Academic literature on the topic 'KL-inequality'

We propose a general variational framework of fair clustering, which integrates an original Kullback-Leibler (KL) fairness term with a large class of clustering objectives, including prototype or graph based. Fundamentally different from the existing combinatorial and spectral solutions, our variational multi-term approach enables to control the trade-off levels between the fairness and clustering objectives. We derive a general tight upper bound based on a concave-convex decomposition of our fairness term, its Lipschitz-gradient property and the Pinsker’s inequality. Our tight upper bound can be jointly optimized with various clustering objectives, while yielding a scalable solution, with convergence guarantee. Interestingly, at each iteration, it performs an independent update for each assignment variable. Therefore, it can be easily distributed for large-scale datasets. This scalability is important as it enables to explore different trade-off levels between the fairness and clustering objectives. Unlike spectral relaxation, our formulation does not require computing its eigenvalue decomposition. We report comprehensive evaluations and comparisons with state-of-the-art methods over various fair clustering benchmarks, which show that our variational formulation can yield highly competitive solutions in terms of fairness and clustering objectives.

The citizenization of migrants has become a challenging practical issue and a significant research topic within the realm of social justice. Nowadays, with a large number of studies concerning the unjust lives of migrants, few researches probe their struggle with inequality. This struggling is reflected in space, in the reterritorialization of the migrants. Reterritorialization involves the reconstruction of the meaning and function of a territory. Deterritorialization meaning the weakening of territoriality. This paper concentrates on the reterritorialization of migrants, and attempts to fill the research gap through an in-depth case study of a representative urban village located in Guangzhou. The data for this study were collected through three rounds of questionnaire surveys conducted with migrants in 2016, 2020, and 2023. Government statistics and semi-structured interviews with local government officials and original villagers are also used in the research. It has been found the initial territorialization of KL Village was driven by the city’s development and the demand for industries and migrant labor. Subsequently, the concentration of migrants led to the transformation of the village into an urban village that caters to professional markets. The distinctive characteristics of the industry and the lifestyles of migrants resulted in the departure of local villagers, making migrants the sole users of the village and facilitating its reterritorialization. Migrants reterritorialized various systems within the village, including the commercial system, public service system, and transportation system. However, the reterritorialization remains uncertain. The urban villages may undergo deterritorialization when industries decline or when significant events with negative societal impacts occur. This paper further emphasizes the importance of governments recognizing the circumstances faced by migrants and ensuring the provision of stable spaces for them. In general, this paper makes an empirical contribution by offering a fresh perspective to the study of migrants and territorialization in China.

The theory of Bayesian learning incorporates the use of Student-t Processes to model heavy-tailed distributions and datasets with outliers. However, despite Student-t Processes having a similar computational complexity as Gaussian Processes, there has been limited emphasis on the sparse representation of this model. This is mainly due to the increased difficulty in modeling and computation compared to previous sparse Gaussian Processes. Our motivation is to address the need for a sparse representation framework that reduces computational complexity, allowing Student-t Processes to be more flexible for real-world datasets. To achieve this, we leverage the conditional distribution of Student-t Processes to introduce sparse inducing points. Bayesian methods and variational inference are then utilized to derive a well-defined lower bound, facilitating more efficient optimization of our model through stochastic gradient descent. We propose two methods for computing the variational lower bound, one utilizing Monte Carlo sampling and the other employing Jensen's inequality to compute the KL regularization term in the loss function. We propose adopting these approaches as viable alternatives to Gaussian processes when the data might contain outliers or exhibit heavy-tailed behavior, and we provide specific recommendations for their applicability. We evaluate the two proposed approaches on various synthetic and real-world datasets from UCI and Kaggle, demonstrating their effectiveness compared to baseline methods in terms of computational complexity and accuracy, as well as their robustness to outliers.

