Framework uses low-dimensional maps to solve high-dimensional Bayesian inference problems.
problem High-dimensional Bayesian inference problems.
method Structure-exploiting lazy maps and flows, focusing on low-dimensional subspace.
result Weak convergence of generated distributions to the posterior.
Study of low dimensional representations of mapping class groups of surfaces, focusing on genus ≥ 7.
problem Classifying (2g+1)-dimensional complex linear representations of mapping class groups. method Using twisted 1-cohomology groups and Morita's computation, a complete classification is given for g≥7. result No irreducible linear representations of dimension 2g+1 for g≥7. Free maps exist on low-dimensional tori and closed surfaces.
problem Embedding closed surfaces in high-dimensional spaces.
method Factorization trick for constructing free immersions.
result Every closed surface embeds freely in \(\mathbb{R}^5\).
Global fixed points in low-dimensional surface group space correspond to trivial representations.
problem Understanding global fixed points in surface group deformation spaces.
method Direct analysis of the deformation space, focusing on the trivial representation.
result Global fixed points in low-dimensional surface group deformation spaces correspond to the trivial representation of the pure mapping class group.
Optimizes optimal transport distances using low-dimensional embeddings.
problem High computational cost of optimal transport distances in high dimensions.
method Approximate OT distances using 1-Lipschitz maps in a lower-dimensional space.
result Efficiently approximates optimal transport distances with lower computational cost.
Efficiently learns sparse low-dimensional Markov chain representations.
problem Learning low-dimensional representations for large-scale Markov chains with sparse structures.
method Formulates as constrained nonnegative matrix factorization and uses gradient descent.
result Proves the effectiveness of the proposed method through convergence analysis.
New algorithm constrains SOMs to create supervised low-dimensional mappings.
problem Creating supervised mappings in neural networks with known internal topology.
method Developed Supervised Topological Maps (STMs) by modifying SOMs to incorporate target distances.
result STMs allow for supervised generation of new data with known internal structure.
We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a tar…
In this survey paper, we give a complete list of known results on the first and the second homology groups of surface mapping class groups. Some known results on higher (co)homology are also mentioned.
We consider the ability of deep neural networks to represent data that lies near a low-dimensional manifold in a high-dimensional space. We show that deep networks can efficiently extract the intrinsic, low-dimensional coordinates of such data. We first show that the first two layers of a deep network can exactly embed…
Manifold learning has been successfully applied to a variety of medical imaging problems. Its use in real-time applications requires fast projection onto the low-dimensional space. To this end, out-of-sample extensions are applied by constructing an interpolation function that maps from the input space to the low-dimen…
Develops interpretable low-dimensional kernels with conic discriminant functions.
problem Improving interpretability in kernel-based classification models.
method Gradually constructs simple feature maps leading to interpretable low-dimensional kernels.
result Obtains high accuracy results without extensive hyperparameter tuning.
Studying SGD on deep neural networks using diffusion maps.
problem Understanding why SGD performs well in deep learning.
method Data-driven approach using diffusion maps to analyze SGD dynamics.
result SGD dynamics may mainly live on a low-dimensional manifold in high-dimensional parameter space.
A new BO method tackles high-dimensional optimization without reconstruction.
problem Optimizing high-dimensional black-box functions is challenging, especially when low-dimensional structures are assumed.
method Tackles the problem in the original high-dimensional space using learned low-dimensional structure.
result Our method explores the high-dimensional space more effectively than existing approaches.
In this paper we provide the small-time heat kernel asymptotics at the cut locus in three relevant cases: generic low-dimensional Riemannian manifolds, generic 3D contact sub-Riemannian manifolds (close to the starting point) and generic 4D quasi-contact sub-Riemannian manifolds (close to a generic starting point). As …
Unique map connects curve pairs to abelian varieties.
problem Uniqueness of the Prym map between curve pairs and abelian varieties.
method Holomorphic map construction, classification of homomorphisms, geometric group theory.
result Prym map is the unique nonconstant holomorphic map for g≥4 and h≤g−1. LASE improves local network structure visualization by targeting locally low-dimensional regions.
problem Global spectral embedding fails to capture local geometric features in sparse, transitive networks.
method Local Adjacency Spectral Embedding (LASE) using weighted spectral decomposition.
result LASE reveals locally low-dimensional structure, improving local reconstruction and visualization.
Word embedding maps words into a low-dimensional continuous embedding space by exploiting the local word collocation patterns in a small context window. On the other hand, topic modeling maps documents onto a low-dimensional topic space, by utilizing the global word collocation patterns in the same document. These two …
We define an equivalence relation called A-isotopy between finitely determined map-germs, which is a strengthened version of A-equivalence. We consider the number of A-isotopy classes of equidimensional Morin singularities, and some other well-known low-dimensional singularities. We also give an application to stable p…
We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of …
Deep networks can adapt to intrinsic dimensionality beyond domain constraints.
problem Approximating functions on low-dimensional manifolds with high-dimensional data.
method Two-layer compositions with ReLU activation, using dimensionality reducing feature maps.
result Near optimal approximation rates depend on the complexity of the dimensionality reducing map, not the ambient dimension.
The paper proposes a method to learn low-dimensional state embeddings from time series data.
problem Finding compact state embeddings from high-dimensional Markov state trajectories.
method The paper introduces a method based on diffusion maps to learn a low-dimensional state embedding and captures the dynamics of the process.
result The method reveals metastable structures in state clustering, providing sharp statistical error bounds and misclassification rates.
