Target Probability Distribution Research Articles

Training normalizing flows with computationally intensive target probability distributions

Machine learning techniques, in particular the so-called normalizing flows, are becoming increasingly popular in the context of Monte Carlo simulations as they can effectively approximate target probability distributions. In the case of lattice field theories (LFT) the target distribution is given by the exponential of the action. The common loss function's gradient estimator based on the “reparametrization trick” requires the calculation of the derivative of the action with respect to the fields. This can present a significant computational cost for complicated, non-local actions like e.g. fermionic action in QCD. In this contribution, we propose an estimator for normalizing flows based on the REINFORCE algorithm that avoids this issue. We apply it to two dimensional Schwinger model with Wilson fermions at criticality and show that it is up to ten times faster in terms of the wall-clock time as well as requiring up to 30% less memory than the reparameterization trick estimator. It is also more numerically stable allowing for single precision calculations and the use of half-float tensor cores. We present an in-depth analysis of the origins of those improvements. We believe that these benefits will appear also outside the realm of the LFT, in each case where the target probability distribution is computationally intensive.

Statistical analysis of the radar cross section of two small fixed‐wing drones using typical flights

AbstractThe presence and importance of small drones (unmanned aerial vehicles) are increasing. With traditional radar detection the target radar cross section and its statistical properties are needed in order to design the detector and to assess detection performance. The authors make a statistical analysis of the radar cross section of two small fixed‐wing drones using measured data from typical flights. Using data from a typical flight instead of measurements or electromagnetic calculations as a function of aspect angle is a convenient method. Work results of the authors in mean values, medians and standard deviations, in analyses of common target probability distributions and in decorrelation times for the radar cross section, are necessary for design and assessment of the detection performance. The authors suggest a new target probability distribution which agrees better with the measured data than the others tested. The authors also note significant differences between the drones.

Variable-Length Resolvability for General Sources and Channels.

We introduce the problem of variable-length (VL) source resolvability, in which a given target probability distribution is approximated by encoding a VL uniform random number, and the asymptotically minimum average length rate of the uniform random number, called the VL resolvability, is investigated. We first analyze the VL resolvability with the variational distance as an approximation measure. Next, we investigate the case under the divergence as an approximation measure. When the asymptotically exact approximation is required, it is shown that the resolvability under two kinds of approximation measures coincides. We then extend the analysis to the case of channel resolvability, where the target distribution is the output distribution via a general channel due to a fixed general source as an input. The obtained characterization of channel resolvability is fully general in the sense that, when the channel is just an identity mapping, it reduces to general formulas for source resolvability. We also analyze the second-order VL resolvability.

Statistical learning of target and distractor spatial probability shape a common attentional priority computation

Converging evidence recently put forward the notion that dedicated neurocognitive mechanisms do exist for the suppression of salient, but irrelevant distractors. Along this line, it is plausible to hypothesize that, in appropriate contexts, experience-dependent forms of attentional learning might selectively induce plastic changes within this dedicated circuitry, thus allowing an independent shaping of priorities at the service of attentional filtering. Conversely, previous work suggested that statistical learning (SL) of both target and distractor spatial probability distributions converge in adjusting only the overall attentional priority of locations: in fact, in the presence of an independent manipulation, either related to the target or to the distractor only, SL induces indirect effects (e.g., changes in filtering efficiency due to an uneven distribution of targets), suggesting that SL-induced plastic changes affect a shared neural substrate. Here we tested whether, when (conflicting) target- and distractor-related manipulations are concurrently applied to the very same locations, dedicated mechanisms might support the selective encoding of spatial priority in relation to the specific attentional operation involved. In three related experiments, human healthy participants discriminated the direction of a target arrow, while ignoring a salient distractor, if present; both target and distractor spatial probability distributions were concurrently manipulated in relation to each single location. Critically, the selection bias produced by the target-related SL was marginally reduced by an adverse distractor contingency, and the suppression bias generated by the distractor-related SL was erased, or even reversed, by an adverse target contingency. Our results suggest that even conflicting target- and distractor-related SL manipulations result in the adjustment of a unique spatial priority computation, likely because the process directly relies on direct plastic alterations of shared spatial priority map(s).

Metaparametric Neural Networks for Survival Analysis.

