Semi-supervised two-dimensional manifold learning based on pair-wise constraints

2014 ◽  
Author(s):  
Wei Xue ◽  
Zheng-qun Wang ◽  
Feng Li ◽  
Zhong-xia Zhou
Keyword(s):  
Manifold Learning ◽  
Dimensional Manifold ◽  
Two Dimensional

scholarly journals Random Assignment Problems on 2d Manifolds

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
D. Benedetto ◽  
E. Caglioti ◽  
S. Caracciolo ◽  
M. D’Achille ◽  
G. Sicuro ◽  
...  
Keyword(s):  
Assignment Problem ◽  
Unit Area ◽  
Constant Term ◽  
Beltrami Operator ◽  
Explicit Calculation ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Random Assignment ◽  
Assignment Problems ◽  
Theoretical Formulation

AbstractWe consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ Ω of unit area. It is known that the average cost scales as $$E_{\Omega }(N)\sim {1}/{2\pi }\ln N$$ E Ω ( N ) ∼ 1 / 2 π ln N with a correction that is at most of order $$\sqrt{\ln N\ln \ln N}$$ ln N ln ln N . In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $$\Omega $$ Ω -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on $$\Omega $$ Ω . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

scholarly journals Classification of Infrared Objects in Manifold Space Using Kullback-Leibler Divergence of Gaussian Distributions of Image Points

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 434 ◽  
Author(s):  
Huilin Ge ◽  
Zhiyu Zhu ◽  
Kang Lou ◽  
Wei Wei ◽  
Runbang Liu ◽  
...  
Keyword(s):  
Manifold Learning ◽  
Gaussian Distribution ◽  
Classification Accuracy ◽  
Infrared Image ◽  
Data Sets ◽  
Dimensional Manifold ◽  
Infrared Images ◽  
Leibler Divergence ◽  
Data Points ◽  
Low Dimensional

Infrared image recognition technology can work day and night and has a long detection distance. However, the infrared objects have less prior information and external factors in the real-world environment easily interfere with them. Therefore, infrared object classification is a very challenging research area. Manifold learning can be used to improve the classification accuracy of infrared images in the manifold space. In this article, we propose a novel manifold learning algorithm for infrared object detection and classification. First, a manifold space is constructed with each pixel of the infrared object image as a dimension. Infrared images are represented as data points in this constructed manifold space. Next, we simulate the probability distribution information of infrared data points with the Gaussian distribution in the manifold space. Then, based on the Gaussian distribution information in the manifold space, the distribution characteristics of the data points of the infrared image in the low-dimensional space are derived. The proposed algorithm uses the Kullback-Leibler (KL) divergence to minimize the loss function between two symmetrical distributions, and finally completes the classification in the low-dimensional manifold space. The efficiency of the algorithm is validated on two public infrared image data sets. The experiments show that the proposed method has a 97.46% classification accuracy and competitive speed in regards to the analyzed data sets.

Study of Fault Diagnosis Based on Manifold Learning

2013 ◽  
Vol 312 ◽  
pp. 650-654 ◽  
Author(s):  
Yi Lin He ◽  
Guang Bin Wang ◽  
Fu Ze Xu
Keyword(s):  
Fault Diagnosis ◽  
Manifold Learning ◽  
Linear Manifold ◽  
Rotating Machinery ◽  
Dimensional Manifold ◽  
Advantages And Disadvantages ◽  
Non Linear ◽  
Future Direction ◽  
Low Dimensional ◽  
Machinery Fault Diagnosis

Characteristic signals in rotating machinery fault diagnosis with the issues of complex and difficult to deal with, while the use of non-linear manifold learning method can effectively extract low-dimensional manifold characteristics embedded in the high-dimensional non-linear data. It greatly maintains the overall geometric structure of the signals and improves the efficiency and reliability of the rotating machinery fault diagnosis. According to the development prospects of manifold learning, this paper describes four classical manifold learning methods and each advantages and disadvantages. It reviews the research status and application of fault diagnosis based on manifold learning, as well as future direction of researches in the field of manifold learning fault diagnosis.

scholarly journals Manifold Learning of Dynamic Functional Connectivity Reliably Identifies Functionally Consistent Coupling Patterns in Human Brains

2019 ◽  
Vol 9 (11) ◽  
pp. 309
Author(s):  
Yuyuan Yang ◽  
Lubin Wang ◽  
Yu Lei ◽  
Yuyang Zhu ◽  
Hui Shen
Keyword(s):  
Functional Connectivity ◽  
Sleep Deprivation ◽  
Manifold Learning ◽  
Temporal Dynamics ◽  
Functional Coupling ◽  
Dimensional Manifold ◽  
Dynamic Functional Connectivity ◽  
Local Linear Embedding ◽  
Human Connectome Project ◽  
Low Dimensional

Most previous work on dynamic functional connectivity (dFC) has focused on analyzing temporal traits of functional connectivity (similar coupling patterns at different timepoints), dividing them into functional connectivity states and detecting their between-group differences. However, the coherent functional connectivity of brain activity among the temporal dynamics of functional connectivity remains unknown. In the study, we applied manifold learning of local linear embedding to explore the consistent coupling patterns (CCPs) that reflect functionally homogeneous regions underlying dFC throughout the entire scanning period. By embedding the whole-brain functional connectivity in a low-dimensional manifold space based on the Human Connectome Project (HCP) resting-state data, we identified ten stable patterns of functional coupling across regions that underpin the temporal evolution of dFC. Moreover, some of these CCPs exhibited significant neurophysiological meaning. Furthermore, we apply this method to HCP rsfMR and tfMRI data as well as sleep-deprivation data and found that the topological organization of these low-dimensional structures has high potential for predicting sleep-deprivation states (classification accuracy of 92.3%) and task types (100% identification for all seven tasks).In summary, this work provides a methodology for distilling coherent low-dimensional functional connectivity structures in complex brain dynamics that play an important role in performing tasks or characterizing specific states of the brain.

