Space Of Probability Distributions Research Articles

A Denseness Property of Stochastic Processes

AbstractIn this work, we show the following general property of stochastic processes: Denseness of probability distributions extends to denseness of stochastic processes. In particular, for every $$d\in \mathbb {N}$$ d ∈ N , we consider any class of probability distributions which is dense in the space of probability distributions on $$\mathbb {R}^{d}$$ R d with respect to convergence in distribution. Using these classes, we construct various explicit families of times series, continuous processes and càdlàg processes that are dense in their respective spaces with respect to convergence in distribution. This is particularly interesting because, for processes, finite-dimensional distributions convergence does not automatically imply convergence in distribution.

Tight conic approximation of testing regions for quantum statistical models and measurements

Quantum statistical models (i.e., families of normalized density matrices) and quantum measurements (i.e., positive operator-valued measures) can be regarded as linear maps: the former, mapping the space of effects to the space of probability distributions; the latter, mapping the space of states to the space of probability distributions. The images of such linear maps are called the testing regions of the corresponding model or measurement. Testing regions are notoriously impractical to treat analytically in the quantum case. Our first result is to provide an implicit outer approximation of the testing region of any given quantum statistical model or measurement in any finite dimension: namely, a region in probability space that contains the desired image, but is defined implicitly, using a formula that depends only on the given model or measurement. The outer approximation that we construct is minimal among all such outer approximations, and close, in the sense that it becomes the maximal inner approximation up to a constant scaling factor. Finally, we apply our approximation to provide sufficient conditions, that can be tested in a semi-device-independent way, for the ability to transform one quantum statistical model or measurement into another.

Solving Stochastic Transportation Programming Problems with Fuzzy Information on Probability Distribution Space Using a New Approach

Solving Stochastic Transportation Programming Problems with Fuzzy Information on Probability Distribution Space Using a New Approach

Coarse Embeddability of Wasserstein Space and the Space of Persistence Diagrams

AbstractWe prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e. Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the p-Wasserstein distance for $$1\le p\le 2$$ 1 ≤ p ≤ 2 remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams is equivalent to embeddability for Wasserstein space on $$\mathbb {R}^2$$ R 2 . When $$p > 1$$ p > 1 , Wasserstein space on $$\mathbb {R}^2$$ R 2 is snowflake universal (an obstruction to embeddability into any Banach space of non-trivial type) if and only if the space of persistence diagrams is snowflake universal.

An essay on semiclassical analysis for microlocal singularities, turbulence intensity and integration of singularities by Schrödinger equation in probabilistic behavior

The Schrödinger equation governs the probabilistic behavior of quantum particles through the wave function. Microlocal singularities denote regions with significantly high probability density or abrupt changes therein. By visualizing the probability distribution in time and space, we discern regions with higher probability density, indicative of potential microlocal singularities. These regions probably correspond to areas with a greater probability of particle presence. Such analysis aligns with Theorem 1, predicting characteristics of microlocal singularities of wave functions. Furthermore, Theorem 2 postulates that semiclassical path integrals along these singularities contribute significantly to solving the Schrödinger equation. Interpreting the temporal evolution of the probability density in the probability distribution visualization reveals the propagation of the particle over time. Regions of high density mean likely presence of particles at specific times, aligning with the predictions of Theorem 2. Consequently, the analysis of the contribution of high-density regions to the temporal evolution of the wave function resembles semi-classical path integral calculations. Thus, our findings demonstrate that visualization of probability distributions obtained from the numerical resolution of the Schrödinger equation allows a comprehensive interpretation of the behavior of quantum particles, consistent with the theorems.

