Study completes financial markets in complex models without external probabilities.
problem Completing financial markets in models without exogenous probability measures.
method Obtained a necessary and sufficient condition for market extension.
result A condition for market extension in complex models is established.
This paper uses probability tensors for efficient path planning in complex scenarios.
problem Efficient path planning in complex environments with obstacles and multiple goals.
method Probability tensors are used to model agent motion and decision-making, incorporating past and future information.
result The model finds solutions in complex scenarios, demonstrating realistic emergent behaviors.
New algorithms estimate and test collision probability with near-optimal sample complexity.
problem Estimating and testing collision probability in discrete distributions.
method Developed algorithms for (α,β)-local differential privacy and sequential testing. result Achieved nearly optimal sample complexity for estimating and testing collision probability.
Proposes IPT for modeling complex joint distributions.
problem Lack of closed-form solutions for complex continuous or mixed distributions.
method Observer-centered framework with three independence axioms; derivation of closed-form solutions.
result Closed-form solutions for complex joint distributions under IPT.
Develops a measure-theoretic framework for complex co-occurrence data.
problem Modeling and interpreting complex co-occurrences in high-dimensional data.
method Introduces measure-theoretic probability and conditional probability, investigates E-integrals.
result Establishes a rigorous measure-theoretic foundation for co-occurrence modeling.
VI approximates complex densities faster than classical methods.
problem Approximating complex probability densities.
method Optimization of a family of probability density functions using KL divergence.
result VI converges faster than Markov Chain Monte Carlo.
New MCMC method for complex models with large variables.
problem Inference on posterior model probabilities in large model spaces.
method Reversible genetically modified mode jumping Markov chain Monte Carlo (GMJMCMC).
result Introduced a proper MCMC with correct limiting distribution.
Method solves complex optimization problems with high probability bounds.
problem Nonlinear equality constrained stochastic optimization problems.
method Step-search sequential quadratic programming method.
result High-probability bound on iteration complexity for first-order stationarity.
Study tests if a probability measure is near a real algebraic variety.
problem Deciding if a probability measure is near a real algebraic variety.
method Proved upper bound on sample complexity, reduced to semialgebraic decision problem, studied Hausdorff geometry of real algebraic varieties.
result Upper bound on sample complexity for testing variety hypothesis.
Algorithm finds safe zones in policy Markov Decision Processes to limit trajectory escape.
problem Finding safe zones in policy Markov Decision Processes to limit trajectory escape.
method Bi-criteria approximation learning algorithm with polynomial sample complexity.
result Achieves almost 2 approximation for both escape probability and safe zone size.
New findings on complexity limits in fixed budget bandit identification.
problem Determining the best possible error rate for fixed budget bandit identification.
method Analyzing the best non-adaptive sampling procedures and showing the existence of complexities.
result No fixed complexity for certain bandit identification tasks.
Optimal testing of discrete distributions with high probability, achieving sample complexity bounds.
problem Testing discrete distributions with high probability accuracy.
method Characterizing sample complexity as a function of parameters like δ, providing sample-optimal testers.
result Optimal algorithms for closeness and independence testing, achieving within constant factors of information-theoretic lower bounds.
The paper introduces optimal transport kernels for comparing cell complexes.
problem Lack of machine learning methods for CW complexes.
method Derives explicit expression for Wasserstein distance, extends Fused Gromov-Wasserstein, introduces novel kernels.
result Introduced novel kernels for comparing probability measures on CW complexes.
Paper simplifies calculating causation probabilities and ranks root causes.
problem Computational challenges in assessing causal relationships.
method Algorithmic simplifications and novel methodological framework for Root Cause Analysis.
result Significantly reduces computational complexity for calculating causation probabilities.
Probability density estimation is a classical and well studied problem, but standard density estimation methods have historically lacked the power to model complex and high-dimensional image distributions. More recent generative models leverage the power of neural networks to implicitly learn and represent probability …
The paper explores statistical and topological properties of sliced probability divergences.
problem Understanding the topological, statistical, and computational consequences of slicing divergences.
method Deriving theoretical properties of sliced probability divergences, including metric axioms preservation and weak continuity.
result Sliced divergences share similar topological properties and have stable sample complexity.
