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In this paper, we discuss some geometrical and topological properties of a parameterized binary order relation. This relation is based on a special set called. We answer the question under which conditions the natural ordering cone ofis a subset of. Moreover, we examine in which cases the setcoincides with a certain half space. Furthermore, we determine the dual set ofand the contingent cone to. [ABSTRACT FROM AUTHOR]

In real linear spaces, partial orderings are usually generated by ordering cones. In many situations, however, such an ordering cone is too small with respect to the whole space. Therefore, in this paper, we extend the concept of ordering cones to a more general concept. For this purpose, we define a parameterized binary relation, based on a convex cone and a binary function. We investigate some geometrical and topological properties of this relation in detail. [ABSTRACT FROM AUTHOR]

Keeping record of daily meal intake is an effective solution for tackling with obesity and overweight. This can be done by developing apps on smartphones that are able to automatically recommend a short list of most probable foods by analyzing the photo taken from food. Then, the user chooses the correct answer from the short list. Hence, the automatic food recognition system must be able to recommend an accurate list. In other words, it is not essential for these apps to have a very high top-1 accuracy. Considering that the app will show the list of 5 most probable foods, the food recognition system must have a high top-5 accuracy. A food recognition system is usually developed by adapting knowledge of state-of-the-art networks such as GoogleNet and ResNet to the domain of food. However, these networks have high number of parameters. In this paper, we propose a 23-layer architecture which has 99.14% and 96.63% fewer parameter compared with ResNet and GoogleNet. Our experiment on Food101 and UECFood-256 datasets shows that although our network reduces the number of parameters dramatically, it produces more accurate results than GoogleNet and its accuracy is comparable with ResNet. [ABSTRACT FROM AUTHOR]

We analyze the accuracy of the discrete least-squares approximation of a function $$u$$ in multivariate polynomial spaces $$\mathbb {P}_\varLambda :=\mathrm{span} \{y\mapsto y^\nu : \nu \in \varLambda \}$$ with $$\varLambda \subset \mathbb {N}_0^d$$ over the domain $$\varGamma :=[-1,1]^d$$ , based on the sampling of this function at points $$y^1,\ldots ,y^m \in \varGamma $$ . The samples are independently drawn according to a given probability density $$\rho $$ belonging to the class of multivariate beta densities, which includes the uniform and Chebyshev densities as particular cases. Motivated by recent results on high-dimensional parametric and stochastic PDEs, we restrict our attention to polynomial spaces associated with downward closed sets $$\varLambda $$ of prescribed cardinality n, and we optimize the choice of the space for the given sample. This implies, in particular, that the selected polynomial space depends on the sample. We are interested in comparing the error of this least-squares approximation measured in $$L^2(\varGamma ,d\rho )$$ with the best achievable polynomial approximation error when using downward closed sets of cardinality n. We establish conditions between the dimension n and the size m of the sample, under which these two errors are proved to be comparable. Our main finding is that the dimension d enters only moderately in the resulting trade-off between m and n, in terms of a logarithmic factor $$\ln (d)$$ , and is even absent when the optimization is restricted to a relevant subclass of downward closed sets, named anchored sets. In principle, this allows one to use these methods in arbitrarily high or even infinite dimension. Our analysis builds upon (Chkifa et al. in ESAIM Math Model Numer Anal 49(3):815-837, 2015), which considered fixed and nonoptimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of efficient numerical methods for computing the solution to high-dimensional parametric or stochastic PDEs, but are not limited to this area. [ABSTRACT FROM AUTHOR]