Embedding Coordinates Research Articles

Online landmark replacement for out-of-sample dimensionality reduction methods

A strategy to assist visualization and analysis of large and complex datasets is dimensionality reduction, with which one maps each data point into a low-dimensional manifold. However, various dimensionality reduction techniques are computationally infeasible for large data. Out-of-sample techniques aim to resolve this difficulty; they only apply the dimensionality reduction technique on a small portion of data, referred to as landmarks, and determine the embedding coordinates of the other points using landmarks as references. Out-of-sample techniques have been applied to online settings, or when data arrive as time series. However, existing online out-of-sample techniques use either all the previous data points as landmarks or the fixed set of landmarks and therefore are potentially not good at capturing the geometry of the entire dataset when the time series is non-stationary. To address this problem, we propose an online landmark replacement algorithm for out-of-sample techniques using geometric graphs and the minimal dominating set on them. We mathematically analyse some properties of the proposed algorithm, particularly focusing on the case of landmark multi-dimensional scaling as the out-of-sample technique, and test its performance on synthetic and empirical time-series data.

Nambu-Goto equation from three-dimensional gravity

We demonstrate that the solutions of three-dimensional gravity obtained by gluing two copies of a spacetime across a junction constituted of a tensile string are in one-to-one correspondence with the solutions of the Nambu-Goto equation in the same spacetime up to a finite number of rigid deformations related to worldsheet and spacetime isometries. The non-linear Nambu-Goto equation satisfied by the average of the embedding coordinates of the junction emerges directly from the junction conditions along with the rigid deformations and corrections due to the tension. Therefore, the equivalence principle generalizes non-trivially to the string. Our results are valid both in three-dimensional flat and AdS spacetimes. In the context of AdS3/CFT2 correspondence, our setup could be used to describe a class of interfaces in the conformal field theory featuring relative time reparametrization at the interface which encodes the solution of the Nambu-Goto equation corresponding to the bulk junction.

Real time holography for higher spin theories

Real time holography is studied in the context of the embedding space formalism. In the first part of this paper, we present matching conditions for on-shell integer spin fields when going from Euclidean to Lorentzian signature on AdS background. Using the BTZ black hole as an example, we discuss various ways of lifting the physical solution from the AdS surface to the whole embedding space. The BTZ propagator for higher spin field is expressed elegantly in terms of the embedding coordinates. In the second part of the paper, we develop the proposed duality between higher spin theory and vector models. We obtain a specific map between the field configurations of these two theories in real time, so called Lorentzian AdS/CFT map. We conclude by exploring the matching conditions for higher spin fields satisfying the proposed bulk quadratic action. The physical and ghost modes can be treated independently during the Wick rotation; only physical modes are considered to be external modes.

GEMS Embeddings of Hayward Regular Black Holes in Massless and Massive Gravities

After finding a solution for the Hayward regular black hole (HRBH) in massive gravity, we embed the (3+1)-dimensional HRBHs both in massless and in massive gravities into (5+2)- and (6+3)-dimensional Minkowski spacetimes, respectively. Here, massive gravity denotes that a graviton acquires a mass holographically by broken momentum conservation in the HRBH. The original HRBH has no holographically added gravitons, which we call ‘massless’. Making use of newly found embedding coordinates, we obtain desired Unruh temperatures and compare them with the Hawking and local fiducial temperatures, showing that the Unruh effect for a uniformly accelerated observer in a higher-dimensional flat spacetime is equal to the Hawking effect for a fiducial observer in a black hole spacetime. We also obtain freely falling temperatures of the HRBHs in massless and massive gravities seen by freely falling observers, which remain finite even at the event horizons while becoming the Hawking temperatures in asymptotic infinity.

Light-cone gauge in non-relativistic AdS5×S5 string theory

We discuss the non-relativistic limit of string theory in AdS5×S5 for different choices of embedding coordinates. We show that, if we consider Cartesian coordinates, the action of fluctuations around the twisted BMN-like string found in arXiv:2109.13240 in uniform light-cone gauge becomes the one of free fields at large string tension and large AdS5 radius.

