arXiv subjects
Publications and source records attributed to Yuxin Dong.
In this paper, inspired by [4, 9], we prove a Sobolev inequality on manifolds with density and asymptotically nonnegative Bakry-Émery Ricci curvature.
We prove the Heisenberg-Pauli-Weyl inequality, Hardy-Sobolev inequality, and Caffarelli-Kohn-Nirenberg (CKN) inequality on manifolds with nonnegative Ricci curvature and Euclidean volume growth, of dimension n>=3.
FRB 20220610A is a high-redshift Fast Radio Burst (FRB) that has not been observed to repeat. Here, we present rest-frame UV and optical $\textit{Hubble Space Telescope}$ observations of the field of FRB 20220610A. The imaging reveals seven extended sources, one of which we identify as the most likely host galaxy with a spectroscopic redshift of $z$=1.017. We spectroscopically confirm at least three additional sources to be at the same redshift, and identify the system as a compact galaxy group with possible signs of interaction among group members. We determine the host of FRB 20220610A to be a star-forming galaxy with stellar mass of $\approx10^{9.7}\,M_{\odot}$, mass-weighted age of $\approx2.6$~Gyr, and star formation rate (integrated over the last 100 Myr) of $\approx1.7$~M$_{\odot}$~yr$^{-1}$. These host properties are commensurate with the star-forming field galaxy population at z~1 and trace their properties analogously to the population of low-$z$ FRB hosts. Based on estimates of the total stellar mass of the galaxy group, we calculate a fiducial contribution to the observed Dispersion Measure (DM) from the intragroup medium of $\approx 110-220$ $\rm pc \, cm^{-3}$ (rest-frame). This leaves a significant excess of $500^{+272}_{-109}$ $\rm pc \, cm^{-3}$ (in the observer frame), with additional sources of DM possibly originating from the circumburst environment, host galaxy interstellar medium, and/or foreground structures along the line of sight. Given the low occurrence rates of galaxies in compact groups, the discovery of an FRB in such a group demonstrates a rare and novel environment in which FRBs can occur.
We present a comprehensive catalog of observations and stellar population properties for 23 highly secure host galaxies of fast radio bursts (FRBs). Our sample comprises six repeating FRBs and 17 apparent non-repeaters. We present 82 new photometric and eight new spectroscopic observations of these hosts. Using stellar population synthesis modeling and employing non-parametric star formation histories (SFHs), we find that FRB hosts have a median stellar mass of $\approx 10^{9.9}\,M_{\odot}$, mass-weighted age $\approx 5.1$ Gyr, and ongoing star formation rate $\approx 1.3\,M_{\odot}$ yr$^{-1}$ but span wide ranges in all properties. Classifying the hosts by degree of star formation, we find that 87% (20/23 hosts) are star-forming, two are transitioning, and one is quiescent. The majority trace the star-forming main sequence of galaxies, but at least three FRBs in our sample originate in less active environments (two non-repeaters and one repeater). Across all modeled properties, we find no statistically significant distinction between the hosts of repeaters and non-repeaters. However, the hosts of repeating FRBs generally extend to lower stellar masses, and the hosts of non-repeaters arise in more optically luminous galaxies. While four of the galaxies with the most clear and prolonged rises in their SFHs all host repeating FRBs, demonstrating heightened star formation activity in the last $\lesssim 100$ Myr, one non-repeating host shows this SFH as well. Our results support progenitor models with short delay channels (i.e., magnetars formed via core-collapse supernova) for most FRBs, but the presence of some FRBs in less active environments suggests a fraction form through more delayed channels.
In this paper, we derive local and global Li-Yau type gradient estimates for the positive solutions of the CR heat equation on complete noncompact pseudo-Hermitian manifolds. As applications of the gradient estimates, we give a Harnack inequality for the positive solutions of the CR heat equation, and then obtain an upper bound estimate for the corresponding heat kernel.
