Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices

Super-resolution is a fundamental task in imaging, where the goal is to extract fine-grained structure from coarse-grained measurements. Here we are interested in a popular mathematical abstraction of this problem that has been widely studied in the statistics, signal processing and machine learning communities. We exactly resolve the threshold at which noisy super-resolution is possible. In particular, we establish a sharp phase transition for the relationship between the cutoff frequency ($m$) and the separation ($\Delta$). If $m > 1/\Delta + 1$, our estimator converges to the true values at an inverse polynomial rate in terms of the magnitude of the noise. And when $m < (1-\epsilon) /\Delta$ no estimator can distinguish between a particular pair of $\Delta$-separated signals even if the magnitude of the noise is exponentially small. Our results involve making novel connections between {\em extremal functions} and the spectral properties of Vandermonde matrices. We establish a sharp phase transition for their condition number which in turn allows us to give the first noise tolerance bounds for the matrix pencil method. Moreover we show that our methods can be interpreted as giving preconditioners for Vandermonde matrices, and we use this observation to design faster algorithms for super-resolution. We believe that these ideas may have other applications in designing faster algorithms for other basic tasks in signal processing.Comment: 19 page

Nominal Unification of Higher Order Expressions with Recursive Let

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems.Comment: Pre-proceedings paper presented at the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh, Scotland UK, 6-8 September 2016 (arXiv:1608.02534

A dependent nominal type theory

Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of beta-eta-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles

Myosin-X functions in polarized epithelial cells

Myosin-X, an unconventional myosin that has been studied primarily in fibroblast-like cells, has been shown to have important functions in polarized epithelial cell junction formation, regulation of paracellular permeability, and epithelial morphogenesis.Myosin-X (Myo10) is an unconventional myosin that localizes to the tips of filopodia and has critical functions in filopodia. Although Myo10 has been studied primarily in nonpolarized, fibroblast-like cells, Myo10 is expressed in vivo in many epithelia-rich tissues, such as kidney. In this study, we investigate the localization and functions of Myo10 in polarized epithelial cells, using Madin-Darby canine kidney II cells as a model system. Calcium-switch experiments demonstrate that, during junction assembly, green fluorescent protein–Myo10 localizes to lateral membrane cell–cell contacts and to filopodia-like structures imaged by total internal reflection fluorescence on the basal surface. Knockdown of Myo10 leads to delayed recruitment of E-cadherin and ZO-1 to junctions, as well as a delay in tight junction barrier formation, as indicated by a delay in the development of peak transepithelial electrical resistance (TER). Although Myo10 knockdown cells eventually mature into monolayers with normal TER, these monolayers do exhibit increased paracellular permeability to fluorescent dextrans. Importantly, knockdown of Myo10 leads to mitotic spindle misorientation, and in three-dimensional culture, Myo10 knockdown cysts exhibit defects in lumen formation. Together these results reveal that Myo10 functions in polarized epithelial cells in junction formation, regulation of paracellular permeability, and epithelial morphogenesis

Positive approximations of the inverse of fractional powers of SPD M-matrices

This study is motivated by the recent development in the fractional calculus and its applications. During last few years, several different techniques are proposed to localize the nonlocal fractional diffusion operator. They are based on transformation of the original problem to a local elliptic or pseudoparabolic problem, or to an integral representation of the solution, thus increasing the dimension of the computational domain. More recently, an alternative approach aimed at reducing the computational complexity was developed. The linear algebraic system $\cal A^\alpha \bf u=\bf f$, $0< \alpha <1$ is considered, where $\cal A$ is a properly normalized (scalded) symmetric and positive definite matrix obtained from finite element or finite difference approximation of second order elliptic problems in $\Omega\subset\mathbb{R}^d$, $d=1,2,3$. The method is based on best uniform rational approximations (BURA) of the function $t^{\beta-\alpha}$ for $0 < t \le 1$ and natural $\beta$. The maximum principles are among the major qualitative properties of linear elliptic operators/PDEs. In many studies and applications, it is important that such properties are preserved by the selected numerical solution method. In this paper we present and analyze the properties of positive approximations of $\cal A^{-\alpha}$ obtained by the BURA technique. Sufficient conditions for positiveness are proven, complemented by sharp error estimates. The theoretical results are supported by representative numerical tests

