High-dimensional Euclidean Space Research Articles

Modeling and rendering non-euclidean spaces approximated with concatenated polytopes

A non-Euclidean space is characterized as a manifold with a specific structure that violates Euclid's postulates. This paper proposes to approximate a manifold with polytopes. Based on the scene designer's specification, the polytopes are automatically concatenated and embedded in a higher-dimensional Euclidean space. Then, the scene is navigated and rendered via novel methods tailored to concatenated polytopes. The proof-of-concept implementation and experiments with it show that the proposed methods bring the virtual-world users unusual and fascinating experiences, which cannot be provided in Euclidean-space applications.

Contagion Dynamics for Manifold Learning.

Contagion maps exploit activation times in threshold contagions to assign vectors in high-dimensional Euclidean space to the nodes of a network. A point cloud that is the image of a contagion map reflects both the structure underlying the network and the spreading behavior of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps and variants thereof as a manifold-learning tool on a number of different synthetic and real-world data sets, and we compare their performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, while Isomap fails due to noise-induced error. This consolidates contagion maps as a technique for manifold learning. We also demonstrate that processing distance estimates between data points before performing methods to determine geometry, topology and dimensionality of a data set leads to clearer results for both Isomap and contagion maps.

On the structure of periodic eigenvalues of the vectorial p-Laplacian

In this paper we will solve an open problem raised by Manásevich and Mawhin twenty years ago on the structure of the periodic eigenvalues of the vectorial p-Laplacian. This is an Euler–Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent p other than 2, the vectorial p-Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to two-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed.

Manifold feature index: A novel index based on high-dimensional data simplification

The current stock index design models typically consider financial factors, such as market capitalization and free-float, regardless of stock price changes. Their constituent selection strategies are based on the ranking and determined by experts in exchanges or financial services. In this paper, we develop a novel stock index model, namely, the manifold feature (MF) index, to reflect the overall price activity of the entire stock market. The MF index is designed using the manifold learning theory and a harmonic analysis method. To our knowledge, it is the first time to do so in expert/intelligent systems to design stock index. Specifically, according to the manifold learning theory, the studied stock dataset is assumed to be a low-dimensional manifold embedded in a higher-dimensional Euclidean space. After data preprocessing, a harmonic analysis method is performed. That is, the discrete Laplace–Beltrami operator (LBO) matrix defined on the stock dataset, which forms a low-dimensional manifold, is firstly constructed. Then, the eigenvectors of LBO are calculated, and the feature points on the eigenvectors are detected. The stocks corresponding to these feature points are considered as the constituent stocks of the MF index. Finally, the MF index is generated by a weighted formula using the price and market capitalization of these constituents. Moreover, we propose four metrics to compare the MF index series and the Shanghai Stock Exchange (SSE) index series (SSE 50, SSE 100, SSE 150, SSE 180 and SSE 380), and the MF indexes are better than the SSE indexes in two aspects: On one hand, from the perspective of data approximation, MF indexes are closer to the stock market than the SSE index series. On the other hand, from the perspective of risk premium, MF indexes have higher stability and lower risk.

Online search for a hyperplane in high-dimensional Euclidean space

We consider the online search problem in which a server starting at the origin of a d-dimensional Euclidean space has to find an arbitrary hyperplane. The best-possible competitive ratio and the length of the shortest curve from which each point on the d-dimensional unit sphere can be seen are within a constant factor of each other. We show that this length is in Ω(d)∩O(d3/2).

Nonhomogeneous Euclidean first-passage percolation and distance learning

Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of the manifold and the underlying density. We define such a sample distance and prove the convergence, as the sample size goes to infinity, to a macroscopic one that we call Fermat distance as it minimizes a path functional, resembling Fermat principle in optics. The proof boils down to the study of geodesics in Euclidean first-passage percolation for nonhomogeneous Poisson point processes.

