Lower Complexity Adaptation for Empirical Entropic Optimal Transport

Entropic optimal transport (EOT) presents an effective and computationally viable alternative to unregularized optimal transport (OT), offering diverse applications for large-scale data analysis. In this work, we derive novel statistical bounds for empirical plug-in estimators of the EOT cost and show that their statistical performance in the entropy regularization parameter $\epsilon$ and the sample size $n$ only depends on the simpler of the two probability measures. For instance, under sufficiently smooth costs this yields the parametric rate $n^{-1/2}$ with factor $\epsilon^{-d/2}$, where $d$ is the minimum dimension of the two population measures. This confirms that empirical EOT also adheres to the lower complexity adaptation principle, a hallmark feature only recently identified for unregularized OT. As a consequence of our theory, we show that the empirical entropic Gromov-Wasserstein distance and its unregularized version for measures on Euclidean spaces also obey this principle. Additionally, we comment on computational aspects and complement our findings with Monte Carlo simulations. Our techniques employ empirical process theory and rely on a dual formulation of EOT over a single function class. Crucial to our analysis is the observation that the entropic cost-transformation of a function class does not increase its uniform metric entropy by much.Comment: 46 pages, 5 figure

Limit Distributions and Sensitivity Analysis for Empirical Entropic Optimal Transport on Countable Spaces

For probability measures on countable spaces we derive distributional limits for empirical entropic optimal transport quantities. More precisely, we show that the empirical optimal transport plan weakly converges to a centered Gaussian process and that the empirical entropic optimal transport value is asymptotically normal. The results are valid for a large class of cost functions and generalize distributional limits for empirical entropic optimal transport quantities on finite spaces. Our proofs are based on a sensitivity analysis with respect to norms induced by suitable function classes, which arise from novel quantitative bounds for primal and dual optimizers, that are related to the exponential penalty term in the dual formulation. The distributional limits then follow from the functional delta method together with weak convergence of the empirical process in that respective norm, for which we provide sharp conditions on the underlying measures. As a byproduct of our proof technique, consistency of the bootstrap for statistical applications is shown.Comment: 68 page

Predicting participants’ attitudes from patterns of event-related potentials during the reading of morally relevant statements – An MVPA investigation

Morality and language are hardly separable, given that morality-related aspects such as knowledge, emotions, or experiences are connected with language on different levels. One question that arises is: How rapidly do neural processes set in when processing statements that reflect moral value containing information? In the current study, participants read sentences about morally relevant statements (e.g., ‘Wars are acceptable’) and expressed their (dis)agreement with the statements while their electroencephalogram (EEG) was recorded. Multivariate pattern classification (MVPA) was used during language processing to predict the individual's response. Our results show that (1) the response (‘yes’ vs. ‘no’) could be predicted from 180 ms following the decision-relevant word (here acceptable), and (2) the attitude (pro vs. contra the topic) could be predicted from 170 ms following the topic word (here wars). We suggest that the successful MVPA classification is due to different brain activity patterns evoked by differences in activated mental representations (e.g. valence, arousal, etc.) depending on whether the attitude towards the topic is positive or negative and whether it is in accordance with the presented decisive word or not

