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Academic literature on the topic 'Optimal embeddings'

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Published: 4 June 2021
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Journal articles on the topic "Optimal embeddings"

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BOZKURT, ILKER NADI, HAI HUANG, BRUCE MAGGS, ANDRÉA RICHA, and MAVERICK WOO. "Mutual Embeddings." Journal of Interconnection Networks 15, no. 01n02 (March 2015): 1550001. http://dx.doi.org/10.1142/s0219265915500012.

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This paper introduces a type of graph embedding called a mutual embedding. A mutual embedding between two n-node graphs [Formula: see text] and [Formula: see text] is an identification of the vertices of V1 and V2, i.e., a bijection [Formula: see text], together with an embedding of G1 into G2 and an embedding of G2 into G1 where in the embedding of G1 into G2, each node u of G1 is mapped to π(u) in G2 and in the embedding of G2 into G1 each node v of G2 is mapped to [Formula: see text] in G1. The identification of vertices in G1 and G2 constrains the two embeddings so that it is not always possible for both to exhibit small congestion and dilation, even if there are traditional one-way embeddings in both directions with small congestion and dilation. Mutual embeddings arise in the context of finding preconditioners for accelerating the convergence of iterative methods for solving systems of linear equations. We present mutual embeddings between several types of graphs such as linear arrays, cycles, trees, and meshes, prove lower bounds on mutual embeddings between several classes of graphs, and present some open problems related to optimal mutual embeddings.
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Vernaeve, H. "OPTIMAL EMBEDDINGS OF DISTRIBUTIONS INTO ALGEBRAS." Proceedings of the Edinburgh Mathematical Society 46, no. 2 (June 2003): 373–78. http://dx.doi.org/10.1017/s0013091500001188.

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AbstractLet $\varOmega$ be a convex, open subset of $\mathbb{R}^n$ and let $\mathcal{D}'(\varOmega)$ be the space of distributions on $\varOmega$. It is shown that there exist linear embeddings of $\mathcal{D}'(\varOmega)$ into a differential algebra that commute with partial derivatives and that embed $\mathcal{C}^{\infty}(\varOmega)$ as a subalgebra. This embedding appears to be the first one after Colombeau’s to possess these properties. We show that many nonlinear operations on distributions can be defined that are not definable in the Colombeau setting.AMS 2000 Mathematics subject classification: Primary 46F30. Secondary 13C11
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Cianchi, Andrea, and Luboš Pick. "Optimal Sobolev trace embeddings." Transactions of the American Mathematical Society 368, no. 12 (January 19, 2016): 8349–82. http://dx.doi.org/10.1090/tran/6606.

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Cianchi, Andrea, and Luboš Pick. "Optimal Gaussian Sobolev embeddings." Journal of Functional Analysis 256, no. 11 (June 2009): 3588–642. http://dx.doi.org/10.1016/j.jfa.2009.03.001.

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Kim, Sook-Yeon, Oh-Heum Kwon, and Kyung-Yong Chwa. "Embeddings of Butterflies into Hypermeshes." Parallel Processing Letters 08, no. 03 (September 1998): 337–50. http://dx.doi.org/10.1142/s0129626498000353.

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Hypermeshes have been given much attention as a versatile interconnection network of parallel computers. A hypermesh is obtained from a mesh by replacing each linear connection with a hyperedge. In this paper, we show how to embed a butterfly or multiple copies of a butterfly into a hypermesh. First, a butterfly B(s) of (s + 1)2s nodes is embedded into a 2s × X hypermesh where X = 2⌊ log 2 s ⌋+ 1. Second, the butterfly B(s) is embedded into a square hypermesh. Third, multiple copies of the butterfly B(s) are embedded into a hypermesh of variable aspect ratio. The efficiency of these embeddings is measured by alignment cost, congestion, and expansion. The alignment cost of all of these embeddings is optimal. The congestion of the first and third embedding is optimal. The expansion of the first and third embedding is one if s = 2k - 1 for some integer k, otherwise, less than two. The expansion of the second embedding is 2 + ∊ (s) where ∊(s) = (2 log (s + 1) + 2)/(s + 1).
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CASSANI, DANIELE, BERNHARD RUF, and CRISTINA TARSI. "GROUP INVARIANCE AND POHOZAEV IDENTITY IN MOSER-TYPE INEQUALITIES." Communications in Contemporary Mathematics 15, no. 02 (March 7, 2013): 1250054. http://dx.doi.org/10.1142/s021919971250054x.

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We study the so-called limiting Sobolev cases for embeddings of the spaces [Formula: see text], where Ω ⊂ ℝn is a bounded domain. Differently from J. Moser, we consider optimal embeddings into Zygmund spaces: we derive related Euler–Lagrange equations, and show that Moser's concentrating sequences are the solutions of these equations and thus realize the best constants of the corresponding embedding inequalities. Furthermore, we exhibit a group invariance, and show that Moser's sequence is generated by this group invariance and that the solutions of the limiting equation are unique up to this invariance. Finally, we derive a Pohozaev-type identity, and use it to prove that equations related to perturbed optimal embeddings do not have solutions.
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Parini, Enea, Bernhard Ruf, and Cristina Tarsi. "Limiting Sobolev inequalities and the 1-biharmonic operator." Advances in Nonlinear Analysis 3, S1 (September 1, 2014): s19—s36. http://dx.doi.org/10.1515/anona-2014-0007.