Entropy-Isomap improves low-dimensional visualization of dynamic processes.
problem Mapping high-dimensional, temporally correlated data to a low-dimensional manifold.
method Entropy-Isomap, a novel method addressing temporal correlations in data.
result Correctly captures process control variables and material morphology evolution.
Cryo-em images are found to be low-dimensional.
problem Understanding the geometric structure of cryo-em data.
method Applied manifold learning techniques to CryoSBI representations.
result Cryo-em data inherently populate low-dimensional manifolds.
UAPCA projects uncertain data to low dimensions using GMMs.
problem Uncertain multidimensional data not well described by normal distributions.
method Model data with Gaussian mixture models, derive UAPCA projection from general formulation.
result Low-dimensional projections better represent multidimensional distributions.
We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…
GDMaps reduces high-dimensional data to lower dimensions for better classification.
problem High-dimensional data classification and representation.
method Grassmannian Diffusion Maps technique for nonlinear dimensionality reduction.
result GDMaps effectively identifies intrinsic subspace structures in high-dimensional data.
Modeling dynamical systems is important in many disciplines, e.g., control, robotics, or neurotechnology. Commonly the state of these systems is not directly observed, but only available through noisy and potentially high-dimensional observations. In these cases, system identification, i.e., finding the measurement map…
We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension …
Model learns cancer tissue images onto a low-dimensional space revealing tissue characteristics.
problem Improving cancer diagnosis through high-fidelity digital pathology.
method Deep generative model using PathologyGAN to map real images onto a latent space.
result Latent space encodes morphological characteristics and reveals distinct tissue clusters.
PCENet reduces uncertainty in high-dimensional data efficiently.
problem Uncertainty quantification in high-dimensional data is computationally expensive.
method Two-stage learning process: variational autoencoder for low-dimensional representation, polynomial chaos expansion for mapping.
result Model captures system dynamics, learns under uncertainty, estimates high-dimensional data uncertainty, matches output distribution moments.
BézierGAN generates smooth curves from low-dimensional parameters.
problem Designing smooth curves for aerodynamic and hydrodynamic shapes.
method Generative model that maps low-dimensional latent representation to Bézier curve points.
result Generates diverse and realistic curves with consistent shape variation.
A new method for manifold learning robust to irregular sampling.
problem Nonlinear dimensionality reduction of high-dimensional datasets with poor sampling and arbitrary topology.
method Parallel transport unfolding (PTU) for quasi-isometric low-dimensional mapping.
result Improved robustness to irregularity and voids in sampling compared to Isomap.
We define, on smooth manifolds, the notions of almost twistorial structure and twistorial map, thus providing a unified framework for all known examples of twistor spaces. The condition of being harmonic morphisms naturally appears among the geometric properties of submersive twistorial maps between low-dimensional Wey…
We characterize those closed 2k-manifolds admitting smooth maps into (k+1)-manifolds with only finitely many critical points, for k∈{2,4}. We compute then the minimal number of critical points of such smooth maps for k=2 and, under some fundamental group restrictions, also for k=4. The main ingredients ar…
Solve-training trains neural nets to map physical solutions efficiently.
problem Representing complex physical solutions with neural networks.
method Variational training using loss functions from physical models.
result Effective neural network representation of solution maps without expensive labels.
ROAD-EnKFs use learned low-dimensional models to improve state reconstruction and forecasting.
problem Reconstructing and forecasting states of unknown or expensive systems.
method Learned low-dimensional surrogate models and ensemble Kalman filter integration.
result ROAD-EnKFs achieve higher accuracy at lower computational cost than existing methods.
Recently, John Franks and Michael Handel proved that, for g≥3 and n≤2g−4, every homomorphism from the mapping class group of an orientable surface of genus g to $\GL (n,\C)$ is trivial. We extend this result to n≤2g−1, also covering the case g=2. As an application, we prove the corresponding resul…
Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line of work and present low distortion embeddings for dot product kernels into linear…
DPA preserves data distribution in reduced dimensions.
problem Loss of data distribution in dimension reduction.
method DPA combines encoder and decoder to match data distribution.
result DPA successfully reconstructs data distribution.
New method optimizes kernel feature maps for better classification.
problem High computational and memory complexity of standard kernel methods.
method Discriminant Information criterion for optimizing kernel feature maps.
result Improved optimization and generalization performances over state-of-the-art methods.
Adaptive ML learns complex time-varying systems without new data.
problem Applying ML to time-varying systems with shifting distributions.
method Mapping high-dimensional inputs to low-dimensional latent space, actively tuning latent space based on feedback.
result Learning correlations and tracking system evolution in real-time without new data.
Kernel-spectral embedding learns low-dim. structures from noisy data.
problem Learning low-dimensional nonlinear structures from high-dimensional noisy data.
method Adaptive bandwidth spectral embedding using integral operators.
result Convergence to noiseless embeddings and eigenfunctions of integral operators.
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.
The paper computes the mapping class group of certain 6-manifolds.
problem Computing the mapping class group of specific 6-manifolds.
method Algebraic properties and homology groups of the mapping class group were determined.
result The mapping class group is residually finite and virtually torsion-free.
SILBO optimizes high-dimensional Bayesian optimization using semi-supervised embedding learning.
problem Bayesian optimization struggles with high-dimensional search spaces.
method SILBO uses semi-supervised dimension reduction to find a low-dimensional space for iterative optimization.
result SILBO outperforms existing methods on high-dimensional Bayesian optimization tasks.
Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map ever…
We introduce the problem of reconstructing a sequence of multidimensional real vectors where some of the data are missing. This problem contains regression and mapping inversion as particular cases where the pattern of missing data is independent of the sequence index. The problem is hard because it involves possibly m…