Daily Papers

We study how to train personalized models for different tasks on decentralized devices with limited local data. We propose "Structured Cooperative Learning (SCooL)", in which a cooperation graph across devices is generated by a graphical model prior to automatically coordinate mutual learning between devices. By choosing graphical models enforcing different structures, we can derive a rich class of existing and novel decentralized learning algorithms via variational inference. In particular, we show three instantiations of SCooL that adopt Dirac distribution, stochastic block model (SBM), and attention as the prior generating cooperation graphs. These EM-type algorithms alternate between updating the cooperation graph and cooperative learning of local models. They can automatically capture the cross-task correlations among devices by only monitoring their model updating in order to optimize the cooperation graph. We evaluate SCooL and compare it with existing decentralized learning methods on an extensive set of benchmarks, on which SCooL always achieves the highest accuracy of personalized models and significantly outperforms other baselines on communication efficiency. Our code is available at https://github.com/ShuangtongLi/SCooL.

Physics is based on probabilities as fundamental entities of a mathematical description. Expectation values of observables are computed according to the classical statistical rule. The overall probability distribution for one world covers all times. The quantum formalism arises once one focuses on the evolution of the time-local probabilistic information. Wave functions or the density matrix allow the formulation of a general linear evolution law for classical statistics. The quantum formalism for classical statistics is a powerful tool which allows us to implement for generalized Ising models the momentum observable with the associated Fourier representation. The association of operators to observables permits the computation of expectation values in terms of the density matrix by the usual quantum rule. We show that probabilistic cellular automata are quantum systems in a formulation with discrete time steps and real wave functions. With a complex structure the evolution operator for automata can be expressed in terms of a Hamiltonian involving fermionic creation and annihilation operators. The time-local probabilistic information amounts to a subsystem of the overall probabilistic system which is correlated with its environment consisting of the past and future. Such subsystems typically involve probabilistic observables for which only a probability distribution for their possible measurement values is available. Incomplete statistics does not permit to compute classical correlation functions for arbitrary subsystem-observables. Bell's inequalities are not generally applicable.

We introduce featom, an open source code that implements a high-order finite element solver for the radial Schr\"odinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the kappa=pm1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10^{-8} Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schr\"odinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (Z) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

It is shown that in the complex trajectory representation of quantum mechanics, the Born's Psi^{\star}Ψprobability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.

The polarization of an X-ray beam that produces electrons with velocity components perpendicular to the beam generates an azimuthal distribution of the ejected electrons. We present methods for simulating and for analyzing the angular dependence of electron detections which enable us to derive simple analytical expressions for useful statistical properties of observable data. The derivations are verified by simulations. While we confirm the results of previous work on this topic, we provide an extension needed for analytical treatment of the full range of possible polarization amplitudes.

We consider N classical particles interacting via the Coulomb potential in spatial dimension d and in the presence of an external trap, at equilibrium at inverse temperature beta. In the large N limit, the particles are confined within a droplet of finite size. We study smooth linear statistics, i.e. the fluctuations of sums of the form {cal L}_N = sum_{i=1}^N f({bf x}_i), where {bf x}_i's are the positions of the particles and where f({bf x}_i) is a sufficiently regular function. There exists at present standard results for the first and second moments of {cal L}_N in the large N limit, as well as associated Central Limit Theorems in general dimension and for a wide class of confining potentials. Here we obtain explicit expressions for the higher order cumulants of {cal L}_N at large N, when the function f({bf x})=f(|{bf x}|) and the confining potential are both rotationnally invariant. A remarkable feature of our results is that these higher cumulants depend only on the value of f'(|{bf x}|) and its higher order derivatives evaluated exactly at the boundary of the droplet, which in this case is a d-dimensional sphere. In the particular two-dimensional case d=2 at the special value beta=2, a connection to the Ginibre ensemble allows us to derive these results in an alternative way using the tools of determinantal point processes. Finally we also obtain the large deviation form of the full probability distribution function of {cal L}_N.

Using an evolutionary algorithm in combination with first-principles density functional theory calculations, we identify two-dimensional (2D) CaP_3 monolayer as a new Dirac semimetal due to inversion and nonsymmorphic spatial symmetries of the structure. This new topological material, composed of light elements, exhibits high structural stability (higher than the phase known in the literature), which is confirmed by thermodynamic and kinetic stability analysis. Moreover, it satisfies the electron filling criteria, so that its Dirac state is located near the Fermi level. The existence of the Dirac state predicted by the theoretical symmetry analysis is also confirmed by first-principles electronic band structure calculations. We find that the energy position of the Dirac state can be tuned by strain, while the Dirac state is unstable against an external electric field since it breaks the spatial inversion symmetry. Our findings should be instrumental in the development of 2D Dirac fermions based on light elements for their application in nanoelectronic devices and topological electronics.