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AbstractIn this article we present recent results on optimal embeddings, and associated PDEs, of the space of functions whose distributional Laplacian belongs to L1. We discuss sharp embedding inequalities which allow to improve the optimal summability results for solutions of Poisson equations with L1-data by Maz'ya (N ≥ 3) and Brezis–Merle (N = 2). Then, we consider optimal embeddings of the mentioned space into L1, for the simply supported and the clamped case, which yield corresponding eigenvalue problems for the 1-biharmonic operator (a higher order analogue of the 1-Laplacian). We derive some properties of the corresponding eigenfunctions, and prove some Faber–Krahn type inequalities.
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Maclachlan, C. "Optimal embeddings in quaternion algebras." Journal of Number Theory 128, no. 10 (October 2008): 2852–60. http://dx.doi.org/10.1016/j.jnt.2007.12.007.

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Arenas, Manuel, Luis Arenas-Carmona, and Jaime Contreras. "On optimal embeddings and trees." Journal of Number Theory 193 (December 2018): 91–117. http://dx.doi.org/10.1016/j.jnt.2018.04.004.

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Chandran, Nishanth, Ryan Moriarty, Rafail Ostrovsky, Omkant Pandey, Mohammad Ali Safari, and Amit Sahai. "Improved algorithms for optimal embeddings." ACM Transactions on Algorithms 4, no. 4 (August 2008): 1–14. http://dx.doi.org/10.1145/1383369.1383376.

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Dissertations / Theses on the topic "Optimal embeddings"

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Clavero, Nadia F. "Optimal Sobolev Embeddings in Spaces with Mixed Norm." Doctoral thesis, Universitat de Barcelona, 2015. http://hdl.handle.net/10803/292613.

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Este proyecto hace referencia a estimaciones, en espacios funcionales, que relacionan la norma de una función y la de sus derivadas. Concretamente, nuestro principal objetivo es estudiar las estimaciones clásicas de las inclusiones de Sobolev, probadas por Gagliardo y Nirenberg, para derivadas de orden superior y espacios más generales. En particular, estamos interesados en describir el dominio y el rango óptimos para estas inclusiones entre los espacios invariantes por reordenamiento (r.i.) y espacios de normas mixtas.
This thesis project concerns estimates, in function spaces, that relate the norm of a function and that of its derivatives. Speci.cally, our main purpose is to study the classical Sobolev-type inequalities due to Gagliardo and Nirenberg for higher order derivatives and more general spaces. In particular, we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between rearrangement-invariant spaces (r.i.) and mixed norm spaces.
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Grant, Elyot. "Dimension reduction algorithms for near-optimal low-dimensional embeddings and compressive sensing." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/84869.

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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 41-42).
In this thesis, we establish theoretical guarantees for several dimension reduction algorithms developed for applications in compressive sensing and signal processing. In each instance, the input is a point or set of points in d-dimensional Euclidean space, and the goal is to find a linear function from Rd into Rk , where k << d, such that the resulting embedding of the input pointset into k-dimensional Euclidean space has various desirable properties. We focus on two classes of theoretical results: -- First, we examine linear embeddings of arbitrary pointsets with the aim of minimizing distortion. We present an exhaustive-search-based algorithm that yields a k-dimensional linear embedding with distortion at most ... is the smallest possible distortion over all orthonormal embeddings into k dimensions. This PTAS-like result transcends lower bounds for well-known embedding teclhniques such as the Johnson-Lindenstrauss transform. -- Next, motivated by compressive sensing of images, we examine linear embeddings of datasets containing points that are sparse in the pixel basis, with the goal of recoving a nearly-optimal sparse approximation to the original data. We present several algorithms that achieve strong recovery guarantees using the near-optimal bound of measurements, while also being highly "local" so that they can be implemented more easily in physical devices. We also present some impossibility results concerning the existence of such embeddings with stronger locality properties.
by Elyot Grant.
S.M.
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Dittner, Mark [Verfasser]. "Globally Optimal Catalysts - Computational Optimization Of Abstract Catalytic Embeddings For Arbitrary Chemical Reactions / Mark Dittner." Kiel : Universitätsbibliothek Kiel, 2019. http://d-nb.info/1194929559/34.

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Muzellec, Boris. "Leveraging regularization, projections and elliptical distributions in optimal transport." Electronic Thesis or Diss., Institut polytechnique de Paris, 2020. http://www.theses.fr/2020IPPAG009.