An Ishihara-style test of animal colour vision

ABSTRACT Colour vision mediates ecologically relevant tasks for many animals, such as mate choice, foraging and predator avoidance. However, our understanding of animal colour perception is largely derived from human psychophysics, and behavioural tests of non-human animals are required to understand how colour signals are perceived. Here, we introduce a novel test of colour vision in animals inspired by the Ishihara colour charts, which are widely used to identify human colour deficiencies. In our method, distractor dots have a fixed chromaticity (hue and saturation) but vary in luminance. Animals can be trained to find single target dots that differ from distractor dots in chromaticity. We provide MATLAB code for creating these stimuli, which can be modified for use with different animals. We demonstrate the success of this method with triggerfish, Rhinecanthus aculeatus, which quickly learnt to select target dots that differed from distractor dots, and highlight behavioural parameters that can be measured, including success of finding the target dot, time to detection and error rate. We calculated discrimination thresholds by testing whether target colours that were of increasing colour distances (ΔS) from distractor dots could be detected, and calculated discrimination thresholds in different directions of colour space. At least for some colours, thresholds indicated better discrimination than expected from the receptor noise limited (RNL) model assuming 5% Weber fraction for the long-wavelength cone. This methodology could be used with other animals to address questions such as luminance thresholds, sensory bias, effects of sensory noise, colour categorization and saliency

Spectral structure and decompositions of optical states, and their applications

We discuss the spectral structure and decomposition of multi-photon states. Ordinarily `multi-photon states' and `Fock states' are regarded as synonymous. However, when the spectral degrees of freedom are included this is not the case, and the class of `multi-photon' states is much broader than the class of `Fock' states. We discuss the criteria for a state to be considered a Fock state. We then address the decomposition of general multi-photon states into bases of orthogonal eigenmodes, building on existing multi-mode theory, and introduce an occupation number representation that provides an elegant description of such states that in many situations simplifies calculations. Finally we apply this technique to several example situations, which are highly relevant for state of the art experiments. These include Hong-Ou-Mandel interference, spectral filtering, finite bandwidth photo-detection, homodyne detection and the conditional preparation of Schr\"odinger Kitten and Fock states. Our techniques allow for very simple descriptions of each of these examples.Comment: 12 page

Paternity alone does not predict long-term investment in juveniles by male baboons

Adult male chacma baboons (Papio hamadryas ursinus) form preferential associations, or friendships, with particular lactating females. Males exhibit high levels of affiliative contact with their friends’ infants and defend them from potentially infanticidal attacks (Palombit et al. 1997). Little is known about males’ associations with juveniles once they have passed the period of infanticidal risk. We conducted an observational, experimental, and genetic study of adult male and juvenile chacma baboons in the Moremi Reserve, Botswana. We identified preferential associations between males and juveniles and used behavioral data and a playback experiment to explore whether those associations have potential fitness benefits for juveniles. We determined whether males preferentially invest in care of their own offspring. We also determined how often males invest in care of their former friends’ offspring. The majority of juveniles exhibited preferential associations with one or two males, who had almost always been their mother’s friend during infancy. However, in only a subset of these relationships was the male the actual father, in part because many fathers died or disappeared before their offspring were weaned. Male caretakers intervened on behalf of their juvenile associates in social conflicts more often than they intervened on behalf of unconnected juveniles, and they did not appear to differentiate between genetic offspring and unrelated associates. Playbacks of juveniles’ distress calls elicited a stronger response from their caretakers than from control males. Chacma males may provide care to unrelated offspring of former friends because the costs associated with such care are low compared with the potentially high fitness costs of refusing aid to a juvenile who is a possible offspring

Field Evaluation of Programmable Thermostats

Prior research suggests that poor programmable thermostats usability may prevent their effective use to save energy. We hypothesized that home occupants with a high-usability thermostats would be more likely to use them to save energy than people with a basic thermostat. We randomly installed a high-usability thermostat in half the 77 apartments of an affordable housing complex, installing a basic thermostat in the other half. During the heating season, we collected space temperature and furnace on-off data to evaluate occupant interaction with the thermostats, foremost nighttime setbacks. We found that thermostat usability did not influence energy-saving behaviors, finding no significant difference in temperature maintained among apartments with high- and low-usability thermostats