The article investigates the phonosemantic features of D. Krien's feminist novel "Die Liebe im Ernstfall" in order to establish the symbolic meanings of the initial consonant phonemes encountered in this text. We analysed the theoretical and methodological basis of contemporary German prose and highlighted the typical labelled features of the feminist literature genre (female authors, female protagonists, subtle authorial humour, hard women's fates, struggle for equal rights, fluidity, etc.) and compared them with the text under study. Based on the opinions of expert critics, the plot, idea, purpose of the novel, and our own reading experience, we argue that D. Krien's novel Die Liebe im Ernstfall meets all the features of a women's novel and contains five separate chapters dedicated to each individual heroine and her difficult life path. D. Krien presents the readers with a veiled problem of women's relationships in society and their relations with men, and also points out the inequality of rights in the modern world. Therefore, the paper investigates the functioning of phonemes in the feminist text Die Liebe im Ernstfall by means of a continuous sample and uses the descriptive method and component-statistical methods, namely the chi-square method and the coefficient of mutual conjugation, to establish the symbolic meanings of the initial phoneme combinations. We processed 5056 units of phonemic vocabulary and published the results of the most frequently used words. We have practically established that the most frequently used vocabulary in the novel belongs to the phoneme systems ∫t-, ∫p-,kl-, kr-, gr-, pr-, tr-, ∫r-, and we have also highlighted adjectives of negative perception, colour adjectives, abstract nouns and nouns of negative meaning. In the course of the study of the novel, we found that the modern feminist text by D. Krien "Die Liebe im Ernstfall" is characterised by the minor tone of the phonemic vocabulary used, which contains the leading phoneme [∫] and [r] in the second or third position and a "negative" emotional impact on the reader, the reliability of the results obtained was confirmed by the statistically significant chi-square criterion and the coefficient of mutual conjugation.

The purpose of this study is to fi nd out the perception of college students' experiences of sexual harassment, types, victims, and countermeasures against sexual harassment in college, and to develop an educational program to improve awareness. As a result of the study, it was found that many students had experiences of sexual harassment as well as experiences of gender inequality in various ways. In this preventive dimension, preventive education in various ways within universities will be of great help in improving college students' gender recognition and concrete responses. Therefore, violence prevention education based on gender equality should be systematically and continuously carried out, and support at the university level would be needed to make various manuals that are of practical help to students. In addition, it will be possible to achieve healthy heterosexual relationships that can respect and care for each other through confl ict resolution education programs that prevent dating violence and are necessary for dating. In everyday life, it is necessary to educate students to critically refl ect on the overall media analysis so that they can have a critical consciousness from a gender equality perspective. For this education, it is necessary to develop education through diversifi cation of teaching methods, such as learner-centered participatory education and media education centered on critical awareness of daily life. Through these various educational programs, it is expected to have an effect on correct gender recognition and improvement for students, and this is expected to contribute to establishing a healthy campus culture within the university.

Background:COVID-19 has been shown to significantly affect the vulnerable population [1,2]. Among them, patients suffering from inflammatory rheumatic diseases, and especially the immunosuppressed [3].Objectives:to assess the effect of SARS-CoV-2 on the course and the treatment of rheumatic inflammatory diseases.Methods:from February to December 2020, 46 patients with inflammatory rheumatic diseases were included (32 female) that got infected with the SARS-CoV-2. Mean age was 65 years old, 17 were smokers, 12 had arterial hypertension, 8 diabetes mellitus, and 3 hypothyroidism. Most of them had their comorbidities well-controlled and their rheumatic disease was in remission. More specifically, 24 patients had rheumatoid arthritis, 13 psoriatic arthritis, and 9 ankylosing spondylitis. All patients were under treatment with conventional synthetic (cs) and/or biological (b) disease-modifying anti-rheumatic drugs (DMARDs), while 7 of them were also on treatment with glucocorticoids (GC) (<5mg/day). Twenty-eight patients were on tumor necrosis alpha (TNF-α) inhibitors (19 as monotherapy), 4 on anti- interleukin (IL)-6 monotherapy, 3 on Janus Kinase (JAK) inhibitors plus on low dose methotrexate (MTX), and the rest (11 patients) were on a csDMARD with or without GCs.Results:positive patients with the SARS-CoV-2, instructed to discontinue their immunosuppressive treatment, except GCs that were adjusted for their disease. Most patients (37 out of 46) had a mild disease course and their symptomatology was nothing more than a simple flu-like syndrome. Furthermore, on 9 of them olfactory dysfunction and gastrointestinal manifestations as well as low grade fever were noted but without the need of a hospital admission. On the other hand, only 5 patients needed hospitalization (2 on MTX monotherapy and 3 on combination therapy) due to dyspnea with low oxygen saturation (hypoxemia) and high fever. From those 5, 3 had a short in-hospital stay, while 2 developed pneumonia and a longer in-hospital stay was required in order to get the appropriate treatment. None of the patients did not require an intensive care unit admission. Finally, in 14 patients that got infected from February to May 2020, viral antibodies had been measured. All patients had high titres of IgG antibodies in their serum for as long as six months after their infection. Of note, none of the infected patients were smokers.Conclusion:patients with rheumatic diseases that are in remission using low doses of GCs and DMARDs, have almost the same chances with the general population to have a serious course of their infection with the SARS-Cov-2. In addition, in these patients, the immune response appears to be adequate, both in the production and maintenance of antibodies, which appear to be maintained for at least 6 months after infection.References:[1]Patel JA, Nielsen FBH, Badiani AA, Assi S, Unadkat VA, Patel B, et al. Poverty, inequality and COVID-19: the forgotten vulnerable. Public Health. 2020;183:110-111. Doi: 10.1016/j.puhe.2020.05.006.[2]Poteat T, Millet GA, Nelson LE, Beyrer C. Understanding COVID-19 risks and vulnerabilities among black communities in America: the lethal force of syndemics. Ann Epidemiol. 2020;47:1-3. Doi: 10.1016/j.annepidem.2020.05.004.[3]Gianfrancesco MA, Hyrich KL, Gossec L, Strangfeld A, Carmona L, Mateus EF, et al. Rheumatic disease and COVID-19: initial data from the COVID-19 Global Rheumatology Alliance provider registries. Lancet Rheumatol. 2020;2(5):e250-e253. Doi: 10.1016/S2665-9913(20)30095-3.Disclosure of Interests:None declared.