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Pouvoir manipuler et de comparer de mesures de probabilité est essentiel pour de nombreuses applications en apprentissage automatique. Le transport optimal (TO) définit des divergences entre distributions fondées sur la géométrie des espaces sous-jacents : partant d'une fonction de coût définie sur l'espace dans lequel elles sont supportées, le TO consiste à trouver un couplage entre les deux mesures qui soit optimal par rapport à ce coût. Par son ancrage géométrique, le TO est particulièrement bien adapté au machine learning, et fait l'objet d'une riche théorie mathématique. En dépit de ces avantages, l'emploi du TO pour les sciences des données a longtemps été limité par les difficultés mathématiques et computationnelles liées au problème d'optimisation sous-jacent. Pour contourner ce problème, une approche consiste à se concentrer sur des cas particuliers admettant des solutions en forme close, ou pouvant se résoudre efficacement. En particulier, le TO entre mesures elliptiques constitue l'un des rares cas pour lesquels le TO admet une forme close, définissant la géométrie de Bures-Wasserstein (BW). Cette thèse s'appuie tout particulièrement sur la géométrie de BW, dans le but de l'utiliser comme outil de base pour des applications en sciences des données. Pour ce faire, nous considérons des situations dans lesquelles la géométrie de BW est tantôt utilisée comme un outil pour l'apprentissage de représentations, étendue à partir de projections sur des sous-espaces, ou régularisée par un terme entropique. Dans une première contribution, la géométrie de BW est utilisée pour définir des plongements sous la forme de distributions elliptiques, étendant la représentation classique sous forme de vecteurs de R^d. Dans une deuxième contribution, nous prouvons l'existence de transports qui extrapolent des applications restreintes à des projections en faible dimension, et montrons que ces plans "sous-espace optimaux" admettent des formes closes dans le cas de mesures gaussiennes. La troisième contribution de cette thèse consiste à obtenir des formes closes pour le transport entropique entre des mesures gaussiennes non-normalisées, qui constituent les premières expressions non triviales pour le transport entropique. Finalement, dans une dernière contribution nous utilisons le transport entropique pour imputer des données manquantes de manière non-paramétrique, tout en préservant les distributions sous-jacentes
Comparing and matching probability distributions is a crucial in numerous machine learning (ML) algorithms. Optimal transport (OT) defines divergences between distributions that are grounded on geometry: starting from a cost function on the underlying space, OT consists in finding a mapping or coupling between both measures that is optimal with respect to that cost. The fact that OT is deeply grounded in geometry makes it particularly well suited to ML. Further, OT is the object of a rich mathematical theory. Despite those advantages, the applications of OT in data sciences have long been hindered by the mathematical and computational complexities of the underlying optimization problem. To circumvent these issues, one approach consists in focusing on particular cases that admit closed-form solutions or that can be efficiently solved. In particular, OT between elliptical distributions is one of the very few instances for which OT is available in closed form, defining the so-called Bures-Wasserstein (BW) geometry. This thesis builds extensively on the BW geometry, with the aim to use it as basic tool in data science applications. To do so, we consider settings in which it is alternatively employed as a basic tool for representation learning, enhanced using subspace projections, and smoothed further using entropic regularization. In a first contribution, the BW geometry is used to define embeddings as elliptical probability distributions, extending on the classical representation of data as vectors in R^d.In the second contribution, we prove the existence of transportation maps and plans that extrapolate maps restricted to lower-dimensional projections, and show that subspace-optimal plans admit closed forms in the case of Gaussian measures.Our third contribution consists in deriving closed forms for entropic OT between Gaussian measures scaled with a varying total mass, which constitute the first non-trivial closed forms for entropic OT and provide the first continuous test case for the study of entropic OT. Finally, in a last contribution, entropic OT is leveraged to tackle missing data imputation in a non-parametric and distribution-preserving way
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Ashley, Michael John Siew Leung, and ashley@gravity psu edu. "Singularity theorems and the abstract boundary construction." The Australian National University. Faculty of Science, 2002. http://thesis.anu.edu.au./public/adt-ANU20050209.165310.

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The abstract boundary construction of Scott and Szekeres has proven a practical classification scheme for boundary points of pseudo-Riemannian manifolds. It has also proved its utility in problems associated with the re-embedding of exact solutions containing directional singularities in space-time. Moreover it provides a model for singularities in space-time - essential singularities. However the literature has been devoid of abstract boundary results which have results of direct physical applicability.¶ This thesis presents several theorems on the existence of essential singularities in space-time and on how the abstract boundary allows definition of optimal em- beddings for depicting space-time. Firstly, a review of other boundary constructions for space-time is made with particular emphasis on the deficiencies they possess for describing singularities. The abstract boundary construction is then pedagogically defined and an overview of previous research provided.¶ We prove that strongly causal, maximally extended space-times possess essential singularities if and only if they possess incomplete causal geodesics. This result creates a link between the Hawking-Penrose incompleteness theorems and the existence of essential singularities. Using this result again together with the work of Beem on the stability of geodesic incompleteness it is possible to prove the stability of existence for essential singularities.¶ Invariant topological contact properties of abstract boundary points are presented for the first time and used to define partial cross sections, which are an generalization of the notion of embedding for boundary points. Partial cross sections are then used to define a model for an optimal embedding of space-time.¶ Finally we end with a presentation of the current research into the relationship between curvature singularities and the abstract boundary. This work proposes that the abstract boundary may provide the correct framework to prove curvature singularity theorems for General Relativity. This exciting development would culminate over 30 years of research into the physical conditions required for curvature singularities in space-time.
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Guo, Gaoyue. "Continuous-time Martingale Optimal Transport and Optimal Skorokhod Embedding." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX038/document.

