Bounding errors of Expectation-Propagation

Expectation Propagation is a very popular algorithm for variational inference, but comes with few theoretical guarantees. In this article, we prove that the approximation errors made by EP can be bounded. Our bounds have an asymptotic interpretation in the number $n$ of datapoints, which allows us to study EP's convergence with respect to the true posterior. In particular, we show that EP converges at a rate of $\mathcal{0}(n^{-2})$ for the mean, up to an order of magnitude faster than the traditional Gaussian approximation at the mode. We also give similar asymptotic expansions for moments of order 2 to 4, as well as excess Kullback-Leibler cost (defined as the additional KL cost incurred by using EP rather than the ideal Gaussian approximation). All these expansions highlight the superior convergence properties of EP. Our approach for deriving those results is likely applicable to many similar approximate inference methods. In addition, we introduce bounds on the moments of log-concave distributions that may be of independent interest.Comment: Accepted and published at NIPS 201

Spatially Invariant Coding of Numerical Information in Functionally Defined Subregions of Human Parietal Cortex

Macaque electrophysiology has revealed neurons responsive to number in lateral (LIP) and ventral (VIP) intraparietal areas. Recently, fMRI pattern recognition revealed information discriminative of individual numbers in human parietal cortex but without precisely localizing the relevant sites or testing for subregions with different response profiles. Here, we defined the human functional equivalents of LIP (feLIP) and VIP (feVIP) using neurophysiologically motivated localizers. We applied multivariate pattern recognition to investigate whether both regions represent numerical information and whether number codes are position specific or invariant. In a delayed number comparison paradigm with laterally presented numerosities, parietal cortex discriminated between numerosities better than early visual cortex, and discrimination generalized across hemifields in parietal, but not early visual cortex. Activation patterns in the 2 parietal regions of interest did not differ in the coding of position-specific or position-independent number information, but in the expression of a numerical distance effect which was more pronounced in feLIP. Thus, the representation of number in parietal cortex is at least partially position invariant. Both feLIP and feVIP contain information about individual numerosities in humans, but feLIP hosts a coarser representation of numerosity than feVIP, compatible with either broader tuning or a summation cod

Bunge’s Mathematical Structuralism Is Not a Fiction

In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involved in mathematical knowledge, in particular its dependence on mental/brain states and material objects

Diagnosis of focal liver lesions from ultrasound using deep learning

PURPOSE: The purpose of this study was to create an algorithm that simultaneously detects and characterizes (benign vs. malignant) focal liver lesion (FLL) using deep learning. MATERIALS AND METHODS: We trained our algorithm on a dataset proposed during a data challenge organized at the 2018 Journées Francophones de Radiologie. The dataset was composed of 367 two-dimensional ultrasound images from 367 individual livers, captured at various institutions. The algorithm was guided using an attention mechanism with annotations made by a radiologist. The algorithm was then tested on a new data set from 177 patients. RESULTS: The models reached mean ROC-AUC scores of 0.935 for FLL detection and 0.916 for FLL characterization over three shuffled three-fold cross-validations performed with the training data. On the new dataset of 177 patients, our models reached a weighted mean ROC-AUC scores of 0.891 for seven different tasks. CONCLUSION: This study that uses a supervised-attention mechanism focused on FLL detection and characterization from liver ultrasound images. This method could prove to be highly relevant for medical imaging once validated on a larger independent cohort

Neural Reuse and the Nature of Evolutionary Constraints

In humans, the reuse of neural structure is particularly pronounced at short, task-relevant timescales. Here, an argument is developed for the claim that facts about neural reuse at task-relevant timescales conflict with at least one characterization of neural reuse at an evolutionary timescale. It is then argued that, in order to resolve the conflict, we must conceptualize evolutionary-scale reuse more abstractly than has been generally recognized. The final section of the paper explores the relationship between neural reuse and human nature. It is argued that neural reuse is not well-described as a process that constrains our present cognitive capacities. Instead, it liberates those capacities from the ancestral tethers that might otherwise have constrained them

Algebraic invariants of five qubits

The Hilbert series of the algebra of polynomial invariants of pure states of five qubits is obtained, and the simplest invariants are computed.Comment: 4 pages, revtex. Short discussion of quant-ph/0506073 include

Mental Addition in Bilinguals: An fMRI Study of Task-Related and Performance-Related Activation

Behavioral studies show that bilinguals are slower and less accurate when performing mental calculation in their nondominant (second; L2) language than in their dominant (first; L1) language. However, little is known about the neural correlates associated with the performance differences observed between bilinguals' 2 languages during arithmetic processing. To address the cortical activation differences between languages, the current study examined task-related and performance-related brain activation during mental addition when problems were presented auditorily in participants' L1 and L2. Eleven Chinese–English bilinguals heard 2-digit addition problems that required exact or approximate calculations. Functional magnetic resonance imaging results showed that auditorily presented multidigit addition in bilinguals activates bilateral inferior parietal and inferior frontal regions in both L1 and L2. Language differences were observed in the form of greater activation for L2 exact addition in the left inferior frontal area. A negative correlation between brain activation and behavioral performance during mental addition in L2 was observed in the left inferior parietal area. Current results provide further evidence for the effects of language-specific experience on arithmetic processing in bilinguals at the cortical level