Continuous tensor categories from quantum groups I: algebraic aspects

We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $\mathcal{P}_\lambda$ of the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of $U_q(\mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of $\mathcal{P}_\lambda \otimes \mathcal{P}_\mu$ into irreducibles.Comment: 55 pages, 38 figure

Goodwillie towers and chromatic homotopy: an overview

This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie's work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic homotopy as in my work and the work of Arone-Mahowald.Comment: This is the version published by Geometry & Topology Monographs on 29 January 200

Role of a plausible nuisance contributor in the declining obesity-mortality risks over time.

CONTEXT: Recent analyses of epidemiological data including the National Health and Nutrition Examination Survey (NHANES) have suggested that the harmful effects of obesity may have decreased over calendar time. The shifting BMI distribution over time coupled with the application of fixed broad BMI categories in these analyses could be a plausible nuisance contributor to this observed change in the obesity-associated mortality over calendar time. OBJECTIVE: To evaluate the extent to which observed temporal changes in the obesity-mortality association may be due to a shifting population distribution for body mass index (BMI), coupled with analyses based on static, broad BMI categories. DESIGN, SETTING, AND PARTICIPANTS: Simulations were conducted using data from NHANES I and III linked with mortality data. Data from NHANES I were used to fit a true model treating BMI as a continuous variable. Coefficients estimated from this model were used to simulate mortality for participants in NHANES III. Hence, the population-level association between BMI and mortality in NHANES III was fixed to be identical to the association estimated in NHANES I. Hazard ratios (HRs) for obesity categories based on BMI for NHANES III with simulated mortality data were compared to the corresponding estimated HRs from NHANES I. MAIN OUTCOME MEASURES: Change in hazard ratios for simulated data in NHANES III compared to observed estimates from NHANES I. RESULTS: On average, hazard ratios for NHANES III based on simulated mortality data were 29.3% lower than the estimates from NHANES I using observed mortality follow-up. This reduction accounted for roughly three-fourths of the apparent decrease in the obesity-mortality association observed in a previous analysis of these data. CONCLUSIONS: Some of the apparent diminution of the association between obesity and mortality may be an artifact of treating BMI as a categorical variable

Symbol grounding and its implications for artificial intelligence

In response to Searle's well-known Chinese room argument against Strong AI (and more generally, computationalism), Harnad proposed that if the symbols manipulated by a robot were sufficiently grounded in the real world, then the robot could be said to literally understand. In this article, I expand on the notion of symbol groundedness in three ways. Firstly, I show how a robot might select the best set of categories describing the world, given that fundamentally continuous sensory data can be categorised in an almost infinite number of ways. Secondly, I discuss the notion of grounded abstract (as opposed to concrete) concepts. Thirdly, I give an objective criterion for deciding when a robot's symbols become sufficiently grounded for "understanding" to be attributed to it. This deeper analysis of what symbol groundedness actually is weakens Searle's position in significant ways; in particular, whilst Searle may be able to refute Strong AI in the specific context of present-day digital computers, he cannot refute computationalism in general

An ADMM Based Framework for AutoML Pipeline Configuration

We study the AutoML problem of automatically configuring machine learning pipelines by jointly selecting algorithms and their appropriate hyper-parameters for all steps in supervised learning pipelines. This black-box (gradient-free) optimization with mixed integer & continuous variables is a challenging problem. We propose a novel AutoML scheme by leveraging the alternating direction method of multipliers (ADMM). The proposed framework is able to (i) decompose the optimization problem into easier sub-problems that have a reduced number of variables and circumvent the challenge of mixed variable categories, and (ii) incorporate black-box constraints along-side the black-box optimization objective. We empirically evaluate the flexibility (in utilizing existing AutoML techniques), effectiveness (against open source AutoML toolkits),and unique capability (of executing AutoML with practically motivated black-box constraints) of our proposed scheme on a collection of binary classification data sets from UCI ML& OpenML repositories. We observe that on an average our framework provides significant gains in comparison to other AutoML frameworks (Auto-sklearn & TPOT), highlighting the practical advantages of this framework

Causal identification with subjective outcomes

Survey questions often elicit responses on ordered scales for which the definitions of the categories are subjective, possibly varying by individual. This paper clarifies what is learned when these subjective reports are used as an outcome in regression-based causal inference. When a continuous treatment variable is statistically independent of both i) potential outcomes; and ii) heterogeneity in reporting styles, a nonparametric regression of integer category numbers on that variable uncovers a positively-weighted linear combination of causal responses among individuals who are on the margin between adjacent response categories. Though the weights do not integrate to one, the ratio of local regression derivatives with respect to two such explanatory variables identifies the relative magnitudes of convex averages of their effects. When results are extended to discrete treatment variables, different weighting schemes apply to different regressors, making comparisons of magnitude less informative. I obtain a partial identification result for comparing the effects of a discrete treatment variable to those of another treatment variable when there are many categories and individual reporting functions are linear. I also provide results for identification using instrumental variables