem in the field of computer vision and machine learning due to its complexity. In this paper, we present a new unsupervised learning method to enforce sparsity on the latent space for the generator model with a gradually sparsified spike and slab distribution as our prior. Our model consists of only one top-down generator network that maps the latent variable to the observed data. Latent variables can be inferred following generator posterior direction using non-persistent gradient based method. Spike and Slab regularization in the inference step can push non-informative latent dimensions towards zero to induce sparsity. Extensive experiments show the model can preserve majority of the information from original images with sparse representations while demonstrating improved results compared to other existing methods. We observe that our model can learn disentangled semantics and increase explainability of the latent codes while boosting the robustness in the task of classification and denoising.

This paper studies the fundamental problem of learning energy-based model (EBM) in the latent space of the generator model. Learning such prior model typically requires running costly Markov Chain Monte Carlo (MCMC). Instead, we propose to use noise contrastive estimation (NCE) to discriminatively learn the EBM through density ratio estimation between the latent prior density and latent posterior density. However, the NCE typically fails to accurately estimate such density ratio given large gap between two densities. To effectively tackle this issue and learn more expressive prior models, we develop the adaptive multi-stage density ratio estimation which breaks the estimation into multiple stages and learn different stages of density ratio sequentially and adaptively. The latent prior model can be gradually learned using ratio estimated in previous stage so that the final latent space EBM prior can be naturally formed by product of ratios in different stages. The proposed method enables informative and much sharper prior than existing baselines, and can be trained efficiently. Our experiments demonstrate strong performances in image generation and reconstruction as well as anomaly detection., Comment: Accepted to NeurIPS 2022

In this paper, we propose a new upper bound on the error probability performance of maximum-likelihood (ML) detection. The proposed approach provides a much tighter upper bound when compared to the traditionally used union bound, especially when the number of pairwise error probabilities (PEPs) is large. In fact, the proposed approach tightens the union bound by first assuming that a detection error always occurs in a deep fading event where the channel gain is lower than a certain threshold. A minimisation is then taken with respect to the gain threshold in order to make the upper bound as tight as possible. We also prove that the objective function has a single minimiser under several general assumptions so that the minimiser can be easily found using optimisation algorithms. The expression of the new upper bound under correlated Rayleigh fading channels is derived and several analytical and numerical examples are provided to show the performance of the proposed bound., Comment: 15 pages, 5 figures

Current pre-training methods in computer vision focus on natural images in the daily-life context. However, abstract diagrams such as icons and symbols are common and important in the real world. This work is inspired by Tangram, a game that requires replicating an abstract pattern from seven dissected shapes. By recording human experience in solving tangram puzzles, we present the Tangram dataset and show that a pre-trained neural model on the Tangram helps solve some mini visual tasks based on low-resolution vision. Extensive experiments demonstrate that our proposed method generates intelligent solutions for aesthetic tasks such as folding clothes and evaluating room layouts. The pre-trained feature extractor can facilitate the convergence of few-shot learning tasks on human handwriting and improve the accuracy in identifying icons by their contours. The Tangram dataset is available at https://github.com/yizhouzhao/Tangram.

With the recent success of deep learning algorithms, many researchers have focused on generative models for human motion animation. However, the research community lacks a platform for training and benchmarking various algorithms, and the animation industry needs a toolkit for implementing advanced motion synthesizing techniques. To facilitate the study of deep motion synthesis methods for skeleton-based human animation and their potential applications in practical animation making, we introduce \genmotion: a library that provides unified pipelines for data loading, model training, and animation sampling with various deep learning algorithms. Besides, by combining Python coding in the animation software \genmotion\ can assist animators in creating real-time 3D character animation. Source code is available at https://github.com/realvcla/GenMotion/.

With the wide application of electronic health records (EHR) in healthcare facilities, health event prediction with deep learning has gained more and more attention. A common feature of EHR data used for deep-learning-based predictions is historical diagnoses. Existing work mainly regards a diagnosis as an independent disease and does not consider clinical relations among diseases in a visit. Many machine learning approaches assume disease representations are static in different visits of a patient. However, in real practice, multiple diseases that are frequently diagnosed at the same time reflect hidden patterns that are conducive to prognosis. Moreover, the development of a disease is not static since some diseases can emerge or disappear and show various symptoms in different visits of a patient. To effectively utilize this combinational disease information and explore the dynamics of diseases, we propose a novel context-aware learning framework using transition functions on dynamic disease graphs. Specifically, we construct a global disease co-occurrence graph with multiple node properties for disease combinations. We design dynamic subgraphs for each patient's visit to leverage global and local contexts. We further define three diagnosis roles in each visit based on the variation of node properties to model disease transition processes. Experimental results on two real-world EHR datasets show that the proposed model outperforms state of the art in predicting health events., Comment: This paper is accepted by AAAI 2022