This work investigates implicit bias in multiclass separable data using a novel geometry-aware optimizer.
problem Understanding implicit bias in overparameterized models on multiclass separable data.
method Introduces NucGD, a geometry-aware optimizer enforcing low-rank structures through nuclear norm constraints.
result NucGD enables scalable training and characterizes the impact of stochastic optimization dynamics.
Paper introduces geometry-aware normalizing flows for improved causal inference.
problem Disparity between sample and population distributions in causal inference.
method Integrates continuous normalizing flows with parametric submodels, employing Wasserstein gradient flows and optimal transport.
result Significantly reduces parameter estimation bias and variance in finite-sample settings.
Geometry-aware noise improves model generalization on complex manifolds.
problem Improving model generalization on highly curved data manifolds.
method Add geometry-aware noise to input space, projecting Gaussian noise onto tangent space of manifold and mapping it via geodesic curve.
result Geometry-aware noise leads to improved generalization and robustness on highly curved manifolds.
Geometry-aware models improve cross-subject EEG decoding accuracy.
problem Strong inter-subject variability in motor imagery decoding.
method Discriminative Congruence Transform (DCT), Deep Linear DCT (DLDCT), Deep DCT-UNet (DDCT-UNet).
result Improves transductive cross-subject accuracy by 2-3%.
GAGA learns a warped metric for geometry-aware data generation and interpolation.
problem Challenges in generating data with meaningful geometry in high-dimensional datasets.
method Combines manifold learning with generative modeling to learn a warped Riemannian metric.
result GAGA improves trajectory inference by 30% in single-cell population-level data.
GeoERM learns shared representations on Riemannian manifolds for multi-task learning.
problem Heterogeneous and adversarial tasks in MTL.
method Geometry-aware MTL framework embedding shared representation on Riemannian manifold, optimizing via manifold operations.
result GeoERM improves estimation accuracy and stability, outperforming alternatives.
New algorithms improve neural architecture search with faster convergence.
problem Improving efficiency and accuracy of neural architecture search.
method Geometry-aware gradient algorithms to optimize continuous relaxation of discrete search spaces.
result Exceeds state-of-the-art results on CIFAR and ImageNet benchmarks.
The lack of proper class discrimination among the Hyperspectral (HS) data points poses a potential challenge in HS classification. To address this issue, this paper proposes an optimal geometry-aware transformation for enhancing the classification accuracy. The underlying idea of this method is to obtain a linear proje…
This paper proposes a geometry-aware active learning framework for spatiotemporal dynamic systems.
problem Challenges in modeling complex dynamic systems with 3D geometries and time evolution.
method Geometry-aware spatiotemporal Gaussian Process (G-ST-GP) and adaptive active learning strategy.
result The proposed framework outperforms traditional methods in predicting high-dimensional dynamic behaviors.
Non-Euclidean BPM extends optimization theory to non-Euclidean norms.
problem Extending BPM's convergence theory to non-Euclidean norms.
method Iteratively minimizing over norm balls in non-Euclidean geometry.
result Most BPM guarantees carry over to non-Euclidean norms.
CDC-FM improves generative model quality-generalization tradeoff by regularizing with geometry-aware noise.
problem Tradeoff between high sample quality and memorization in deep generative models.
method Introduces Carré du champ flow matching (CDC-FM) that replaces homogeneous noise with anisotropic Gaussian noise capturing latent data manifold geometry.
result CDC-FM consistently offers better quality-generalization tradeoff across diverse datasets and architectures.
GH-PID uses guided harmonic paths for efficient SOT with interpretable diagnostics.
problem Efficiently solving Stochastic Optimal Transport with hard terminal distributions and soft costs.
method Guided Harmonic Path-Integral Diffusion (GH-PID) framework with low-dimensional guidance.
result GH-PID generates geometry-aware, cost-reducing trajectories that match terminal distributions.
Sparse GEMINI selects relevant features for clustering without assumptions.
problem Feature selection in clustering with relevant clusters and variables.
method Discriminative clustering model maximizing GEMINI with l1 penalty.
result Sparse GEMINI selects relevant subsets of variables without prior hypotheses.
New estimators for intrinsic dimension and Wasserstein distance improve OT accuracy.
problem Intrinsic dimension estimation and Wasserstein distance estimation in large-scale OT.
method Introduces novel estimators for intrinsic dimension and Wasserstein distance.
result Simple, tuning-free estimator of OT and fast intrinsic dimension estimator.
CWGD measures gradient diversity weighted by curvature, improving SGD convergence.
problem Gradient noise in high-curvature directions is underestimated by standard methods.
method CWGD weights gradient diversity by the inverse square root of the Hessian.
result CWGD-Cosine reduces optimization error by up to 20% compared to standard cosine annealing.
