PAC-Bayesian Bound for the Conditional Value at Risk

Conditional Value at Risk (CVaR) is a family of "coherent risk measures" which generalize the traditional mathematical expectation. Widely used in mathematical finance, it is garnering increasing interest in machine learning, e.g., as an alternate approach to regularization, and as a means for ensuring fairness. This paper presents a generalization bound for learning algorithms that minimize the CVaR of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical CVaR is small. We achieve this by reducing the problem of estimating CVaR to that of merely estimating an expectation. This then enables us, as a by-product, to obtain concentration inequalities for CVaR even when the random variable in question is unbounded

Green Currents for Meromorphic Maps of Compact K\"ahler Manifolds

We consider the dynamics of meromorphic maps of compact K\"ahler manifolds. In this work, our goal is to locate the non-nef locus of invariant classes and provide necessary and sufficient conditions for existence of Green currents in codimension one.Comment: Statement of Theorem 1.5 is slightly improved. Proposition 5.2 and Theorem 5.3 are adde

Increase of precuneus metabolism correlates with reduction of PTSD symptoms after EMDR therapy in military veterans: an 18F-FDG PET study during virtual reality exposure to war

International audienc

Univalent Foundations and the UniMath Library

We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander

PAC-Bayesian Bound for the Conditional Value at Risk

International audienceConditional Value at Risk (CVAR) is a family of "coherent risk measures" which generalize the traditional mathematical expectation. Widely used in mathematical finance, it is garnering increasing interest in machine learning, e.g., as an alternate approach to regularization, and as a means for ensuring fairness. This paper presents a generalization bound for learning algorithms that minimize the CVAR of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical CVAR is small. We achieve this by reducing the problem of estimating CVAR to that of merely estimating an expectation. This then enables us, as a by-product, to obtain concentration inequalities for CVAR even when the random variable in question is unbounded

Focused Bayesian Prediction

We propose a new method for conducting Bayesian prediction that delivers accurate predictions without correctly specifying the unknown true data generating process. A prior is defined over a class of plausible predictive models. After observing data, we update the prior to a posterior over these models, via a criterion that captures a user-specified measure of predictive accuracy. Under regularity, this update yields posterior concentration onto the element of the predictive class that maximizes the expectation of the accuracy measure. In a series of simulation experiments and empirical examples we find notable gains in predictive accuracy relative to conventional likelihood-based prediction

Sur les exposants de Lyapounov des applications meromorphes

Let f be a dominating meromorphic self-map of a compact Kahler manifold. We give an inequality for the Lyapounov exponents of some ergodic measures of f using the metric entropy and the dynamical degrees of f. We deduce the hyperbolicity of some measures.Comment: 27 pages, paper in french, final version: to appear in Inventiones Mat

Clinical characteristics and brain PET findings in 3 cases of dissociative amnesia : Disproportionate retrograde devicit and posterior middle temporal gyrus hypometabolism

Background Precipitated by psychological stress, dissociative amnesia occurs in the absence of identifiable brain damage. Its clinical characteristics and functional neural basis are still a matter of controversy. Methods In the present paper, we report 3 cases of retrograde autobiographical amnesia, characterized by an acute onset concomitant with emotional/neurological precipitants. We present 2 cases of dissociative amnesia with fugue (cases 1 and 2), and one case of focal dissociative amnesia after a minor head trauma (case 3). The individual case histories and neuropsychological characteristics are reported, as well as the whole-brain voxel-based 18FDG-PET metabolic findings obtained at group-level in comparison to 15 healthy subjects. Results All patients suffered from autobiographical memory loss, in the absence of structural lesion. They had no significant impairment of anterograde memory or of executive function. Impairment of autobiographical memory was complete for two of the three patients, with loss of personal identity (cases 1 and 2). A clinical recovery was found for the two patients in whom follow-up was available (cases 2 and 3). Voxel-based group analysis highlighted a metabolic impairment of the right posterior middle temporal gyrus. 18FDG-PET was repeated in case 3, and showed a complete functional brain recovery. Conclusion The situation of dissociative amnesia with disproportionate retrograde amnesia is clinically heterogeneous between individuals. Our findings may suggest that impairment of high-level integration of visual and/or emotional information processing involving dysfunction of the right posterior middle temporal gyrus could reduce triggering of multi-modal visual memory traces, thus impeding reactivation of aversive memories

Post-critical set and non existence of preserved meromorphic two-forms

We present a family of birational transformations in $CP_2$ depending on two, or three, parameters which does not, generically, preserve meromorphic two-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called ``post-critical set'', we get some new structures, some "non-analytic" two-form which reduce to meromorphic two-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the \emph{degrees of the parameters}, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in $CP_2$ is first carried out using Diller-Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, one more time, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic two-form can be preserved for this mapping. These birational transformations in $CP_2$, which, generically, do not preserve any meromorphic two-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic two-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more ``probabilistic'' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic two-form explains most of the (numerical) discrepancy between the topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure