Adapts category theory to manifold learning for better understanding and stability.
problem Understanding and improving manifold learning algorithms.
method Develops a functorial perspective on manifold learning, proving bounds and constructing new algorithms.
result Derives new manifold learning algorithms that are competitive with existing methods.
Equivariant flows learn symmetrical distributions on manifolds.
problem Learning symmetrical distributions on arbitrary manifolds.
method Equivariant manifold flows.
result Learned gauge invariant densities over SU(n) in quantum field theory.
M-flows learn data manifolds and densities, improving manifold learning and inference.
problem Representing datasets with manifold structure more faithfully.
method Combining normalizing flows, GANs, autoencoders, and energy-based models, with a new training algorithm.
result M-flows learn data manifolds better than standard flows and provide handles for dimensionality reduction.
A new metric-based principal curve method learns 1D manifolds from spatial data.
problem Learning 1D manifolds from spatial data.
method Metric-based Principal Curve (MPC) approach.
result The method effectively learns the shape of 1D manifolds from synthetic and real datasets.
Proposes learning manifold implicitly via heat kernel.
problem Direct manifold learning methods lack flexibility for down-stream applications.
method Implicit manifold learning using heat kernel.
result Framework achieves state-of-the-art results for data generation and Bayesian inference.
Generative mixture models of VAEs learn manifolds for inverse problems.
problem Representing high-dimensional data manifolds efficiently and accurately.
method Mixture model of variational autoencoders (VAEs) with Riemannian gradient descent.
result Learned manifold enables solving inverse problems with data fidelity.
Paper proposes a new method for supervised manifold learning using random forest proximities.
problem Existing supervised manifold learning methods fail to uncover meaningful embeddings due to using class-conditional distances.
method Proposes a data-geometry-preserving variant of random forest proximities as an initialization for manifold learning methods.
result Local and global structure preservation is near universal across manifold learning approaches using diffusion-based algorithms.
New method for manifold topological learning avoids remeshing issues.
problem Persistent homology on manifolds is numerically inconsistent.
method Persistent de Rham-Hodge Laplacians in Eulerian representation.
result Avoids numerical inconsistency over multiscale manifolds.
A method for learning distributions on complex manifolds using normalizing flows.
problem Learning distributions on non-Euclidean manifolds with high efficiency and accuracy.
method Learning a distribution on a manifold by combining local models that form an open cover.
result The method achieves better sample efficiency and competitive performance on manifolds of unknown topology.
Simplifies denoising score matching for manifold learning.
problem Learning distributions on manifolds is computationally intensive.
method Modifies denoising score matching to implicitly account for the manifold.
result Reduces computational burden while maintaining efficiency.
Paper investigates hardness of learning neural networks under manifold hypothesis.
problem Hardness of learning neural networks under the manifold hypothesis.
method Extending proofs of hardness in the SQ and cryptographic settings to the geometric setting.
result Learning is hard under input manifolds of bounded curvature but learnable with additional assumptions on manifold volume.
Paper proposes methods to learn sub-manifolds and estimate densities in normalizing flows.
problem Normalizing flows struggle with finding sub-manifolds in high-dimensional data.
method Introduces per-pixel penalized log-likelihood and hierarchical training approaches.
result Validated superior performance in manifold learning and density estimation.
Semi-supervised learning algorithms typically construct a weighted graph of data points to represent a manifold. However, an explicit graph representation is problematic for neural networks operating in the online setting. Here, we propose a feed-forward neural network capable of semi-supervised learning on manifolds w…
TMDA aligns subdomain data distribution discrepancies across domains using manifold representations.
problem Transfer learning challenges due to domain divergence.
method TMDA uses low-dimensional manifolds to represent subdomains and aligns local data distribution discrepancies across domains using M3D.
result TMDA is a promising method for various transfer learning tasks.
DMT enhances deep neural networks to better preserve data structures.
problem Preserving geometric, topological, and distributional structures of data in NLDR.
method Deep manifold transformation (DMT) using cross-layer LGP constraints.
result DMT networks outperform existing NLDR methods in preserving data structures.
