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.
A new method for group invariant machine learning using geometric projections.
problem Supervised group invariant and equivariant machine learning.
method Geometric topology approach involving projection of input data into a geometric space parametrizing symmetry group orbits.
result Improvement in accuracy compared to existing methods.
FiberNet integrates geometry into machine learning for clearer classification.
problem Lack of interpretability in traditional deep learning.
method Reformulates classification as geometric optimization on fiber bundles, introducing learnable Riemannian metrics and variational prototype optimization.
result Clear geometric interpretability and efficiency in classification.
Enhances quantum circuit synthesis using deep learning and geometric methods.
problem Optimizing quantum circuits for time efficiency.
method Combining deep learning with geometric control techniques.
result Improved time-optimal control in quantum circuit synthesis.
Machine learning methods struggle with geometric data, but shape space analysis provides a framework for studying and analyzing geometric variability.
problem Machine learning methods struggle with geometric data
method Shape space analysis provides a mathematical and computational framework
result Characterizes shape variability, compares geometric objects, and analyzes structural trajectories
New method uses geometric moments for accurate machine learning potentials.
problem Creating high-dimensional potential energy surfaces efficiently.
method Feed-forward neural networks with invariant local molecular descriptors based on geometric moments.
result Accuracy comparable to established models, high efficiency.
Geometric framework explains and controls implicit bias in machine learning.
problem Understanding and controlling the selection of solutions in overparameterized models.
method Developed a theoretical and constructive framework based on geometric corrections induced by gradient noise and continuous symmetries of the loss.
result Computed the induced bias across various architectures and enabled inverse design to shape the bias.
A new geometric shaping method is proposed, leveraging unsupervised machine learning to optimize the constellation design. The learned constellation mitigates nonlinear effects with gains up to 0.13 bit/4D when trained with a simplified fiber channel model.
MCBP detects boundaries in high-dimensional data using curvature.
problem Boundary detection in high-dimensional data.
method MCBP uses mean curvature to model data manifold curvature.
result MCBP improves clustering performance in complex scenarios.
GQML uses symmetries from representation theory to improve quantum machine learning.
problem Creating quantum models with symmetries to improve performance.
method Introduction to representation theory for quantum learning, focusing on group actions and symmetries.
result Effective implementation of GQML requires knowledge of group representation theory.
The volume of a credal set correlates with epistemic uncertainty in binary classification but not in multi-class.
problem Representing and quantifying epistemic uncertainty in machine learning.
method Examined the geometric representation of credal sets as d-dimensional polytopes and their volume as a measure of uncertainty. result The volume of a credal set is a meaningful measure of epistemic uncertainty in binary classification but not in multi-class.
Machine learning applied to algebraic geometry for physics problems.
problem Reformulating algebraic geometry problems as tensor mappings for machine learning.
method Supervised and unsupervised machine learning techniques applied to algebraic geometry problems.
result Machine learning provides insights into the structure of algebraic geometry data.
This review explores the use of machine learning in discovering collective variables for biomolecular dynamics.
problem Understanding the conformational dynamics and molecular recognition in biomolecules.
method Statistical analysis of high-dimensional spatiotemporal data generated from molecular dynamics simulations.
result Machine learning algorithms can be used to discover abstract collective variables that describe biomolecular dynamics.
New method learns operators with geometric singularities from few samples.
problem Learning operators with geometric singularities from limited data.
method Double fibration transforms and cross-attention architectures.
result Operators can be learned superalgebraically from few samples.
In this paper, we are concerned with the 2D and 3D geometric shape generation by prescribing a set of characteristic values of a specific geometric body. One of the major motivations of our study is the 3D human body generation in various applications. We develop a novel method that can generate the desired body with c…
Introduces q-paths for generalizing geometric annealing paths in machine learning.
problem Limited applicability of existing path methods in machine learning.
method Develops a family of paths derived from a generalized mean, including geometric and arithmetic mixtures.
result Empirical gains in Bayesian inference and generative model evaluation.