Recently, information theoretic analysis has become a popular framework for understanding the generalization behavior of deep neural networks. It allows a direct analysis for stochastic gradient/Langevin descent (SGD/SGLD) learning algorithms without strong assumptions such as Lipschitz or convexity conditions. However, the current generalization error bounds within this framework are still far from optimal, while substantial improvements on these bounds are quite challenging due to the intractability of high-dimensional information quantities. To address this issue, we first propose a novel information theoretical measure: kernelized Renyi's entropy, by utilizing operator representation in Hilbert space. It inherits the properties of Shannon's entropy and can be effectively calculated via simple random sampling, while remaining independent of the input dimension. We then establish the generalization error bounds for SGD/SGLD under kernelized Renyi's entropy, where the mutual information quantities can be directly calculated, enabling evaluation of the tightness of each intermediate step. We show that our information-theoretical bounds depend on the statistics of the stochastic gradients evaluated along with the iterates, and are rigorously tighter than the current state-of-the-art (SOTA) results. The theoretical findings are also supported by large-scale empirical studies1.
The recently developed matrix based Renyi's entropy enables measurement of information in data simply using the eigenspectrum of symmetric positive semi definite (PSD) matrices in reproducing kernel Hilbert space, without estimation of the underlying data distribution. This intriguing property makes the new information measurement widely adopted in multiple statistical inference and learning tasks. However, the computation of such quantity involves the trace operator on a PSD matrix $G$ to power $α$(i.e., $tr(G^α)$), with a normal complexity of nearly $O(n^3)$, which severely hampers its practical usage when the number of samples (i.e., $n$) is large. In this work, we present computationally efficient approximations to this new entropy functional that can reduce its complexity to even significantly less than $O(n^2)$. To this end, we leverage the recent progress on Randomized Numerical Linear Algebra, developing Taylor, Chebyshev and Lanczos approximations to $tr(G^α)$ for arbitrary values of $α$ by converting it into matrix-vector multiplications problem. We also establish the connection between the matrix-based Renyi's entropy and PSD matrix approximation, which enables exploiting both clustering and block low-rank structure of $G$ to further reduce the computational cost. We theoretically provide approximation accuracy guarantees and illustrate the properties of different approximations. Large-scale experimental evaluations on both synthetic and real-world data corroborate our theoretical findings, showing promising speedup with negligible loss in accuracy.
The matrix-based Rényi's entropy allows us to directly quantify information measures from given data, without explicit estimation of the underlying probability distribution. This intriguing property makes it widely applied in statistical inference and machine learning tasks. However, this information theoretical quantity is not robust against noise in the data, and is computationally prohibitive in large-scale applications. To address these issues, we propose a novel measure of information, termed low-rank matrix-based Rényi's entropy, based on low-rank representations of infinitely divisible kernel matrices. The proposed entropy functional inherits the specialty of of the original definition to directly quantify information from data, but enjoys additional advantages including robustness and effective calculation. Specifically, our low-rank variant is more sensitive to informative perturbations induced by changes in underlying distributions, while being insensitive to uninformative ones caused by noises. Moreover, low-rank Rényi's entropy can be efficiently approximated by random projection and Lanczos iteration techniques, reducing the overall complexity from $\mathcal{O}(n^3)$ to $\mathcal{O}(n^2 s)$ or even $\mathcal{O}(ns^2)$, where $n$ is the number of data samples and $s \ll n$. We conduct large-scale experiments to evaluate the effectiveness of this new information measure, demonstrating superior results compared to matrix-based Rényi's entropy in terms of both performance and computational efficiency.
We present the stellar population properties of 69 short gamma-ray burst (GRB) host galaxies, representing the largest uniformly-modeled sample to-date. Using the Prospector stellar population inference code, we jointly fit photometry and/or spectroscopy of each host galaxy. We find a population median redshift of $z=0.64^{+0.83}_{-0.32}$ ($68\%$ confidence), including 10 new or revised photometric redshifts at $z\gtrsim1$. We further find a median mass-weighted age of $t_m=0.8^{+2.71}_{-0.53}$Gyr, stellar mass of $\log(M_*/M_\odot)=9.69^{+0.75}_{-0.65}$, star formation rate of SFR=$1.44^{+9.37}_{-1.35}M_\odot$yr$^{-1}$, stellar metallicity of $\log(Z_*/Z_\odot)=-0.38^{+0.44}_{-0.42}$, and dust attenuation of $A_V=0.43^{+0.85}_{-0.36}$~mag (68\% confidence). Overall, the majority of short GRB hosts are star-forming ($\approx84\%$), with small fractions that are either transitioning ($\approx6\%$) or quiescent ($\approx10\%$); however, we observe a much larger fraction ($\approx40\%$) of quiescent and transitioning hosts at $z\lesssim0.25$, commensurate with galaxy evolution. We find that short GRB hosts populate the star-forming main sequence of normal field galaxies, but do not include as many high-mass galaxies, implying that their binary neutron star (BNS) merger progenitors are dependent on a combination of host star formation and stellar mass. The distribution of ages and redshifts implies a broad delay-time distribution, with a fast-merging channel at $z>1$ and a decreased BNS formation efficiency at lower redshifts. If short GRB hosts are representative of BNS merger hosts within the horizon of current gravitational wave detectors, these results can inform future searches for electromagnetic counterparts. All of the data and modeling products are available on the BRIGHT website.
