Central limit theorems for Gaussian polytopes

Choose $n$ random, independent points in $\R^d$ according to the standard normal distribution. Their convex hull $K_n$ is the {\sl Gaussian random polytope}. We prove that the volume and the number of faces of $K_n$ satisfy the central limit theorem, settling a well known conjecture in the field.Comment: to appear in Annals of Probabilit

On the deformation morphology of bulk metallic glasses underneath a Vickers indentation

16th International Symposium on Metastable, Amorphous and Nanostructured Materials, Beijing, PEOPLES R CHINA, JUL 05-09, 2009International audienceThe techniques commonly used for observing the deformation mechanisms underneath a Vickers indentation in metallic glasses (chemical etching, bonded interface) induce artefacts such as cracks or semi-circular shear-bands. We propose an alternative technique based on the propagation of indentation corner cracks through a pre-existing imprint, which is possible in metallic glasses such as iron-based compositions. With this procedure, only radial shear-bands are observed. Comparisons between the chemical etching or the bonded interface techniques and the new technique are made. (C) 2010 Elsevier B.V. All rights reserved

Circular strings in Kerr-AdS5AdS_5 black holes

The quest for extension of holographic correspondence to the case of finite temperature naturally includes Kerr-AdS black holes and their field theory duals. We probe the five-dimensional Kerr-AdS space time by pulsating strings. First we find particular pulsating string solutions and then semi-classically quantize the theory. For the string with large values of energy, we use the Bohr-Sommerfeld analysis to find the energy of the string as a function of a large quantum number. We obtain the wave function of the problem and thoroughly study the corrections to the energy, which according to the holographic dictionary are related to anomalous dimensions of certain operators in the dual gauge theory. The interpretation of results from holographic point of view is not straightforward since the dual theory is at finite temperature. Nevertheless, near or at conformal point the expressions can be thought of as the dispersion relations of stationary states.Comment: 32 pp, 1 figure; v2: Sec.3 improved, Sec.4 recalculated for a general case, typos corrected, Appendix B included; v3: Sec.3 corrected, new figure of an effective potential added, typos correcte

Accounting-based variables as an early warning indicator of financial distress in crisis and non-crisis periods

Financial integration in the Association of Southeast Asian Nations (ASEAN) region is a key focus of the ASEAN Economic Community. Whereas many studies focus on modelling corporate default, this paper identifies early warning indicators of financial distress before a default, using multiple discriminant analysis (MDA) models with a sample of listed and delisted companies in the ASEAN region. The analysis examines 720 companies in 10 different industries across six ASEAN countries from 1997 to 2016. The study constructs individual models for each country as well as an overall model for the entire region, using both in-sample and out-of-sample approaches. This overall model could be useful for an integrated banking system. To ensure robustness, the study also separately examines the predictive performance of the MDA models across different economic crises: the Asian financial crisis (AFC) from 1997 to 2000, the global financial crisis (GFC) from 2007 to 2009 and their pre- and post-crisis periods. We find that profitability ratios are the best indicators of financial distress in the ASEAN region, followed by liquidity and leverage ratios. In addition, our findings reveal common indicators that can be used to predict financial distress across ASEAN countries. The single model performs reasonably well in predicting financial distress 1 year ahead. In addition, the model is extended to incorporate a market-based indicator into the MDA models, the distance to default. However, the inclusion of this indicator does not significantly improve the accuracy of the models in predicting financial distress at listed firms in the ASEAN region

Zooming in on local level statistics by supersymmetric extension of free probability

We consider unitary ensembles of Hermitian NxN matrices H with a confining potential NV where V is analytic and uniformly convex. From work by Zinn-Justin, Collins, and Guionnet and Maida it is known that the large-N limit of the characteristic function for a finite-rank Fourier variable K is determined by the Voiculescu R-transform, a key object in free probability theory. Going beyond these results, we argue that the same holds true when the finite-rank operator K has the form that is required by the Wegner-Efetov supersymmetry method of integration over commuting and anti-commuting variables. This insight leads to a potent new technique for the study of local statistics, e.g., level correlations. We illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering.Comment: 38 pages, 3 figures, published version, minor changes in Section

Empirical Evaluation of Pre-trained Transformers for Human-Level NLP: The Role of Sample Size and Dimensionality

In human-level NLP tasks, such as predicting mental health, personality, or demographics, the number of observations is often smaller than the standard 768+ hidden state sizes of each layer within modern transformer-based language models, limiting the ability to effectively leverage transformers. Here, we provide a systematic study on the role of dimension reduction methods (principal components analysis, factorization techniques, or multi-layer auto-encoders) as well as the dimensionality of embedding vectors and sample sizes as a function of predictive performance. We first find that fine-tuning large models with a limited amount of data pose a significant difficulty which can be overcome with a pre-trained dimension reduction regime. RoBERTa consistently achieves top performance in human-level tasks, with PCA giving benefit over other reduction methods in better handling users that write longer texts. Finally, we observe that a majority of the tasks achieve results comparable to the best performance with just $\frac{1}{12}$ of the embedding dimensions.Comment: 2021 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL-HLT