We discuss an unfortunate mistake, for a Dirac free particle, in the last Fermi lecture notes on quantum mechanics, in a course given at the University of Chicago in winter and spring of 1954. As is demonstrated, the correct result can be obtained by a simple matrix multiplication. An attempt to collect a relevant bibliography is made.

Molecules are frequently represented as graphs, but the underlying 3D molecular geometry (the locations of the atoms) ultimately determines most molecular properties. However, most molecules are not static and at room temperature adopt a wide variety of geometries or conformations. The resulting distribution on geometries p(x) is known as the Boltzmann distribution, and many molecular properties are expectations computed under this distribution. Generating accurate samples from the Boltzmann distribution is therefore essential for computing these expectations accurately. Traditional sampling-based methods are computationally expensive, and most recent machine learning-based methods have focused on identifying modes in this distribution rather than generating true samples. Generating such samples requires capturing conformational variability, and it has been widely recognized that the majority of conformational variability in molecules arises from rotatable bonds. In this work, we present VonMisesNet, a new graph neural network that captures conformational variability via a variational approximation of rotatable bond torsion angles as a mixture of von Mises distributions. We demonstrate that VonMisesNet can generate conformations for arbitrary molecules in a way that is both physically accurate with respect to the Boltzmann distribution and orders of magnitude faster than existing sampling methods.

It is well known that there is a particle-hole symmetry for spin-polarized electrons with two-body interactions in a partially filled Landau level, which becomes exact in the limit where the cyclotron energy is large compared to the interaction strength, so one can ignore mixing between Landau levels. This symmetry is explicit in the description of a half-filled Landau level recently introduced by D. T. Son, using Dirac fermions, but it was thought to be absent in the older fermion-Chern- Simons approach, developed by Halperin, Lee, and Read and subsequent authors. We show here, however, that when properly evaluated, the Halperin, Lee, Read (HLR) theory gives results for long-wavelength low-energy physical properties, including the Hall conductance in the presence of impurities and the positions of minima in the magnetoroton spectra for fractional quantized Hall states close to half-filling, that are identical to predictions of the Dirac formulation. In fact, the HLR theory predicts an emergent particle-hole symmetry near half filling, even when the cyclotron energy is finite.

Astrophysical uncertainties in dark matter direct detection experiments are typically addressed by parametrizing the velocity distribution in terms of a few uncertain parameters that vary around some central values. Here we propose a method to optimize over all velocity distributions lying within a given distance measure from a central distribution. We discretize the dark matter velocity distribution as a superposition of streams, and use a variety of information divergences to parametrize its uncertainties. With this, we bracket the limits on the dark matter-nucleon and dark matter-electron scattering cross sections, when the true dark matter velocity distribution deviates from the commonly assumed Maxwell-Boltzmann form. The methodology pursued is general and could be applied to other physics scenarios where a given physical observable depends on a function that is uncertain.

According to the generalized Polya theorem, the Gaussian distribution on the real line is characterized by the property of equidistribution of a monomial and a linear form of independent identically distributed random variables. We give a complete description of a-adic solenoids for which an analog of this theorem is true. The proof of the main theorem is reduced to solving some functional equation in the class of continuous positive definite functions on the character group of an a-adic solenoid

In this work, we derive sharp non-asymptotic deviation bounds for weighted sums of Dirichlet random variables. These bounds are based on a novel integral representation of the density of a weighted Dirichlet sum. This representation allows us to obtain a Gaussian-like approximation for the sum distribution using geometry and complex analysis methods. Our results generalize similar bounds for the Beta distribution obtained in the seminal paper Alfers and Dinges [1984]. Additionally, our results can be considered a sharp non-asymptotic version of the inverse of Sanov's theorem studied by Ganesh and O'Connell [1999] in the Bayesian setting. Based on these results, we derive new deviation bounds for the Dirichlet process posterior means with application to Bayesian bootstrap. Finally, we apply our estimates to the analysis of the Multinomial Thompson Sampling (TS) algorithm in multi-armed bandits and significantly sharpen the existing regret bounds by making them independent of the size of the arms distribution support.

We search for dark matter (DM) annihilating subhalos of the Milky Way halo among the Fermi Large Area Telescope (LAT) unassociated sources. We construct, for the first time, a statistical model of the unassociated sources at latitudes above 10 degrees. The latter is built as a combination of both DM annihilation subhalos as well as Galactic and extragalactic astrophysical components. The astrophysical components are constructed based on distributions of associated sources, while the distribution of DM subhalos is derived from Monte Carlo simulations. In this model we take into account the differences in the distributions of associated and unassociated sources including both covariate and prior probability shifts (both being forms of ``dataset shifts''). Previous searches of DM subhalos were based on classify-and-count strategies, while the approach adopted in this work is based on quantification learning, which allows one to determine a well-defined statistical interpretation of the contribution of a population of DM subhalos to the unassociated Fermi-LAT sources. In the bb annihilation channel and for a range of DM masses from 10 GeV to 1 TeV, we don't find a significant contribution from DM subhalos and derive a statistical 95% confidence upper limit on the DM annihilation cross section in this channel. While the derived limits are consistent with previous classify-and-count approaches, our generative statistical model opens new avenues for population studies of Fermi-LAT sources and, more generally, for searches of anomalies on top of backgrounds in presence of statistical and systematic uncertainties.

