Survey of Locally Linear Embedding and its variants.
problem Representing high-dimensional data in a lower-dimensional space while preserving local structure.
method Explains various LLE and variant methods, including kernel LLE, inverse LLE, feature fusion, out-of-sample embedding, incremental LLE, landmark LLE, supervised LLE, robust LLE, fusion with other methods, and weighted LLE.
result Comprehensive overview of LLE and its variants.
Generative LLE modifies LLE to generate stochastic embeddings.
problem Nonlinear dimensionality reduction and manifold learning.
method Generative LLE modifies LLE by using stochastic linear reconstruction.
result Generative LLE can generate various LLE embeddings stochastically.
Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
A new method simplifies HLLE for better robustness.
problem Improving robustness of Hessian locally linear embedding.
method Replacing Hessian with arbitrary weights and modifying manifold dimension.
result Achieved a new LLE-type method called tangential LLE.
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.
LLE produces unwanted results without regularization, which can be prevented with regularization.
problem LLE's inherent unwanted results without regularization.
method Mathematical proof and numerical examples of regularization effectiveness.
result Regularization prevents unwanted results in LLE.
Proves local isometric embedding of low-differentiability metrics in 3D space.
problem Isometric embedding of metrics of low differentiability in Euclidean 3-space.
method Simplified notation, geodesic and level parameters, solutions of initial value problems for first order non-linear PDEs, classical linear algebraic systems.
result Local isometric embedding exists for metrics of C1 differentiability.
Local model for Poisson manifolds around submanifolds.
problem Linearization of Poisson manifolds around submanifolds.
method Constructing a first order local model and giving conditions for it to be a normal form.
result Includes known linearization theorems for fixed points and symplectic leaves.
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…
Local Linear embedding (LLE) is a popular dimension reduction method. In this paper, we first show LLE with nonnegative constraint is equivalent to the widely used Laplacian embedding. We further propose to iterate the two steps in LLE repeatedly to improve the results. Thirdly, we relax the kNN constraint of LLE and p…
Background: Functional magnetic resonance imaging (fMRI) provides non-invasive measures of neuronal activity using an endogenous Blood Oxygenation-Level Dependent (BOLD) contrast. This article introduces a nonlinear dimensionality reduction (Locally Linear Embedding) to extract informative measures of the underlying ne…
Proposes a boundary detection method inspired by LLE for high-dimensional data.
problem Identifying boundary points from data on an embedded manifold.
method Inspired by locally linear embedding, uses nearest neighbor search schemes and spectral properties of local covariance matrix.
result Enhanced boundary detection in noisy data.
LNPE enhances local connections in embeddings using extended neighbor propagation.
problem Improving local connections and interactions in nonlinear dimensionality reduction.
method Inspired by GCN, LNPE extends 1-hop neighbors to n-hop neighbors in LLE.
result LNPE produces more faithful and robust embeddings with better topological and geometrical properties.
LOCA learns standardized data coordinates from measurements.
problem Learning invariant data coordinates from non-linearly deformed manifolds.
method LOCA, a LOcal Conformal Autoencoder, learns an isometric embedding.
result LOCA preserves geometric information while learning invariant coordinates.
We reduce the question of local nonsolvability of the Darboux equation, and hence of the isometric embedding problem for surfaces, to the local nonsolvability of a simple linear equation whose type is explicitly determined by the Gaussian curvature.
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…
Locally-contextual CRFs improve sequence labeling performance.
problem Improving sequence labeling with contextual embeddings.
method Locally-contextual nonlinear CRFs using deep neural networks.
result Consistently outperforms linear chain CRF and previous state of the art.
In this paper, we propose and study random maxout features, which are constructed by first projecting the input data onto sets of randomly generated vectors with Gaussian elements, and then outputing the maximum projection value for each set. We show that the resulting random feature map, when used in conjunction with …
SA-REMBO adapts to nonstationary high-dimensional optimization.
problem Bayesian Optimization in high-dimensional spaces is limited by the curse of dimensionality and rigidity of global assumptions.
method SA-REMBO uses multiple random Gaussian embeddings and an index variable to adaptively select the best embedding for the optimization problem.
result SA-REMBO outperforms traditional REMBO and other low-rank BO methods across synthetic and real-world benchmarks.
A new method for creating simpler models from complex ones.
problem Creating accurate approximations of complex models at reduced costs.
method Sequential adaptive surrogate modeling based on locally spectral expansions.
result Stochastic spectral embedding (SSE) shows good approximation capabilities and scalability.
A new approach predicts next observations without explicit decoding for better control.
problem High-dimensional observations and unknown dynamics in real-world control tasks.
method Proposes a novel information-theoretic LCE approach using predictive coding to develop a decoder-free model.
result The model reliably learns a controllable latent space leading to superior performance.
Survey of Laplacian-based methods for data dimensionality reduction and embedding.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving important features and structures.
method Laplacian-based methods including spectral clustering, Laplacian eigenmap, locality preserving projection, graph embedding, and diffusion map.
result Comprehensive overview of various optimization variants and applications of Laplacian-based techniques.
The paper proves local isometric embeddings for singular metrics near a point.
problem Existence of local isometric embeddings for singular Riemannian metrics.
method Ramified local isometric embeddings using Leray's ramified Cauchy-Kovalevskaya Theorem.
result Existence of local analytic isometric embeddings into Euclidean space.
First explicit isometric immersion of a flat Klein bottle in 3D space.
problem Finding an isometric embedding of a Klein bottle in 3D.
method Piecewise-linear map from a Klein bottle to Euclidean 3-space.
result Explicit numerical data for a flat Klein bottle isometrically immersed in 3D.