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Cette thèse présente trois principaux sujets de recherche, les deux premiers étant indépendants et le dernier indiquant la relation des deux premières problématiques dans un cas concret.Dans la première partie nous nous intéressons au problème de transport optimal martingale dans l’espace de Skorokhod, dont le premier but est d’étudier systématiquement la tension des plans de transport martingale. On s’intéresse tout d’abord à la semicontinuité supérieure du problème primal par rapport aux distributions marginales. En utilisant la S-topologie introduite par Jakubowski, on dérive la semicontinuité supérieure et on montre la première dualité. Nous donnons en outre deux problèmes duaux concernant la surcouverture robuste d’une option exotique, et nous établissons les dualités correspondantes, en adaptant le principe de la programmation dynamique et l’argument de discrétisation initie par Dolinsky et Soner.La deuxième partie de cette thèse traite le problème du plongement de Skorokhod optimal. On formule tout d’abord ce problème d’optimisation en termes de mesures de probabilité sur un espace élargi et ses problèmes duaux. En utilisant l’approche classique de la dualité; convexe et la théorie d’arrêt optimal, nous obtenons les résultats de dualité. Nous rapportons aussi ces résultats au transport optimal martingale dans l’espace des fonctions continues, d’où les dualités correspondantes sont dérivées pour une classe particulière de fonctions de paiement. Ensuite, on fournit une preuve alternative du principe de monotonie établi par Beiglbock, Cox et Huesmann, qui permet de caractériser les optimiseurs par leur support géométrique. Nous montrons à la fin un résultat de stabilité qui contient deux parties: la stabilité du problème d’optimisation par rapport aux marginales cibles et le lien avec un autre problème du plongement optimal.La dernière partie concerne l’application de contrôle stochastique au transport optimal martingale avec la fonction de paiement dépendant du temps local, et au plongement de Skorokhod. Pour le cas d’une marginale, nous retrouvons les optimiseurs pour les problèmes primaux et duaux via les solutions de Vallois, et montrons en conséquence l’optimalité des solutions de Vallois, ce qui regroupe le transport optimal martingale et le plongement de Skorokhod optimal. Quand au cas de deux marginales, on obtient une généralisation de la solution de Vallois. Enfin, un cas spécial de plusieurs marginales est étudié, où les temps d’arrêt donnés par Vallois sont bien ordonnés
This PhD dissertation presents three research topics, the first two being independent and the last one relating the first two issues in a concrete case.In the first part we focus on the martingale optimal transport problem on the Skorokhod space, which aims at studying systematically the tightness of martingale transport plans. Using the S-topology introduced by Jakubowski, we obtain the desired tightness which yields the upper semicontinuity of the primal problem with respect to the marginal distributions, and further the first duality. Then, we provide also two dual formulations that are related to the robust superhedging in financial mathematics, and we establish the corresponding dualities by adapting the dynamic programming principle and the discretization argument initiated by Dolinsky and Soner.The second part of this dissertation addresses the optimal Skorokhod embedding problem under finitely-many marginal constraints. We formulate first this optimization problem by means of probability measures on an enlarged space as well as its dual problems. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results. We also relate these results to the martingale optimal transport on the space of continuous functions, where the corresponding dualities are derived for a special class of reward functions. Next, We provide an alternative proof of the monotonicity principle established in Beiglbock, Cox and Huesmann, which characterizes the optimizers by their geometric support. Finally, we show a stability result that is twofold: the stability of the optimization problem with respect to target marginals and the relation with another optimal embedding problem.The last part concerns the application of stochastic control to the martingale optimal transport with a payoff depending on the local time, and the Skorokhod embedding problem. For the one-marginal case, we recover the optimizers for both primal and dual problems through Vallois' solutions, and show further the optimality of Vallois' solutions, which relates the martingale optimal transport and the optimal Skorokhod embedding. As for the two-marginal case, we obtain a generalization of Vallois' solution. Finally, a special multi-marginal case is studied, where the stopping times given by Vallois are well ordered
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Perinelli, Alessio. "A new approach to optimal embedding of time series." Doctoral thesis, Università degli studi di Trento, 2020. http://hdl.handle.net/11572/280754.