Schrödinger from Wheeler–DeWitt: The issues of time and inner product in canonical quantum gravity

The wave-function in quantum gravity is supposed to obey the Wheeler–DeWitt (WDW) equation, however there is neither a satisfactory probability interpretation nor a successful solution to the problem of time in the WDW framework. To gain some insight on these issues we compare quantization of ordinary systems, first in the usual way having the Schrödinger equation and second by promoting them as parametrized theories by introducing embedding coordinate fields, which yields first class constraints and the WDW equation. We observe that the time evolution in the WDW framework can be described with respect to the embedding coordinates, where the WDW equation becomes Schrödinger like, i.e. it involves first order timelike functional derivatives. Moreover, the equivalence with the ordinary quantization procedure determines a suitable Hilbert space with a viable probability interpretation. We then apply the same construction to general relativity by adding embedding fields without any prior coordinate choice. The reparametrized general relativity has two different types of diffeomorphism invariance, which arises from world-volume and target-space reparametrizations. As in the case of ordinary systems, the time evolution can be described with respect to the embedding fields and the WDW equation becomes Schrödinger like; the construction is almost identical to an ordinary parametrized field theory in terms of time evolution and Hilbert space structure. However, this time, the constraint algebra enforces the wave-function to be in a subspace of states annihilated by an operator that can be identified as the Hamiltonian. The implications of these results for the canonical quantization program, and in particular for the minisuperspace quantum cosmology, are discussed.

Spatiotemporal patterns of firearm acquisition in the United States in different presidential terms.

This study develops mathematical tools and approaches to investigate spatiotemporal patterns of firearm acquisition in the U.S. complemented by hypothesis testing and statistical analysis. First, state-level and nation-level instant background check (BC) data are employed as proxy of firearm acquisition corresponding to 1999-2021. The relative-phase time-series of BC in each U.S. state is recovered and utilized to calculate the time-series of the U.S. states' synchronization degree. We reveal that U.S. states present a high-level degree of synchronization except in 2010-2011 and after 2018. Comparing these results with respect to a sitting U.S. president provides additional information: specifically, any two presidential terms are characterized by statistically different synchronization degrees except G. W. Bush's first term and B. H. Obama's second term. Next, to detail variations of BC, short-time Fourier transform, dimensionality reduction techniques, and diffusion maps are implemented within a time-frequency representation. Firearm acquisition in the high frequency band is described by a low-dimensional embedding, in the form of a plane with two embedding coordinates. Data points on the embedding plane identify separate clusters that signify state transitions in the original BC data with respect to different time windows. Through this analysis, we reveal that the frequency content of the BC data has a time-dependent characteristic. By comparing the diffusion map at hand with respect to a presidential term, we find that at least one of the embedding coordinates presents statistically significant variations between any two presidential terms except B. H. Obama's first term and D. J. Trump's pre-COVID term. The results point at a possible interplay between firearm acquisition in the U.S. and a presidential term.

Feature extraction and fault detection scheme via improved locality preserving projection and SVDD

Manifold learning is widely adopted for the fault detection of industrial processes. However, the quality of low-dimensional embedding coordinates can be adversely affected by ill-constructed graph Laplacian. An improved locality preserving projection (ILPP) scheme is proposed. ILPP is built on a geometrically inspired Laplacian, and the Riemannian metric is used to find the suitable bandwidth parameter. The proposed approach combines the advantages of ILPP in preserving manifold data structures and those of support vector data description (SVDD) in handling complex process data distributions. Case studies on helix data, hot strip mill, and Pensim benchmark processes demonstrate the utility and feasibility of the proposed approach. The average fault detection rate for proposed ILPP is 99%, which is higher than locality preserving projection (LPP; 87.8%), local tangent space alignment (LTSA; 74.9%), and principal component analysis (PCA; 90.6%).