We provide a concise framework for generalized ensemble theory through a matrix-based approach. By introducing an observation matrix, any discrete probability distribution, including those for non-equilibrium steady states, can be expressed as a generalized Boltzmann distribution, with observables and conjugate variables as the basis and coordinates in a linear space. In this framework, we identify the minimal sufficient statistics required for inferring the Boltzmann distribution. Furthermore, we show that the Hadamard and Vandermonde matrices are suitable observation matrices for spin systems and random walks. In master equation systems, the probability flux observation matrix facilitates the identification of detailed balance violations. Our findings provide a new approach to developing generalized ensemble theory for non-equilibrium steady-state systems.

Hamiltonian Monte Carlo has proven a remarkable empirical success, but only recently have we begun to develop a rigorous understanding of why it performs so well on difficult problems and how it is best applied in practice. Unfortunately, that understanding is confined within the mathematics of differential geometry which has limited its dissemination, especially to the applied communities for which it is particularly important. In this review I provide a comprehensive conceptual account of these theoretical foundations, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor. Whether a practitioner or a statistician, the dedicated reader will acquire a solid grasp of how Hamiltonian Monte Carlo works, when it succeeds, and, perhaps most importantly, when it fails.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Collections of probability distributions arise in a variety of applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions can be defined over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of distributions over such general domains. To this end, we propose the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, allowing us to reduce the distribution collection comparison problem to a more familiar mean testing problem in a Hilbert space. We provide two testing procedures one based on resampling and another on combining p-values from coordinate-wise tests. Our experiments in various synthetic and real data settings show that the resulting tests are powerful and the p-values are well-calibrated.

The superfluid phase transition dynamics and associated spontaneous vortex formation with the crossing of the critical temperature in a disk geometry is studied in the framework of the AdS/CFT correspondence by solving the Einstein-Abelian-Higgs model in an AdS_4 black hole. For a slow quench, the vortex density admits a universal scaling law with the cooling rate as predicted by the Kibble-Zurek mechanism (KZM), while for fast quenches, the density shows a universal scaling behavior as a function of the final temperature, that lies beyond the KZM prediction. The vortex number distribution in both the power-law and saturation regimes can be approximated by a normal distribution. However, the study of the universal scaling of the cumulants reveals non-normal features and indicates that vortex statistics in the newborn superfluid is best described by the Poisson binomial distribution, previously predicted in the KZM regime [Phys. Rev. Lett. 124, 240602 (2020)]. This is confirmed by studying the cumulant scalings as a function of the quench time and the quench depth. Our work supports the existence of a universal defect number distribution that accommodates the KZM scaling, its breakdown at fast quenches, and the additional universal scaling laws as a function of the final value of the control parameter.

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance significantly.

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as ``the simple method'') optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi--Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfeld's original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.

This paper studies the scale and redshift variation of density and velocity distributions in self-gravitating collisionless dark matter flow by a halo-based non-projection approach. All particles are divided into halo and out-of-halo particles for redshift variation of distributions. Without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales r. We demonstrate that: i) Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained, modeled, and compared with the N-body simulation; ii) the velocity distributions are symmetric on both small and large scales and are non-symmetric with a negative skewness on intermediate scales due to the inverse energy cascade at a constant rate varepsilon_u; iii) On small scales, the even order moments of pairwise velocity Delta u_L follow a two-thirds law (-varepsilon_ur)^{2/3}, while the odd order moments follow a linear scaling langle(Delta u_L)^{2n+1}rangle=(2n+1)langle(Delta u_L)^{2n}ranglelangleDelta u_Lrangler; iv) The scale variation of the velocity distributions was studied for longitudinal velocities u_L or u_L^{'}, pairwise velocity (velocity difference) Delta u_L=u_L^{'}-u_L and velocity sum Sigma u_L=u^{'}_L+u_L. Fully developed velocity fields are never Gaussian on any scale, despite that they can initially be Gaussian; v) On small scales, u_L and Sigma u_L can be modeled by a X distribution to maximize the system entropy; vi) On large scales, Delta u_L and Sigma u_L can be modeled by a logistic or a X distribution; vii) the redshift variation of the velocity distributions follows the evolution of the X distribution involving a shape parameter alpha(z) decreasing with time.

