New method classifies manifold-valued data using Riemannian geometry.
problem Classifying data on curved Riemannian manifolds.
method Probabilistic Learning Vector Quantization on Symmetric Positive Definite Matrices.
result The method outperforms traditional Euclidean methods on manifold-valued data.
Discusses new probabilistic morphisms and geometric methods in machine and statistical learning.
problem Addressing challenges in statistical, machine, and manifold learning.
method Introduces category of probabilistic morphisms and geometric methods.
result New insights and applications in various learning fields.
In a recent paper, the authors proposed a general methodology for probabilistic learning on manifolds. The method was used to generate numerical samples that are statistically consistent with an existing dataset construed as a realization from a non-Gaussian random vector. The manifold structure is learned using diffus…
Enhanced probabilistic sampling on manifolds using Double Diffusion Maps and Geometric Harmonics.
problem Overfitting and loss of generalization in PLoM when N is small and dimensionality approaches N.
method Extending PLoM with Double Diffusion Maps and Geometric Harmonics to handle small N and high-dimensional data.
result Effective and robust method for generating statistically consistent realizations from limited data.
Normal-bundle bootstrap generates new data preserving geometric structure.
problem Probabilistic models often exhibit salient geometric structure.
method NBB method decomposes probability measure into manifold and normal spaces, estimates manifold as density ridge, and generates new data by bootstrapping projection vectors.
result NBB generates new data that preserves the geometric structure of a given data set.
We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on a set of neighbourhood relationships. The model allows straightforward variatio…
We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space. We first analyze kernel-based algorithms and show that under the usual regularizations, non-probabilistic methods cannot recover the differential geometric structure, but instead find mostl…
New approach combines geometric and probabilistic methods to estimate manifold dimension in high-dimensional data.
problem Estimating the dimension of manifolds in high-dimensional data.
method Combines a modified box-counting algorithm (geometric) and a new probabilistic method (nearest neighbor distance analysis).
result The combined method is robust, fast, and effective in estimating manifold dimension.
A novel GPUM constructs Gaussian Processes for unknown manifolds with probabilistic metrics.
problem High-dimensional data on unknown manifolds with non-Euclidean geometry.
method Bayesian Gaussian Processes latent variable models (BGPLVM), Riemannian geometry, probabilistic metric tensor, Brownian Motion.
result GPUM provides more accurate predictions on unknown manifolds compared to traditional methods.
Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
The paper presents a probabilistic framework for SPD matrices in machine learning.
problem Machine learning on SPD matrices is fragmented; this paper aims to unify it.
method Unified probabilistic framework using Gaussian distributions and Bayes classifiers.
result Different SPD machine learning tools can be reinterpreted and extended using Gaussian distributions.
PGPCA improves PCA for nonlinear data in neuroscience.
problem Nonlinear data distribution in neuroscience.
method Developed PGPCA for nonlinear manifolds, incorporating EM algorithm.
result PGPCA outperforms PPCA in modeling data around nonlinear manifolds.
Latent variable models (LVMs) learn probabilistic models of data manifolds lying in an \emph{ambient} Euclidean space. In a number of applications, a priori known spatial constraints can shrink the ambient space into a considerably smaller manifold. Additionally, in these applications the Euclidean geometry might induc…
GP+ is a Python library for Gaussian process learning.
problem Probabilistic learning and inference for complex models.
method Integration of nonlinear manifold learning with Gaussian processes.
result GP+ enables probabilistic data fusion and inverse parameter estimation.
Probabilistic deep learning uses neural networks and models to handle uncertainty.
problem Handling uncertainty in deep learning models.
method Two approaches: probabilistic neural networks and deep probabilistic models.
result TensorFlow Probability library supports both approaches.
