On the Information Dimension of Stochastic Processes

In 1959, Rényi proposed the information dimension and the d-dimensional entropy to measure the information content of general random variables. This paper proposes a generalization of information dimension to stochastic processes by defining the information dimension rate as the entropy rate of the uniformly quantized stochastic process divided by minus the logarithm of the quantizer step size 1/m in the limit as m to infty. It is demonstrated that the information dimension rate coincides with the rate-distortion dimension, defined as twice the rate-distortion function R(D) of the stochastic process divided by -log (D) in the limit as D downarrow 0 . It is further shown that among all multivariate stationary processes with a given (matrix-valued) spectral distribution function (SDF), the Gaussian process has the largest information dimension rate and the information dimension rate of multivariate stationary Gaussian processes is given by the average rank of the derivative of the SDF. The presented results reveal that the fundamental limits of almost zero-distortion recovery via compressible signal pursuit and almost lossless analog compression are different in general.The work of Bernhard C. Geiger has partly been funded by the Erwin Schrödinger Fellowship J 3765 of the Austrian Science Fund and by the German Ministry of Education and Research in the framework of an Alexander von Humboldt Professorship. The Know-Center is funded within the Austrian COMET Program - Competence Centers for Excellent Technologies - under the auspices of the Austrian Federal Ministry of Transport, Innovation and Technology, the Austrian Federal Ministry of Digital and Economic Affairs, and by the State of Styria. COMET is managed by the Austrian Research Promotion Agency FFG. The work of Tobias Koch has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 714161), from the 7th European Union Framework Programme under Grant 333680, from the Ministerio de EconomÍa y Competitividad of Spain under Grants TEC2013-41718-R, RYC-2014-16332, and TEC2016-78434-C3-3-R (AEI/FEDER, EU), and from the Comunidad de Madrid under Grant S2103/ICE-2845

A Transformer Architecture for Online Gesture Recognition of Mathematical Expressions

The Transformer architecture is shown to provide a powerful framework as an end-to-end model for building expression trees from online handwritten gestures corresponding to glyph strokes. In particular, the attention mechanism was successfully used to encode, learn and enforce the underlying syntax of expressions creating latent representations that are correctly decoded to the exact mathematical expression tree, providing robustness to ablated inputs and unseen glyphs. For the first time, the encoder is fed with spatio-temporal data tokens potentially forming an infinitely large vocabulary, which finds applications beyond that of online gesture recognition. A new supervised dataset of online handwriting gestures is provided for training models on generic handwriting recognition tasks and a new metric is proposed for the evaluation of the syntactic correctness of the output expression trees. A small Transformer model suitable for edge inference was successfully trained to an average normalised Levenshtein accuracy of 94%, resulting in valid postfix RPN tree representation for 94% of predictions.Comment: 12 pages, 3 Figures, 4 Table

On the information dimension rate of stochastic processes

Proceeding of: 2017 IEEE International Symposium on Information Theory, Aachen, Germany, 25-30 June 2017Jalali and Poor ("Universal compressed sensing," arXiv:1406.7807v3, Jan. 2016) have recently proposed a generalization of Rényi's information dimension to stationary stochastic processes by defining the information dimension of the stochastic process as the information dimension of k samples divided by k in the limit as k →∞ to. This paper proposes an alternative definition of information dimension as the entropy rate of the uniformly-quantized stochastic process divided by minus the logarithm of the quantizer step size 1/m in the limit as m →∞ ; to. It is demonstrated that both definitions are equivalent for stochastic processes that are ψ*-mixing, but that they may differ in general. In particular, it is shown that for Gaussian processes with essentially-bounded power spectral density (PSD), the proposed information dimension equals the Lebesgue measure of the PSD's support. This is in stark contrast to the information dimension proposed by Jalali and Poor, which is 1 if the process's PSD is positive on a set of positive Lebesgue measure, irrespective of its support size.The work of Bernhard C. Geiger has been funded by the Erwin Schrödinger Fellowship J 3765 of the Austrian Science Fund and by the German Ministry of Education and Research in the framework of an Alexander von Humboldt Professorship. The work of Tobias Koch has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 714161), from the 7th European Union Framework Programme under Grant 333680, from the Spanish Ministerio de Economía y Competitividad under Grants TEC2013- 41718-R, RYC-2014-16332 and TEC2016-78434-C3-3-R (AEI/FEDER, EU), and from the Comunidad de Madrid under Grant S2103/ICE-2845

Differential Dynamics at Glycosidic Linkages of an Oligosaccharide as Revealed by 13C NMR Spin Relaxation and Stochastic Modeling

