Discrepancy of Symmetric Products of Hypergraphs

For a hypergraph ${\mathcal H} = (V,{\mathcal E})$, its $d$--fold symmetric product is $\Delta^d {\mathcal H} = (V^d,\{E^d |E \in {\mathcal E}\})$. We give several upper and lower bounds for the $c$-color discrepancy of such products. In particular, we show that the bound ${disc}(\Delta^d {\mathcal H},2) \le {disc}({\mathcal H},2)$ proven for all $d$ in [B. Doerr, A. Srivastav, and P. Wehr, Discrepancy of {C}artesian products of arithmetic progressions, Electron. J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than $c = 2$ colors. In fact, for any $c$ and $d$ such that $c$ does not divide $d!$, there are hypergraphs having arbitrary large discrepancy and ${disc}(\Delta^d {\mathcal H},c) = \Omega_d({disc}({\mathcal H},c)^d)$. Apart from constant factors (depending on $c$ and $d$), in these cases the symmetric product behaves no better than the general direct product ${\mathcal H}^d$, which satisfies ${disc}({\mathcal H}^d,c) = O_{c,d}({disc}({\mathcal H},c)^d)$.Comment: 12 pages, no figure

What determines Financial Development? Culture, Institutions, or Trade

This paper endeavours to explain the vast differences in the size of capital markets across countries, by drawing together theories emphasising cultural values, dysfunctional institutions, or impediments to trade as obstacles to financial development. To account for endogeneity, instrumental variables pertaining to culture, geography, and colonial history are employed. We find that trade openness and institutions constraining the political elite from expropriating financiers exhibit a strong positive effect on the size of capital markets. Conversely, cultural beliefs and the cost of enforcing financial contracts seem not to introduce significant obstacles for financial development.Financial Development, Culture, Institutional Quality, Trade

Virtual Rephotography: Novel View Prediction Error for 3D Reconstruction

The ultimate goal of many image-based modeling systems is to render photo-realistic novel views of a scene without visible artifacts. Existing evaluation metrics and benchmarks focus mainly on the geometric accuracy of the reconstructed model, which is, however, a poor predictor of visual accuracy. Furthermore, using only geometric accuracy by itself does not allow evaluating systems that either lack a geometric scene representation or utilize coarse proxy geometry. Examples include light field or image-based rendering systems. We propose a unified evaluation approach based on novel view prediction error that is able to analyze the visual quality of any method that can render novel views from input images. One of the key advantages of this approach is that it does not require ground truth geometry. This dramatically simplifies the creation of test datasets and benchmarks. It also allows us to evaluate the quality of an unknown scene during the acquisition and reconstruction process, which is useful for acquisition planning. We evaluate our approach on a range of methods including standard geometry-plus-texture pipelines as well as image-based rendering techniques, compare it to existing geometry-based benchmarks, and demonstrate its utility for a range of use cases.Comment: 10 pages, 12 figures, paper was submitted to ACM Transactions on Graphics for revie