High-dimensional Embedding Research Articles

A priori anisotropic mesh adaptation driven by a higher dimensional embedding

We generalize the higher embedding approach proposed in Lévy and Bonneel (2013) to generate an adapted mesh matching the intrinsic directionalities of an assigned function. In more detail, the original embedding map between the physical (lower dimensional) and the embedded (higher dimensional) setting is modified to include information associated with the function and with its gradient. Then, we set an adaptive procedure, driven by the embedded metric but performed in the lower dimensional setting, which results into an anisotropic adapted mesh of the physical domain. The effectiveness of the proposed procedure is extensively investigated on several two-dimensional test cases, involving both analytical functions and finite element approximations of differential problems. The preliminary verification in three dimensions corroborates the robustness of the method.

E 7(7) invariant non-extremal entropy

The entropy of generic non-extremal dyonic black holes in the STU model has been shown to admit a remarkably universal form. The missing invariant in the formula was recently identified by S\'arosi using the formalism of quantum entanglement as well as a higher dimensional embedding of the U-duality group. Here, we express the non-extremal black hole entropy in the STU model in terms of U-duality covariant tensors. We then provide the extension to the most general non-extremal black hole of ungauged $\mathcal N= 8$ supergravity using $E_{7(7)}$ invariants. We also conjecture a generalization for ungauged $\mathcal N = 2$ supergravity coupled to vector multiplets with arbitrary cubic prepotential. The most general rotating dyonic black hole solution of the STU model with all scalar moduli turned on is provided in an appendix.

An Anisoptropic Surface Remeshing Strategy Combining Higher Dimensional Embedding with Radial Basis Functions

Many applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy.

Anisotropic Finite Element Mesh Adaptation via Higher Dimensional Embedding

In this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1–4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in R3. In the context of adaptive finite element simulation, the solution (which is an unknown function f : Ω ⊂ ▪d → ▪) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φf (x):= (x1, …, xd, s f (x1, …, xd), s ▿ f (x1, …, xd))t, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function f itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function f. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φf (x) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial differential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG – a metric-based adaptive mesh generator. The errors measured in the L2 norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG.

Consistent Kaluza-Klein truncations via exceptional field theory

We present the generalized Scherk-Schwarz reduction ansatz for the full supersymmetric exceptional field theory in terms of group valued twist matrices subject to consistency equations. With this ansatz the field equations precisely reduce to those of lower-dimensional gauged supergravity parametrized by an embedding tensor. We explicitly construct a family of twist matrices as solutions of the consistency equations. They induce gauged supergravities with gauge groups SO(p,q) and CSO(p,q,r). Geometrically, they describe compactifications on internal spaces given by spheres and (warped) hyperboloides $H^{p,q}$, thus extending the applicability of generalized Scherk-Schwarz reductions beyond homogeneous spaces. Together with the dictionary that relates exceptional field theory to D=11 and IIB supergravity, respectively, the construction defines an entire new family of consistent truncations of the original theories. These include not only compactifications on spheres of different dimensions (such as AdS$_5\times S^5$), but also various hyperboloid compactifications giving rise to a higher-dimensional embedding of supergravities with non-compact and non-semisimple gauge groups.

Strata of vector spaces of forms in R = k[x, y], and of rational curves in ℙ k

Consider the polynomial ring R = k[x, y] over an infinite field k and the subspace R j of degree- j homogeneous polynomials. The Grassmanian G = Grass(R j , d) parametrizes the vector spaces V ⊂ R j having dimension d. The strata Grass H (R j , d) ⊂ G determined by the Hilbert functions H = H(R/(V)) or, equivalently, by the Betti numbers of the algebras R/(V), are locally closed and irreducible of known dimension. They satisfy a frontier property that the closure of a stratum is its union with lower strata [I1, I2]. The strata are determined also by the decomposition of the restricted tangent bundle $$\mathcal{T}_V = \varphi _V^* (\mathcal{T})$$ where $$\mathcal{T}$$ is the tangent bundle on ℙ d −1 and ϕ V is the rational curve determined by V. Each stratum corresponds to a partition, and the the poset of strata under closure is isomorphic to the poset of corresponding partitions in the Bruhat order. They are coarser than the strata defined by D. Cox, A. Kustin, C. Polini, and B. Ulrich [CKPU] and studied further in [KPU], that are determined in part by singularities of ϕ V . We explain these results and give examples to make them more accessible. We also generalize a result of D. Cox, T. Sederberg, and F. Chen and another of C. D’Andrea concerning the dimension and closure of μ families of parametrized rational curves from planar [CSC, D] to higher dimensional embeddings (Theorem 1.11).

Novel steganographic method based on generalized K-distance N-dimensional pixel matching

In this paper, a steganographic scheme adopting the concept of the generalized Kd-distance N-dimensional pixel matching is proposed. The generalized pixel matching embeds a B-ary digit (B is a function of K and N) into a cover vector of length N, where the order-d Minkowski distance-measured embedding distortion is no larger than K. In contrast to other pixel matching-based schemes, a N-dimensional reference table is used. By choosing d, K, and N adaptively, an embedding strategy which is suitable for arbitrary relative capacity can be developed. Additionally, an optimization algorithm, namely successive iteration algorithm (SIA), is proposed to optimize the codeword assignment in the reference table. Benefited from the high dimensional embedding and the optimization algorithm, nearly maximal embedding efficiency is achieved. Compared with other content-free steganographic schemes, the proposed scheme provides better image quality and statistical security. Moreover, the proposed scheme performs comparable to state-of-the-art content-based approaches after combining with image models.

Particle-based anisotropic surface meshing

This paper introduces a particle-based approach for anisotropic surface meshing. Given an input polygonal mesh endowed with a Riemannian metric and a specified number of vertices, the method generates a metric-adapted mesh. The main idea consists of mapping the anisotropic space into a higher dimensional isotropic one, called "embedding space". The vertices of the mesh are generated by uniformly sampling the surface in this higher dimensional embedding space, and the sampling is further regularized by optimizing an energy function with a quasi-Newton algorithm. All the computations can be re-expressed in terms of the dot product in the embedding space, and the Jacobian matrices of the mappings that connect different spaces. This transform makes it unnecessary to explicitly represent the coordinates in the embedding space, and also provides all necessary expressions of energy and forces for efficient computations. Through energy optimization, it naturally leads to the desired anisotropic particle distributions in the original space. The triangles are then generated by computing the Restricted Anisotropic Voronoi Diagram and its dual Delaunay triangulation. We compare our results qualitatively and quantitatively with the state-of-the-art in anisotropic surface meshing on several examples, using the standard measurement criteria.

SU-E-J-146: Time Series Prediction of Lung Cancer Patients' Breathing Pattern Based on Nonlinear Dynamics.

Prediction methods for breathing patterns, which are crucial to deal with system latency in treatments of moving lung tumors using state-space methodologies based on non-linear dynamics are contrasted to linear predictive methods. In our previous work we established that breathing patterns can be described as a 5-6 dimensional nonlinear, stationary and deterministic system that exhibits sensitive dependence on initial conditions. In this work, nonlinear prediction methods are used to predict the short-term evolution of the respiratory system for 3 patients. Single step and N-point multi step prediction are performed for sampling rates of 5Hz, 10Hz, and 30Hz. We compare the employed nonlinear prediction methods with respect to prediction accuracy to Infinite Impulse Response (IIR) prediction filters. The simplest form of local prediction is finding similar segments of scalar time series data in a higher dimensional embedding space. Hence, we predict the future value x(t)of N-time steps ahead by simply finding the average of nearest neighbor points to the point x(t) in the past and using them to estimate x(t+N), yielding a local average model (LAM). Local linear models (LLM) which are linear autoregressive models that hold only for a region around the target point formed by the nearest neighbor points is combined with a set of linear regularization techniques to solve ill-posed regression problems are also implemented. For all sampling frequencies, both single step and N-point multi step prediction results obtained using LAM and LLM with regularization methods are better than IIR prediction filters for the selected sample patients. The use of non-linear prediction methods for predicting the breathing pattern of lung cancer patients may lead to improved, robust and accurate long-term prediction to account for system latencies.

Lifshitz singularities

Lifshitz spacetimes are possible gravitational duals to strongly coupled field theories with an anisotropic scaling symmetry. These spacetimes however, have a null curvature singularity. We find that higher dimensional embeddings of Lifshitz also have a similar singularity. We study the propagation of test strings in this background and find that they become infinitely excited if they try to propagate through the singularity. This means that the Lifshitz geometry is unstable and will receive large corrections in string theory.

Non-relativistic solutions of $ \mathcal{N} = 2 $ gauged supergravity

We find infinite families of supersymmetric solutions of four dimensional, N=2 gauged supergravity with Lifshitz, Schrodinger and also AdS symmetries. We focus on the canonical example of a single hypermultiplet and a single vector multiplet and find that the spectrum of solutions depends crucially on whether the gaugings are electric or magnetic but to a far milder extent on the strength of the gaugings. For purely electric or purely magnetic gaugings we generically find Lifshitz solutions, while for a mixed gauging we find Schrodinger and AdS solutions. For some of the gaugings the theory has a known lift to string/M-theory thus giving a higher dimensional embedding of our solutions.

Comparison of phylogenetic trees through alignment of embedded evolutionary distances

BackgroundThe understanding of evolutionary relationships is a fundamental aspect of modern biology, with the phylogenetic tree being a primary tool for describing these associations. However, comparison of trees for the purpose of assessing similarity and the quantification of various biological processes remains a significant challenge.ResultsWe describe a novel approach for the comparison of phylogenetic distance information based on the alignment of representative high-dimensional embeddings (xCEED: Comparison of Embedded Evolutionary Distances). The xCEED methodology, which utilizes multidimensional scaling and Procrustes-related superimposition approaches, provides the ability to measure the global similarity between trees as well as incongruities between them. We demonstrate the application of this approach to the prediction of coevolving protein interactions and demonstrate its improved performance over the mirrortree, tol-mirrortree, phylogenetic vector projection, and partial correlation approaches. Furthermore, we show its applicability to both the detection of horizontal gene transfer events as well as its potential use in the prediction of interaction specificity between a pair of multigene families.ConclusionsThese approaches provide additional tools for the study of phylogenetic trees and associated evolutionary processes. Source code is available at http://gomezlab.bme.unc.edu/tools.

