A novel GP architecture, Thin and Deep GP, learns lower-dimensional representations without losing interpretability.
problem Challenges in selecting appropriate kernel for Gaussian processes.
method Proposes a novel synthesis of deep and shallow GP approaches, parameterizing lengthscale in a way that maintains interpretability and learns lower-dimensional embeddings.
result TDGP discovers lower-dimensional manifolds in input data, performs well in benchmark datasets, and behaves well with increasing layers.
Many features of dimensional reduction schemes are determined by the breaking of higher dimensional general covariance associated with the selection of a particular subset of coordinates. By investigating residual covariance we introduce lower dimensional tensors --generalizing to one side Kaluza-Klein gauge fields and…
Paper shows graphs can be embedded in lower dimensions than expected.
problem Choosing the right embedding dimension for graph analysis.
method Utilizes hidden manifold structure to predict lower-dimensional embedding.
result Graphs can be embedded in much lower dimensions than previously thought.
A method models nonlinear dynamics from data using barycentric coordinates and memory.
problem Modeling complex dynamical systems from data.
method SPA for data projection, barycentric coordinates, delay-embedding theorem for memory.
result Stable models of chaotic dynamics and attractors are reproduced.
Hyperbolic embeddings have recently gained attention in machine learning due to their ability to represent hierarchical data more accurately and succinctly than their Euclidean analogues. However, multi-relational knowledge graphs often exhibit multiple simultaneous hierarchies, which current hyperbolic models do not c…
A new method combines multiple node embeddings using tensor decomposition.
problem Generating accurate node embeddings for complex networks.
method TenSemble2Vec: combines multiple embeddings via tensor decomposition.
result Improves node embeddings by leveraging complementary information from different methods.
Two methods monitor high-dimensional processes via manifold fitting or learning.
problem Monitoring high-dimensional, dynamic industrial processes.
method Manifold fitting and learning approaches for online SPC.
result Manifold-fitting approach achieves performance competitive with classical methods.
This paper describes an approach for automatic construction of dictionaries for Named Entity Recognition (NER) using large amounts of unlabeled data and a few seed examples. We use Canonical Correlation Analysis (CCA) to obtain lower dimensional embeddings (representations) for candidate phrases and classify these phra…
Bayesian optimization (BO) is a powerful approach for seeking the global optimum of expensive black-box functions and has proven successful for fine tuning hyper-parameters of machine learning models. However, BO is practically limited to optimizing 10--20 parameters. To scale BO to high dimensions, we usually make str…
A new method improves generative models by learning lower-dimensional representations.
problem Normalizing flows cannot learn lower-dimensional representations of data.
method Noisy injective flows (NIF) that map latent space to a learnable manifold in high-dimensional data space using injective transformations and an additive noise model.
result Simple application of NIF to existing flow architectures significantly improves sample quality and yields separable data embeddings.
Bayesian optimization (BO) is a popular approach to optimize expensive-to-evaluate black-box functions. A significant challenge in BO is to scale to high-dimensional parameter spaces while retaining sample efficiency. A solution considered in existing literature is to embed the high-dimensional space in a lower-dimensi…
Ensembling word embeddings to improve distributed word representations has shown good success for natural language processing tasks in recent years. These approaches either carry out straightforward mathematical operations over a set of vectors or use unsupervised learning to find a lower-dimensional representation. Th…
We present a method for training multi-label, massively multi-class image classification models, that is faster and more accurate than supervision via a sigmoid cross-entropy loss (logistic regression). Our method consists in embedding high-dimensional sparse labels onto a lower-dimensional dense sphere of unit-normed …
Manifold embedding algorithms map high-dimensional data down to coordinates in a much lower-dimensional space. One of the aims of dimension reduction is to find intrinsic coordinates that describe the data manifold. The coordinates returned by the embedding algorithm are abstract, and finding their physical or domain-r…
Spectral embedding of adjacency or Laplacian matrices of undirected graphs is a common technique for representing a network in a lower dimensional latent space, with optimal theoretical guarantees. The embedding can be used to estimate the community structure of the network, with strong consistency results in the stoch…
In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume Vol(2,2) for 5-dimensional and 6-dimensional spin manifolds with boundary and we also get the Kastler-Kalau-Walze type theorem in this case.
Improves visualization of high-dimensional data by correcting misleading artifacts in neighbor embedding methods.
problem Misleading visual artifacts in t-SNE and UMAP due to lack of data-independent manifold learning interpretations.
method LOO-map framework that extends embedding maps to the entire input space, identifying and correcting map discontinuities.
result Developed point-wise diagnostic scores to detect unreliable embedding points and improve hyperparameter selection.
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.
New model estimates higher-order interactions in stochastic processes using lower-dimensional projections.
problem Estimating higher-order interaction effects in stochastic processes with limited data.
method Additive Poisson Process (APP) combines information geometry and generalized additive models to model intensity functions in lower dimensions.
result The model can estimate higher-order intensity functions with sparse data.
Proposes generating virtual data points to overcome the curse of dimensionality.
problem Increased intrinsic dimensionality requires large data sets for local sampling.
method Manifold embedding motivated super sampling (MESS) framework.
result Generates virtual data points that faithfully represent the manifold.
Paper proposes DMGD for integrating outlier and community detection in graph embedding.
problem Outlier nodes affect graph embedding of regular nodes, especially in networks with multiple communities.
method DMGD integrates outlier and community detection with node embedding using multiclass graph description.
result DMGD detects outliers relative to their communities and achieves better node embedding compared to state-of-the-arts.