Active learning improves GP regression on complex, high-dimensional data.
problem Improving Gaussian Process regression in high-dimensional spaces with discontinuous functions.
method Combines manifold learning with active learning to optimize data selection and reduce dimensionality.
result Superior performance over random learning in synthetic data experiments.
Paper uses 'manifold attack' to improve model accuracy and robustness.
problem Improving model performance with limited training data and large model parameters.
method Enforces manifold preservation from original data into latent space using adversarial learning.
result Improves accuracy rate and robustness to adversarial examples.
Few-shot learning algorithms aim to learn model parameters capable of adapting to unseen classes with the help of only a few labeled examples. A recent regularization technique - Manifold Mixup focuses on learning a general-purpose representation, robust to small changes in the data distribution. Since the goal of few-…
Most of existing manifold learning methods rely on Mean Squared Error (MSE) or ℓ2 norm. However, for the problem of image quality assessment, these are not promising measure. In this paper, we introduce the concept of an image structure manifold which captures image structure features and discriminates image dist…
The paper proposes a method to learn 3D object pose manifolds using GANs and elasticae.
problem Learning image manifolds of 3D objects with limited data.
method Geom-SGAN and elasticae for geometry-preserving image interpolation.
result The method outperforms state-of-the-art GANs and VAEs in learning rotation paths.
This work proposes a new neural implicit manifold model for more accurate density estimation on manifolds.
problem Current generative models struggle with representing manifolds accurately and learning densities within them.
method Proposes a neural implicit manifold model and a constrained energy-based model to learn manifold-supported distributions.
result The proposed model can learn manifold-supported distributions with complex topologies more accurately than pushforward models.
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.
Improves LSTM performance by initializing states via manifold learning.
problem Improving LSTM performance through better initialization.
method Learning an intrinsic data manifold to initialize LSTM internal states.
result Improved LSTM performance through consistent initialization.
A new method for learning manifolds efficiently using canonical basis functions.
problem Learning manifolds in high-dimensional data with efficient and distinct latent dimensions.
method Proposes a novel optimization objective to enforce a transformation matrix with a few prominent and non-degenerate basis functions.
result Demonstrates that minimizing the off-diagonal manifold metric elements ℓ1-norm results in a more efficient latent space representation. MMCGAN uses explicit manifold learning to improve GAN performance.
problem GAN mode collapse and unstable training.
method Introduces Minimum Manifold Coding (MMC) as a prior to guide GAN training.
result MMCGAN effectively alleviates mode collapse and stabilizes GAN training.
CLAMP uses neural manifold packing to improve self-supervised learning.
problem Improving self-supervised learning for vision tasks.
method CLAMP recasts representation learning as a manifold packing problem, introducing a loss function inspired by particle systems.
result CLAMP achieves competitive performance with state-of-the-art models and separates neural manifolds effectively.
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.
In this paper, we introduce McTorch, a manifold optimization library for deep learning that extends PyTorch. It aims to lower the barrier for users wishing to use manifold constraints in deep learning applications, i.e., when the parameters are constrained to lie on a manifold. Such constraints include the popular orth…
Researchers extend ResNets to Riemannian manifolds, improving performance over existing methods.
problem Learning on Riemannian manifolds, especially for hierarchical graphs and manifold-valued data.
method Geometrically principled extension of ResNets to general Riemannian manifolds.
result Riemannian ResNets outperform existing manifold neural networks in relevant metrics and training dynamics.
Unified framework for Riemannian deep learning across manifold-valued representations.
problem Deep learning on manifold-valued representations often relies on Euclidean approximations or costly geometric operations.
method Develops reusable neural modules, manifold-specific network architectures, and geometric designs.
result Generalizes batch normalization and multinomial logistic regression to broader classes of manifolds.
Paper warns of metric deformation in manifold learning, leading to incorrect answers.
problem Metric deformation in manifold learning.
method Analysis of manifold learning techniques.
result Metric deformation can lead to incorrect answers in manifold learning.