In this paper, we study the local variational geometry of the optimal solution set of the trust region subproblem (TRS), which minimizes a general, possibly nonconvex, quadratic function over the unit ball. Specifically, we demonstrate that a Hölderian error bound holds globally for the TRS with modulus 1/4, and the Kurdyka-Łojasiewicz (KL) inequality holds locally for the TRS with a KL exponent 3/4 at any optimal solution. We further prove that, unless in a special case, the Hölderian error bound modulus and the KL exponent is 1/2. Finally, as a byproduct, we further apply the obtained KL property to show that projected gradient methods studied elsewhere for solving the TRS achieve a local sublinear or even linear rate of convergence with probability 1 by choosing a proper initial point.

AbstractThe Polyak–Łojasiewicz inequality (PŁI) in $$\mathbb {R}^d$$ R d is a natural condition for proving convergence of gradient descent algorithms (Karimi et al. in: Frasconi et al. (eds) Machine learning and knowledge discovery in databases, Springer International Publishing, Cham, pp 795–811, 2016). In the present paper, we study an analogue of PŁI on the space of probability measures $$\mathcal {P}(\mathbb {R}^d)$$ P ( R d ) and show that it is a natural condition for showing exponential convergence of a class of birth-death processes related to certain mean-field optimization problems. We verify PŁI for a broad class of such problems for energy functions regularised by the KL-divergence.