Complexity of fermionic states

How much information does a fermionic state contain? To address this fundamental question, we define the complexity of a particle-conserving many-fermion state as the entropy of its Fock space probability distribution, minimized over all Fock representations. The complexity characterizes the minimum computational and physical resources required to represent the state and store the information obtained from it by measurements. Alternatively, the complexity can be regarded as a Fock space entanglement measure describing the intrinsic many-particle entanglement in the state. We establish a universal lower bound for the complexity in terms of the single-particle correlation matrix eigenvalues and formulate a finite-size complexity scaling hypothesis. Remarkably, numerical studies on interacting lattice models suggest a general model-independent complexity hierarchy: ground states are exponentially less complex than average excited states, which, in turn, are exponentially less complex than generic states in the Fock space. Our work has fundamental implications on how much information is encoded in fermionic states. Published by the American Physical Society 2024

Dissimilarity functions co-symmetry property: a focus on probability distributions with involutive negation

In today’s data-rich era, there is a growing need for developing effective similarity and dissimilarity measures to compare vast datasets. It is desirable that these measures reflect the intrinsic structure of the domain of these measures. Recently, it was shown that the space of finite probability distributions has a symmetric structure generated by involutive negation mapping probability distributions into their “opposite” probability distributions and back, such that the correlation between opposite distributions equals –1. An important property of similarity and dissimilarity functions reflecting such symmetry of probability distribution space is the co-symmetry of these functions when the similarity between probability distributions is equal to the similarity between their opposite distributions. This article delves into the analysis of five well-known dissimilarity functions, used for creating new co-symmetric dissimilarity functions. To conduct this study, a random dataset of one thousand probability distributions is employed. From these distributions, dissimilarity matrices are generated that are used to determine correlations similarity between different dissimilarity functions. The hierarchical clustering is applied to better understand the relationships between the studied dissimilarity functions. This methodology aims to identify and assess the dissimilarity functions that best match the characteristics of the studied probability distribution space, enhancing our understanding of data relationships and patterns. The study of these new measures offers a valuable perspective for analyzing and interpreting complex data, with the potential to make a significant impact in various fields and applications.

Information-response inequality in the small noise limit

The invariant response was defined from a formulation of the fluctuation-response theorem in the space of probability distributions. An inequality which sets the mutual information as a limiting factor to the invariant response is here derived in the small noise limit based on Stam's isoperimetric inequality. Beyond the small noise limit, numerical violations exclude its general validity, however, a strong distribution bias is observed. Applications to the thermodynamics of feedback control and to estimation theory are discussed.

SOLVING STOCHASTIC TRANSPORTATION ELECTRICITY PROBLEM WITH FUZZY INFORMATION ON PROBABILITY DISTRIBUTION USING MATLAB PROGRAM

This study focuses on MATLAB code programs of the entire stages of solving Stochastic Transportation Linear Programming Problems with Fuzzy Uncertainty Information on Probability Distribution Space (STLPPFI) with its algorithm outlines. A MATLAB code program of STLPPFI problem solver with algorithm outlines are proposed to solve STLPPFI model problems, and it utilizes many concepts as Alpha-Cut technique, Truth Degrees technique, Linear Fuzzy Membership Function (LFMF), Trapezoidal Fuzzy Number , Triangular Fuzzy Number , Linear Fuzzy Ranking Function (LFRF), Expectation Weighted Summation technique (EWS) and analyzing cases via second condition test of alpha-cut technique. The STLPPFI problem solver is utlized to convert STLPPFI into its corresponding equivalent Deterministic Transportation Linear Programming Problem (DTLPP) via defuzzifying from fuzziness on probability distribution space and derandomization randomness of problem formulation respectively. In addition, Dual-Simplex algorithm method with Vogel Approximation Algorithm Method (VAM) are used to obtain optimal solution from DTLPP. All MATLAB code programs with their proposed algorithm outlines are new except Dual-Simplex and VAM. The MATLAB code program of STLPPFI problem solver are more efficient along with a numerical example on electricity field illustrating practicability of this proposed MATLAB code program with its algorithm. Finally, the solution procedure illustrates the MATLAB code program of the proposed method is practical and applicable in the fields of energy and industry as it facilitates the method of transforming the energy at the lowest cost, least time running and is commercially applicable. Comparative comments are provided between Dual-Simplex and VAM in solution process.