Modified LPP based on Riemannian metric for feature extraction and fault detection

Dimensionality reduction methods based on manifold learning are widely adopted for industrial process monitoring. However, in many situations, these methods fail to preserve manifold intrinsic features in low-dimensional space, resulting in reduced process monitoring efficacy. To overcome this problem, a modified locality preserving projection (MLPP) based on the Riemannian metric is put forward. First, the Riemannian metric, which embodies a manifold’s geometric information, is estimated from process data. Then, the low dimensional embedding coordinates obtained from LPP are supplemented with an estimate of the Riemannian metric. Finally, a process monitoring model is developed, and kernel density estimation is utilized to approximate confidence bounds for T2 and SPE statistics. The proposed MLPP method is applied to the feature extraction of Twin-Peaks dataset, fault detection of hot strip mill, steam turbine system and Tennessee Eastman processes. The effectiveness of MLPP method is compared with both the manifold learning and deep learning approaches. In addition, the proposed method is evaluated under various noisy conditions. The average fault detection rate of 98.9%, 99.6% and 84.4% in hot strip mill, steam turbine system and Tennessee Eastman processes, respectively, are higher than the existing methods. Quantitative results indicate the superiority of the proposed MLPP method.

Hessian Locally Linear Embedding of PMU Data for Efficient Fault Detection in Power Systems

This article develops a computationally efficient fault-detection method for power systems, by exploiting phasor measurement unit (PMU) data and Hessian locally linear embedding (HLLE) technique. First, via HLLE technique, high-dimensional PMU data are transformed into low-dimensional embedding coordinates, which effectively captures the fluctuations of PMU data. Next, based on the feature space of embedding coordinates, T-squared statistic is employed for online fault detection. The method is evaluated on IEEE 39-bus system and a real-world system, exhibiting decent fault-detection performance as well as a considerably lower computational complexity when compared with existent methods.

Bladder tumor ILC1s undergo Th17‐like differentiation in human bladder cancer

PurposeHuman innate lymphoid cells (hILCs) are lineage‐negative immune cells that do not express rearranged adaptive antigen receptors. Natural killer (NK) cells are hILCs that contribute to cancer defense. The role of non‐NK hILCs in cancer is unclear. Our study aimed to characterize non‐NK hILCs in bladder cancer.Experimental designMass cytometry was used to characterize intratumoral non‐NK hILCs based on 35 parameters, including receptors, cytokines, and transcription factors from 21 muscle‐invasive bladder tumors. Model‐based clustering was performed on t‐distributed stochastic neighbor embedding (t‐SNE) coordinates of hILCs, and the association of hILCs with tumor stage was analyzed.ResultsMost frequent among intratumoral non‐NK hILCs were hILC1s, which were increased in higher compared with lower stage tumors. Intratumoral hILC1s were marked by Th17‐like phenotype with high RORγt, IL‐17, and IL‐22 compared to Th1 differentiation markers, including Tbet, perforin, and IFN‐γ. Compared with intratumoral hILC2s and hILC3s, hILC1s also had lower expression of activation markers (NKp30, NKp46, and CD69) and increased expression of exhaustion molecules (PD‐1 and Tim3). Unsupervised clustering identified nine clusters of bladder hILCs, which were not defined by the primary hILC subtypes 1–3. hILC1s featured in all the nine clusters indicating that intratumoral hILC1s displayed the highest phenotypic heterogeneity among all hILCs.ConclusionshILC1s are increased in higher stage tumors among patients with muscle‐invasive bladder cancer. These intratumoral hILC1s exhibit an exhausted phenotype and Th17‐like differentiation, identifying them as potential targets for immunotherapy.