Polynomial approximate discretization of geometric centers in high-dimensional Euclidean space

Polynomial approximate discretization of geometric centers in high-dimensional Euclidean space

Accelerating exact nearest neighbor search in high dimensional Euclidean space via block vectors

The nearest neighbor search is an essential operation for many computer vision, data mining, and machine learning problems. Since it is so widely used, the nearest neighbor search should be as fast as possible. This paper explores lower bound-based approaches to speed up the exact nearest neighbor search in high dimensional Euclidean space. We compute the lower bound of Euclidean Distance by using the block vectors and Cauchy–Schwartz inequality. The proposed lower bound is calculated efficiently and is close to the real Euclidean Distance. Besides, the preprocessing step of the proposal has linear time complexity. Given a query, during the procedure of identifying the nearest neighbor, our method can eliminate many expensive actual distance computations using the lower bound to approximate Euclidean Distance. In addition, we develop a multilevel lower bound strategy, which calculates the lower bound step by step and utilizes the multistep filtering mechanism to improve the searching process further. Theoretical analysis is provided to show that the proposals can guarantee to obtain the same result as the brute-force search. Comprehensive experiments on 16 public data sets collected from various domains demonstrate that our approach performs well in finding the exact nearest neighbor compared to related competitors. The experimental results also illustrate that the multilevel lower bound strategy is effective.

XGest: Enabling Cross-Label Gesture Recognition with RF Signals

Extensive efforts have been devoted to human gesture recognition with radio frequency (RF) signals. However, their performance degrades when applied to novel gesture classes that have never been seen in the training set. To handle unseen gestures, extra efforts are inevitable in terms of data collection and model retraining. In this article, we present XGest, a cross-label gesture recognition system that can accurately recognize gestures outside of the predefined gesture set with zero extra training effort. The key insight of XGest is to build a knowledge transfer framework between different gesture datasets. Specifically, we design a novel deep neural network to embed gestures into a high-dimensional Euclidean space. Several techniques are designed to tackle the spatial resolution limits imposed by RF hardware and the specular reflection effect of RF signals in this model. We implement XGest on a commodity mmWave device, and extensive experiments have demonstrated the significant recognition performance.

Beyond ITQ: Efficient binary multi-view subspace learning for instance retrieval

The existing hashing methods mainly handle either the feature based nearest-neighbor search or the category-level image retrieval, whereas a few efforts are devoted to instance retrieval problem. In this paper, we propose a binary multi-view fusion framework for directly recovering a latent Hamming subspace from the multi-view features for instance retrieval. More specifically, the multi-view subspace reconstruction and the binary quantization are integrated in a unified framework so as to minimize the discrepancy between the original multi-view high-dimensional Euclidean space and the resulting compact Hamming subspace. Besides, our method is essentially an unsupervised learning scheme without any labeled data involved, and thus can be used in the cases when the supervised information is unavailable or insufficient. Experiments on public benchmark and large-scale datasets reveal that our method achieves competitive retrieval performance comparable to the state-of-the-arts and has excellent scalability in large-scale scenario.

Weak curvatures of irregular curves in high-dimensional Euclidean spaces

Weak curvatures of irregular curves in high-dimensional Euclidean spaces

Solutions of Inhomogeneous Perturbed Generalized Moisil–Teodorescu System and Maxwell’s Equations in Euclidean Space

In this paper, based on a proposed notion of generalized conjugate harmonic pairs in the framework of complex Clifford analysis, necessary and sufficient conditions for the solvability of inhomogeneous perturbed generalized Moisil-Teodorescu systems in higher dimensional Euclidean spaces are proved. As an application, we derive corresponding solvability conditions for the inhomogeneous Maxwell's equations.

Wormhole solutions in embedding class 1 space–time

The present work looks for new spherically symmetric wormhole solutions of the Einstein field equations based on the well-known embedding class 1, i.e. Karmarkar condition. The embedding theorems have an interesting property that connects an [Formula: see text]-dimensional space–time to the higher-dimensional Euclidean flat space–time. The Einstein field equations yield the wormhole solution by violating the null energy condition (NEC). Here, wormholes solutions are obtained corresponding to three different redshift functions: rational, logarithm, and inverse trigonometric functions, in embedding class 1 space–time. The obtained shape function in each case satisfies the flare-out condition after the throat radius, i.e. good enough to represents wormhole structure. In cases of WH1 and WH2, the solutions violate the NEC as well as strong energy condition (SEC), i.e. here the exotic matter content exists within the wormholes and strongly sustains wormhole structures. In the case of WH3, the solution violates NEC but satisfies SEC, so for violating the NEC wormhole preserve due to the presence of exotic matter. Moreover, WH1 and WH2 are asymptotically flat while WH3 is not asymptotically flat. So, indeed, WH3 cutoff after some radial distance [Formula: see text], the Schwarzschild radius, and match to the external vacuum solution.

Solving Approximate Nearest Neighbor by Hash Aggregate and Clustering in High Dimensional Euclidean Space

Approximate Nearest Neighbor search has attracted great attention in the field of information search. The method based on Locality-Sensitive Hashing has good performance with probability guarantee, which needs m hash functions for collision detection. In the hash table corresponding to each hash function, collision counts are performed respectively during querying, which will result in additional collision counts and longer response time. To avoid the shortcoming, we aggregate the m hash values as the representation of the original data object. Then the aggregate values of all data objects in the dataset are clustered. In the process of querying, it is needed to find the class of the aggregate value of the query for searching. Experiments show the effectiveness of our method.

The mathematical foundations of anelasticity: existence of smooth global intermediate configurations.

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

Morphogenetic systems: Models and experiments

M systems are mathematical models of morphogenesis developed to gain insights into its relations to phenomena such as self-assembly, self-controlled growth, homeostasis, self-healing and self-reproduction, in both natural and artificial systems. M systems rely on basic principles of membrane computing and self-assembly, as well as explicit emphasis on geometrical structures (location and shape) in 2D, 3D or higher dimensional Euclidean spaces. They can be used for principled studies of these phenomena, both theoretically and experimentally, at a computational level abstracted from their detailed implementation. In particular, they afford 2D and 3D models to explore biological morphogenetic processes. Theoretical studies have shown that M systems are powerful tools (e.g., computational universal, i.e. can become as complex as any computer program) and their parallelism allows for trading space for time in solving efficiently problems considered infeasible on conventional computers (NP-hard problems). In addition, they can also exhibit properties such as robustness to injuries and degrees of self-healing.This paper focuses on the experimental side of M systems. To this end, we have developed a high-level morphogenetic simulator, Cytos, to implement and visualize M systems in silico in order to verify theoretical results and facilitate research in M systems. We summarize the software package and make a brief comparison with some other simulators of membrane systems. The core of the article is a description of a range of experiments inspired by aspects of morphogenesis in both prokaryotic and eukaryotic cells. The experiments explore the regulatory role of the septum and of the cytoskeleton in cell fission, the robustness of cell models against injuries, and, finally, the impact of changing nutrient concentration on population growth.

Novel metrics for computing semantic similarity with sense embeddings

In the last years many efforts have been spent to build word embeddings, a representational device in which word meanings are described through dense unit vectors of real numbers over a continuous, high-dimensional Euclidean space, where similarity can be interpreted as a metric. Afterwards, sense-level embeddings have been proposed to describe the meaning of senses, rather than terms. More recently, additional intermediate representations have been designed, providing a vector description for pairs 〈term,sense〉, and mapping both term and sense descriptions onto a shared semantic space. However, surprisingly enough, this wealth of approaches and resources has not been supported by a parallel refinement in the metrics used to compute semantic similarity: to date, the semantic similarity featuring two input entities is mostly computed as the maximization of some angular distance intervening between vector pairs, typically cosine similarity. In this work we introduce two novel similarity metrics to compare sense-level representations, and show that by exploiting the features of sense-embeddings it is possible to substantially improve on existing strategies, by obtaining enhanced correlation with human similarity ratings. Additionally, we argue that semantic similarity needs to be complemented by another task, involving the identification of the senses at the base of the similarity rating. We experimentally verified that the proposed metrics are beneficial when dealing with both semantic similarity task and sense identification task. The experimentation also provides a detailed how-to illustrating how six important sets of sense embeddings can be used to implement the proposed similarity metrics.