Classifying marine mammals signal using cubic splines interpolation combining with triple loss variational auto-encoder

In practical applications of passive sonar principles for extracting characteristic frequencies of acoustic signals, scientists typically employ traditional time-frequency domain transformation methods such as Mel-frequency, Short time Fourier transform (STFT), and Wavelet transform (WT). However, these solutions still face limitations in resolution and information loss when transforming data collected over extended periods. In this paper, we present a study using a two-stage approach that combines pre-processing by Cubic-splines interpolation (CSI) with a probability distribution in the hidden space with Siamese triple loss network model for classifying marine mammal (MM) communication signals. The Cubic-splines interpolation technique is tested with the STFT transformation to generate STFT-CSI spectrograms, which enforce stronger relationships between characteristic frequencies, enhancing the connectivity of spectrograms and highlighting frequency-based features. Additionally, stacking spectrograms generated by three consecutive methods, Mel, STFT-CSI, and Wavelet, into a feature spectrogram optimizes the advantages of each method across different frequency bands, resulting in a more effective classification process. The proposed solution using an Siamese Neural Network-Variational Auto Encoder (SNN-VAE) model also overcomes the drawbacks of the Auto-Encoder (AE) structure, including loss of discontinuity and loss of completeness during decoding. The classification accuracy of marine mammal signals using the SNN-VAE model increases by 11% and 20% compared to using the AE model (2013), and by 6% compared to using the Resnet model (2022) on the same actual dataset NOAA from the National Oceanic and Atmospheric Administration - United State of America.

Random hypergraphs, random simplicial complexes and their Künneth-type formulae

Random hypergraphs and random simplicial complexes on finite vertices were studied by [M. Farber, L. Mead and T. Nowik, Random simplicial complexes, duality and the critical dimension, J. Topol. Anal. 41(1) (2022) 1–32]. The map algebra on random sub-hypergraphs of a fixed simplicial complex, which detects relations between random sub-hypergraphs and random simplicial sub-complexes, was studied by the authors of this paper. In this paper, we study the map algebra on random sub-hypergraphs of a fixed hypergraph. We give some algorithms generating random hypergraphs and random simplicial complexes by considering the actions of the map algebra on the space of probability distributions. We prove some Künneth-type formulae for random hypergraphs and random simplicial complexes on finite vertices.

Bayesian renormalization

In this note we present a fully information theoretic approach to renormalization inspired by Bayesian statistical inference, which we refer to as Bayesian renormalization. The main insight of Bayesian renormalization is that the Fisher metric defines a correlation length that plays the role of an emergent renormalization group (RG) scale quantifying the distinguishability between nearby points in the space of probability distributions. This RG scale can be interpreted as a proxy for the maximum number of unique observations that can be made about a given system during a statistical inference experiment. The role of the Bayesian renormalization scheme is subsequently to prepare an effective model for a given system up to a precision which is bounded by the aforementioned scale. In applications of Bayesian renormalization to physical systems, the emergent information theoretic scale is naturally identified with the maximum energy that can be probed by current experimental apparatus, and thus Bayesian renormalization coincides with ordinary renormalization. However, Bayesian renormalization is sufficiently general to apply even in circumstances in which an immediate physical scale is absent, and thus provides an ideal approach to renormalization in data science contexts. To this end, we provide insight into how the Bayesian renormalization scheme relates to existing methods for data compression and data generation such as the information bottleneck and the diffusion learning paradigm. We conclude by designing an explicit form of Bayesian renormalization inspired by Wilson’s momentum shell renormalization scheme in quantum field theory. We apply this Bayesian renormalization scheme to a simple neural network and verify the sense in which it organizes the parameters of the model according to a hierarchy of information theoretic importance.