In this paper, we investigate the stability problem of subelliptic harmonic maps with potential. First, we derive the first and second variation formulas for subelliptic harmonic maps with potential. As a result, it is proved that a subelliptic harmonic map with potential is stable if the target manifold has nonpositive curvature and the Hessian of the potential is nonpositive definite. We also give Leung type results which involve the instability of subelliptic harmonic maps with potential when the target manifold is a sphere of dimension $\geq 3$.
Using the ABP-method as in a recent work by Brendle, we establish some sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds in a complete Riemannian manifold with asymptotically nonnegative curvature. These inequalities generalize those given by Brendle in the case of complete Riemannian manifolds with nonnegative curvature.
We present a comprehensive optical and near-infrared census of the fields of 90 short gamma-ray bursts (GRBs) discovered in 2005-2021, constituting all short GRBs for which host galaxy associations are feasible ($\approx$ 60% of the total Swift short GRB population). We contribute 245 new multi-band imaging observations across 49 distinct GRBs and 25 spectra of their host galaxies. Supplemented by literature and archival survey data, the catalog contains 335 photometric and 40 spectroscopic data sets. The photometric catalog reaches $3σ$ depths of $\gtrsim 24-27$ mag and $\gtrsim 23-26$ mag for the optical and near-infrared bands, respectively. We identify host galaxies for 84 bursts, in which the most robust associations make up 54% (49/90) of events, while only a small fraction, 6.7%, have inconclusive host associations. Based on new spectroscopy, we determine 17 host spectroscopic redshifts with a range of $z\approx 0.15-1.6$ and find that $\approx$ 25-44% of Swift short GRBs originate from $z>1$. We also present the galactocentric offset catalog for 83 short GRBs. Taking into account the large range of individual measurement uncertainties, we find a median of projected offset of $\approx 7.9$ kpc, for which the bursts with the most robust associations have a smaller median of $\approx 4.9$ kpc. Our catalog captures more high-redshift and low-luminosity hosts, and more highly-offset bursts than previously found, thereby diversifying the population of known short GRB hosts and properties. In terms of locations and host luminosities, the populations of short GRBs with and without detectable extended emission are statistically indistinguishable. This suggests that they arise from the same progenitors, or from multiple progenitors which form and evolve in similar environments. All of the data products are available on the BRIGHT website.
The Matrix-based Renyi's entropy enables us to directly measure information quantities from given data without the costly probability density estimation of underlying distributions, thus has been widely adopted in numerous statistical learning and inference tasks. However, exactly calculating this new information quantity requires access to the eigenspectrum of a semi-positive definite (SPD) matrix $A$ which grows linearly with the number of samples $n$, resulting in a $O(n^3)$ time complexity that is prohibitive for large-scale applications. To address this issue, this paper takes advantage of stochastic trace approximations for matrix-based Renyi's entropy with arbitrary $α\in R^+$ orders, lowering the complexity by converting the entropy approximation to a matrix-vector multiplication problem. Specifically, we develop random approximations for integer order $α$ cases and polynomial series approximations (Taylor and Chebyshev) for non-integer $α$ cases, leading to a $O(n^2sm)$ overall time complexity, where $s,m \ll n$ denote the number of vector queries and the polynomial order respectively. We theoretically establish statistical guarantees for all approximation algorithms and give explicit order of s and m with respect to the approximation error $\varepsilon$, showing optimal convergence rate for both parameters up to a logarithmic factor. Large-scale simulations and real-world applications validate the effectiveness of the developed approximations, demonstrating remarkable speedup with negligible loss in accuracy.