A novel AdaBoost ensemble model based on the reconstruction of local tangent space alignment and its application to multiple faults recognition

In order to recognize the coupling faults of complex industrial processes effectively, this paper proposed an AdaBoost ensemble (AdBE) model based on the reconstruction of local tangent space alignment (RLTSA). First, to obtain the low-dimensional manifold structure embedded in original data space, RLTSA algorithm is designed by constructing the tangent space in the neighborhood of each data point to represent the local geometry and then aligning them to obtain the embedding coordinates. Secondly, to solve the loss of global feature information, an affine matrix is used to inversely map the low-dimensional coordinates to restore the global structure information. Thirdly, based on the above reconstruction of local tangent space alignment, an AdaBoost ensemble (AdBE) classifier is constructed for multiple faults recognition in which the AdaBoost algorithm is used to improve the performance of Decision Tree (DT), and One vs. Rest (OvR) ensemble strategy is introduced to establish the RLTSA-AdBE model. Case studies are conducted using a three-dimensional S_curve data set and the Tennessee Eastman process (TEP) to respectively verify the performance of the RLTSA algorithm and the proposed RLTSA-AdBE model. The simulation results indicate that the proposed method guarantees high diagnosis accuracy and macro_F1 Score of coupling faults recognition.

String integrability of defect CFT and dynamical reflection matrices

The D3-D5 probe-brane system is holographically dual to a defect CFT which is known to be integrable. The evidence comes mainly from the study of correlation functions at weak coupling. In the present work we shed light on the emergence of integrability on the string theory side. We do so by constructing the double row transfer matrix which is conserved when the appropriate boundary conditions are imposed. The corresponding reflection matrix turns out to be dynamical and depends both on the spectral parameter and the string embedding coordinates.

Lattice AdS geometry and continuum limit

We construct the lattice AdS geometry. The lattice AdS2 geometry and AdS3 geometry can be extended from the lattice AdS2 induced metric, which provided the lattice Schwarzian theory at the classical limit. Then we use the lattice embedding coordinates to rewrite the lattice AdS2 geometry and AdS3 geometry with the manifest isometry. The lattice AdS2 geometry can be obtained from the lattice AdS3 geometry through the compactification without the lattice artifact. The lattice embedding coordinates can also be used in the higher-dimensional AdS geometry. Because the lattice Schwarzian theory does not suffer from the issue of the continuum limit, the lattice AdS2 geometry can be obtained from the higher-dimensional AdS geometry through the compactification, and the lattice AdS metric does not depend on the angular coordinates, we expect that the continuum limit should exist in the lattice Einstein gravity theory from this geometric lattice AdS geometry. Finally, we apply this lattice construction to construct the holographic tensor network without the issue of a continuum limit.

Automatic identification of relevant genes from low-dimensional embeddings of single-cell RNA-seq data.

MotivationDimensionality reduction is a key step in the analysis of single-cell RNA-sequencing data. It produces a low-dimensional embedding for visualization and as a calculation base for downstream analysis. Nonlinear techniques are most suitable to handle the intrinsic complexity of large, heterogeneous single-cell data. However, with no linear relation between gene and embedding coordinate, there is no way to extract the identity of genes driving any cell’s position in the low-dimensional embedding, making it difficult to characterize the underlying biological processes.ResultsIn this article, we introduce the concepts of local and global gene relevance to compute an equivalent of principal component analysis loadings for non-linear low-dimensional embeddings. Global gene relevance identifies drivers of the overall embedding, while local gene relevance identifies those of a defined sub-region. We apply our method to single-cell RNA-seq datasets from different experimental protocols and to different low-dimensional embedding techniques. This shows our method’s versatility to identify key genes for a variety of biological processes.Availability and implementationTo ensure reproducibility and ease of use, our method is released as part of destiny 3.0, a popular R package for building diffusion maps from single-cell transcriptomic data. It is readily available through Bioconductor.Supplementary information Supplementary data are available at Bioinformatics online.

GEMS embeddings and freely falling temperatures of Schwarzschild(-AdS) black holes in massive gravity

We globally embed (3+1)-dimensional Schwarzschild and Schwarzschild-AdS black holes in massive gravity into (5+2)-dimensional flat spacetimes. Making use of embedding coordinates, we directly obtain the generalized Hawking, Unruh and freely falling temperatures in Schwarzschild and Schwarzschild-AdS black holes due to massive graviton effects.