Self-similarity guided probabilistic embedding matching based on transformer for occluded person re-identification

In real scenes, the occluded situation caused by different factors is the key to the person re-identification (Re-ID) problem. Many methods retrieve the corresponding images in the gallery by employing feature matching. However, the occluded region is still not well addressed, and the noise generated inevitably increases the difficulty of Re-ID. Therefore, we propose a Self-Similarity guided Probabilistic Embedding Matching (SSPEM) method to solve the problem of occluded person Re-ID. Specifically, we first design a Feature Similarity Enhancement (FSE) module to perform self-similarity calculation based on the extracted features of the same pedestrians by Vision Transformer (ViT) and complete feature enhancement by suppressing irrelevant information. Then, we design a probabilistic embedding matching (PEM) module to represent the features as probability distributions in a common embedding space, which is able to learn more feature structures. The uncertainty for the occluded region can maximize the metric computation to obtain features with better matching. Extensive experiments on five challenging datasets for occluded and holistic person Re-ID tasks are conducted, and the results show the effectiveness of our proposed SSPEM method.

Generalized Bayes approach to inverse problems with model misspecification

We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This assumption is difficult to justify in many inverse problems, where the specification of the data generation process is not obvious. We adopt a Gibbs posterior framework that directly posits a regularized variational problem on the space of probability distributions of the parameter. We propose a novel model comparison framework that evaluates the optimality of a given loss based on its ‘predictive performance’. We provide cross-validation procedures to calibrate the regularization parameter of the variational objective and compare multiple loss functions. Some novel theoretical properties of Gibbs posteriors are also presented. We illustrate the utility of our framework via a simulated example, motivated by dispersion-based wave models used to characterize arterial vessels in ultrasound vibrometry.

Geometric approach to nonequilibrium hasty shortcuts.

Complex and even non-monotonic responses to external control can be found in many thermodynamic systems. In such systems, nonequilibrium shortcuts can rapidly drive the system from an initial state to a desired final state. One example is the Mpemba effect, where preheating a system allows it to cool faster. We present nonequilibrium hasty shortcuts-externally controlled temporal protocols that rapidly steer a system from an initial steady state to a desired final steady state. The term "hasty" indicates that the shortcut only involves fast dynamics without relying on slow relaxations. We provide a geometric analysis of such shortcuts in the space of probability distributions by using timescale separation and eigenmode decomposition. We further identify the necessary and sufficient condition for the existence of nonequilibrium hasty shortcuts in an arbitrary system. The geometric analysis within the probability space sheds light on the possible features of a system that can lead to hasty shortcuts, which can be classified into different categories based on their temporal pattern. We also find that the Mpemba-effect-like shortcuts only constitute a small fraction of the diverse categories of hasty shortcuts. This theory is validated and illustrated numerically in the self-assembly model inspired by viral capsid assembly processes.

Robust 4D climate-optimal flight planning in structured airspace using parallelized simulation on GPUs: ROOST V1.0

Abstract. The climate impact of non-CO2 emissions, which are responsible for two-thirds of aviation radiative forcing, highly depends on the atmospheric chemistry and weather conditions. Hence, by planning aircraft trajectories to reroute areas where the non-CO2 climate impacts are strongly enhanced, called climate-sensitive regions, there is a potential to reduce aviation-induced non-CO2 climate effects. Weather forecast is inevitably uncertain, which can lead to unreliable determination of climate-sensitive regions and aircraft dynamical behavior and, consequently, inefficient trajectories. In this study, we propose robust climate-optimal aircraft trajectory planning within the currently structured airspace considering uncertainties in standard weather forecasts. The ensemble prediction system is employed to characterize uncertainty in the weather forecast, and climate-sensitive regions are quantified using the prototype algorithmic climate change functions. As the optimization problem is constrained by the structure of airspace, it is associated with hybrid decision spaces. To account for discrete and continuous decision variables in an integrated and more efficient manner, the optimization is conducted on the space of probability distributions defined over flight plans instead of directly searching for the optimal profile. A heuristic algorithm based on the augmented random search is employed and implemented on graphics processing units to solve the proposed stochastic optimization computationally fast. An open-source Python library called ROOST (V1.0) is developed based on the aircraft trajectory optimization technique. The effectiveness of our proposed strategy to plan robust climate-optimal trajectories within the structured airspace is analyzed through two scenarios: a scenario with a large contrail climate impact and a scenario with no formation of persistent contrails. It is shown that, for a nighttime flight from Frankfurt to Kyiv, a 55 % reduction in climate impact can be achieved at the expense of a 4 % increase in the operating cost.