Survival analysis is a critical tool for the modeling of time-to-event data, such as life expectancy after a cancer diagnosis or optimal maintenance scheduling for complex machinery. However, current neural network models provide an imperfect solution for survival analysis as they either restrict the shape of the target probability distribution or restrict the estimation to predetermined times. As a consequence, current survival neural networks lack the ability to estimate a generic function without prior knowledge of its structure. In this article, we present the metaparametric neural network framework that encompasses the existing survival analysis methods and enables their extension to solve the aforementioned issues. This framework allows survival neural networks to satisfy the same independence of generic function estimation from the underlying data structure that characterizes their regression and classification counterparts. Furthermore, we demonstrate the application of the metaparametric framework using both simulated and large real-world datasets and show that it outperforms the current state-of-the-art methods in: 1) capturing nonlinearities and 2) identifying temporal patterns, leading to more accurate overall estimations while placing no restrictions on the underlying function structure.

Gradient-based adaptive importance samplers

Importance sampling (IS) is a powerful Monte Carlo methodology for the approximation of intractable integrals, very often involving a target probability distribution. The performance of IS heavily depends on the appropriate selection of the proposal distributions where the samples are simulated from. In this paper, we propose an adaptive importance sampler, called GRAMIS, that iteratively improves the set of proposals. The algorithm exploits geometric information of the target to adapt the location and scale parameters of those proposals. Moreover, in order to allow for a cooperative adaptation, a repulsion term is introduced that favors a coordinated exploration of the state space. This translates into a more diverse exploration and a better approximation of the target via the mixture of proposals. Moreover, we provide a theoretical justification of the repulsion term. We show the good performance of GRAMIS in two problems where the target has a challenging shape and cannot be easily approximated by a standard uni-modal proposal.

Markov Chain Monte Carlo for Gaussian: A Linear Control Perspective

Drawing samples from a given target probability distribution is a fundamental task in many science and engineering applications. A commonly used method for sampling is the Markov chain Monte Carlo (MCMC) which simulates a Markov chain whose stationary distribution coincides with the target one. In this work, we study the convergence and complexity of MCMC algorithms from a dynamic system point of view. We focus on the special cases with Gaussian target distributions and provide a Lyapunov perspective to them using tools from linear control theory. In particular, we systematically analyze two popular MCMC algorithms: Langevin Monte Carlo (LMC) and kinetic Langevin Monte Carlo (KLMC). By applying Lyapunov theory we derive impressive complexity bounds to these algorithms: for LMC, our result is better than all existing results, and for KLMC, ours matches the best known bound. Our analysis also highlights subtle differences between sampling and optimization that could inform the more challenging task to sample from general distributions. Overall, our findings offer valuable insights for improving MCMC algorithms.

Determining reference ranges and sample sizes in parallel-group studies.

Reference ranges are widely used to locate the major range of the target probability distribution. When future measurements fall outside the reference range, they are classified as atypical and require further investigation. The fundamental principles and statistical properties of reference ranges are closely related to those of tolerance interval procedures. Existing investigations of reference ranges and tolerance intervals mainly devoted to the primitive cases of one- and paired-sample designs. Although reference ranges hold considerable promise for parallel group designs, the corresponding methodological and computational issues for determining reference limits and sample sizes have not been adequately addressed. This paper describes a complete collection of one- and two-sided reference ranges for assessing measurement differences in parallel-group studies that assume variance homogeneity. The problem of sample size determination for precise reference ranges is also examined under the expected half-width and assurance probability considerations. Unlike the current methods, the suggested sample size criteria explicitly accommodate desired interval width in precise interval estimation. Theoretical examinations and empirical assessments are presented to validate the usefulness of the proposed reference range and sample size procedures. To enhance the usages of the recommended techniques in practical applications, computer programs are developed for efficient calculation and exact analysis. A real data example regarding tablet absorption rate and extent is presented to illustrate the suggested assessments between two drug formulations.