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The analysis of signals stemming from a physical system is crucial for the experimental investigation of the underlying dynamics that drives the system itself. The field of time series analysis comprises a wide variety of techniques developed with the purpose of characterizing signals and, ultimately, of providing insights on the phenomena that govern the temporal evolution of the generating system. A renowned example in this field is given by spectral analysis: the use of Fourier or Laplace transforms to bring time-domain signals into the more convenient frequency space allows to disclose the key features of linear systems. A more complex scenario turns up when nonlinearity intervenes within a system's dynamics. Nonlinear coupling between a system's degrees of freedom brings about interesting dynamical regimes, such as self-sustained periodic (though anharmonic) oscillations ("limit cycles"), or quasi-periodic evolutions that exhibit sharp spectral lines while lacking strict periodicity ("limit tori"). Among the consequences of nonlinearity, the onset of chaos is definitely the most fascinating one. Chaos is a dynamical regime characterized by unpredictability and lack of periodicity, despite being generated by deterministic laws. Signals generated by chaotic dynamical systems appear as irregular: the corresponding spectra are broad and flat, prediction of future values is challenging, and evolutions within the systems' state spaces converge to strange attractor sets with noninteger dimensionality. Because of these properties, chaotic signals can be mistakenly classified as noise if linear techniques such as spectral analysis are used. The identification of chaos and its characterization require the assessment of dynamical invariants that quantify the complex features of a chaotic system's evolution. For example, Lyapunov exponents provide a marker of unpredictability; the estimation of attractor dimensions, on the other hand, highlights the unconventional geometry of a chaotic system's state space. Nonlinear time series analysis techniques act directly within the state space of the system under investigation. However, experimentally, full access to a system's state space is not always available. Often, only a scalar signal stemming from the dynamical system can be recorded, thus providing, upon sampling, a scalar sequence. Nevertheless, by virtue of a fundamental theorem by Takens, it is possible to reconstruct a proxy of the original state space evolution out of a single, scalar sequence. This reconstruction is carried out by means of the so-called embedding procedure: m-dimensional vectors are built by picking successive elements of the scalar sequence delayed by a lag L. On the other hand, besides posing some necessary conditions on the integer embedding parameters m and L, Takens' theorem does not provide any clue on how to choose them correctly. Although many optimal embedding criteria were proposed, a general answer to the problem is still lacking. As a matter of fact, conventional methods for optimal embedding are flawed by several drawbacks, the most relevant being the need for a subjective evaluation of the outcomes of applied algorithms. Tackling the issue of optimally selecting embedding parameters makes up the core topic of this thesis work. In particular, I will discuss a novel approach that was pursued by our research group and that led to the development of a new method for the identification of suitable embedding parameters. Rather than most conventional approaches, which seek a single optimal value for m and L to embed an input sequence, our approach provides a set of embedding choices that are equivalently suitable to reconstruct the dynamics. The suitability of each embedding choice m, L is assessed by relying on statistical testing, thus providing a criterion that does not require a subjective evaluation of outcomes. The starting point of our method are embedding-dependent correlation integrals, i.e. cumulative distributions of embedding vector distances, built out of an input scalar sequence. In the case of Gaussian white noise, an analytical expression for correlation integrals is available, and, by exploiting this expression, a gauge transformation of distances is introduced to provide a more convenient representation of correlation integrals. Under this new gauge, it is possible to test—in a computationally undemanding way—whether an input sequence is compatible with Gaussian white noise and, subsequently, whether the sequence is compatible with the hypothesis of an underlying chaotic system. These two statistical tests allow ruling out embedding choices that are unsuitable to reconstruct the dynamics. The estimation of correlation dimension, carried out by means of a newly devised estimator, makes up the third stage of the method: sets of embedding choices that provide uniform estimates of this dynamical invariant are deemed to be suitable to embed the sequence.The method was successfully applied to synthetic and experimental sequences, providing new insight into the longstanding issue of optimal embedding. For example, the relevance of the embedding window (m-1)L, i.e. the time span covered by each embedding vector, is naturally highlighted by our approach. In addition, our method provides some information on the adequacy of the sampling period used to record the input sequence.The method correctly distinguishes a chaotic sequence from surrogate ones generated out of it and having the same power spectrum. The technique of surrogate generation, which I also addressed during my Ph. D. work to develop new dedicated algorithms and to analyze brain signals, allows to estimate significance levels in situations where standard analytical algorithms are unapplicable. The novel embedding approach being able to tell apart an original sequence from surrogate ones shows its capability to distinguish signals beyond their spectral—or autocorrelation—similarities.One of the possible applications of the new approach concerns another longstanding issue, namely that of distinguishing noise from chaos. To this purpose, complementary information is provided by analyzing the asymptotic (long-time) behaviour of the so-called time-dependent divergence exponent. This embedding-dependent metric is commonly used to estimate—by processing its short-time linearly growing region—the maximum Lyapunov exponent out of a scalar sequence. However, insights on the kind of source generating the sequence can be extracted from the—usually overlooked—asymptotic behaviour of the divergence exponent. Moreover, in the case of chaotic sources, this analysis also provides a precise estimate of the system's correlation dimension. Besides describing the results concerning the discrimination of chaotic systems from noise sources, I will also discuss the possibility of using the related correlation dimension estimates to improve the third stage of the method introduced above for the identification of suitable embedding parameters. The discovery of chaos as a possible dynamical regime for nonlinear systems led to the search of chaotic behaviour in experimental recordings. In some fields, this search gave plenty of positive results: for example, chaotic dynamics was successfully identified and tamed in electronic circuits and laser-based optical setups. These two families of experimental chaotic systems eventually became versatile tools to study chaos and its possible applications. On the other hand, chaotic behaviour is also looked for in climate science, biology, neuroscience, and even economics. In these fields, nonlinearity is widespread: many smaller units interact nonlinearly, yielding a collective motion that can be described by means of few, nonlinearly coupled effective degrees of freedom. The corresponding recorded signals exhibit, in many cases, an irregular and complex evolution. A possible underlying chaotic evolution—as opposed to a stochastic one—would be of interest both to reveal the presence of determinism and to predict the system's future states. While some claims concerning the existence of chaos in these fields have been made, most results are debated or inconclusive. Nonstationarity, low signal-to-noise ratio, external perturbations and poor reproducibility are just few among the issues that hinder the search of chaos in natural systems. In the final part of this work, I will briefly discuss the problem of chasing chaos in experimental recordings by considering two example sequences, the first one generated by an electronic circuit and the second one corresponding to recordings of brain activity. The present thesis is organized as follows. The core concepts of time series analysis, including the key features of chaotic dynamics, are presented in Chapter 1. A brief review of the search for chaos in experimental systems is also provided; the difficulties concerning this quest in some research fields are also highlighted. Chapter 2 describes the embedding procedure and the issue of optimally choosing the related parameters. Thereupon, existing methods to carry out the embedding choice are reviewed and their limitations are pointed out. In addition, two embedding-dependent nonlinear techniques that are ordinarily used to characterize chaos, namely the estimation of correlation dimension by means of correlation integrals and the assessment of maximum Lyapunov exponent, are presented. The new approach for the identification of suitable embedding parameters, which makes up the core topic of the present thesis work, is the subject of Chapter 3 and 4. While Chapter 3 contains the theoretical outline of the approach, as well as its implementation details, Chapter 4 discusses the application of the approach to benchmark synthetic and experimental sequences, thus illustrating its perks and its limitations. The study of the asymptotic behaviour of the time-dependent divergent exponent is presented in Chapter 5. The alternative estimator of correlation dimension, which relies on this asymptotic metric, is discussed as a possible improvement to the approach described in Chapters 3, 4. The search for chaos out of experimental data is discussed in Chapter 6 by means of two examples of real-world recordings. Concluding remarks are finally drawn in Chapter 7.
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Desai, Palash. "Embeddings of a cray T3D supercomputer onto a free-space optical backplane." Thesis, McGill University, 1995. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=23743.