Geometry of sample spaces

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 Mn 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 Fréchet 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.

Neural-network solutions to stochastic reaction networks

The stochastic reaction network in which chemical species evolve through a set of reactions is widely used to model stochastic processes in physics, chemistry and biology. To characterize the evolving joint probability distribution in the state space of species counts requires solving a system of ordinary differential equations, the chemical master equation, where the size of the counting state space increases exponentially with the type of species. This makes it challenging to investigate the stochastic reaction network. Here we propose a machine learning approach using a variational autoregressive network to solve the chemical master equation. Training the autoregressive network employs the policy gradient algorithm in the reinforcement learning framework, which does not require any data simulated previously by another method. In contrast with simulating single trajectories, this approach tracks the time evolution of the joint probability distribution, and supports direct sampling of configurations and computing their normalized joint probabilities. We apply the approach to representative examples in physics and biology, and demonstrate that it accurately generates the probability distribution over time. The variational autoregressive network exhibits plasticity in representing the multimodal distribution, cooperates with the conservation law, enables time-dependent reaction rates and is efficient for high-dimensional reaction networks, allowing a flexible upper count limit. The results suggest a general approach to study stochastic reaction networks based on modern machine learning. Stochastic reaction networks involve solving a system of ordinary differential equations, which becomes challenging as the number of reactive species grows, but a new approach based on evolving a variational autoregressive neural network provides an efficient way to track time evolution of the joint probability distribution for general reaction networks.

Unified, Geometric Framework for Nonequilibrium Protocol Optimization.

Controlling thermodynamic cycles to minimize the dissipated heat is a long-standing goal in thermodynamics, and more recently, a central challenge in stochastic thermodynamics for nanoscale systems. Here, we introduce a theoretical and computational framework for optimizing nonequilibrium control protocols that can transform a system between two distributions in a minimally dissipative fashion. These protocols optimally transport a system along paths through the space of probability distributions that minimize the dissipative cost of a transformation. Furthermore, we show that the thermodynamic metric-determined via a linear response approach-can be directly derived from the same objective function that is optimized in the optimal transport problem, thus providing a unified perspective on thermodynamic geometries. We investigate this unified geometric framework in two model systems and observe that our procedure for optimizing control protocols is robust beyond linear response.

Sensitivity of nonequilibrium relaxation to interaction potentials: Timescales of response from Boltzmann's H function.

We investigate, by simulations and analytic theory, the sensitivity of nonequilibrium relaxation to interaction potential and dimensionality by using Boltzmann's H function H(t). We evaluate H(t) for three different intermolecular potentials in all three dimensions and find that the well-known H theorem is valid and that the H function exhibits rather strong sensitivity to all these factors. The relaxation of H(t) is long in one dimension, but short in three dimensions, longer for the Lennard-Jones potential than for the hard spheres. The origin of the ultraslow approach to the equilibrium of H(t) in one-dimensional systems is discussed. Importantly, we obtain a closed-form analytic expression for H(t) using the solution of the Fokker-Planck equation for velocity space probability distribution and compare its predictions with the simulation results. Interestingly, H(t) is found to exhibit a linear response when vastly different initial nonequilibrium conditions are employed. The microscopic origin of this linear response is discussed. The oft-quoted relation of H function with Clausius's entropy theorem is discussed.