Geometric method improves uncertainty estimation in real-time.
problem Improving uncertainty estimation in machine learning models.
method Geometric distance from training inputs for uncertainty estimation, post-hoc calibration.
result Method yields better uncertainty estimations than existing approaches.
Improved method for computing Fréchet means on SPD matrices.
problem Computing Fréchet means on the manifold of SPD matrices.
method Random matrix theory-based approach for estimating Fréchet means.
result Significantly outperforms state-of-the-art methods in experiments.
BGNNs model particle-boundary interactions efficiently.
problem Efficiently modeling geometric boundaries in 3D simulations.
method Introduce Boundary Graph Neural Networks (BGNNs) to dynamically modify graph structures.
result BGNNs accurately reproduce 3D granular flows without handcrafted conditions.
Predictive geometric models deliver excellent results for many Machine Learning use cases. Despite their undoubted performance, neural predictive algorithms can show unexpected degrees of instability and variance, particularly when applied to large datasets. We present an approach to measure changes in geometric models…
In this paper, an unsupervised machine learning method for geometric constellation shaping is investigated. By embedding a differentiable fiber channel model within two neural networks, the learning algorithm is optimizing for a geometric constellation shape. The learned constellations yield improved performance to sta…
For binary classification we establish learning rates up to the order of n−1 for support vector machines (SVMs) with hinge loss and Gaussian RBF kernels. These rates are in terms of two assumptions on the considered distributions: Tsybakov's noise assumption to establish a small estimation error, and a new geometr…
Geometric theory connects machine learning classifiers to differential geometry.
problem Classifying data points in machine learning.
method Mapping binary classification to vector bundles and differential geometry.
result Harmonic interpolation solves RKHS interpolation problems.
New geometric structures defined on SPD matrices for better understanding.
problem Understanding SPD matrices and their geometric properties.
method Introducing Finslerian and dual information-geometric structures on James' bicone domain.
result Geodesics correspond to straight lines in coordinate systems, and new dissimilarities generalize existing ones.
Geometric vector perceptrons improve protein structure learning.
problem Learning from protein structure with efficient and natural representations.
method Introducing geometric vector perceptrons to extend dense layers for Euclidean vectors, integrating geometric and relational reasoning.
result Improves model quality assessment and computational protein design over existing methods.
SVarM uses varifold representations for shape classification and regression.
problem Challenges in analyzing geometric data due to non-Euclidean shape spaces.
method Develops a neural network-based framework for varifold representations of shapes.
result Demonstrates strong performance and robustness in shape classification and regression.
The paper outlines future work in random sets theory.
problem Developing a theory of statistical reasoning with random sets.
method Generalizing logistic regression, probability laws, and geometric uncertainty.
result A new geometric approach to uncertainty with general random sets.
SpatialSim benchmarks machine learning in recognizing object spatial configurations.
problem Machine learning in recognizing precise geometrical configurations of groups of objects.
method SpatialSim benchmark with tasks of Identification and Comparison, using Graph Neural Networks (MPGNNs).
result MPGNNs outperform baselines in recognizing spatial configurations, highlighting current limits.
Accelerates MCMC sampling for large-scale problems using machine learning.
problem Efficiently sampling large-scale Bayesian inference problems with high computational cost.
method Integrates low-fidelity machine learning models into a multilevel MCMC framework.
result Significantly accelerates multilevel sampling by a factor of two with similar accuracy.
New optimization algorithms on orthogonal group for machine learning.
problem Efficient optimization on the orthogonal group for machine learning tasks.
method Stochastic geometric algorithms on Lie groups.
result Strong performance on diverse machine learning tasks.
Logifold improves ensemble machine learning by identifying fuzzy domains.
problem Improving ensemble machine learning accuracy.
method Formulating logifold structure and interpreting local charts of datasets.
result Logifold improves accuracy compared to averaging model outputs.
Machine learning maps knots to embeddings, revealing topological invariants.
problem Learning topological invariance in knot theory.
method Contrastive and generative machine learning techniques, auto-regressive decoder Transformer network.
result Neural networks can map different knots to the same point in an embedding vector space.