A new method estimates rare failure events in complex systems.
problem Estimating the probability of rare failure events in non-linear systems.
method Stochastic Spectral Embedding (SSE) combined with modifications for efficient rare event estimation.
result Rare failure probability decomposed into conditional probabilities for easier computation.
We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley's behavioural theory of imprecise probabilities, and Shafer and Vovk's game-theoretic account of probability. We show that the two theories are more closely related than would be …
This work proposes a new method to estimate joint probability from pairwise marginals, reducing sample complexity.
problem Direct nonparametric estimation of high-dimensional joint probability is infeasible due to the curse of dimensionality.
method Developed a coupled nonnegative matrix factorization (CNMF) framework using only pairwise marginals.
result The method provably recovers the joint probability mass function up to bounded error in finite iterations under reasonable conditions.
New complexity analysis for estimating normalizing constants in high dimensions.
problem Estimating the normalizing constant of unnormalized probability densities in high dimensions.
method Analyze and derive the oracle complexity of annealed importance sampling.
result Oracle complexity of $\widetilde{O}\left(\frac{dβ^2{\mathcal{A}}^2}{\varepsilon^4}
ight)$ for estimating Z within ε relative error. PAC learning sample complexity is decidable with finite support bounds.
problem Determining the exact sample complexity for PAC learning concepts.
method Observation and proof of decidability with a-priori bounds.
result Sample complexity can be exactly determined for various concepts with finite support bounds.
We provide theoretical complexity analysis for new algorithms to compute the optimal transport (OT) distance between two discrete probability distributions, and demonstrate their favorable practical performance over state-of-art primal-dual algorithms and their capability in solving other problems in large-scale, such …
There are many advantages to use probability method for nonlinear system identification, such as the noises and outliers in the data set do not affect the probability models significantly; the input features can be extracted in probability forms. The biggest obstacle of the probability model is the probability distribu…
LITE efficiently estimates Gaussian PoM with linear time and memory complexity.
problem Estimating the probability of maximality (PoM) of Gaussian vectors efficiently.
method LITE: entropy-regularized UCB approach for almost-linear time and memory complexity.
result Achieves state-of-the-art accuracy with significantly faster performance than existing methods.
We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities…
Research reveals simplicity bias in random logistic map, impacting data analysis and forecasting.
problem Simplicity bias in dynamical systems and its impact on data analysis and prediction.
method Examined the logistic map and random logistic map, focusing on simplicity bias and noise effects.
result Simplicity bias is observable in the random logistic map, persisting even with small noise levels.
This paper establishes for the first time the predictive performance of speed priors and their computational complexity. A speed prior is essentially a probability distribution that puts low probability on strings that are not efficiently computable. We propose a variant to the original speed prior (Schmidhuber, 2002),…
This paper applies quantum probability theory to model asset returns, avoiding assumptions about quantum effects.
problem Modeling asset returns with classical probability theory.
method Derives a Schrödinger-like trading equation using quantum probability, linking it to traders' decisions and market behaviors.
result Quantum probability can describe multimodal distributions of asset returns without assuming quantum effects.
We propose a statistical model for natural language that begins by considering language as a monoid, then representing it in complex matrices with a compatible translation invariant probability measure. We interpret the probability measure as arising via the Born rule from a translation invariant matrix product state.
MEP-Net uses MEP to generate solutions from limited data.
problem Generating solutions to scientific problems with incomplete information.
method Combines MEP with neural networks to learn complex distributions from moment constraints.
result Demonstrates MEP-Net's effectiveness in modeling biochemical reaction networks and generating complex distributions.
NCP uses neural networks to efficiently learn conditional distributions.
problem Learning conditional distributions for statistical inference.
method Neural Conditional Probability (NCP) approach.
result NCP efficiently handles complex probability distributions and matches leading methods.
We present a framework and analysis of consistent binary classification for complex and non-decomposable performance metrics such as the F-measure and the Jaccard measure. The proposed framework is general, as it applies to both batch and online learning, and to both linear and non-linear models. Our work follows recen…
GOCPD detects change points by maximizing the probability of two independent models.
problem Large false discovery rates in online change point detection methods.
method GOCPD uses ternary search to find change points by maximizing the probability of two independent models.
result GOCPD accelerates CPD with logarithmic complexity for single change point detection.