scholarly journals Two-dimensional Manifold with Point-like Defects

2015 ◽  
Vol 74 ◽  
pp. 32-35 ◽  
Author(s):  
V.A. Gani ◽  
A.E. Dmitriev ◽  
S.G. Rubin
Keyword(s):  
Dimensional Manifold ◽  
Two Dimensional

Energy transfer in rotating turbulence

1997 ◽  
Vol 337 ◽  
pp. 303-332 ◽  
Author(s):  
CLAUDE CAMBON ◽  
N. N. MANSOUR ◽  
F. S. GODEFERD
Keyword(s):  
Energy Transfer ◽  
Rotation Axis ◽  
Physical Space ◽  
Nonlinear Effects ◽  
Spectral Energy ◽  
Rossby Number ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Plane Normal

The influence of rotation on the spectral energy transfer of homogeneous turbulence is investigated in this paper. Given the fact that linear dynamics, e.g. the inertial waves regime found in an RDT (rapid distortion theory) analysis, cannot affect a homogeneous isotropic turbulent flow, the study of nonlinear dynamics is of prime importance in the case of rotating flows. Previous theoretical (including both weakly nonlinear and EDQNM theories), experimental and DNS (direct numerical simulation) results are collected here and compared in order to give a self-consistent picture of the nonlinear effects of rotation on turbulence.The inhibition of the energy cascade, which is linked to a reduction of the dissipation rate, is shown to be related to a damping of the energy transfer due to rotation. A model for this effect is quantified by a model equation for the derivative-skewness factor, which only involves a micro-Rossby number Roω=ω′/(2Ω) – ratio of r.m.s. vorticity and background vorticity – as the relevant rotation parameter, in accordance with DNS and EDQNM results.In addition, anisotropy is shown also to develop through nonlinear interactions modified by rotation, in an intermediate range of Rossby numbers (RoL<1 and Roω>1), which is characterized by a macro-Rossby number RoL based on an integral lengthscale L and the micro-Rossby number previously defined. This anisotropy is mainly an angular drain of spectral energy which tends to concentrate energy in the wave-plane normal to the rotation axis, which is exactly both the slow and the two-dimensional manifold. In addition, a polarization of the energy distribution in this slow two-dimensional manifold enhances horizontal (normal to the rotation axis) velocity components, and underlies the anisotropic structure of the integral length-scales. Finally a generalized EDQNM (eddy damped quasi-normal Markovian) model is used to predict the underlying spectral transfer structure and all the subsequent developments of classic anisotropy indicators in physical space. The results from the model are compared to recent LES results and are shown to agree well. While the EDQNM2 model was developed to simulate ‘strong’ turbulence, it is shown that it has a strong formal analogy with recent weakly nonlinear approaches to wave turbulence.

scholarly journals Unsupervised Metric Learning Using Low Dimensional Embedding

2018 ◽  
Author(s):  
Parag Jain
Keyword(s):  
Manifold Learning ◽  
Euclidean Distance ◽  
Metric Learning ◽  
Special Property ◽  
Dimensional Manifold ◽  
Laplacian Eigenmaps ◽  
Learning Techniques ◽  
Out Of Sample ◽  
Low Dimensional ◽  
Project Data

Unsupervised metric learning has been generally studied as a byproduct of dimensionality reduction or manifold learning techniques. Manifold learning techniques like Diusion maps, Laplacian eigenmaps has a special property that embedded space is Euclidean. Although laplacian eigenmaps can provide us with some (dis)similarity information it does not provide with a metric which can further be used on out-of-sample data. On other hand supervised metric learning technique like ITML which can learn a metric needs labeled data for learning. In this work propose methods for incremental unsupervised metric learning. In rst approach Laplacian eigenmaps is used along with Information Theoretic Metric Learning(ITML) to form an unsupervised metric learning method. We rst project data into a low dimensional manifold using Laplacian eigenmaps, in embedded space we use euclidean distance to get an idea of similarity between points. If euclidean distance between points in embedded space is below a threshold t1 value we consider them as similar points and if it is greater than a certain threshold t2 we consider them as dissimilar points. Using this we collect a batch of similar and dissimilar points which are then used as a constraints for ITML algorithm and learn a metric. To prove this concept we have tested our approach on various UCI machine learning datasets. In second approach we propose Incremental Diusion Maps by updating SVD in a batch-wise manner.

scholarly journals Growing two-dimensional manifold of nonlinear maps based on generalized Foliation condition

2012 ◽  
Vol 61 (2) ◽  
pp. 029501
Author(s):  
Li Hui-Min ◽  
Fan Yang-Yu ◽  
Sun Heng-Yi ◽  
Zhang Jing ◽  
Jia Meng
Keyword(s):  
Dimensional Manifold ◽  
Two Dimensional ◽  
Nonlinear Maps

scholarly journals Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

2019 ◽  
Vol 11 (4) ◽  
pp. 72-79
Author(s):  
Anna Kravchenko ◽  
Sergiy Maksymenko
Keyword(s):  
Homotopy Type ◽  
Morse Function ◽  
Previous Article ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Reeb Graph ◽  
Natural Right ◽  
Morse Functions ◽  
Path Component ◽  
Unique Vertex

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M, R)$ be a Morse function, and $\Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h \in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{\infty}$, and by $S(f)={h\in D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $h\in S(f)$ induces an automorphism of the graph $\Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) \cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$. In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere.  The present paper is devoted to the case $M = S^2$. In this situation $\Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 \to R$ whose fixed subtree $Fix(G)$ consists of more than one point.

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