Training shapes the geometry of neural network feature maps, revealing local area magnification.
problem Understanding how training affects the geometric structure of neural network feature maps.
method Analyzing the Riemannian geometry induced by neural network feature maps at infinite width and after training.
result Training breaks the symmetry of the geometry induced by random neural network feature maps, magnifying local areas along decision boundaries.
Machine learning reveals hidden features in knot classification.
problem Classifying the topology of closed curves.
method Investigating shortcut methods used by ML for knot classification.
result Developed a dataset and code to remove non-topological features.
QCML uses quantum geometry to represent data.
problem Data representation and the curse of dimensionality.
method QCML encodes data as Hermitian matrices in Hilbert space.
result Data geometry reveals intrinsic dimension and topological properties.
Geometric framework detects concept frustration between human concepts and machine representations.
problem Aligning human concepts with machine learning representations.
method Geometric framework and similarity measures for detecting concept frustration.
result Concept frustration affects machine learning model performance and reorganizes learned concept representations.
New geometric perspective for optimal learning on hexagonal structures.
problem Optimal learning process on hexagonal structures.
method Local trivial fibrations and Ceva's theorem.
result Learning can be defined on hexagonal structures.
In high dimensions, the mean and geometric median are nearly identical.
problem Understanding the relationship between mean and geometric median in high-dimensional spaces.
method Analytical derivation and simulation of the distance between mean and geometric median.
result The distance between mean and geometric median vanishes with dimensionality in high dimensions.
New geometric methods improve optimization and data science problems.
problem Improving optimization and data science problems.
method Geometric tools for high-dimensional optimization and statistical data science.
result New algorithms and statistical guarantees for optimization and data science.
Randomized Geometric Algebra for Convex Neural Networks Optimizes Transfer Learning.
problem Training neural networks to global optimality via convex optimization.
method Randomized algorithms in Clifford's Geometric Algebra for hypercomplex vector spaces.
result Convex optimization and geometric algebra improve LLMs' robustness and reliability in transfer learning.
Machine learning and topological data analysis identify unique geometric and topological features of human papillae.
problem Identifying unique features of human papillae across individuals.
method 3D microscopic scans, machine learning, discrete differential geometry, computational topology, persistent homology.
result Persistent homology features of papillae shape predict papillae type with high accuracy and can identify individuals with high accuracy.
Locally adaptive federated learning improves convergence in distributed machine learning.
problem Balancing local updates in federated learning leads to slow convergence.
method Locally adaptive federated learning algorithms that use uncoordinated stepsizes based on local geometric information.
result Locally adaptive methods can be particularly efficient in overparameterized settings and outperform standard federated algorithms.
Develops Riemannian geometry for optimization on manifolds with detailed derivations.
problem Abstract high-level optimization on nonlinear spaces like matrix manifolds.
method Systematic derivation of geometric structures and constructions in coordinates and matrix form.
result Unified treatment of Riemannian geometry for optimization on manifolds.
Machine learning classifies complex geometric patterns with high accuracy.
problem Classifying extension degree of dessins d'enfants over the rationals.
method Deep feed-forward neural network trained on machine learning.
result 0.92 accuracy in classification with 0.03 standard error.
GeometricKernels package implements kernels for uncertain data on graphs, manifolds, and meshes.
problem Defining and computing kernels for structured data on graphs, manifolds, and meshes.
method Implementation of geometric analogs of Euclidean kernels (heat and Matérn) with automatic differentiation support.
result Ability to compute Fourier-feature-type expansions on geometric spaces.
New approach connects robust learning to privacy issues.
problem Vulnerability of deep neural networks to adversarial examples.
method Privacy-utility tradeoff and rate-distortion theory.
result Found saddle point via maximum conditional entropy problem.
Proposes a new neural network architecture combining MLP and basis functions.
problem Function approximation and operator learning in scientific machine learning.
method Combines robust MLP inner functions with flexible basis functions outer functions.
result KKAN outperforms MLPs and KANs in function approximation and operator learning tasks.
Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great successes. Unfortunately, the understanding on how it works remains unclear. It has th…