Learning of low dimensional structure in multidimensional data is a canonical problem in machine learning. One common approach is to suppose that the observed data are close to a lower-dimensional smooth manifold. There are a rich variety of manifold learning methods available, which allow mapping of data points to the…
Probabilistic pseudo knots model uncertain knot diagrams.
problem Modeling knots with unresolved crossings.
method Assign probabilities to undetermined crossings; define probabilistic equivalence and extend classical knot invariants.
result Capture uncertainty in physical, biological, and computational contexts.
Probabilistic ML improves healthcare data analysis.
problem Insufficient understanding and incomplete data in healthcare.
method Examination of probabilistic machine learning models for healthcare challenges.
result Probabilistic models enhance healthcare data analysis and model building.
This work connects LLE, factor analysis, and probabilistic PCA through a stochastic perspective.
problem Exploring the theoretical connection between LLE, factor analysis, and probabilistic PCA.
method Solving the stochastic linear reconstruction of LLE using expectation maximization.
result LLE, factor analysis, and probabilistic PCA are shown to be connected through a stochastic perspective.
Probabilistic method proves gap estimates on sphere.
problem Proving fundamental gap estimates for Schrödinger operators on spheres.
method Reflection coupling method on Riemannian manifolds.
result Extends probabilistic proof to sphere, generalizing previous results.
We introduce a novel approach for predicting the progression of adolescent idiopathic scoliosis from 3D spine models reconstructed from biplanar X-ray images. Recent progress in machine learning have allowed to improve classification and prognosis rates, but lack a probabilistic framework to measure uncertainty in the …
Wide accessibility of imaging and profile sensors in modern industrial systems created an abundance of high-dimensional sensing variables. This led to a a growing interest in the research of high-dimensional process monitoring. However, most of the approaches in the literature assume the in-control population to lie on…
Neural network fusion reduces data acquisition costs for multi-fidelity sources.
problem Reducing cost in acquiring information from multiple data sources with varying fidelity.
method Employing a novel neural network architecture for nonlinear manifold learning of multi-fidelity data.
result Our approach provides high predictive power and quantifies various sources uncertainties.
We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients. The system of polynomials is assumed to be triangularized. An application of Whitney's embedding theorem allows us to work in a reduced dimensional embedding space. A numerical continuation metho…
Probabilistic modeling enables combining domain knowledge with learning from data, thereby supporting learning from fewer training instances than purely data-driven methods. However, learning probabilistic models is difficult and has not achieved the level of performance of methods such as deep neural networks on many …
We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the per…
New method learns PDE solutions from low-fidelity data.
problem Challenges in learning PDE surrogates with scarce data.
method Flow matching in infinite-dimensional space with conditional neural operators.
result Accurately learns PDE solutions across different resolutions and fidelities.
Unified probabilistic foundation for fuzzy simplicial sets in dimensionality reduction.
problem Lack of clear probabilistic interpretation in fuzzy simplicial sets.
method Introducing a probabilistic framework explaining fuzzy simplicial sets as marginals of probability measures on simplicial sets.
result Unified probabilistic theoretical foundation for fuzzy simplicial sets.
Proposes a method to use generators as EBM foundations without latent inference.
problem Training EBMs from generator outputs without latent variables.
method Formulates a Hat EBM using generator outputs and residual variables.
result Strong performance on various generator tasks.
New bound on partition function proves Kähler-Einstein stability.
problem Proving Kähler-Einstein metrics on complex manifolds.
method Quantitative bound on partition function, connecting probabilistic and quantization approaches.
result Direct analytic proof of Kähler-Einstein stability for uniformly Gibbs stable manifolds.
Probabilistic models with discrete latent variables naturally capture datasets composed of discrete classes. However, they are difficult to train efficiently, since backpropagation through discrete variables is generally not possible. We present a novel method to train a class of probabilistic models with discrete late…
D2PCCA integrates deep learning and probabilistic modeling for nonlinear dynamical systems.
problem Analyzing nonlinear dynamical systems with probabilistic understanding.
method Combines deep learning and probabilistic modeling, using KL annealing and normalizing flows.
result Captures latent dynamics in sequential datasets with improved convergence and flexibility.
