Search

We establish a deep learning theory for distribution regression with deep convolutional neural networks (DCNNs). Deep learning based on structured deep neural networks has been powerful in practical applications. Generalization analysis for regression with DCNNs has been carried out very recently. However, for the distribution regression problem in which the input variables are probability measures, there is no mathematical model or theoretical analysis of DCNN-based learning theory. One of the difficulties is that the classical neural network structure requires the input variable to be a Euclidean vector. When the input samples are probability distributions, the traditional neural network structure cannot be directly used. A well-defined DCNN framework for distribution regression is desirable. In this paper, we overcome the difficulty and establish a novel DCNN-based learning theory for a two-stage distribution regression model. Firstly, we realize an approximation theory for functionals defined on the set of Borel probability measures with the proposed DCNN framework. Then, we show that the hypothesis space is well-defined by rigorously proving its compactness. Furthermore, in the hypothesis space induced by the general DCNN framework with distribution inputs, by using a two-stage error decomposition technique, we derive a novel DCNN-based two-stage oracle inequality and optimal learning rates (up to a logarithmic factor) for the proposed algorithm for distribution regression. [ABSTRACT FROM AUTHOR]

In this note, I present my personal view on the interaction of the three areas Automated Programming, Symbolic Computation, and Machine Learning. Programming is the activity of finding a (hopefully) correct program (algorithm) for a given problem. Programming is central to automation in all areas and is considered one of the most creative human activities. However, already very early in the history of programming, people started to "jump to the meta-level" of programming, i.e., started to develop procedures that automate, or semi-automate, (various aspects or parts of) the process of programming. This area has various names like "Automated Programming", "Automated Algorithm Synthesis", etc. Developing compilers can be considered an early example of a problem in automated programming. Automated reasoners for proving the correctness of programs with respect to a specification is an advanced example of a topic in automated programming. ChatGPT producing (amazingly good) programs from problem specifications in natural language is a recent example of automated programming. Programming tends to become the most important activity as the level of technological sophistication increases. Therefore, automating programming is maybe the most exciting and relevant technological endeavor today. It also will have enormous impact on the global job market in the software industry. Roughly, I see two main approaches to automated programming: symbolic computation and machine learning. In this note, I explain how the two approaches work and that they are fundamentally different because they address two completely different ways of how problems are specified. Together, the two approaches constitute (part of) what some people like to call "artificial intelligence". In my analysis, both approaches are just part of (algorithmic) mathematics. The approaches, like all non-trivial mathematical methods, need quite some intelligence on the side of the human inventors of the methods. However, applying the methods is just "machine execution" of algorithms. It is misleading to call the application "machine intelligence" or "artificial intelligence". The analysis of the two approaches to automated programming also suggests that the two approaches, in the future, should be combined to achieve even higher levels of sophistication. At the end of this note, I propose some research questions for this new direction. [ABSTRACT FROM AUTHOR]

In this work, we investigate the influence of general damping and potential terms on the blow-up and lifespan estimates for energy solutions to power-type semilinear wave equations. The space-dependent damping and potential functions are assumed to be critical or short range, spherically symmetric perturbation. The blow up results and the upper bound of lifespan estimates are obtained by the so-called test function method. The key ingredient is to construct special positive solutions to the linear dual problem with the desired asymptotic behavior, which is reduced, in turn, to constructing solutions to certain elliptic "eigenvalue" problems. [ABSTRACT FROM AUTHOR]

Purpose of Review: Aging leads to decline in bone mass and quality starting at age 30 in humans. All mammals undergo a basal age-dependent decline in bone mass. Osteoporosis is characterized by low bone mass and changes in bone microarchitecture that increases the risk of fracture. About a third of men over the age of 50 years are osteoporotic because they have higher than basal bone loss. In women, there is an additional acute decrement in bone mass, atop the basal rate, associated with loss of ovarian function (menopause) causing osteoporosis in about half of the women. Both genetics and environmental factors such as smoking, chronic infections, diet, microbiome, and metabolic disease can modulate basal age-dependent bone loss and eventual osteoporosis. Here, we review recent studies on the etiology of age-dependent decline in bone mass and propose a mechanism that integrates both genetic and environmental factors. Recent Findings: Recent findings support that aging and menopause dysregulate the immune system leading to sterile low-grade inflammation. Both animal models and human studies demonstrate that certain kinds of inflammation, in both men and women, mediate bone loss. Senolytics, meant to block a wide array of age-induced effects by preventing cellular senescence, have been shown to improve bone mass in aged mice. Based on a synthesis of the recent data, we propose that aging activates long-lived tissue resident memory T-cells to become senescent and proinflammatory, leading to bone loss. Targeting this population may represent a promising osteoporosis therapy. Summary: Emerging data indicates that there are several mechanisms that lead to sterile low-grade chronic inflammation, inflammaging, that cause age- and estrogen-loss dependent osteoporosis in men and women. [ABSTRACT FROM AUTHOR]

We define renormalised energies for maps that describe the first-order asymptotics of harmonic maps outside of singularities arising due to obstructions generated by the boundary data and the mutliple connectedness of the target manifold. The constructions generalise the definition by Bethuel et al. (Ginzburg–Landau vortices, progress in nonlinear differential equations and their applications, vol 13, Birkhäuser, Boston, 1994) for the circle. In general, the singularities are geometrical objects and the dependence on homotopic singularities can be studied through a new notion of synharmony. The renormalised energies are showed to be coercive and Lipschitz-continuous. The renormalised energies are associated to minimising renormalisable singular harmonic maps and minimising configurations of points can be characterised by the flux of the stress–energy tensor at the singularities. We compute the singular energy and the renormalised energy in several particular cases. [ABSTRACT FROM AUTHOR]

We consider weakly pseudoconvex hypersurfaces with polynomial models in C N and their symmetry algebras. In the most prominent case of special models, given by sums of squares of polynomials, we give their complete classification. In particular, we prove that such manifolds do not admit any nonlinear symmetries, depending only on complex tangential variables, nor do they admit real or nilpotent linear symmetries. This leads to a sharp 2-jet determination result for local automorphisms. We also give partial results in the general case and a more detailed description of the graded components in complex dimension three. The results also provide an important necessary step for solving the local equivalence problem on such manifolds. [ABSTRACT FROM AUTHOR]