Lifting to the spectral ball with interpolation

We give necessary and sufficient conditions for solving the spectral Nevanlinna--Pick lifting problem. This reduces the spectral Nevanlinna--Pick problem to a jet interpolation problem into the symmetrized polydisc.Comment: 7 pages; corrections to references and typo

Stein Spaces Characterized by their Endomorphisms

Finite dimensional Stein spaces admitting a proper holomorphic embedding of the complex line are characterized, among all complex spaces, by their holomorphic endomorphism semigroup in the sense that any semigroup isomorphism induces either a biholomorphic or an antibiholomorphic map between them.Comment: 15 pages; additional references in the introduction; corrected typo

Breakout character of islet amyloid polypeptide hydrophobic mutations at the onset of type-2 diabetes

Toxic fibrillar aggregates of Islet Amyloid PolyPeptide (IAPP) appear as the physical outcome of a peptidic phase-transition signaling the onset of type-2 diabetes mellitus in different mammalian species. In particular, experimentally verified mutations on the amyloidogenic segment 20-29 in humans, cats and rats are highly correlated with the molecular aggregation propensities. Through a microcanonical analysis of the aggregation of IAPP_{20-29} isoforms, we show that a minimalist one-bead hydrophobic-polar continuum model for protein interactions properly quantifies those propensities from free-energy barriers. Our results highlight the central role of sequence-dependent hydrophobic mutations on hot spots for stabilization, and so for the engineering, of such biological peptides.Comment: 8 pages, 6 figures, 1 table; final version to appear in Physical Review

Nonextensive lattice gauge theories: algorithms and methods

High-energy phenomena presenting strong dynamical correlations, long-range interactions and microscopic memory effects are well described by nonextensive versions of the canonical Boltzmann-Gibbs statistical mechanics. After a brief theoretical review, we introduce a class of generalized heat-bath algorithms that enable Monte Carlo lattice simulations of gauge fields on the nonextensive statistical ensemble of Tsallis. The algorithmic performance is evaluated as a function of the Tsallis parameter q in equilibrium and nonequilibrium setups. Then, we revisit short-time dynamic techniques, which in contrast to usual simulations in equilibrium present negligible finite-size effects and no critical slowing down. As an application, we investigate the short-time critical behaviour of the nonextensive hot Yang-Mills theory at q- values obtained from heavy-ion collision experiments. Our results imply that, when the equivalence of statistical ensembles is obeyed, the long-standing universality arguments relating gauge theories and spin systems hold also for the nonextensive framework.Comment: to appear in Comput. Phys. Com

Teaching decision theory proof strategies using a crowdsourcing problem

Teaching how to derive minimax decision rules can be challenging because of the lack of examples that are simple enough to be used in the classroom. Motivated by this challenge, we provide a new example that illustrates the use of standard techniques in the derivation of optimal decision rules under the Bayes and minimax approaches. We discuss how to predict the value of an unknown quantity, $\theta \! \in \! \{0,1\}$, given the opinions of $n$ experts. An important example of such crowdsourcing problem occurs in modern cosmology, where $\theta$ indicates whether a given galaxy is merging or not, and $Y_1, \ldots, Y_n$ are the opinions from $n$ astronomers regarding $\theta$. We use the obtained prediction rules to discuss advantages and disadvantages of the Bayes and minimax approaches to decision theory. The material presented here is intended to be taught to first-year graduate students.Comment: 21 pages, 2 figures. This is an Accepted Manuscript of an article published by Taylor & Francis Group in The American Statistician, available online: https://amstat.tandfonline.com/doi/abs/10.1080/00031305.2016.126431

The fibred density property and the automorphism group of the spectral ball

We generalize the notion of the density property for complex manifolds to holomorphic fibrations, and introduce the notion of the fibred density property. We prove that the natural fibration of the spectral ball over the symmetrized polydisc enjoys the fibred density property and describe the automorphism group of the spectral ball.Comment: 23 pages; major changes in Section 5 concerning the dimension estimates of the kernel of homogeneous derivation