We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtual…
Probabilistic learning for binary classification with categorical variables.
problem Binary classification with categorical covariates.
method Probabilistic analysis and two algorithms for learning boolean functions.
result Effective learning of boolean functions from binary data.
PLoM learns stochastic solutions to PDEs with limited data.
problem Synthesizing solutions to nonlinear PDEs with scarce data.
method Probabilistic Learning on Manifolds constrained by PDEs.
result Learned stochastic solutions minimize PDE residuals.
This paper improves probabilistic latent models on hyperbolic spaces.
problem Uncertainty in predictions due to geodesics crossing low-data regions.
method Augmenting hyperbolic manifold with a pullback metric for probabilistic pullback metrics.
result Geodesics on pullback metric respect both geometry and data distribution, reducing uncertainty.
Probabilistic programming is a powerful abstraction for statistical machine learning. Applying static analysis methods to probabilistic programs could serve to optimize the learning process, automatically verify properties of models, and improve the programming interface for users. This field of static analysis for pro…
Study efficient interactive learning for structured outputs with reliable computation.
problem Interactive learning with noisy labels and structured output spaces.
method Identify and utilize CRISPs (probabilistic models) that guarantee reliable and efficient computation of probabilistic quantities.
result CRISPs enable robust and efficient active and skeptical learning in large structured output spaces.
A deep learning method for probabilistic weather forecasting.
problem Probabilistic forecasting of weather.
method Two chained machine-learning steps: dimension reduction and density estimation using normalizing flows.
result The method produces accurate conditional forecast distributions for weather.
pRSL combines probabilistic rules to improve multi-label classification.
problem Modeling the structure between multi-label classes for better performance.
method Uses probabilistic propositional logic rules and belief propagation to combine predictions from multiple classifiers.
result pRSL achieves state-of-the-art performance on various benchmark datasets.
We review recent probabilistic results on covariant Schrödinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian m…
GPMI method interpolates uncertain atrial conduction velocity on non-Euclidean manifolds.
problem Uncertainty in atrial conduction velocity calculations.
method Gaussian Process Manifold Interpolation (GPMI) on human atrial manifolds.
result GPMI accounts for atrial topology and calculates CV uncertainty.
ProbRes calibrates probabilistic forecasts by learning volatility dynamics.
problem Quantifying risk and uncertainty in time series forecasting.
method ProbRes learns conditional mean and volatility separately, generating well-calibrated prediction intervals.
result ProbRes accurately captures predictive distributions and produces well-calibrated prediction intervals.
Probabilistic methods improve SHM by learning from noisy, incomplete data.
problem Noisy and incomplete SHM data, lack of prior labels.
method Probabilistic algorithms for semi-supervised, active, and multi-task learning.
result Probabilistic methods enhance SHM by incorporating new data.
New probabilistic invariants bound classical topological complexity and category.
problem Bounding classical topological complexity and category.
method Developed probabilistic variants of one-category and diagonal topological complexity.
result Identified new invariants with distributional category and complexity on Eilenberg-Mac Lane spaces.
Proposes a deep probabilistic multi-view model for multi-view learning.
problem Learning from multiple related views with shared latent structure.
method Probabilistic Canonical Correlation Analysis (CCA) in latent space, deep generative networks, variational inference.
result Efficient variational inference approximates posterior distributions of latent multi-view layer.
Paper proposes a probabilistic alignment method for domain adaptation.
problem Latent distribution mismatch and miscalibrated uncertainty in adapting large-scale models.
method Bayesian latent transport framework with PAC-Bayesian regularization.
result Reduction in latent manifold discrepancy and improved uncertainty calibration.
Unified framework for differentiable graph partitioning with probabilistic cuts.
problem Lack of general guarantees and principled gradients in prior probabilistic relaxations of graph cuts.
method Unified probabilistic framework covering a wide class of cuts, including Normalized Cut, with tight analytic upper bounds.
result Rigorous, numerically stable foundation for scalable, differentiable graph partitioning.