Neuronale Korrelate moralischer Entscheidungsprozesse

Die Ergründung moralischer Urteilsprozesse ist für ein soziales Zusammenleben bedeutsam und beschäftigt Philosophen und Wissenschaftler seit jeher. In den letzten Jahrzehnten eröffneten neurowissenschaftliche Untersuchungsmethoden ganz neue Perspektiven zur Erforschung dieser Thematik. Mittels Aufzeichnung hirnelektrischer Spannungsschwankungen lassen sich beispielsweise genaue zeitliche Prozessabläufe untersuchen. Obwohl dies eine wichtige und interessante Komponente in der Ergründung moralischer Entscheidungen darstellt, wurde der Versuch bislang wenig unternommen. Daher lag das Hauptinteresse der vorliegenden Arbeit darin, einen Beitrag zu diesem Forschungsfeld zu leisten und moralische Entscheidungsprozesse mit Hilfe der Elektroenzephalografie zu untersuchen. In zwei Experimenten wurden Teilnehmende (N=84) gebeten, ihre Zustimmung beziehungsweise Ablehnung gegenüber Aussagen zu verschiedenen moralischen Themen (z. B. „Geschwisterinzest ist akzeptabel“) abzugeben. Anschließend erhielten sie zu jedem Thema einen Text mit Sachinformationen und wurden hierauf erneut abgefragt. Um jedoch die letzteren moralischen Entscheidungsprozesse unterschiedlich zu beeinflussen, beinhaltete jeweils die Hälfte der Texte Informationen, die der zuvor abgegebenen Einstellung entsprachen oder aber dieser entgegenstanden. Während dessen wurden die elektrische Gehirnaktivität und die Antwortzeiten erfasst und in Abhängigkeit wertesystemkongruenter bzw. -inkongruenter Satzendungen (akzeptabel versus inakzeptabel) gesetzt. Die Ergebnisse können wie folgt zusammengefasst werden. Hinsichtlich der hirnelektrischen Potentiale konnten in beiden Experimenten während der Sprachverarbeitung Unterschiede zwischen wertekongruenten und werteinkongruenten Satzendungen nachgewiesen werden. Zudem zeigte sich im ersten Experiment nach dem Darbieten von Sachinformationen zu den moralischen Themen ein vergrößerter Kongruenzeffekt im Falle einstellungswiedersprechender Textinhalte. Insgesamt lässt sich aus den Ergebnissen schließen, dass Aussagen, die mit dem eigenen Wertesystem unvereinbar sind, mit einer erschwerten Sprachverarbeitung und/oder intensiveren Aufmerksamkeitsressourcen einhergehen. Die divergierenden Befunde der beiden Experimente werden insbesondere im Zusammenhang mit zeitlichen und räumlichen Überlappungen der Gehirnpotentiale diskutiert

Neural correlates of error detection during complex response selection: Introduction of a novel eight-alternative response task

Error processing in complex decision tasks should be more difficult compared to a simple and commonly used two-choice task. We developed an eight-alternative response task (BART), which allowed us to investigate different aspects of error detection. We analysed event-related potentials (ERP; N = 30). Interestingly, the response time moderated several findings. For example, only for fast responses, we observed the well-known effect of larger error negativity (N-e) in signalled and non-signalled errors compared to correct responses, but not for slow responses. We identified at least two different error sources due to post-experimental reports and certainty ratings: impulsive (fast) errors and (slow) memory errors. Interestingly, the participants were able to perform the task and to identify both, impulsive and memory errors successfully. Preliminary evidence indicated that early (N-e-related) error processing was not sensitive to memory errors but to impulsive errors, whereas the error positivity seemed to be sensitive to both error types

Brain indices of disagreement with one's social values predict EU referendum voting behavior

Pre-electoral surveys typically attempt, and sometimes fail, to predict voting behavior on the basis of explicit measures of agreement or disagreement with a candidate or political position. Here, we assessed whether a specific brain signature of disagreement with one's social values, the event-related potential component N400, could be predictive of voting behavior. We examined this possibility in the context of the EU referendum in the United Kingdom. In the five weeks preceding the referendum, we recorded the N400 while participants with different vote intentions expressed their agreement or disagreement with pro- and against-EU statements. We showed that the N400 responded to statements incongruent with one's view regarding the EU. Crucially, this effect predicted actual voting behavior in decided as well as undecided voters. The N400 was a better predictor of voting choice than an explicit index of preference based on the behavioral responses. Our findings demonstrate that well-defined patterns of brain activity can forecast future voting behavior

Genetic variation at twentythree microsatellite loci in sixteen human populations

Artículo científico -- Universidad de Costa Rica, Instituto de Investigaciones en Salud. 1999We have analysed genetic variation at 23 microsatellite loci in a global sample of 16 ethnically and geographically diverse human populations. On the basis of their ancestral heritage and geographic locations, the studied populations can be divided into five major groups, viz. African, Caucasian, Asian Mongoloid, American Indian and Pacific Islander. With respect to the distribution of alleles at the 23 loci, large variability exists among the examined populations. However, with the exception of the American Indians and the Pacific Islanders, populations within a continental group show a greater degree of similarity. Phylogenetic analyses based on allele frequencies at the examined loci show that the first split of the present-day human populations had occurred between the Africans and all of the non-African populations, lending support to an African origin of modern human populations. Gene diversity analyses show that the coefficient of gene diversity estimated from the 23 loci is, in general, larger for populations that have remained isolated and probably of smaller effective sizes, such as the American Indians and the Pacific Islanders. These analyses also demonstrate that the component of total gene diversity, which is attributed to variation between groups of populations, is significantly larger than that among populations within each group. The empirical data presented in this work and their analyses reaffirm that evolutionary histories and the extent of genetic variation among human populations can be studied using microsatellite loci.Universidad de Costa Rica. Instituto de Investigaciones en SaludUCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias de la Salud::Instituto de Investigaciones en Salud (INISA