Geometry-aware KDE model improves multiclass quantification.
problem Accurately estimating class prevalence for label shift adaptation.
method Log-ratio representations and Aitchison geometry for compositional data, shrinkage regularization.
result Competitive with state-of-the-art quantifiers, often improving over standard KDE-based baselines.
New framework improves experimental design using integral probability metrics.
problem Challenges in Bayesian Optimal Experimental Design (BOED) with KL divergence.
method Integrates integral probability metrics (IPMs) for stability and flexibility.
result IPM-based designs yield more robust and accurate credible sets.
This paper tackles high-dimensional Bayesian optimization by projecting a manifold into a lower space.
problem High-dimensional optimization of expensive functions with limited labeled data.
method Random linear projection of a manifold embedded in high-dimensional space, combined with semi-supervised learning of the manifold's geometry.
result Our approach outperforms existing high-dimensional BO methods in various synthetic and real-world applications.
While matrix factorisation models are ubiquitous in large scale recommendation and search, real time application of such models requires inner product computations over an intractably large set of item factors. In this manuscript we present a novel framework that uses the inverted index representation to exploit struct…
Hybrid method improves SABR implied volatility approximation.
problem Improving SABR implied volatility approximation.
method Combining analytical structure with machine learning, using geometric features and residual correction.
result Hybrid model improves accuracy and robustness compared to analytical and neural-network approaches.
Develops a curvature-corrected tangent space method for manifold-valued data.
problem Generalizing real-valued data approximation to manifold-valued data.
method Systematic approach to developing global-geometry aware, computationally feasible approximation schemes.
result Proposes CC-tHOSVD for low-rank approximation of manifold-valued data.
A new algorithm improves sampling for graph learning models.
problem Euclidean proposals struggle near the boundary of PSD matrices.
method ConeMALA, a geometry-aware Langevin algorithm.
result ConeMALA achieves higher ESS/sec and stable diagnostics.
The notion of task similarity is at the core of various machine learning paradigms, such as domain adaptation and meta-learning. Current methods to quantify it are often heuristic, make strong assumptions on the label sets across the tasks, and many are architecture-dependent, relying on task-specific optimal parameter…
A neural network method tackles high-dimensional diffeomorphic mapping problems.
problem High-dimensional diffeomorphic mapping struggles with the curse of dimensionality.
method Combines variational principles with quasi-conformal theory for accurate, bijective mappings.
result Validated accuracy, robustness, and effectiveness in complex registration scenarios.
Geometric approach for unsupervised word embedding alignment.
problem Learning alignment between word embeddings of source and target languages.
method Formulates alignment as domain adaptation on the manifold of doubly stochastic matrices, employing Riemannian conjugate gradient algorithm.
result Empirically outperforms state-of-the-art methods on bilingual lexicon induction tasks.
OrthoGrad improves neural calibration by constraining gradient updates orthogonally.
problem Overconfidence in neural networks, leading to poor uncertainty estimates.
method Orthogonal gradient updates to optimize for decision boundaries and reduce overconfidence.
result Significant improvements in test loss, predictive entropy, and confidence measures.
New framework models neural systems with random architecture on manifolds.
problem Complex, uncertain systems with non-Gaussian outputs.
method Latent random field on compact manifold generates neural architecture and weights.
result Synthetic neural systems can produce stochastic outputs for deterministic inputs.
GTA improves transformer-based NVS models by encoding geometric structure.
problem Suboptimal positional encoding for 3D vision tasks.
method Geometry-aware attention mechanism encoding geometric structure of tokens.
result GTA improves learning efficiency and performance of NVS models.
This paper introduces GEMINI, a new mutual information metric for unsupervised neural network training.
problem The mutual information (MI) as a clustering objective does not lead to satisfactory clusters.
method The authors generalised MI by changing its core distance, introducing GEMINIs that do not require regularizations and can automatically select the number of clusters.
result GEMINIs can automatically select the number of clusters without requiring a priori knowledge of the number of clusters.
Proposes a robust IV estimator using optimal transport for corrupted or adversarial data.
problem Lack of robustness in traditional IV estimators for corrupted or adversarial data.
method Integrates data-derivative information through optimal transport to address geometric aspects of data.
result Improves robustness against data corruption and adversarial attacks.
FedSPDnet improves federated learning for SPD matrices, outperforming existing methods.
problem Federated learning for SPD matrices with orthogonality constraints.
method Two efficient aggregation strategies: ProjAvg and RLAvg, preserving geometric structure.
result FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation.