On Orthogonal Polynomials and Finite Moment Problem

Background: This paper is an improvement of a previous work on the problem recovering a function or probability density function from a finite number of its geometric moments, [1]. The previous worked solved the problem with the help of the B-Spline theory which is a great approach as long as the resulting linear system is not very large. In this work, two solution algorithms based on the approximate representation of the target probability distribution function via an orthogonal expansion are provided. One primary application of this theory is the reconstruction of the Particle Size Distribution (PSD), occurring in chemical engineering applications. Another application of this theory is the reconstruction of the Radon transform of an image at an unknown angle using the moments of the transform at known angles which leads to the reconstruction of the image form limited data. Objective: The aim is to recover a probability density function from a finite number of its geometric moments. Methods: The tool is the orthogonal expansion approach. The Shifted-Legendre Polynomials and the Chebyshev Polynomials as bases for the orthogonal expansion are used in this study. Results: A high degree of accuracy has been obtained in recovering a function without facing a possible ill-conditioned linear system, which is the case with many typical approaches of solving the problem. In fact, for a normalized template function f on the interval [0, 1], and a reconstructed function ; the reconstruction accuracy is measured in two domains. One is the moment domain, in which the error (difference between the moments of f and the moments of ) is zero. The other measure is the standard difference in the norm -space ||f- || which can be ≈ 10-6 or less. Conclusion: This paper discusses the problem of recovering a function from a finite number of its geometric moments for the PSD application. Linear transformations were used, as needed, so that the function is supported on the unit interval [0, 1], or on [0, α] for some choice of α. This transformation forces the sequence of moments to vanish. Then, an orthogonal expansion of the Scaled Shifted Legendre Polynomials, as well as the Chebyshev Polynomials, are developed. The result shows good accuracy in recovering different types of synthetic functions. It is believed that up to fifteen moments, this approach is safe and reliable.

Three learning stages and accuracy–efficiency tradeoff of restricted Boltzmann machines

Restricted Boltzmann Machines (RBMs) offer a versatile architecture for unsupervised machine learning that can in principle approximate any target probability distribution with arbitrary accuracy. However, the RBM model is usually not directly accessible due to its computational complexity, and Markov-chain sampling is invoked to analyze the learned probability distribution. For training and eventual applications, it is thus desirable to have a sampler that is both accurate and efficient. We highlight that these two goals generally compete with each other and cannot be achieved simultaneously. More specifically, we identify and quantitatively characterize three regimes of RBM learning: independent learning, where the accuracy improves without losing efficiency; correlation learning, where higher accuracy entails lower efficiency; and degradation, where both accuracy and efficiency no longer improve or even deteriorate. These findings are based on numerical experiments and heuristic arguments.

Path reinforcement learning planning of nuclear power plant UAVs based on dynamic target probability distribution

This work develops a path reinforcement learning planning algorithm for unmanned aerial vehicles (UAVs) based on the probability distribution of dynamic targets. The proposed algorithm enables the effective interception of aerial dynamic invasive targets into nuclear power plants. In accordance with the status information of the invasive target, the target probability distribution is calculated by the probability diffusion principle and the target's possible location is deduced based on the obtained distribution. On this basis, the action space based on path point transfer rules is established and the dynamic updating mechanism of the reward function based on the target probability distribution is designed. Using $Q$-learning for continuous path optimization, a UAV path planning framework based on the target probability distribution and reinforcement learning is constructed, so as to realize the UAV path planning. Simulation results indicate that the proposed method can enable the autonomous path planning of UAVs for intercepting aerial dynamic invasive targets into nuclear power plants when the target point changes.

How much BiGAN and CycleGAN-learned hidden features are effective for COVID-19 detection from CT images? A comparative study

Bidirectional generative adversarial networks (BiGANs) and cycle generative adversarial networks (CycleGANs) are two emerging machine learning models that, up to now, have been used as generative models, i.e., to generate output data sampled from a target probability distribution. However, these models are also equipped with encoding modules, which, after weakly supervised training, could be, in principle, exploited for the extraction of hidden features from the input data. At the present time, how these extracted features could be effectively exploited for classification tasks is still an unexplored field. Hence, motivated by this consideration, in this paper, we develop and numerically test the performance of a novel inference engine that relies on the exploitation of BiGAN and CycleGAN-learned hidden features for the detection of COVID-19 disease from other lung diseases in computer tomography (CT) scans. In this respect, the main contributions of the paper are twofold. First, we develop a kernel density estimation (KDE)-based inference method, which, in the training phase, leverages the hidden features extracted by BiGANs and CycleGANs for estimating the (a priori unknown) probability density function (PDF) of the CT scans of COVID-19 patients and, then, in the inference phase, uses it as a target COVID-PDF for the detection of COVID diseases. As a second major contribution, we numerically evaluate and compare the classification accuracies of the implemented BiGAN and CycleGAN models against the ones of some state-of-the-art methods, which rely on the unsupervised training of convolutional autoencoders (CAEs) for attaining feature extraction. The performance comparisons are carried out by considering a spectrum of different training loss functions and distance metrics. The obtained classification accuracies of the proposed CycleGAN-based (resp., BiGAN-based) models outperform the corresponding ones of the considered benchmark CAE-based models of about 16% (resp., 14%).

Polar encoded probability shaping for highly reliable 7-core fiber optical communication system based on prefix-free code distribution matching

A highly reliable optical orthogonal frequency division multiplexing (OFDM) system based on systematic polar encoded prefix-free codebook distribution matching (PCDM) is proposed. The PCDM is used to convert the uniform binary data into the unipolar coded amplitude symbols with the target probability distribution, which are then encoded by systematic polar code to maintain non-uniform probability distribution. The redundant parity bits generated by polar encoding are used as signs, map to the bipolar amplitude symbols as the real or imaginary part of the final constellation symbols when combined with the unipolar amplitude symbols. The proposed scheme generates a polar encoded probability-shaped 16 quadrature amplitude modulation (PS-Polar-16QAM) OFDM signal, which improves the bit error rate (BER) performance of the optical transmission system. The transmission of a 70 Gb/s polar encoded PS-Polar-16QAM-OFDM signal over a 2 km seven-core fiber is demonstrated experimentally. In comparison to normal-16QAM-OFDM without polar code and PCDM, the polar encoded PS-Polar-16QAM-OFDM achieves about 3.65 dB gain in receiver sensitivity at a BER of 10−4. The proposed polar encoded PS-Polar-16QAM-OFDM based on PCDM has a broad application prospect in a low-cost and reliable optical access system due to its good BER performance.

Transferring model structure in Bayesian transfer learning for Gaussian process regression

Bayesian transfer learning (BTL) is defined in this paper as the task of conditioning a target probability distribution on a transferred source distribution. The target globally models the interaction between the source and target, and conditions on a probabilistic data predictor made available by an independent local source modeller. Fully probabilistic design is adopted to solve this optimal decision-making problem in the target. By successfully transferring higher moments of the source, the target can reject unreliable source knowledge (i.e. it achieves robust transfer). This dual-modeller framework means that the source’s local processing of raw data into a transferred predictive distribution – with compressive possibilities – is enriched by (the possible expertise of) the local source model. In addition, the introduction of the global target modeller allows correlation between the source and target tasks – if known to the target – to be accounted for. Important consequences emerge. Firstly, the new scheme attains the performance of fully modelled (i.e. conventional) multitask learning schemes in (those rare) cases where target model misspecification is avoided. Secondly, and more importantly, the new dual-modeller framework is robust to the model misspecification that undermines conventional multitask learning. We thoroughly explore these issues in the key context of interacting Gaussian process regression tasks. Experimental evidence from both synthetic and real data settings validates our technical findings: that the proposed BTL framework enjoys robustness in transfer while also being robust to model misspecification.

Point source regularization of the finite source reflector problem

We address the “freeform optics” inverse problem of designing a reflector surface mapping a prescribed source distribution of light to a prescribed far-field distribution, for a finite light source. When the finite source reduces to a point source, the light source distribution has support only on the optics ray directions. In this setting, the inverse problem is well-posed for arbitrary source and target probability distributions. It can be recast as an Optimal Transportation problem and has been studied both mathematically and numerically. We are not aware of any similar mathematical formulation in the finite source case: i.e. the source has an “étendue” with support both in space and directions. We propose to leverage the well-posed variational formulation of the point source problem to build a smooth parameterization of the reflector and the reflection map. Under this parameterization, we can construct a smooth loss/misfit function to optimize for the best solution in this class of reflectors. Both steps, the parameterization and the loss, are related to Optimal Transportation distances. We also take advantage of recent progress in the numerical approximation and resolution of these mathematical objects to perform a numerical study.

Improved polytope volume calculations based on Hamiltonian Monte Carlo with boundary reflections and sweet arithmetics

Computing the volume of a high dimensional polytope is a fundamental problem in geometry, also connected to the calculation of densities of states in statistical physics, and a central building block of such algorithms is the method used to sample a target probability distribution. This paper studies Hamiltonian Monte Carlo (HMC) with reflections on the boundary of a domain, providing an enhanced alternative to Hit-and-run (HAR) to sample a target distribution restricted to the polytope. We make three contributions. First, we provide a convergence bound, paving the way to more precise mixing time analysis. Second, we present a robust implementation based on multi-precision arithmetic, a mandatory ingredient to guarantee exact predicates and robust constructions. We however allow controlled failures to happen, introducing the Sweeten Exact Geometric Computing (SEGC) paradigm. Third, we use our HMC random walk to perform H-polytope volume calculations, using it as an alternative to HAR within the volume algorithm by Cousins and Vempala. The systematic tests conducted up to dimension $n$ = 100 on the cube, the isotropic and the standard simplex show that HMC significantly outperforms HAR both in terms of accuracy and running time. Additional tests show that calculations may be handled up to dimension $n$ = 500. These tests also establish that multiprecision is mandatory to avoid exits from the polytope.