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The rapid increase in the demands for high bandwidth systems has motivated research in optoelectronic technologies and architectures. At McGill University, a five year five Major Project in Photonic Devices and Systems has been undertaken, with funding from the Canadian Institute for Telecommunications Research. One of the main goals of the project is to develop an optical backplane architecture capable of interconnecting several electronic printed circuit boards with an aggregate bandwidth on the order of 1 Terabit per second. Currently, the project is in its third year in which a representative subset of a Terabit Photonic Backplane is under development.
The objectives of this thesis are three fold. First, we motivate the study of optical backplanes by demonstrating that the interconnection network of a Cray T3D Supercomputer can be embedded onto the optical backplane. In particular, we demonstrate that the 3D mesh interconnect of the Cray T3D can be embedded into the "Dual Stream Linear HyperPlane" (9). Secondly, having established a motivation we then provide a detailed review of the functional specifications of an optical backplane. In particular, we provide a detailed review of the June 1995 backplane design (31). Thirdly, having established a motivation and a detailed design we then develop a executable software model of the June 1995 backplane using the VHDL hardware description language. The VHDL model is used to establish the functional correctness of the design. In addition, the VHDL model is used to develop a real-time simulator for the photonic backplane using 4013 Field Programmable Gate Arrays (FPGAs). The real time simulator can operate at a 4 MHz clock rate and can be used to test other electronic components such as the Message-Processors at realistic clock rates. (Abstract shortened by UMI.)
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Zaman, Faisal Ameen. "VN Embedding in SDN-based Metro Optical Network for Multimedia Services." Thesis, Université d'Ottawa / University of Ottawa, 2017. http://hdl.handle.net/10393/35933.

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Currently a growing number of users depend on the Edge Cloud Computing Paradigm in a Metro Optical Network (MON). This has led to increased competition among the Cloud Service Providers (CPs) to supply incentives for the user through guaranteed Quality of Service (QoS). If the CP fails to guarantee the QoS for the accepted request, then the user will move to another CP. Making an informed decision dynamically in such a sensitive situation demands that the CP knows the user's application requirements. The Software Defined Networking (SDN) paradigm enabled the CP to achieve such desired requirement. Therefore, a framework called Virtual Network Embedding on SDN-based Metro Optical Network (VNE-MON) is proposed in this Thesis. The use of SDN paradigm in the framework guarantees profit to the CP as well as QoS to the user.\par The design concept of the SDN control plane, raises concerns regarding its scalability, reliability and performance compared to a traditionally distributed network. To justify concerns regarding the SDN, the performance of VNE-MON and its possible dependancy on the controller location is investigated. Several strategies are proposed and formulated using Integer Linear Programming to determine the controller location in a MON. Performance results from the assessment of the VNE-MON illustrates that it is more stable compare to GMPLS-based network. It is evident that the controller location's attributes have a significant effect on the efficacy of the accepted VN request.
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Raheem-Kizchery, Ayesha Rubiath. "Ceramic coatings for silica and sapphire optical waveguides for high temperature embedding and sensing." Thesis, This resource online, 1990. http://scholar.lib.vt.edu/theses/available/etd-09052009-040217/.