SinSim improves self-supervised learning by integrating optimal transport into contrastive learning.
problem Lack of explicit regularization in contrastive learning methods leads to suboptimal generalization.
method Integrates Sinkhorn regularization from optimal transport theory into SimCLR.
result SinSim outperforms SimCLR and other self-supervised methods on various datasets.
Proposes Gromov-Wasserstein methods for multi-view embedding.
problem Integrating multiple representations of the same samples in heterogeneous geometries.
method Gromov-Wasserstein optimal transport for multi-view embedding.
result Preserves intrinsic relational structure across views effectively.
Proposes a geometry-aware VAE for better latent space modeling.
problem Lack of meaningful latent space structure in VAEs for small datasets.
method Introduces a Riemannian Hamiltonian VAE with a learned metric.
result Improves latent space structure leading to better interpolations and data generation.
A new method for faster optimization on statistical manifolds.
problem Slow convergence of first-order methods in manifold optimization.
method Dual Riemannian Newton method on manifolds with dual connections.
result Local quadratic convergence of the dual Riemannian Newton method.
New method uses geometric properties for better density estimation.
problem Uncertainty quantification in ambiguous tasks.
method Winner-takes-all training with centroidal Voronoi tessellations.
result Improved quantization and density estimation.
A new algorithm computes Wasserstein barycenters without entropic regularization.
problem Computing Wasserstein barycenters efficiently and accurately.
method Free-support algorithm based on particle flow and Riemannian geometry.
result The algorithm avoids entropic regularization and is computationally tractable.
S-GAI initializes MLPs using spectral geometry from data, improving performance.
problem Lack of guidance on initial weights encoding data geometry.
method S-GAI uses SVD to estimate spectral class geometry, initializing MLPs from training data.
result S-GAI-initialized MLPs start from a more informative hidden state and achieve comparable accuracy.
Optimizes neural network training by dynamically updating Tucker decomposition ranks.
problem Redundant parameters in neural network architectures.
method Geometry-aware training of factorized layers in tensor Tucker format.
result Optimal locally approximating the original dynamics without initial rank knowledge.
GABI learns geometry from diverse systems to improve Bayesian inference.
problem Bayesian inversion of physical systems with varying geometries.
method Geometric Autoencoders for Bayesian Inversion (GABI) learns geometry-aware priors from large datasets.
result GABI yields comparable predictive accuracy to deterministic methods and well-calibrated uncertainty quantification.
Understanding the 3-dimensional structure of the world is a core challenge in computer vision and robotics. Neural rendering approaches learn an implicit 3D model by predicting what a camera would see from an arbitrary viewpoint. We extend existing neural rendering to more complex, higher dimensional scenes than previo…
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
Develops a Riemannian archetypal analysis for interpretable non-linear data.
problem Limited performance of classical archetypal analysis on non-linear data.
method Riemannian geometry for data-driven pullback, geodesic convex combinations, convex relaxation followed by non-convex refinement.
result Combines interpretability of classical archetypal analysis with expressive power of modern non-linear models.
We advocate the use of a notion of entropy that reflects the relative abundances of the symbols in an alphabet, as well as the similarities between them. This concept was originally introduced in theoretical ecology to study the diversity of ecosystems. Based on this notion of entropy, we introduce geometry-aware count…
PSLR classifies functional data with scalar covariates using path signatures.
problem Classical functional logistic regression models have limitations in capturing nonlinear and cross-channel dependencies.
method PSLR uses truncated path signatures to create a basis-free representation of functional data.
result PSLR outperforms traditional functional classifiers in accuracy and robustness, especially under non-uniform sampling.
New method aligns brain data across individuals for better brain decoding.
problem Inter-individual variability in brain response patterns limits decoder generalization.
method SpectralOT method that embeds cortical geometry into Laplace-Beltrami eigenmodes.
result SpectralOT strikes balance between aligning functional features and preserving anatomical structure.
New optimizers control network width scaling, improving stability and transfer across different model sizes.
problem Designing stable optimizers for networks of varying widths.
method Interpreting optimizers as steepest descent under mean-normalized operator norms, enabling layerwise composability and width-independent bounds.
result New optimizers like row normalization and column normalization provide stable learning-rate transfer across different model widths.
ManifoldMind uses adaptive-curvature probabilistic spheres for trustworthy recommendations in semantic hierarchies.
problem Sparse and abstract recommendation domains where users explore diverse conceptual paths.
method Adaptive-curvature probabilistic spheres, soft multi-hop inference, and curvature-aware semantic kernel.
result Superior NDCG, calibration, and diversity compared to baselines on public benchmarks.