Quantum unary approach to option pricing

We present a quantum algorithm for European option pricing in finance, where the key idea is to work in the unary representation of the asset value. The algorithm needs novel circuitry and is divided in three parts: first, the amplitude distribution corresponding to the asset value at maturity is generated using a low depth circuit; second, the computation of the expected return is computed with simple controlled gates; and third, standard Amplitude Estimation is used to gain quantum advantage. On the positive side, unary representation remarkably simplifies the structure and depth of the quantum circuit. Amplitude distributions uses quantum superposition to bypass the role of classical Monte Carlo simulation. The unary representation also provides a post-selection consistency check that allows for a substantial mitigation in the error of the computation. On the negative side, unary representation requires linearly many qubits to represent a target probability distribution, as compared to the logarithmic scaling of binary algorithms. We compare the performance of both unary vs. binary option pricing algorithms using error maps, and find that unary representation may bring a relevant advantage in practice for near-term devices.

Training Image Free High-Order Stochastic Simulation Based on Aggregated Kernel Statistics

A training image free, high-order sequential simulation method is proposed herein, which is based on the efficient inference of high-order spatial statistics from the available sample data. A statistical learning framework in kernel space is adopted to develop the proposed simulation method. Specifically, a new concept of aggregated kernel statistics is proposed to enable sparse data learning. The conditioning data in the proposed high-order sequential simulation method appear as data events corresponding to the attribute values associated with the so-called spatial templates of various geometric configurations. The replicates of the data events act as the training data in the learning framework for inference of the conditional probability distribution and generation of simulated values. These replicates are mapped into spatial Legendre moment kernel spaces, and the kernel statistics are computed thereafter, encapsulating the high-order spatial statistics from the available data. To utilize the incomplete information from the replicates, which partially match the spatial template of a given data event, the aggregated kernel statistics combine the ensemble of the elements in different kernel subspaces for statistical inference, embedding the high-order spatial statistics of the replicates associated with various spatial templates into the same kernel subspace. The aggregated kernel statistics are incorporated into a learning algorithm to obtain the target probability distribution in the underlying random field, while preserving in the simulations the high-order spatial statistics from the available data. The proposed method is tested using a synthetic dataset, showing the reproduction of the high-order spatial statistics of the sample data. The comparison with the corresponding high-order simulation method using TIs emphasizes the generalization capacity of the proposed method for sparse data learning.

Controllability and decentralized stabilization of the Kolmogorov forward equation for Markov chains

In this paper, we provide several results on controllability and stabilizability properties of the Kolmogorov forward equation of a continuous-time Markov chain (CTMC) evolving on a finite state space, with the transition rates defined as the control parameters. First, we show that any target probability distribution can be reached asymptotically using time-varying control parameters. Second, we characterize all stationary distributions that are stabilizable using time-independent control parameters. For bidirected graphs, we construct rational and polynomial density feedback laws that stabilize stationary distributions while satisfying the additional constraint that the feedback law takes zero value at equilibrium. This last result enables the construction of decentralized density feedback controllers, using tools from linear systems theory and sum-of-squares based polynomial optimization, that stabilize a swarm of robots modeled as a CTMC to a target state distribution with no state-switching at equilibrium. In addition to these results, we prove a sufficient condition under which the classical rank conditions for controllability can be generalized to forward equations with non-negativity constraints on the control inputs. We apply this result to prove local controllability of a forward equation in which only a small subset of the transition rates are the control inputs. Lastly, we extend our feedback stabilization results to stationary distributions that have a strongly connected support.

Asymptotically Exact Data Augmentation: Models, Properties, and Algorithms

Data augmentation, by the introduction of auxiliary variables, has become an ubiquitous technique to improve convergence properties, simplify the implementation or reduce the computational time of inference methods such as Markov chain Monte Carlo ones. Nonetheless, introducing appropriate auxiliary variables while preserving the initial target probability distribution and offering a computationally efficient inference cannot be conducted in a systematic way. To deal with such issues, this article studies a unified framework, coined asymptotically exact data augmentation (AXDA), which encompasses both well-established and more recent approximate augmented models. In a broader perspective, this article shows that AXDA models can benefit from interesting statistical properties and yield efficient inference algorithms. In non-asymptotic settings, the quality of the proposed approximation is assessed with several theoretical results. The latter are illustrated on standard statistical problems. Supplementary materials including computer code for this article are available online.