A new method estimates rare failure events in complex systems.
problem Estimating the probability of rare failure events in non-linear systems.
method Stochastic Spectral Embedding (SSE) combined with modifications for efficient rare event estimation.
result Rare failure probability decomposed into conditional probabilities for easier computation.
The ability to characterize the color content of natural imagery is an important application of image processing. The pixel by pixel coloring of images may be viewed naturally as points in color space, and the inherent structure and distribution of these points affords a quantization, through clustering, of the color i…
New subgroups found in CAT(0) groups with exotic finiteness properties.
problem Constructing subgroups with specific finiteness properties.
method Building subgroups L of CAT(0) groups G with tailored finiteness types. result Found subgroups L of finiteness type Fn−1 but not Fn. Study reveals how manifold geometry impacts linear regression solutions.
problem Impact of manifold geometry on linear regression solutions.
method Linear regression applied to manifold-structured data, focusing on extrinsic geometry.
result Linear regression does not have a unique solution on flat manifolds.
In this paper, we propose a semi-supervised dictionary learning method that uses both the information in labelled and unlabelled data and jointly trains a linear classifier embedded on the sparse codes. The manifold structure of the data in the sparse code space is preserved using the same approach as the Locally Linea…
The local linear embedding algorithm (LLE) is a non-linear dimension-reducing technique, widely used due to its computational simplicity and intuitive approach. LLE first linearly reconstructs each input point from its nearest neighbors and then preserves these neighborhood relations in the low-dimensional embedding. W…
The paper studies which branched covers can be lifted to braided embeddings.
problem Which branched covers lift to braided embeddings?
method Analyzes conditions for liftability using Hansen's criterion and examples in different dimensions.
result Not all branched covers lift to braided embeddings; examples are provided in various dimensions.
The objective in extreme multi-label learning is to train a classifier that can automatically tag a novel data point with the most relevant subset of labels from an extremely large label set. Embedding based approaches make training and prediction tractable by assuming that the training label matrix is low-rank and hen…
Adding attributes for nodes to network embedding helps to improve the ability of the learned joint representation to depict features from topology and attributes simultaneously. Recent research on the joint embedding has exhibited a promising performance on a variety of tasks by jointly embedding the two spaces. Howeve…
Classifies linear embeddings of grassmannians and ind-grassmannians.
problem Understanding linear embeddings of grassmannians and ind-grassmannians.
method Classification through isomorphism of Picard groups and direct limits.
result Most linear embeddings of grassmannians are equivariant.
Two log-linear approximations speed up optimal transport for deep learning applications.
problem Computing optimal transport in high dimensions is computationally expensive.
method Locality-sensitive hashing (LSH) and Nyström approximation with LSH-based sparse corrections.
result Log-linear time algorithms for entropy-regularized OT perform well in high-dimensional spaces.
Word embedding models offer continuous vector representations that can capture rich contextual semantics based on their word co-occurrence patterns. While these word vectors can provide very effective features used in many NLP tasks such as clustering similar words and inferring learning relationships, many challenges …
We present function preserving projections (FPP), a scalable linear projection technique for discovering interpretable relationships in high-dimensional data. Conventional dimension reduction methods aim to maximally preserve the global and/or local geometric structure of a dataset. However, in practice one is often mo…
Flexible Kernels for Protein Property Prediction
problem Predicting protein properties from sparse experimental data
method Sequence kernels using evolutionary substitution matrices and local linearity
result Data-efficient models of protein property landscapes
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.
Sketch-GNN reduces GNN training time and memory usage to sublinear scales.
problem Training GNNs on large graphs is computationally expensive and memory-intensive.
method Develops a sketch-based algorithm that trains GNNs on compact sketches of graph adjacency and node embeddings.
result Training time and memory usage grow sublinearly with respect to graph size.
Complex textual information extraction tasks are often posed as sequence labeling or \emph{shallow parsing}, where fields are extracted using local labels made consistent through probabilistic inference in a graphical model with constrained transitions. Recently, it has become common to locally parametrize these models…
In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic m…
Adaptive framework improves nonparametric dimensionality reduction.
problem Optimal hyper-parameter tuning for nonparametric dimensionality reduction.
method Adaptive framework using intrinsic dimension estimator and optimal local neighbourhood sizes.
result Significant improvements in various learning tasks through better low-dimensional visualizations.
We present ARU, an Adaptive Recurrent Unit for streaming adaptation of deep globally trained time-series forecasting models. The ARU combines the advantages of learning complex data transformations across multiple time series from deep global models, with per-series localization offered by closed-form linear models. Un…
We demonstrate new applications of the trace embedding lemma to the study of piecewise-linear surfaces and the detection of exotic phenomena in dimension four. We provide infinitely many pairs of homeomorphic 4-manifolds W and W′ homotopy equivalent to S2 which have smooth structures distinguished by several for…
Researchers solve a complex equation to embed graphs with negative curvature.
problem Embedding graphs in Rn+1 with negative Gauss curvature. method Solving a fully nonlinear Monge-Ampère equation using energy estimates and Nash-Moser iteration.
result Local solvability of the fully nonlinear equation for negative curvature.
Notes a flaw in a proof about embedding graphs.
problem A flaw in proving Sachs' conjecture about graph embeddings.
method Analyzing Stanfield's proof for gaps.
result Identifies a significant error in the proof.
Topological manifolds can be embedded flatly in high-dimensional Euclidean space and are locally retracts.
problem Embedding and retraction of topological manifolds in Euclidean spaces.
method Locally flat embedding and retraction of manifolds in high-dimensional Euclidean space.
result Every topological n-manifold can be embedded locally flatly in R2n+1 and is a retract of some neighborhood in R2n+1.