In contrast to classical reinforcement learning, distributional reinforcement learning algorithms aim to learn the distribution of returns rather than their expected value. Since the nature of the return distribution is generally unknown a priori or arbitrarily complex, a common approach finds approximations within a set of representable, parametric distributions. Typically, this involves a projection of the unconstrained distribution onto the set of simplified distributions. We argue that this projection step entails a strong inductive bias when coupled with neural networks and gradient descent, thereby profoundly impacting the generalization behavior of learned models. In order to facilitate reliable uncertainty estimation through diversity, this work studies the combination of several different projections and representations in a distributional ensemble. We establish theoretical properties of such projection ensembles and derive an algorithm that uses ensemble disagreement, measured by the average 1-Wasserstein distance, as a bonus for deep exploration. We evaluate our algorithm on the behavior suite benchmark and find that diverse projection ensembles lead to significant performance improvements over existing methods on a wide variety of tasks with the most pronounced gains in directed exploration problems.

A fully differential calculation in perturbative quantum chromodynamics is presented for the production of massive photon pairs at hadron colliders. All next-to-leading order perturbative contributions from quark-antiquark, gluon-(anti)quark, and gluon-gluon subprocesses are included, as well as all-orders resummation of initial-state gluon radiation valid at next-to-next-to-leading logarithmic accuracy. The region of phase space is specified in which the calculation is most reliable. Good agreement is demonstrated with data from the Fermilab Tevatron, and predictions are made for more detailed tests with CDF and DO data. Predictions are shown for distributions of diphoton pairs produced at the energy of the Large Hadron Collider (LHC). Distributions of the diphoton pairs from the decay of a Higgs boson are contrasted with those produced from QCD processes at the LHC, showing that enhanced sensitivity to the signal can be obtained with judicious selection of events.

We reconstruct the extra-galactic gamma-ray source-count distribution, or dN/dS, of resolved and unresolved sources by adopting machine learning techniques. Specifically, we train a convolutional neural network on synthetic 2-dimensional sky-maps, which are built by varying parameters of underlying source-counts models and incorporate the Fermi-LAT instrumental response functions. The trained neural network is then applied to the Fermi-LAT data, from which we estimate the source count distribution down to flux levels a factor of 50 below the Fermi-LAT threshold. We perform our analysis using 14 years of data collected in the (1,10) GeV energy range. The results we obtain show a source count distribution which, in the resolved regime, is in excellent agreement with the one derived from catalogued sources, and then extends as dN/dS sim S^{-2} in the unresolved regime, down to fluxes of 5 cdot 10^{-12} cm^{-2} s^{-1}. The neural network architecture and the devised methodology have the flexibility to enable future analyses to study the energy dependence of the source-count distribution.

Zipf's law appears in many application areas but does not have a closed form expression, which may make its use cumbersome. Since it coincides with the truncated version of the Zeta distribution, in this paper we propose three approximate closed form expressions for the truncated Zeta distribution, which may be employed for Zipf's law as well. The three approximations are based on the replacement of the sum occurring in Zipf's law with an integral, and are named respectively the integral approximation, the average integral approximation, and the trapezoidal approximation. While the first one is shown to be of little use, the trapezoidal approximation exhibits an error which is typically lower than 1\%, but is as low as 0.1\% for the range of values of the Zipf parameter below 1.

Turbulent magnetic fields are to some extent a universal feature in astrophysical phenomena. Charged particles that encounter these turbulence get on average accelerated according to the so-called second-order Fermi process. However, in most astrophysical environments there are additional competing processes, such as different kinds of first-order energy changes and particle escape, that effect the resulting momentum distribution of the particles. In this work we provide to our knowledge the first semi-analytical solution of the isotropic steady-state momentum diffusion equation including continuous and catastrophic momentum changes that can be applied to any arbitrary astrophysical system of interest. Here, we adopt that the assigned magnetic turbulence is constrained on a finite range and the particle flux vanishes beyond these boundaries. Consequently, we show that the so-called pile-up bump -- that has for some special cases long been established -- is a universal feature of stochastic acceleration that emerges around the momentum chi_{rm eq} where acceleration and continuous loss are in equilibrium if the particle's residence time in the system is sufficient at chi_{rm eq}. In general, the impact of continuous and catastrophic momentum changes plays a crucial role in the shape of the steady-state momentum distribution of the accelerated particles, where simplified unbroken power-law approximations are often not adequate.

In this paper we argue for the fundamental importance of the value distribution: the distribution of the random return received by a reinforcement learning agent. This is in contrast to the common approach to reinforcement learning which models the expectation of this return, or value. Although there is an established body of literature studying the value distribution, thus far it has always been used for a specific purpose such as implementing risk-aware behaviour. We begin with theoretical results in both the policy evaluation and control settings, exposing a significant distributional instability in the latter. We then use the distributional perspective to design a new algorithm which applies Bellman's equation to the learning of approximate value distributions. We evaluate our algorithm using the suite of games from the Arcade Learning Environment. We obtain both state-of-the-art results and anecdotal evidence demonstrating the importance of the value distribution in approximate reinforcement learning. Finally, we combine theoretical and empirical evidence to highlight the ways in which the value distribution impacts learning in the approximate setting.