SBMs learn manifold-like structures by mixing samples with a non-conservative field.
problem How SBMs learn data distributions on low-dimensional manifolds.
method Investigating linear approximations and subspaces of local feature vectors during diffusion.
result SBMs mix samples by a non-conservative field within the manifold, maintaining manifold-like structure.
Manifold Learning is a class of algorithms seeking a low-dimensional non-linear representation of high-dimensional data. Thus manifold learning algorithms are, at least in theory, most applicable to high-dimensional data and sample sizes to enable accurate estimation of the manifold. Despite this, most existing manifol…
MLDL preserves manifold geometry in vector transformations.
problem Geometric deterioration in neural network transformations.
method Locally isometric smoothness (LIS) and Markov random field (MRF) encoding.
result Enhanced vector transformations into well-behaved metric homeomorphisms.
This paper tackles learning functions on manifolds using parallel distributed learning.
problem Learning real-valued functions on manifolds from input-output data pairs.
method Filtered hyperinterpolation and parallel distributed learning.
result Optimal approximation order for non-distributed case, and quantitative relations for distributed case.
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.
GAMLA learns manifold structures with auto-encoding for global insights.
problem Limited global insight and lack of interpretable analytical descriptions in manifold learning.
method Two-round auto-encoding process to derive character and complementary representations.
result GAMLA provides global and analytical descriptions of smooth manifolds.
PFM generates novel samples on data manifolds using pullback geometry.
problem Generating novel samples on complex data manifolds.
method Pullback Flow Matching framework leveraging pullback geometry and isometric learning.
result PFM achieves improved manifold learning and generative performance.
We introduce a novel co-learning paradigm for manifolds naturally equipped with a group action, motivated by recent developments on learning a manifold from attached fibre bundle structures. We utilize a representation theoretic mechanism that canonically associates multiple independent vector bundles over a common bas…
Proposes a method to cluster multi-aspect data using manifold learning with NMF.
problem Clustering multi-aspect data with diverse features and views.
method Includes inter-manifold learning in NMF framework to handle different data types.
result The method improves clustering accuracy and efficiency on various datasets.
Contagion maps detect network structure in noisy data.
problem Detecting underlying manifold structure in noisy data.
method Using activation times in threshold contagions to map network nodes to high-dimensional space.
result Contagion maps reliably detect manifold structure in noisy data, while Isomap fails.
New method learns identity-preserving transformations on data manifolds without labels.
problem Learning identity-preserving transformations on natural variations without supervision.
method Introduces a learning strategy that does not require transformation labels and learns local regions for operators.
result Trains on MNIST and Fashion MNIST, and CelebA, learning transformations without labels.
Quantum dynamics algorithm learns manifold from data.
problem Learning manifolds from high-dimensional datasets.
method Simulation of quantum dynamics on a graph embedding of data.
result Algorithm reveals connections between data sampling and quantization.
Paper introduces Manifold Probe for discovering representation manifolds in superposition.
problem Discovering representation manifolds in complex superposition representations.
method Generalizes linear regression probes to learn feature spaces and directions in superposition representations.
result Demonstrates Manifold Probe on Llama 2-7b representations, finding causally involved manifolds in model behaviour.
New neural networks flatten and reconstruct manifolds from samples.
problem Learning from high-dimensional data embedded in submanifolds.
method Flattening Networks (FlatNet) that linearize and reconstruct embedded submanifolds.
result FlatNet achieves balance of interpretability, feasibility, and generalization.
We present a robust multiple manifolds structure learning (RMMSL) scheme to robustly estimate data structures under the multiple low intrinsic dimensional manifolds assumption. In the local learning stage, RMMSL efficiently estimates local tangent space by weighted low-rank matrix factorization. In the global learning …
A new method for functional data classification using supervised manifold learning.
problem Improving classification of functional data.
method Supervised manifold learning approach considering label information.
result The proposed method achieves highly competitive performance compared to existing methods.
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.