The Geometry and Calculus of Losses

Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are binary and multiclass classification and class probability estimation. Central to their definition is the choice of loss function, which is the means by which the quality of a solution is evaluated. In this paper we systematically develop the theory of loss functions for such problems from a novel perspective whose basic ingredients are convex sets with a particular structure. The loss function is defined as the subgradient of the support function of the convex set. It is consequently automatically proper (calibrated for probability estimation). This perspective provides three novel opportunities. It enables the development of a fundamental relationship between losses and (anti)-norms that appears to have not been noticed before. Second, it enables the development of a calculus of losses induced by the calculus of convex sets which allows the interpolation between different losses, and thus is a potential useful design tool for tailoring losses to particular problems. In doing this we build upon, and considerably extend existing results on $M$-sums of convex sets. Third, the perspective leads to a natural theory of ``polar'' loss functions, which are derived from the polar dual of the convex set defining the loss, and which form a natural universal substitution function for Vovk's aggregating algorithm.Comment: 65 pages, 17 figure

Spur-Gear-System Efficiency at Part and Full Load

A simple method for predicting the part- and full-load power loss of a steel spur gearset of arbitrary geometry supported by ball bearings is described. The analysis algebraically accounts for losses due to gear sliding, rolling traction, and windage in addition to support-ball-bearing losses. The analysis compares favorably with test data. A theoretical comparison of the component losses indicates that losses due to gear rolling traction, windage, and support bearings are significant and should be included along with gear sliding loss in a calculation of gear-system power loss

Hyperparameter-free losses for model-based monocular reconstruction

This work proposes novel hyperparameter-free losses for single view 3D reconstruction with morphable models (3DMM). We dispense with the hyperparameters used in other works by exploiting geometry, so that the shape of the object and the camera pose are jointly optimized in a sole term expression. This simplification reduces the optimization time and its complexity. Moreover, we propose a novel implicit regularization technique based on random virtual projections that does not require additional 2D or 3D annotations. Our experiments suggest that minimizing a shape reprojection error together with the proposed implicit regularization is especially suitable for applications that require precise alignment between geometry and image spaces, such as augmented reality. We evaluate our losses on a large scale dataset with 3D ground truth and publish our implementations to facilitate reproducibility and public benchmarking in this field.Peer ReviewedPostprint (author's final draft

Statistical analysis of factors affecting the flow characteristics and thermal efficiency of a building integrated thermal (bit) solar collector

Previous research has identified four factors (array geometry, manifold to riser channel ratio, flow direction in manifold, and the mass flow rate) which will influence the distribution of internal fluid flow within a solar thermal collector. In this study, a two level full factorial (2k) experiment was designed in order to statistically rank their impact and also to identify any significant interactions between these factors. The thermal efficiency of the array, calculated by means of a fluid and heat transfer analysis was taken to be the experiment response. During the heat transfer analysis we approximated the fin efficiency of a BIT collector using the finite difference method which considered the heat losses through the structural ribs of the collector. A statistical analysis of the results revealed that all four main effects had a statistical influence on thermal efficiency of the array at 5 per cent significance level. The main effects ranked from highest to lowest in impact were found to be; geometry, manifold to riser fluid channel diameter, mass flow rate, and the direction of flow in the manifolds. Additionally, two secondary interactions were found to have a statistical influence on the experiment response; the array geometry and the direction of flow in the manifold followed by the array geometry and the ratio of manifold to fluid channel diameter. As the geometry of the BIT collector will vary from customer to customer due to its custom nature, these results indicate that the design of a BIT system should consider the effects of flow distribution. Finally, our numerical analysis of the fin efficiency revealed an approximate 5% drop due to additional heat losses through the structural ribs

Electromagnetic losses in magnetic shields for buried high voltage cables

The electromagnetic losses and shielding efficiency of shields for a buried three phase high voltage cable are studied for several shielding configurations. The shields are U-shaped gutters covered with plates, and the power cables are positioned either in trefoil or in flat configuration. The shielding efficiency and the losses are compared for shields with the same geometry but several shielding materials: aluminium, and two ferromagnetic steel grades. The numerical models are validated with experimental results. From the experiments, it is observed that the average reducing factor of the flux density is about 7 with the flat cable configuration while the average reducing factor of the flux density is about 5 with the trefoil cable configuration. But the power losses in the DX52 shield for trefoil configuration is about 40% lower compared to the flat configuration. In case of trefoil configuration, the losses are 12.14 W/m per meter length in the shield for a current of 750 A. Next to the shield material and the cable configuration, the paper investigates the influence of several parameters on both the shielding efficiency and the losses: the size of the shield, the current amplitude in the cable and the thickness of the shield