Modal regression, a widely used regression protocol, has been extensively investigated in statistical and machine learning communities due to its robustness to outliers and heavy-tailed noises. Understanding modal regression's theoretical behavior can be fundamental in learning theory. Despite significant progress in characterizing its statistical property, the majority of the results are based on the assumption that samples are independent and identical distributed (i.i.d.), which is too restrictive for real-world applications. This paper concerns the statistical property of regularized modal regression (RMR) within an important dependence structure - Markov dependent. Specifically, we establish the upper bound for RMR estimator under moderate conditions and give an explicit learning rate. Our results show that the Markov dependence impacts on the generalization error in the way that sample size would be discounted by a multiplicative factor depending on the spectral gap of underlying Markov chain. This result shed a new light on characterizing the theoretical underpinning for robust regression.
Let $(M,H,g_H;g)$ be a sub-Riemannian manifold and $(N,h)$ be a Riemannian manifold. For a smooth map $u: M \to N$, we consider the energy functional $E_G(u) = \frac{1}{2} \int_M[|\mathrm{d}u_H|^2-2G(u)] \mathrm{d}V_M$, where $\mathrm{d}u_H$ is the horizontal differential of $u$, $G:N\to \mathbb{R}$ is a smooth function on $N$. The critical maps of $E_G(u)$ are referred to as subelliptic harmonic maps with potential $G$. In this paper, we investigate the existence problem for subelliptic harmonic maps with potentials by a subelliptic heat flow. Assuming that the target Riemannian manifold has non-positive sectional curvature and the potential $G$ satisfies various suitable conditions, we prove some Eells-Sampson type existence results when the source manifold is either a step-$2$ sub-Riemannian manifold or a step-$r$ sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation.
Subsampling is an important technique to tackle the computational challenges brought by big data. Many subsampling procedures fall within the framework of importance sampling, which assigns high sampling probabilities to the samples appearing to have big impacts. When the noise level is high, those sampling procedures tend to pick many outliers and thus often do not perform satisfactorily in practice. To tackle this issue, we design a new Markov subsampling strategy based on Huber criterion (HMS) to construct an informative subset from the noisy full data; the constructed subset then serves as a refined working data for efficient processing. HMS is built upon a Metropolis-Hasting procedure, where the inclusion probability of each sampling unit is determined using the Huber criterion to prevent over scoring the outliers. Under mild conditions, we show that the estimator based on the subsamples selected by HMS is statistically consistent with a sub-Gaussian deviation bound. The promising performance of HMS is demonstrated by extensive studies on large scale simulations and real data examples.
We present the Australian Square Kilometre Array Pathfinder (ASKAP) localization and follow-up observations of the host galaxy of the repeating fast radio burst (FRB) source, FRB20201124A, the fifth such extragalactic repeating FRB with an identified host. From spectroscopic observations using the 6.5-m MMT Observatory, we derive a redshift of $z=0.0979 \pm 0.0001$, a star formation rate inferred from H$α$ emission of SFR(H$α$) $\approx 2.1 M_{\odot}$ yr$^{-1}$, and a gas-phase metallicity of 12+log(O/H)$\approx 9.0$. By jointly modeling the 12-filter optical-mid-infrared (MIR) photometry and spectroscopy of the host, we infer a median stellar mass of $\approx 2 \times 10^{10} M_{\odot}$, internal dust extinction of $A_V\approx 1-1.5$ mag, and a mass-weighted stellar population age of $\approx 5-6$ Gyr. Connecting these data to the radio and X-ray observations, we cannot reconcile the broad-band behavior with strong AGN activity and instead attribute the dominant source of persistent radio emission to star formation, likely originating from the circumnuclear region of the host. The modeling also indicates a hot dust component contributing to the MIR luminosity at a level of $\approx 10-30\%$. We model the host galaxy's star formation and mass assembly histories, finding that the host assembled $>90\%$ of its mass by 1 Gyr ago and exhibited a fairly constant SFR for most of its existence, with no clear evidence of past star-burst activity.
In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.