Among biomolecules, carbohydrates are unique in that not only can linkages be formed through different positions but the structures may also be branched. The trisaccharide \uf062-D-Glcp-(1\uf0ae3)[\uf062-D-Glcp-(1\uf0ae2)]-\uf061-D-Manp-OMe represents a model of a branched vicinally disubstituted structure. A 13C site-specific isotopologue with labeling in each of the two terminal glucosyl residues enabled acquisition of high-quality 13C NMR relaxation parameters T1, T2 and heteronuclear NOE, with standard deviations of \uf0a3 0.5%. For interpretation of the experimental NMR data a diffusive chain model was used in which the dynamics of the glycosidic linkages is coupled to the global reorientation motion of the trisaccharide. Brownian dynamics simulations relying on the potential of mean force at the glycosidic linkages were employed to evaluate spectral densities of the spin probes. Calculated NMR relaxation parameters showed very good agreement with experimental data, deviating < 3%. The resulting dynamics is described by correlation times of 196 ps and 174 ps for the \uf062-(1\uf0ae2)- and \uf062-(1\uf0ae3)-linked glucosyl residues, respectively, i.e., different and linkage dependent. Notably, the devised computational protocol was performed without any fitting of parameters

Iron-catalyzed regioselective synthesis of 2-arylbenzoxazoles and 2-arylbenzothiazoles via alternative reaction pathways

A one‐pot regioselective method for the preparation of 2‐arylbenzoxazoles from N‐arylbenzamides has been developed using iron(III)‐catalyzed bromination of the aryl ring, followed by copper(I)‐catalyzed O‐cyclization with the benzamide side chain. In contrast, reaction of N‐arylthiobenzamides with N‐bromosuccinimide and iron triflimide led directly to the isolation of the corresponding 2‐arylbenzothiazoles via intramolecular C–S bond formation. Mechanistic and control experiments suggest that in this case, bromination occurs at the sulfur atom, resulting in a reactive intermediate that can undergo electrophilic aromatic substitution and S‐cyclization. The scope of both processes was explored yielding a range of structural analogues, including a pharmaceutically active compound for the treatment of Duchenne muscular dystrophy and an affinity agent of the amyloid‐beta protein in Alzheimer's disease

Simplicial models of social aggregation I

This paper presents the foundational ideas for a new way of modeling social aggregation. Traditional approaches have been using network theory, and the theory of random networks. Under that paradigm, every social agent is represented by a node, and every social interaction is represented by a segment connecting two nodes. Early work in family interactions, as well as more recent work in the study of terrorist organizations, shows that network modeling may be insufficient to describe the complexity of human social structures. Specifically, network theory does not seem to have enough flexibility to represent higher order aggregations, where several agents interact as a group, rather than as a collection of pairs. The model we present here uses a well established mathematical theory, the theory of simplicial complexes, to address this complex issue prevalent in interpersonal and intergroup communication. The theory enables us to provide a richer graphical representation of social interactions, and to determine quantitative mechanisms to describe the robustness of a social structure. We also propose a methodology to create random simplicial complexes, with the purpose of providing a new method to simulate computationally the creation and disgregation of social structures. Finally, we propose several measures which could be taken and observed in order to describe and study an actual social aggregation occurring in interpersonal and intergroup contexts.Comment: 31 page

A survey experiment on information, inattention and online privacy

Personal data lie at the forefront of different business models and constitute the main source of revenue of several online companies. In many cases, consumers may have incomplete information or may be inattentive about the digital transactions of their data. This paper investigates whether highlighting positive or negative aspects of online privacy policies, thereby mitigating the informational problem, can affect consumers’ privacy actions and attitudes. Results of an online survey experiment indicate that participants adopt a more conservative stance on disclosing sensitive and identifiable information, even when positive attitudes of companies towards their privacy are made salient, compared to when privacy is not mentioned. On the other hand, they do not change their attitudes and social actions towards privacy. These findings suggest that privacy behavior is not necessarily sensitive to exposure to objective threats or benefits of disclosing personal information. Rather, people are inattentive and their dormant privacy concerns may manifest only when consumers are asked to think about privacy.publishersversionpublishe

Lagrangian planetary equations in Schwarzschild space--time

We have developed a method to study the effects of a perturbation to the motion of a test point--like object in a Schwarzschild spacetime. Such a method is the extension of the Lagrangian planetary equations of classical celestial mechanics into the framework of the full theory of general relativity. The method provides a natural approach to account for relativistic effects in the unperturbed problem in an exact way.Comment: 7 pages; revtex; accepted for publication in Class. Quantum Gra