We construct an exact analytic solution of the revised small-x helicity evolution equations, where the contributions of the quark-to-gluon and gluon-to-quark transition operators were newly included. These evolution equations are written in the large-N_c&N_f limit and are double-logarithmic, resumming powers of alpha_sln^2(1/x). Here N_c and N_f are the numbers of quark colors and flavors, while alpha_s is the strong coupling constant and x is the Bjorken-x variable. Using our solution, we obtain analytic expressions for the flavor singlet quark and gluon helicity parton distribution functions (PDFs) and for the g_1 structure function as double-inverse Laplace transforms. We also extract analytic expressions for the four DGLAP polarized anomalous dimensions Delta gamma_{qq}, Delta gamma_{qG}, Delta gamma_{Gq}, and Delta gamma_{GG}: these expressions resum powers of alpha_s/omega^2 to all orders at large-N_c&N_f (with omega the Mellin moment variable). We extract the leading small-x growth of the helicity distributions, align \Delta\Sigma(x,Q^2) \sim \Delta G(x,Q^2)\sim g_1(x,Q^2) \sim \left(1{x}\right)^{\alpha_h}, align where the intercept alpha_h satisfies an algebraic equation. We determine alpha_h numerically for various values of N_c and N_f. We further obtain the explicit asymptotic expressions for the helicity distributions, which yield numerical values for the ratio of the gluon helicity PDF to the flavor singlet quark helicity PDF in the small-x asymptotic limit (for different N_f/N_c). We find that all our predictions for polarized DGLAP anomalous dimensions are fully consistent with the existing finite-order calculations. Similar to the large-N_c case, our intercept alpha_h exhibits a very slight disagreement with the predictions made within the infrared evolution equations framework.

We reanalyze the spectral lag data for GRB 160625B using frequentist inference in order to constrain the energy scale (E_{QG}) of Lorentz Invariance Violation (LIV). For this purpose, we use profile likelihood to deal with the astrophysical nuisance parameters. This is in contrast to Bayesian inference implemented in previous works, where marginalization was carried out over the nuisance parameters. We show that with profile likelihood, we do not find a global minimum for chi^2 as a function of E_{QG} below the Planck scale for both linear and quadratic models of LIV, whereas bounded credible intervals were previously obtained using Bayesian inference. Therefore, we can set one-sided lower limits in a straightforward manner. We find that E_{QG} geq 2.55 times 10^{16} GeV and E_{QG} geq 1.85 times 10^7 GeV at 95\% c.l., for linear and quadratic LIV, respectively. Therefore, this is the first proof-of-principles application of profile likelihood method to the analysis of GRB spectral lag data to constrain LIV.

In recent years, there has been an emerging trend of combining two innovations in computer science and physics to achieve better computation capability. Exploring the potential of quantum computation to achieve highly efficient performance in various tasks is a vital development in engineering and a valuable question in sciences, as it has a significant potential to provide exponential speedups for technologically complex problems that are specifically advantageous to quantum computers. However, one key issue in unleashing this potential is constructing an efficient approach to load classical data into quantum states that can be executed by quantum computers or quantum simulators on classical hardware. Therefore, the split-step quantum walks (SSQW) algorithm was proposed to address this limitation. We facilitate SSQW to design parameterized quantum circuits (PQC) that can generate probability distributions and optimize the parameters to achieve the desired distribution using a variational solver. A practical example of implementing SSQW using Qiskit has been released as open-source software. Showing its potential as a promising method for generating desired probability amplitude distributions highlights the potential application of SSQW in option pricing through quantum simulation.

A sizeable fraction of gamma-ray burst (GRB) light curves (LCs) features a sequence of peaks, which holds information on the unknown way energy is dissipated into gamma-rays over time. Traditional searches for periodic signals in GRB LCs turned out to be inconclusive, partly because they are challenging as a consequence of the short-lived, coloured-noise, and non-stationary nature of the LCs themselves. Yet, recent claims have revived the issue. We searched for periodic components in GRB LCs through a new approach to GRBs, that avoids most of the issues faced by traditional techniques. We identified peaks through a well tested algorithm and selected GRBs with at least 10 peaks out of 5 GRB catalogues (Swift/BAT, CGRO/BATSE, Fermi/GBM, Insight-HXMT, BeppoSAX/GRBM). Each GRB was simply treated as a discrete point process, whose realisation coincides with the sequence of peak times. We searched for possible periodic recurrences based on the multinomial distribution, after accounting for the clustering of peaks due to the non-stationarity of the GRB signals. The best candidate has a p-value of 3e-4 that there is no periodic recurrence. However, accounting for the multiple trials of 555 searched GRBs, its statistical significance is demoted to 17%. The overall distribution of the p-values obtained for all GRBs is compatible with a uniform distribution in [0,1]. We found no robust evidence for multi-peaked GRBs with periodic recurrences. We can exclude that a sizeable fraction (>~ 0.75) of peaks of each GRB with at least 10 peaks are periodic. While our result does not necessarily clash with claimed periodicities based on Fourier techniques, it constrains the putative recurrent behaviour, which would not manifest itself through the sequence of peaks, but, evidently, in a more elusive way.