It is known that describing or calculating the conditional probabilities of multiple events is exponentially expensive. In this work, Bayesian tensor network (BTN) is proposed to efficiently capture the conditional probabilities of multiple sets of events with polynomial complexity. BTN is a directed acyclic graphical …
Paper develops a TR-SSQP method for noisy optimization with heavy-tailed noise.
problem Optimization problems with stochastic objectives and heavy-tailed noise.
method Trust-Region Stochastic Sequential Quadratic Programming (TR-SSQP) method.
result Achieves high-probability first-order and second-order stationarity bounds for heavy-tailed noise.
Adaptive Multilevel Monte Carlo improves probability estimation for complex random variables.
problem Estimating probabilities of complex random variables with multiple approximations.
method Adaptive Multilevel Monte Carlo framework for discontinuous functionals.
result Achieves optimal computational complexities for both smooth and discontinuous functionals.
New method reduces variance in stochastic optimization with high confidence.
problem Achieving high-probability guarantees in stochastic optimization with weaker noise assumptions.
method Stochastic proximal point method combining proximal subproblem solver and probability booster.
result Demonstrates convergence with low sample complexity under bounded variance assumptions.
The paper explores geometric calculations on probability manifolds derived from master equations.
problem Understanding geometric properties of probability manifolds from master equations.
method Deriving geometric quantities like Levi-Civita connection, gradient, Hessian, parallel transport, and curvatures on probability manifolds.
result Calculation of geometric quantities in probability manifolds, including curvatures and connections.
The problem of population recovery refers to estimating a distribution based on incomplete or corrupted samples. Consider a random poll of sample size n conducted on a population of individuals, where each pollee is asked to answer d binary questions. We consider one of the two polling impediments: (a) in lossy pop…
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
New method identifies Condorcet winner in dueling bandits with improved sample complexity.
problem Identifying Condorcet winner in noisy pairwise comparisons.
method Exploits full gap matrix Δ to improve sample complexity.
result Improves sample complexity guarantees by leveraging informative comparisons.
The fundamental group of the 2-dimensional Linial-Meshulam random simplicial complex Y2(n,p) was first studied by Babson, Hoffman and Kahle. They proved that the threshold probability for simple connectivity of Y2(n,p) is about p≈n−1/2. In this paper, we show that this threshold probability is at mo…
We solve a complex optimization problem for Wasserstein barycenters using stochastic methods.
problem Optimizing the average of multiple probability distributions in a streaming data setting.
method We reformulate the problem as a convex-concave saddle-point problem and propose a stochastic optimization algorithm.
result Our algorithm has better complexity than existing methods for arbitrary distributions.
RS-NSGD improves SGD convergence for heavy-tailed noise.
problem Nonconvex optimization with heavy-tailed noise.
method Integrates direction normalization into subspace updates.
result Achieves better oracle complexity than full-dimensional normalized SGD.
Improved algorithm for selecting a hypothesis locally privately with fewer queries.
problem Locally private hypothesis selection with minimal privacy-preserving queries.
method Introduces a Scheffé graph to reduce query complexity for hypothesis selection.
result Algorithm performs O~(k3/2) queries, improving on previous methods. This paper introduces AdaSDCA: an adaptive variant of stochastic dual coordinate ascent (SDCA) for solving the regularized empirical risk minimization problems. Our modification consists in allowing the method adaptively change the probability distribution over the dual variables throughout the iterative process. AdaSD…
Improved MLMC method for robust and efficient probability and density estimation.
problem Stability and poor complexity of MLMC for low-regularity functionals.
method Numerical smoothing combined with MLMC for deterministic quadrature methods.
result Significant improvement in strong convergence and robustness of MLMC method.
Flow-based models use ODEs to generate complex data distributions.
problem Generating high-dimensional data with complex probability distributions.
method Flow-based models use invertible mappings governed by ODEs to capture these distributions.
result Flow-based models provide exact likelihood estimation and efficient sampling.