Dynamic programming optimization in line of sight networks

Line of Sight (LoS) networks were designed to model wireless communication in settings which may contain obstacles. For fixed positive integer d, and positive integer ω, a graph G=(V,E) is a (d-dimensional) LoS network with range parameter ω if it can be embedded in a finite cube of the d-dimensional integer grid so that each pair of vertices in V are adjacent if and only if their embedding coordinates differ only in one position and such difference is less than ω.In this paper we investigate a dynamic programming (DP) approach which can be used to obtain efficient algorithmic solutions for various combinatorial problems in LoS networks. In particular DP solves the Maximum Independent Set (MIS) problem in LoS networks optimally, for any ω, on narrow LoS networks (i.e. networks which can be embedded in a k×k…×k×n region, for some fixed k independent of n). In the unrestricted case it has been shown that the (decision version of the) problem is NP-hard when 2<ω=O(n1−δ), for fixed 0<δ<1. We describe how DP can be used as a building block in the design of good approximation algorithms in this case. In particular we present a semi-online polynomial-time approximation scheme for the MIS problem in narrow d-dimensional LoS networks, as well as a polynomial-time 2-approximation algorithm and a fast polynomial time approximation scheme for the MIS problem in arbitrary d-dimensional LoS networks. Finally we comment on how the approach can be adapted to prove similar results for a number of important optimization problems in LoS networks.

Correlators on non-supersymmetric Wilson line in mathcal{N}=4 SYM and AdS2/CFT1

Correlators of local operators inserted on a straight or circular Wilson loop in a conformal gauge theory have the structure of a one-dimensional “defect” CFT. As was shown in arXiv:1706.00756, in the case of supersymmetric Wilson-Maldacena loop in mathcal{N}=4 SYM one can compute the leading strong-coupling contributions to 4-point correlators of the simplest “protected” operators by starting with the AdS5 × S5 string action expanded near the AdS2 minimal surface and evaluating the corresponding tree-level AdS2 Witten diagrams. Here we perform the analogous computations in the non-supersymmetric case of the standard Wilson loop with no coupling to the scalars. The corresponding non-supersymmetric “defect” CFT1 should have an unbroken SO(6) global symmetry. The elementary bosonic operators (6 SYM scalars and 3 components of the SYM field strength) are dual respectively to the S5 embedding coordinates and AdS5 coordinates transverse to the minimal surface ending on the line at the boundary. The SO(6) symmetry is preserved on the string side provided the 5-sphere coordinates satisfy Neumann boundary conditions (as opposed to Dirichlet in the supersymmetric case); this implies that one should integrate over the S5 expansion point. The massless S5 fluctuations then have logarithmic propagator, corresponding to the boundary scalar operator having dimension Delta =frac{5}{sqrt{lambda }}+dots at strong coupling. The resulting functions of 1d cross-ratio appearing in the 4-point functions turn out to have a more complicated structure than in the supersymmetric case, involving polylog (Li3 and Li2) functions. We also discuss consistency with the operator product expansion which allows extracting the leading strong coupling corrections to the anomalous dimensions of the operators appearing in the intermediate channels.

Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings

A major challenge in the study of dynamical systems is that of model discovery: turning data into models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. This problem is made more difficult by the fact that many systems of interest exhibit diverse behaviors across multiple time scales. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales. Specifically, we can discover distinct governing equations at slow and fast scales. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates, can be used to obtain a linear model and Koopman invariant measurement system that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. We introduce two strategies for using HAVOK on systems with multiple time scales. Together, our approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.

Discreteness of Fuzzy de Sitter Space

We discuss properties of fuzzy de Sitter space defined within the algebra of de Sitter group $$SO(1,4).$$ We find that the embedding coordinates have discrete spectra in the $$\left( {\rho ,s = \tfrac{1}{2}} \right)$$ unitary irreducible representation of the principal continuous series.