Nonlinear noise reduction of chaotic time series based on multidimensional recurrent LS-SVM

In order to resolve the noise reduction in chaotic time series, a novel method based on multidimensional recurrent least squares support vector machine (MDRLS-SVM) is proposed in this paper. Considering the evolvement feature of the chaotic system, we utilize the recurrent version of least squares support vector machines (LS-SVM) to manipulate the iterative problem. From the high-dimensional phase space point of view, the function approximation in the high-dimensional embedding phase space is carried out and the noise reduction achieved simultaneously based on the reconstructed embedding phase theory. We show by means of simulation of Ikeda map that the proposed method is able to provide accurate results in noise reduction of chaotic system.

Parent form for higher spin fields on anti-de Sitter space

We construct a first order parent field theory for free higher spin gauge fields on constant curvature spaces. As in the previously considered flat case, both Fronsdal's and Vasiliev's unfolded formulations can be reached by two different straightforward reductions. The parent theory itself is formulated using a higher dimensional embedding space and turns out to be geometrically extremely transparent and free of the intricacies of both of its reductions.

A multi-dimensional approach to force-directed layouts of large graphs

We present a novel hierarchical force-directed method for drawing large graphs. Given a graph G=( V, E), the algorithm produces an embedding for G in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E . Such projections typically result in drawings that are “smoother” and more symmetric than direct drawings in 2D and 3D. In order to obtain fast placement of the vertices of the graph our algorithm employs a multi-scale technique based on a maximal independent set filtration of vertices of the graph. While most existing force-directed algorithms begin with an initial random placement of all the vertices, our algorithm attempts to place vertices “intelligently”, close to their final positions. Other notable features of our approach include a fast energy function minimization strategy and efficient memory management. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a 550 MHz Pentium PC.

Simplifying Schmidt number witnesses via higher-dimensional embeddings

We apply the generalised concept of witness operators to arbitrary convex sets, and review the criteria for the optimisation of these general witnesses. We then define an embedding of state vectors and operators into a higher-dimensional Hilbert space. This embedding leads to a connection between any Schmidt number witness in the original Hilbert space and a witness for Schmidt number two (i.e. the most general entanglement witness) in the appropriate enlarged Hilbert space. Using this relation we arrive at a conceptually simple method for the construction of Schmidt number witnesses in bipartite systems.

Graph Drawing by High-Dimensional Embedding

We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2-D plane using principal components analysis. Running time is linear in the graph size, and experiments we have carried out show the ability of the method to draw graphs of 10 5 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.

The structure of the big bang from higher-dimensional embeddings

We give relations for the embedding of spatially-flat Friedmann–Robertson–Walker cosmological models of Einstein's theory in flat manifolds of the type used in Kaluza–Klein theory. We present embedding diagrams that depict different 4D universes as hypersurfaces in a higher-dimensional flat manifold. The morphology of the hypersurfaces is found to depend on the equation of state of the matter. The hypersurfaces possess a line-like curvature singularity infinitesimally close to the t = 0+ 3-surface, where t is the time expired since the big bang. The family of timelike comoving geodesics on any given hypersurface is found to have a caustic on the singular line, which we conclude is the 5D position of the point-like big bang.

Noise reduction for human speech signals by local projections in embedding spaces

A local projective noise-reduction scheme, originally developed for low-dimensional deterministic stationary signals, is successfully applied to human speech. This is possible by exploiting properties of the speech signal which mimic structure exhibited by deterministic chaotic systems. In high-dimensional embedding spaces, the strong nonstationarity is resolved as a sequence of different dynamical regimes of moderate complexity. This filtering technique does not make use of the spectral contents of the signal and is superior to the Ephraim-Malah adaptive filter.

Bertotti-Robinson-type solutions to dilaton-axion gravity

We present a new solution to dilaton-axion gravity which looks like a rotating Bertotti-Robinson (BR) Universe. It is supported by an homogeneous Maxwell field and a linear axion and can be obtained as a near-horizon limit of extremal rotating dilaton-axion black holes. It has the isometry $SL(2,R)\times U(1)$ where U(1) is the remnant of the SO(3) symmetry of BR broken by rotation, while $SL(2,R)$ corresponds to the $AdS_2$ sector which no longer factors out of the full spacetime. Alternatively our solution can be obtained from the D=5 vacuum counterpart to the dyonic BR with equal electric and magnetic field strengths. The derivation amounts to smearing it in D=6 and then reducing to D=4 with dualization of one Kaluza-Klein two-form in D=5 to produce an axion. Using a similar dualization procedure, the rotating BR solution is uplifted to D=11 supergravity. We show that it breaks all supersymmetries of N=4 supergravity in D=4, and that its higher dimensional embeddings are not supersymmetric either. But, hopefully it may provide a new arena for corformal mechanics and holography. Applying a complex coordinate transformation we also derive a BR solution endowed with a NUT parameter.