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Books on the topic "Optimal embeddings"

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Milman, Mario. Extrapolation and optimal decompositions withapplications to analysis. Berlin: Springer-Verlag, 1994.

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Milman, Mario. Extrapolation and optimal decompositions: With applications to analysis. Berlin: Springer-Verlag, 1994.

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Riesen, Kaspar. Graph classification and clustering based on vector space embedding. New Jersey: World Scientific, 2010.

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Jakobson, Dmitry, Pierre Albin, and Frédéric Rochon. Geometric and spectral analysis. Providence, Rhode Island: American Mathematical Society, 2014.

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1943-, Gossez J. P., and Bonheure Denis, eds. Nonlinear elliptic partial differential equations: Workshop in celebration of Jean-Pierre Gossez's 65th birthday, September 2-4, 2009, Université libre de Bruxelles, Belgium. Providence, R.I: American Mathematical Society, 2011.

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Extrapolation and Optimal Decompositions: With Applications to Analysis. Springer, 1994.

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Delipetrev, Blagoj. Nested Algorithms for Optimal Reservoir Operation and Their Embedding in a Decision Support Platform. Taylor & Francis Group, 2020.

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Delipetrev, Blagoj. Nested Algorithms for Optimal Reservoir Operation and Their Embedding in a Decision Support Platform. Taylor & Francis Group, 2020.

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Nested Algorithms for Optimal Reservoir Operation and Their Embedding in a Decision Support Platform. Taylor & Francis Group, 2020.

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Delipetrev, Blagoj. Nested Algorithms for Optimal Reservoir Operation and Their Embedding in a Decision Support Platform. Taylor & Francis Group, 2016.

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Book chapters on the topic "Optimal embeddings"

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Voight, John. "Optimal embeddings." In Graduate Texts in Mathematics, 541–67. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-56694-4_30.

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Mutzel, Petra, and René Weiskircher. "Computing Optimal Embeddings for Planar Graphs." In Lecture Notes in Computer Science, 95–104. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/3-540-44968-x_10.

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Bläsius, Thomas, Tobias Friedrich, Maximilian Katzmann, and Anton Krohmer. "Hyperbolic Embeddings for Near-Optimal Greedy Routing." In 2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX), 199–208. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2018. http://dx.doi.org/10.1137/1.9781611975055.17.

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Das, Sajal K., and Aisheng Mao. "Optimal embeddings in the Hamming cube networks." In Lecture Notes in Computer Science, 205–16. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/bfb0020466.

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Forster, Jürgen, Niels Schmitt, and Hans Ulrich Simon. "Estimating the Optimal Margins of Embeddings in Euclidean Half Spaces." In Lecture Notes in Computer Science, 402–15. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-44581-1_26.

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Heun, Volker, and Ernst W. Mayr. "A general method for efficient embeddings of graphs into optimal hypercubes." In Lecture Notes in Computer Science, 222–33. Berlin, Heidelberg: Springer Berlin Heidelberg, 1996. http://dx.doi.org/10.1007/3-540-61626-8_29.

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Dessmark, Anders, Andrzej Lingas, and Eva-Marta Lundell. "Subexponential-Time Framework for Optimal Embeddings of Graphs in Integer Lattices." In Algorithm Theory - SWAT 2004, 248–59. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-27810-8_22.

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Mchedlidze, Tamara, and Antonios Symvonis. "Crossing-Optimal Acyclic Hamiltonian Path Completion and Its Application to Upward Topological Book Embeddings." In WALCOM: Algorithms and Computation, 250–61. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-00202-1_22.

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Ghaffari, Mohsen, and Christoph Lenzen. "Near-Optimal Distributed Tree Embedding." In Lecture Notes in Computer Science, 197–211. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-662-45174-8_14.

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Eppstein, David, and Kevin A. Wortman. "Optimal Embedding into Star Metrics." In Lecture Notes in Computer Science, 290–301. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-03367-4_26.

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Conference papers on the topic "Optimal embeddings"

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Xu, Hu, Bing Liu, Lei Shu, and Philip S. Yu. "Lifelong Domain Word Embedding via Meta-Learning." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. California: International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/627.

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Learning high-quality domain word embeddings is important for achieving good performance in many NLP tasks. General-purpose embeddings trained on large-scale corpora are often sub-optimal for domain-specific applications. However, domain-specific tasks often do not have large in-domain corpora for training high-quality domain embeddings. In this paper, we propose a novel lifelong learning setting for domain embedding. That is, when performing the new domain embedding, the system has seen many past domains, and it tries to expand the new in-domain corpus by exploiting the corpora from the past domains via meta-learning. The proposed meta-learner characterizes the similarities of the contexts of the same word in many domain corpora, which helps retrieve relevant data from the past domains to expand the new domain corpus. Experimental results show that domain embeddings produced from such a process improve the performance of the downstream tasks.
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Huang, Shou-Hsuan S., Hongfei Liu, and Rakesh M. Verma. "On Embeddings of Rectangles into Optimal Squares." In 1993 International Conference on Parallel Processing - ICPP'93 Vol3. IEEE, 1993. http://dx.doi.org/10.1109/icpp.1993.123.