Enhances multi-tag classification using low-dimensional vector representations and virtual data.
problem Improving the performance of multi-tag classifiers.
method Embedding raw data into a low-dimensional feature space, then generating virtual data from linear operations on these vectors, to train multi-tag classifiers.
result Significant improvement in F1 scores (up to 224%) compared to training directly with raw data.
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…
In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential …
This paper tackles deep clustering evaluation challenges in high-dimensional data.
problem Evaluation of deep clustering methods is problematic due to the curse of dimensionality and variations in embedding spaces.
method Develops a theoretical framework to highlight the ineffectiveness of internal validation measures and proposes a systematic approach to applying clustering validity indices in deep learning.
result The proposed framework reduces misguidance from improper use of clustering validity indices in deep learning.
This work reduces the dimensionality of text data using SVD, improving performance and computational efficiency.
problem High-dimensional input spaces in text classification lead to excessive parameter count and computational infeasibility.
method Singular Value Decomposition (SVD) is applied to transform the input space into a lower-dimensional latent space.
result Neural networks trained on the lower-dimensional latent space achieve comparable or better performance with reduced computational complexity.
Network embedding algorithms are able to learn latent feature representations of nodes, transforming networks into lower dimensional vector representations. Typical key applications, which have effectively been addressed using network embeddings, include link prediction, multilabel classification and community detectio…
Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data representation possibly characterised by convenient geometric properties. Euclidean spaces…
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.
Framework uses RL with dynamic embedding to outperform benchmarks in volatile markets.
problem Challenges in high-dimensional, non-stationary, and noisy market information.
method Dynamic embedding of market information using generative autoencoders and online meta-learning in a reinforcement learning framework.
result Framework outperforms common portfolio benchmarks and PTO approach during market stress.
Denote by ΔM the M-dimensional simplex. A map f:ΔM→Rd is an almost r-embedding if fσ1∩…∩fσr=∅ whenever σ1,…,σr are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d≥2r+1, then th…
Dimensionality reduction (DR) on the manifold includes effective methods which project the data from an implicit relational space onto a vectorial space. Regardless of the achievements in this area, these algorithms suffer from the lack of interpretation of the projection dimensions. Therefore, it is often difficult to…
In this paper, we study the problem of author identification under double-blind review setting, which is to identify potential authors given information of an anonymized paper. Different from existing approaches that rely heavily on feature engineering, we propose to use network embedding approach to address the proble…
Motivated by vision tasks such as robust face and object recognition, we consider the following general problem: given a collection of low-dimensional linear subspaces in a high-dimensional ambient (image) space and a query point (image), efficiently determine the nearest subspace to the query in ℓ1 distance. We …
This work improves understanding of dimension reduction algorithms and their probabilistic embeddings.
problem Improving theoretical understanding of non-linear dimension reduction algorithms.
method Analytical investigation of a generalized multidimensional scaling optimization problem.
result Probabilistic formulation of the problem leads to deterministic embeddings, contrary to standard implementations.
Clustering is a fundamental unsupervised learning approach. Many clustering algorithms -- such as k-means -- rely on the euclidean distance as a similarity measure, which is often not the most relevant metric for high dimensional data such as images. Learning a lower-dimensional embedding that can better reflect the …
In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential geometry tools such as moving frame and parallel transportation are successfully…
In this paper, we define lower dimensional volumes associated to sub-Dirac operators for foliations. In some cases, we compute these lower dimensional volumes. We also prove the Kastler-Kalau-Walze type theorems for foliations with or without boundary. As a corollary, we give an explanation of the gravitational action …
Monitoring physiological responses to hemodynamic stress can help in determining appropriate treatment and ensuring good patient outcomes. Physicians' intuition suggests that the human body has a number of physiological response patterns to hemorrhage which escalate as blood loss continues, however the exact etiology a…
Modern techniques simplify complex high-dimensional data.
problem Complex, high-dimensional data.
method Unsupervised dimension reduction techniques.
result Simplified representation of high-dimensional data.
In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume Vol6(1,3) for 6-dimensional spin manifolds with boundary and the gravity on boundary is derived by the noncommutative residue associated with Dirac operators.For 6-dimensional manifo…
Change detection in dynamic networks is an important problem in many areas, such as fraud detection, cyber intrusion detection and health care monitoring. It is a challenging problem because it involves a time sequence of graphs, each of which is usually very large and sparse with heterogeneous vertex degrees, resultin…
Algorithm learns stock correlation matrix embedding using graph machine learning.
problem Understanding complex relationships among stocks based on their correlation matrix.
method Proposes a graph machine learning approach called Node2Vec to compress the correlation network into an embedding.
result The algorithm can learn an embedding from the correlation network of S&P 500 stock data.
Computing optimal transport (OT) between measures in high dimensions is doomed by the curse of dimensionality. A popular approach to avoid this curse is to project input measures on lower-dimensional subspaces (1D lines in the case of sliced Wasserstein distances), solve the OT problem between these reduced measures, a…
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.
Generative Topological Networks (GTNs) simplify latent space generation.
problem Complex latent space generation in generative models.
method GTNs use topology theory for a simple, supervised learning approach.
result GTNs improve upon VAEs and converge quickly, generating realistic samples.
We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, t…
Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain …