1,721,649 research outputs found
Robust Radical Sylvester-Gallai Theorem for Quadratics
Full text linkWe prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials. More precisely, given a parameter 0 < δ ≤ 1 and a finite collection ℱ of irreducible and pairwise independent polynomials of degree at most 2, we say that ℱ is a (δ, 2)-radical Sylvester-Gallai configuration if for any polynomial F_i ∈ ℱ, there exist δ(|ℱ|-1) polynomials F_j such that |rad (F_i, F_j) ∩ ℱ| ≥ 3, that is, the radical of F_i, F_j contains a third polynomial in the set. We prove that any (δ, 2)-radical Sylvester-Gallai configuration ℱ must be of low dimension: that is dim span_ℂ{ℱ} = poly(1/δ)
Uniform Bounds on Product Sylvester-Gallai Configurations
Full text linkIn this work, we explore a non-linear extension of the classical Sylvester-Gallai configuration. Let be an algebraically closed field of characteristic zero, and let ℱ = {F_1, …, F_m} ⊂ [x_1, …, x_N] denote a collection of irreducible homogeneous polynomials of degree at most d, where each F_i is not a scalar multiple of any other F_j for i ≠ j. We define ℱ to be a product Sylvester-Gallai configuration if, for any two distinct polynomials F_i, F_j ∈ ℱ, the following condition is satisfied: ∏_{k≠i, j} F_k ∈ rad (F_i, F_j) .
We prove that product Sylvester-Gallai configurations are inherently low dimensional. Specifically, we show that there exists a function λ : ℕ → ℕ, independent of , N, and m, such that any product Sylvester-Gallai configuration must satisfy: dim(span_(ℱ)) ≤ λ(d).
This result generalizes the main theorems from (Shpilka 2019, Peleg and Shpilka 2020, Oliveira and Sengupta 2023), and gets us one step closer to a full derandomization of the polynomial identity testing problem for the class of depth 4 circuits with bounded top and bottom fan-in
Radical Sylvester-Gallai Theorem for Tuples of Quadratics
Full text linkWe prove a higher codimensional radical Sylvester-Gallai type theorem for quadratic polynomials, simultaneously generalizing [Hansen, 1965; Shpilka, 2020]. Hansen’s theorem is a high-dimensional version of the classical Sylvester-Gallai theorem in which the incidence condition is given by high-dimensional flats instead of lines. We generalize Hansen’s theorem to the setting of quadratic forms in a polynomial ring, where the incidence condition is given by radical membership in a high-codimensional ideal. Our main theorem is also a generalization of the quadratic Sylvester-Gallai Theorem of [Shpilka, 2020].
Our work is the first to prove a radical Sylvester-Gallai type theorem for arbitrary codimension k ≥ 2, whereas previous works [Shpilka, 2020; Shir Peleg and Amir Shpilka, 2020; Shir Peleg and Amir Shpilka, 2021; Garg et al., 2022] considered the case of codimension 2 ideals. Our techniques combine algebraic geometric and combinatorial arguments. A key ingredient is a structural result for ideals generated by a constant number of quadratics, showing that such ideals must be radical whenever the quadratic forms are far apart. Using the wide algebras defined in [Garg et al., 2022], combined with results about integral ring extensions and dimension theory, we develop new techniques for studying such ideals generated by quadratic forms. One advantage of our approach is that it does not need the finer classification theorems for codimension 2 complete intersection of quadratics proved in [Shpilka, 2020; Garg et al., 2022]
Applications of Fog Computing, IoT, and Cloud Computing in Healthcare Multilayer Networks
No full textFog computing and the IoT are increasingly vital technologies in developing smart healthcare solutions. This synergy offers a promising avenue to enhance healthcare delivery, patient monitoring, and health data management, especially with the growing demand for efficient, real-time healthcare services.
The impact of leveraging multilayer network theory systems, where fog computing acts as the copula nodes linking different layers, on smart healthcare is profound and multifaceted. This approach enhances the architecture and functionality of healthcare systems, leading to significant improvements in patient care, system efficiency, and innovation.
This chapter focuses on the main features of telemedicine platforms rotating around fog computing, showing the main cost/benefit improvements, even regarding quality of care and patient-centric considerations. Blockchains and artificial intelligence scale up the capacity of interconnected ecosystems, with value-co-creating patterns shared by patients, doctors, and other stakeholders
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Robust and Automated Variational Inference
No full textBayesian inference offers a sound and consistent framework to analyze data under uncertainty. Good decision making under uncertainty requires writing elaborate probabilistic models to model and simulate the systems we find around us. The hope is to obtain calibrated probabilistic predictions on unseen conditions. The challenge is that, inference for these models is intractable in general. This necessitates application of approximate inference techniques which can perform fast and accurate inference on probabilistic models.
The success of Bayesian methods and any application which builds on it, depends to a large extent on the success of approximate inference algorithm chosen by the user.
Variational inference has emerged as a popular approximate inference algorithm. It can be seen as an optimization problem where the task is to find an optimal distribution as an approximation to the true intractable posterior. This optimization requires computing fast and unbiased gradients.
The contributions of this thesis can be broadly divided into two themes. The first part is the application of variational inference to the task of fitting a Gaussian process model where we have access to batches of observations, which do not have a numerical value, but are available as rankings in a set. Interestingly, the approximation of softmax link function for multi-class Gaussian Process classification can also be seen as a pairwise comparison of classes. This viewpoint helps in deriving a similar variational inference algorithm for scaling Gaussian Process classification to settings where the number of classes and data points is very large compared to existing algorithms.
The second part of thesis deals with automated variational inference as a general tool of inference for probabilistic programs in context of modern programming frameworks which use automatic differentiation to compute gradients. The recent innovations in automatic differentiation software and algorithmic improvements in the form of computation of noisy unbiased gradients with Monte Carlo and mini-batching has made it possible to use model agnostic and standardized automated stochastic optimization-based algorithms and scale it to large datasets.
In settings where accurate posterior inference is important, this work shows some potential pitfalls of current practices which may lead to incorrect conclusions. This work provides a wide set of diagnostic tools to evaluate if the stochastic optimization has worked well enough and the obtained solution is accurate enough to be used as approximation to the true posterior. This work concludes by providing a set of recommendations to the end user which is- either to use a more expressive approximating distribution or to reparameterize the model itself to hopefully end up with an easier posterior distribution
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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