A Spectral Series Approach to High-Dimensional Nonparametric Regression

A key question in modern statistics is how to make fast and reliable inferences for complex, high-dimensional data. While there has been much interest in sparse techniques, current methods do not generalize well to data with nonlinear structure. In this work, we present an orthogonal series estimator for predictors that are complex aggregate objects, such as natural images, galaxy spectra, trajectories, and movies. Our series approach ties together ideas from kernel machine learning, and Fourier methods. We expand the unknown regression on the data in terms of the eigenfunctions of a kernel-based operator, and we take advantage of orthogonality of the basis with respect to the underlying data distribution, P, to speed up computations and tuning of parameters. If the kernel is appropriately chosen, then the eigenfunctions adapt to the intrinsic geometry and dimension of the data. We provide theoretical guarantees for a radial kernel with varying bandwidth, and we relate smoothness of the regression function with respect to P to sparsity in the eigenbasis. Finally, using simulated and real-world data, we systematically compare the performance of the spectral series approach with classical kernel smoothing, k-nearest neighbors regression, kernel ridge regression, and state-of-the-art manifold and local regression methods

Converting High-Dimensional Regression to High-Dimensional Conditional Density Estimation

There is a growing demand for nonparametric conditional density estimators (CDEs) in fields such as astronomy and economics. In astronomy, for example, one can dramatically improve estimates of the parameters that dictate the evolution of the Universe by working with full conditional densities instead of regression (i.e., conditional mean) estimates. More generally, standard regression falls short in any prediction problem where the distribution of the response is more complex with multi-modality, asymmetry or heteroscedastic noise. Nevertheless, much of the work on high-dimensional inference concerns regression and classification only, whereas research on density estimation has lagged behind. Here we propose FlexCode, a fully nonparametric approach to conditional density estimation that reformulates CDE as a non-parametric orthogonal series problem where the expansion coefficients are estimated by regression. By taking such an approach, one can efficiently estimate conditional densities and not just expectations in high dimensions by drawing upon the success in high-dimensional regression. Depending on the choice of regression procedure, our method can adapt to a variety of challenging high-dimensional settings with different structures in the data (e.g., a large number of irrelevant components and nonlinear manifold structure) as well as different data types (e.g., functional data, mixed data types and sample sets). We study the theoretical and empirical performance of our proposed method, and we compare our approach with traditional conditional density estimators on simulated as well as real-world data, such as photometric galaxy data, Twitter data, and line-of-sight velocities in a galaxy cluster

Logically-consistent hypothesis testing in the hexagon of oppositions

Although logical consistency is desirable in scientific research, standard statistical hypothesis tests are typically logically inconsistent. In order to address this issue, previous work introduced agnostic hypothesis tests and proved that they can be logically consistent while retaining statistical optimality properties. This paper characterizes the credal modalities in agnostic hypothesis tests and uses the hexagon of oppositions to explain the logical relations between these modalities. Geometric solids that are composed of hexagons of oppositions illustrate the conditions for these modalities to be logically consistent. Prisms composed of hexagons of oppositions show how the credal modalities obtained from two agnostic tests vary according to their threshold values. Nested hexagons of oppositions summarize logical relations between the credal modalities in these tests and prove new relations.Comment: 25 pages, 9 figure

Pragmatic hypotheses in the evolution of science

This paper introduces pragmatic hypotheses and relates this concept to the spiral of scientific evolution. Previous works determined a characterization of logically consistent statistical hypothesis tests and showed that the modal operators obtained from this test can be represented in the hexagon of oppositions. However, despite the importance of precise hypothesis in science, they cannot be accepted by logically consistent tests. Here, we show that this dilemma can be overcome by the use of pragmatic versions of precise hypotheses. These pragmatic versions allow a level of imprecision in the hypothesis that is small relative to other experimental conditions. The introduction of pragmatic hypotheses allows the evolution of scientific theories based on statistical hypothesis testing to be interpreted using the narratological structure of hexagonal spirals, as defined by Pierre Gallais