Motivé par l'omniprésence de l'optimisation dans de nombreux domaines de la science et de l'ingénierie, en particulier dans la science des données, ce manuscrit de thèse exploite le lien étroit entre les systèmes dynamiques dissipatifs à temps continu et les algorithmes d'optimisation pour fournir une analyse systématique du comportement global et local de plusieurs systèmes du premier et du second ordre, en se concentrant sur le cadre convexe, stochastique et en dimension infinie d'une part, et le cadre non convexe, déterministe et en dimension finie d'autre part. Pour les problèmes de minimisation convexe stochastique dans des espaces de Hilbert réels séparables de dimension infinie, notre proposition clé est de les analyser à travers le prisme des équations différentielles stochastiques (EDS) et des inclusions différentielles stochastiques (IDS), ainsi que de leurs variantes inertielles. Nous considérons d'abord les problèmes convexes différentiables lisses et les EDS du premier ordre, en démontrant une convergence faible presque sûre vers les minimiseurs sous hypothèse d'intégrabilité du bruit et en fournissant une analyse globale et locale complète de la complexité. Nous étudions également des problèmes convexes non lisses composites utilisant des IDS du premier ordre et montrons que, sous des conditions d'intégrabilité du bruit, la convergence faible presque sûre des trajectoires vers les minimiseurs, et avec la régularisation de Tikhonov la convergence forte presque sûre des trajectoires vers la solution de norme minimale. Nous développons ensuite un cadre mathématique unifié pour analyser la dynamique inertielle stochastique du second ordre via la reparamétrisation temporelle et le moyennage de la dynamique stochastique du premier ordre, ce qui permet d'obtenir une convergence faible presque sûre des trajectoires vers les minimiseurs et une convergence rapide des valeurs et des gradients. Ces résultats sont étendus à des EDS plus générales du second ordre avec un amortissement visqueux et Hessien, en utilisant une analyse de Lyapunov spécifique pour prouver la convergence et établir de nouveaux taux de convergence. Enfin, nous étudions des problèmes d'optimisation déterministes non convexes et proposons plusieurs algorithmes inertiels pour les résoudre, dérivés d'équations différentielles ordinaires (EDO) du second ordre combinant à la fois un amortissement visqueux sans vanité et un amortissement géométrique piloté par le Hessien, sous des formes explicites et implicites. Nous prouvons d'abord la convergence des trajectoires en temps continu des EDO vers un point critique pour des objectives vérifiant la propriété de Kurdyka-Lojasiewicz (KL) avec des taux explicites, et génériquement vers un minimum local si l'objective est Morse. De plus, nous proposons des schémas algorithmiques par une discrétisation appropriée de ces EDO et montrons que toutes les propriétés précédentes des trajectoires en temps continu sont toujours valables dans le cadre discret sous réserve d'un choix approprié de la taille du pas
Motivated by the ubiquity of optimization in many areas of science and engineering, particularly in data science, this thesis exploits the close link between continuous-time dissipative dynamical systems and optimization algorithms to provide a systematic analysis of the global and local behavior of several first- and second-order systems, focusing on convex, stochastic, and infinite-dimensional settings on the one hand, and non-convex, deterministic, and finite-dimensional settings on the other hand. For stochastic convex minimization problems in infinite-dimensional separable real Hilbert spaces, our key proposal is to analyze them through the lens of stochastic differential equations (SDEs) and inclusions (SDIs), as well as their inertial variants. We first consider smooth differentiable convex problems and first-order SDEs, demonstrating almost sure weak convergence towards minimizers under integrability of the noise and providing a comprehensive global and local complexity analysis. We also study composite non-smooth convex problems using first-order SDIs, and show under integrability conditions on the noise, almost sure weak convergence of the trajectory towards a minimizer, with Tikhonov regularization almost sure strong convergence of trajectory to the minimal norm solution. We then turn to developing a unified mathematical framework for analyzing second-order stochastic inertial dynamics via time scaling and averaging of stochastic first-order dynamics, achieving almost sure weak convergence of trajectories towards minimizers and fast convergence of values and gradients. These results are extended to more general second-order SDEs with viscous and Hessian-driven damping, utilizing a dedicated Lyapunov analysis to prove convergence and establish new convergence rates. Finally, we study deterministic non-convex optimization problems and propose several inertial algorithms to solve them derived from second-order ordinary differential equations (ODEs) combining both non-vanishing viscous damping and geometric Hessian-driven damping in explicit and implicit forms. We first prove convergence of the continuous-time trajectories of the ODEs to a critical point under the Kurdyka-Lojasiewicz (KL) property with explicit rates, and generically to a local minimum under a Morse condition. Moreover, we propose algorithmic schemes by appropriate discretization of these ODEs and show that all previous properties of the continuous-time trajectories still hold in the discrete setting under a proper choice of the stepsize

Abstract A computational study of the temperature and flow fields of solidifying Cd1-xMnxTe in a vertical Bridgeman furnace is conducted. Influence of furnace temperature profile, pulling velocity, Grashof number Gr, Prandtl number Pr, ratio of thermal to liquid conductivity Kl/Ks, and Stefan number is examined. Convection consistently causes an undesirable convex interface (as seen from the solid). The solid liquid interface is found to be sensitive to only the temperature profile and Kl/Ks. The flow field is significantly affected by Kl/Ks, the furnace temperature profile, and Gr. Due to the pseudo state model employed, the pulling velocity shows little effect for the parameter values of this study. The flow is bicellular, with a strong upper cell (caused by the particular furnace temperature profile) and a weak lower cell (caused by the inequality of Kl and Ks). These cells are competitive, with the upper cell often absorbing the lower one. When the lower cell is present the interface is more convex. At high Gr, the flow is bicellular with mass transport between the two cells occurring by diffusion.

This paper proposed a surrogate model to simplify the process of estimating the axial compressive capacity of cold-formed steel (CFS) open built-up sections composed of lipped channels with different section sizes, thickness, length, and connector spacing. The surrogate model was developed based on the current design methods, i.e., the Effective Width Method (EWM) and Direct Strength Method (DSM), which are codified in the North American Specification AISI S100-16. This new model features two surface regression equations with a boundary inequality criteria, anchored on two important parameters, i.e., modified slenderness ratio, (KL/r)m and minimum thickness-to-width ratio (t/w)min of the built-up sections. The model was validated with 1089 sets of the experimental results data collected from previous research tested on the axial capacity of CFS open built-up sections with the different design configurations. The proposed surrogate model is aimed to simplify the design process among practising engineers for a quick preliminary calculation of the axial compressive capacity of these new CFS open built-up sections.