Recently, the statistical effect of fermionic superradiance is approved by series of experiments both in free space and in a cavity. The Pauli blocking effect can be visualized by a 1/2 scaling of Dicke transition critical pumping strength against particle number Nat for fermions in a trap. However, the Fermi surface nesting effect, which manifests the enhancement of superradiance by Fermi statistics is still very hard to be identified. Here we studied the influence of localized fermions on the trap edge when both pumping optical lattice and the trap are presented. We find due to localization, the statistical effect in superradiant transition is enhanced. Two new scalings of critical pumping strength are observed as 4/3, and 2/3 for mediate particle number, and the Pauli blocking scaling 1/3 (2d case) in large particle number limit is unaffected. Further, we find the 4/3 scaling is subject to a power law increasing with rising ratio between recoil energy and trap frequency in pumping laser direction. The divergence of this scaling of critical pumping strength against N_{rm at} in E_R/omega_xrightarrow+infty limit can be identified as the Fermi surface nesting effect. Thus we find a practical experimental scheme for visualizing the long-desired Fermi surface nesting effect with the help of trap induced localization in a two-dimensional Fermi gas in a cavity.

In reinforcement learning and imitation learning, an object of central importance is the state distribution induced by the policy. It plays a crucial role in the policy gradient theorem, and references to it--along with the related state-action distribution--can be found all across the literature. Despite its importance, the state distribution is mostly discussed indirectly and theoretically, rather than being modeled explicitly. The reason being an absence of appropriate density estimation tools. In this work, we investigate applications of a normalizing flow-based model for the aforementioned distributions. In particular, we use a pair of flows coupled through the optimality point of the Donsker-Varadhan representation of the Kullback-Leibler (KL) divergence, for distribution matching based imitation learning. Our algorithm, Coupled Flow Imitation Learning (CFIL), achieves state-of-the-art performance on benchmark tasks with a single expert trajectory and extends naturally to a variety of other settings, including the subsampled and state-only regimes.

In this paper, we propose a unified approach to harness quantum conformal methods for multi-output distributions, with a particular emphasis on two experimental paradigms: (i) a standard 2-qubit circuit scenario producing a four-dimensional outcome distribution, and (ii) a multi-basis measurement setting that concatenates measurement probabilities in different bases (Z, X, Y) into a twelve-dimensional output space. By combining a multioutput regression model (e.g., random forests) with distributional conformal prediction, we validate coverage and interval-set sizes on both simulated quantum data and multi-basis measurement data. Our results confirm that classical conformal prediction can effectively provide coverage guarantees even when the target probabilities derive from inherently quantum processes. Such synergy opens the door to next-generation quantum-classical hybrid frameworks, providing both improved interpretability and rigorous coverage for quantum machine learning tasks. All codes and full reproducible Colab notebooks are made available at https://github.com/detasar/QECMMOU.

The radiation physics of repeating fast radio bursts (FRBs) remains enigmatic. Motivated by the observed narrow-banded emission spectrum and ambiguous fringe pattern of the spectral peak frequency (nu_{rm pk}) distribution of some repeating FRBs, such as FRB 20121102A, we propose that the bursts from repeating FRBs arise from synchrotron maser radiation in localized blobs within weakly magnetized plasma that relativistically moves toward observers. Assuming the plasma moves toward the observers with a bulk Lorentz factor of Gamma=100 and the electron distribution in an individual blob is monoenergetic (gamma_{rm e}sim300), our analysis shows that bright and narrow-banded radio bursts with peak flux density sim 1 {rm Jy} at peak frequency (nu_{rm pk}) sim 3.85 GHz can be produced by the synchrotron maser emission if the plasma blob has a magnetization factor of sigmasim10^{-5} and a frequency of nu_{rm P}sim 4.5 MHz. The spectrum of bursts with lower nu_{rm pk} tends to be narrower. Applying our model to the bursts of FRB 20121102A, the distributions of both the observed nu_{rm pk} and isotropic energy E_{rm iso} detected by the Arecibo telescope at the L band and the Green Bank Telescope at the C band are successfully reproduced. We find that the nu_{rm P} distribution exhibits several peaks, similar to those observed in the nu_{rm pk} distribution of FRB 20121102A. This implies that the synchrotron maser emission in FRB 20121102A is triggered in different plasma blobs with varying nu_{rm P}, likely due to the inhomogeneity of relativistic electron number density.