Unsupervised Congestion Status Identification Using LMP Data

Having a better understanding of how locational marginal prices (LMPs) change helps in price forecasting and market strategy making. This paper investigates the fundamental distribution of the congestion part of LMPs in high-dimensional Euclidean space using an unsupervised approach. LMP models based on the lossless and lossy DC optimal power flow (DC-OPF) are analyzed to show the overlapping subspace property of the LMP data. The congestion part of LMPs is spanned by certain row vectors of the power transfer distribution factor (PTDF) matrix, and the subspace attributes of an LMP vector uniquely are found to reflect the instantaneous congestion status of all the transmission lines. The proposed method searches for the basis vectors that span the subspaces of congestion LMP data in hierarchical ways. In the bottom-up search, the data belonging to 1-dimensional subspaces are detected, and other data are projected on the orthogonal subspaces. This procedure is repeated until all the basis vectors are found or the basis gap appears. Top-down searching is used to address the basis gap by hyperplane detection with outliers. Once all the basis vectors are detected, the congestion status can be identified. Numerical experiments based on the IEEE 30-bus system, IEEE 118-bus system, Illinois 200-bus system, and Southwest Power Pool are conducted to show the performance of the proposed method.

The Cauchy Integral Formula in Hermitian, Quaternionic and $$\mathfrak {osp}(4|2)$$ Clifford Analysis

As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a cornerstone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as to how hermitian Clifford analysis in euclidean space $${\mathbb {R}}^{2n}$$ of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on $${\mathbb {R}}^{2n}$$ , quaternionic Clifford analysis arose as a further refinement by introducing a so-called hypercomplex structure $${\mathbb {Q}}$$ , i.e. three complex structures ( $${\mathbb {I}}$$ , $${\mathbb {J}}$$ , $${\mathbb {K}}$$ ) which follow the quaternionic multiplication rules, on $${\mathbb {R}}^{4p}$$ , the dimension now being a fourfold. Two, respectively four, differential operators lead to first order systems invariant under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are called hermitian monogenic and quaternionic monogenic functions respectively. In this contribution we further elaborate on the Cauchy Integral Formula for hermitian and quaternionic monogenic functions. Moreover we establish Caychy integral formulae for $$\mathfrak {osp}(4|2)$$ -monogenic functions, the newest branch of Clifford analysis refining quaternionic monogenicity by taking the underlying symplectic symmetry fully into account.

The Language of Innovation.

Predicting innovation is a peculiar problem in data science. Following its definition, an innovation is always a never-seen-before event, leaving no room for traditional supervised learning approaches. Here we propose a strategy to address the problem in the context of innovative patents, by defining innovations as never-seen-before associations of technologies and exploiting self-supervised learning techniques. We think of technological codes present in patents as a vocabulary and the whole technological corpus as written in a specific, evolving language. We leverage such structure with techniques borrowed from Natural Language Processing by embedding technologies in a high dimensional euclidean space where relative positions are representative of learned semantics. Proximity in this space is an effective predictor of specific innovation events, that outperforms a wide range of standard link-prediction metrics. The success of patented innovations follows a complex dynamics characterized by different patterns which we analyze in details with specific examples. The methods proposed in this paper provide a completely new way of understanding and forecasting innovation, by tackling it from a revealing perspective and opening interesting scenarios for a number of applications and further analytic approaches.