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Figiel, Aleksander, Leon Kellerhals, Rolf Niedermeier, Matthias Rost, Stefan Schmid, and Philipp Zschoche. "Optimal Virtual Network Embeddings for Tree Topologies." In SPAA '21: 33rd ACM Symposium on Parallelism in Algorithms and Architectures. New York, NY, USA: ACM, 2021. http://dx.doi.org/10.1145/3409964.3461787.

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Grant, Elyot, Chinmay Hegde, and Piotr Indyk. "Nearly optimal linear embeddings into very low dimensions." In 2013 IEEE Global Conference on Signal and Information Processing (GlobalSIP). IEEE, 2013. http://dx.doi.org/10.1109/globalsip.2013.6737055.

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Vyavahare, Pooja, and Akhil Shetty. "On optimal embeddings for distributed computation of arbitrary functions." In 2014 International Conference on Signal Processing and Communications (SPCOM). IEEE, 2014. http://dx.doi.org/10.1109/spcom.2014.6984006.

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Zhang, Hanyuan, Xinyu Zhang, Qize Jiang, Baihua Zheng, Zhenbang Sun, Weiwei Sun, and Changhu Wang. "Trajectory Similarity Learning with Auxiliary Supervision and Optimal Matching." In Twenty-Ninth International Joint Conference on Artificial Intelligence and Seventeenth Pacific Rim International Conference on Artificial Intelligence {IJCAI-PRICAI-20}. California: International Joint Conferences on Artificial Intelligence Organization, 2020. http://dx.doi.org/10.24963/ijcai.2020/444.

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Trajectory similarity computation is a core problem in the field of trajectory data queries. However, the high time complexity of calculating the trajectory similarity has always been a bottleneck in real-world applications. Learning-based methods can map trajectories into a uniform embedding space to calculate the similarity of two trajectories with embeddings in constant time. In this paper, we propose a novel trajectory representation learning framework Traj2SimVec that performs scalable and robust trajectory similarity computation. We use a simple and fast trajectory simplification and indexing approach to obtain triplet training samples efficiently. We make the framework more robust via taking full use of the sub-trajectory similarity information as auxiliary supervision. Furthermore, the framework supports the point matching query by modeling the optimal matching relationship of trajectory points under different distance metrics. The comprehensive experiments on real-world datasets demonstrate that our model substantially outperforms all existing approaches.
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Rost, Matthias, Stefan Schmid, and Anja Feldmann. "It's About Time: On Optimal Virtual Network Embeddings under Temporal Flexibilities." In 2014 IEEE International Parallel & Distributed Processing Symposium (IPDPS). IEEE, 2014. http://dx.doi.org/10.1109/ipdps.2014.14.

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Doan, Khoa D., Saurav Manchanda, Suchismit Mahapatra, and Chandan K. Reddy. "Interpretable Graph Similarity Computation via Differentiable Optimal Alignment of Node Embeddings." In SIGIR '21: The 44th International ACM SIGIR Conference on Research and Development in Information Retrieval. New York, NY, USA: ACM, 2021. http://dx.doi.org/10.1145/3404835.3462960.

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Guo, Guibing, Shichang Ouyang, Fajie Yuan, and Xingwei Wang. "Approximating Word Ranking and Negative Sampling for Word Embedding." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. California: International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/569.

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CBOW (Continuous Bag-Of-Words) is one of the most commonly used techniques to generate word embeddings in various NLP tasks. However, it fails to reach the optimal performance due to uniform involvements of positive words and a simple sampling distribution of negative words. To resolve these issues, we propose OptRank to optimize word ranking and approximate negative sampling for bettering word embedding. Specifically, we first formalize word embedding as a ranking problem. Then, we weigh the positive words by their ranks such that highly ranked words have more importance, and adopt a dynamic sampling strategy to select informative negative words. In addition, an approximation method is designed to efficiently compute word ranks. Empirical experiments show that OptRank consistently outperforms its counterparts on a benchmark dataset with different sampling scales, especially when the sampled subset is small. The code and datasets can be obtained from https://github.com/ouououououou/OptRank.
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Bäckström, Christer, Peter Jonsson, and Sebastian Ordyniak. "Novel Structural Parameters for Acyclic Planning Using Tree Embeddings." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. California: International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/647.

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We introduce two novel structural parameters for acyclic planning (planning restricted to instances with acyclic causal graphs): up-depth and down-depth. We show that cost-optimal acyclic planning restricted to instances with bounded domain size and bounded up- or down-depth can be solved in polynomial time. For example, many of the tractable subclasses based on polytrees are covered by our result. We analyze the parameterized complexity of planning with bounded up- and down-depth: in a certain sense, down-depth has better computational properties than up-depth. Finally, we show that computing up- and down-depth are fixed-parameter tractable problems, just as many other structural parameters that are used in computer science. We view our results as a natural step towards understanding the complexity of acyclic planning with bounded treewidth and other parameters.
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