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

Initial states of dense matter with nonzero electron chiral imbalance could potentially give rise to strong magnetic fields through chiral plasma instability. Previous work indicated that unless chiral chemical potential is as large as the electron vector chemical potential, the growth of magnetic fields due to the instability is washed out by chirality flipping rate enabled by electron mass. We re-examine this claim in a broader range of parameters and find that at higher temperatures the hierarchy is reversed supporting a growing magnetic field for an initial electron chiral chemical potential much smaller than the electron vector chemical potential. Further, we identify a qualitatively new effect relevant for magnetized hot and dense medium where chiral magnetic effect (CME) sourced by density fluctuation acts as a powerful source of Joule heating. Remarkably, even modest chiral chemical potentials (keV) in such environment can deposit energy densities set by the QCD scale in a relatively short time of the order of a few milliseconds or seconds. We speculate how this mechanism makes CME-driven Joule heating a potentially critical ingredient in the dynamics of turbulent density fluctuation of supernovae and neutron star mergers.

We explore a generative machine learning-based approach for estimating multi-dimensional probability density functions (PDFs) in a target sample using a statistically independent but related control sample - a common challenge in particle physics data analysis. The generative model must accurately reproduce individual observable distributions while preserving the correlations between them, based on the input multidimensional distribution from the control sample. Here we present a conditional normalizing flow model (CNF) based on a chain of bijectors which learns to transform unpaired simulation events to data events. We assess the performance of the CNF model in the context of LHC Higgs to diphoton analysis, where we use the CNF model to convert a Monte Carlo diphoton sample to one that models data. We show that the CNF model can accurately model complex data distributions and correlations. We also leverage the recently popularized Modified Differential Multiplier Method (MDMM) to improve the convergence of our model and assign physical meaning to usually arbitrary loss-function parameters.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

We develop a deep variational free energy framework to compute the equation of state of hydrogen in the warm dense matter region. This method parameterizes the variational density matrix of hydrogen nuclei and electrons at finite temperature using three deep generative models: a normalizing flow model that represents the Boltzmann distribution of the classical nuclei, an autoregressive transformer that models the distribution of electrons in excited states, and a permutational equivariant flow model that constructs backflow coordinates for electrons in Hartree-Fock orbitals. By jointly optimizing the three neural networks to minimize the variational free energy, we obtain the equation of state and related thermodynamic properties of dense hydrogen. We compare our results with other theoretical and experimental results on the deuterium Hugoniot curve, aiming to resolve existing discrepancies. The calculated results provide a valuable benchmark for deuterium in the warm dense matter region.

This article introduces Regression Discontinuity Design (RDD) with Distribution-Valued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise when treatment is assigned at a higher level of aggregation than the outcome-for example, when a subsidy is allocated based on a firm-level revenue cutoff while the outcome of interest is the distribution of employee wages within the firm. Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. The target estimand is a "local average quantile treatment effect", which averages across random quantiles. To estimate this target, I introduce two related approaches: one that extends local polynomial regression to random quantiles and another based on local Fr\'echet regression, a form of functional regression. For both estimators, I establish asymptotic normality and develop uniform, debiased confidence bands together with a data-driven bandwidth selection procedure. Simulations validate these theoretical properties and show existing methods to be biased and inconsistent in this setting. I then apply the proposed methods to study the effects of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality-efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

We develop a novel way to probe subgalactic-scale matter distribution with diffractive lensing on gravitational waves. Five-year observations from Einstein Telescope and DECIGO are expected to probe k= 10^5sim 10^8 ,{rm Mpc}^{-1} down to P(k) = 10^{-16} sim 10^{-14} ,{rm Mpc}^3 level. These results can be interpreted in terms of primordial black holes in the range M_{rm PBH} gtrsim 10^{-3}M_odot down to f_{rm PBH} = 10^{-6} level, or QCD axion minihalos in the range m_a = 10^{-3} sim 10^{-12} ,{rm eV}. A key result of the paper is the approximate relation between the scale k and the gravitational wave frequency f, derived in an ensemble of `multi-lensing' events. This relation enables direct measurement of the power spectrum at specific scales, with sensitivities characterized by model-independent kernels delta P(k). Additionally, we delineate the statistical properties of `multi-lensing' based on the `Fresnel number' N_F. When N_F cal O(1), the statistical significance can be approximately calculated by Variance of lensing effects, which is directly related to the power spectrum among other moments of matter distribution.

Classical methods to simulate quantum systems are not only a key element of the physicist's toolkit for studying many-body models but are also increasingly important for verifying and challenging upcoming quantum computers. Pauli propagation has recently emerged as a promising new family of classical algorithms for simulating digital quantum systems. Here we provide a comprehensive account of Pauli propagation, tracing its algorithmic structure from its bit-level implementation and formulation as a tree-search problem, all the way to its high-level user applications for simulating quantum circuits and dynamics. Utilising these observations, we present PauliPropagation.jl, a Julia software package that can perform rapid Pauli propagation simulation straight out-of-the-box and can be used more generally as a building block for novel simulation algorithms.

When the performance of a machine learning model varies over groups defined by sensitive attributes (e.g., gender or ethnicity), the performance disparity can be expressed in terms of the probability distributions of the input and output variables over each group. In this paper, we exploit this fact to reduce the disparate impact of a fixed classification model over a population of interest. Given a black-box classifier, we aim to eliminate the performance gap by perturbing the distribution of input variables for the disadvantaged group. We refer to the perturbed distribution as a counterfactual distribution, and characterize its properties for common fairness criteria. We introduce a descent algorithm to learn a counterfactual distribution from data. We then discuss how the estimated distribution can be used to build a data preprocessor that can reduce disparate impact without training a new model. We validate our approach through experiments on real-world datasets, showing that it can repair different forms of disparity without a significant drop in accuracy.

We review Bohr's atomic model and its extension by Sommerfeld from a mathematical perspective of wave mechanics. The derivation of quantization rules and energy levels is revisited using semiclassical methods. Sommerfeld-type integrals are evaluated by elementary techniques, and connections with the Schr\"{o}dinger and Dirac equations are established. Historical developments and key transitions from classical to quantum theory are discussed to clarify the structure and significance of the old quantum mechanics.

We study stellar core growth in simulations of merging massive (M_star>10^{11},M_odot) elliptical galaxies by a supermassive black hole (SMBH) displaced by gravitational wave induced recoil velocity. With controlled, dense sampling of the SMBH recoil velocity, we find the core radius originally formed by SMBH binary scouring can grow by a factor of 2-3 when the recoil velocity exceeds sim50 per cent of the central escape velocity, and the mass deficit grows by up to a factor of sim4. Using Bayesian inference we predict the distribution of stellar core sizes formed through this process to peak at sim1,kpc. An orbital decomposition of stellar particles within the core reveals that radial orbits dominate over tube orbits when the recoil velocity exceeds the velocity dispersion of the core, whereas tube orbits dominate for the lowest recoil kicks. A change in orbital structure is reflected in the anisotropy parameter, with a central tangential bias present only for recoil velocities less than the local stellar velocity dispersion. Emulating current integral field unit observations of the stellar line-of-sight velocity distribution, we uncover a distinct signature in the Gauss-Hermite symmetric deviation coefficient h_4 that uniquely constrains the core size due to binary scouring. This signature is insensitive to the later evolution of the stellar mass distribution due to SMBH recoil. Our results provide a novel method to estimate the SMBH recoil magnitude from observations of local elliptical galaxies, and implies these galaxies primarily experienced recoil velocities less than the stellar velocity dispersion of the core.

We give a classical analogue to Kerenidis and Prakash's quantum recommendation system, previously believed to be one of the strongest candidates for provably exponential speedups in quantum machine learning. Our main result is an algorithm that, given an m times n matrix in a data structure supporting certain ell^2-norm sampling operations, outputs an ell^2-norm sample from a rank-k approximation of that matrix in time O(poly(k)log(mn)), only polynomially slower than the quantum algorithm. As a consequence, Kerenidis and Prakash's algorithm does not in fact give an exponential speedup over classical algorithms. Further, under strong input assumptions, the classical recommendation system resulting from our algorithm produces recommendations exponentially faster than previous classical systems, which run in time linear in m and n. The main insight of this work is the use of simple routines to manipulate ell^2-norm sampling distributions, which play the role of quantum superpositions in the classical setting. This correspondence indicates a potentially fruitful framework for formally comparing quantum machine learning algorithms to classical machine learning algorithms.

Boltzmann generators approach the sampling problem in many-body physics by combining a normalizing flow and a statistical reweighting method to generate samples in thermodynamic equilibrium. The equilibrium distribution is usually defined by an energy function and a thermodynamic state. Here we propose temperature-steerable flows (TSF) which are able to generate a family of probability densities parametrized by a choosable temperature parameter. TSFs can be embedded in generalized ensemble sampling frameworks to sample a physical system across multiple thermodynamic states.

Crystal symmetry plays a fundamental role in determining its physical, chemical, and electronic properties such as electrical and thermal conductivity, optical and polarization behavior, and mechanical strength. Almost all known crystalline materials have internal symmetry. However, this is often inadequately addressed by existing generative models, making the consistent generation of stable and symmetrically valid crystal structures a significant challenge. We introduce WyFormer, a generative model that directly tackles this by formally conditioning on space group symmetry. It achieves this by using Wyckoff positions as the basis for an elegant, compressed, and discrete structure representation. To model the distribution, we develop a permutation-invariant autoregressive model based on the Transformer encoder and an absence of positional encoding. Extensive experimentation demonstrates WyFormer's compelling combination of attributes: it achieves best-in-class symmetry-conditioned generation, incorporates a physics-motivated inductive bias, produces structures with competitive stability, predicts material properties with competitive accuracy even without atomic coordinates, and exhibits unparalleled inference speed.