new dissipated energy and partial regularity for thin film equations abstract we study the thin film equation h t h n h x x x x and prove that the energy h p h x 2 d x dissipates for suitable weak solutions when n p lies in the whole domain e defined by laugesen which extends partially the corresponding results of laugesen chugunova and taranets and then improve some previous results of partial regularity by chou and the first author

quasistationary trajectories of the mean field xy hamiltonian model a topological perspective we employ a topological approach to investigate the nature of quasistationary states of the mean field xy hamiltonian model we focus on the quasistationary states reached when the system is initially prepared in a fully magnetized configuration by means of numerical simulations and analytical considerations we show that along the quasistationary trajectories the system evolves in a manifold of critical points of the potential energy function although these critical points are maxima the large number of directions with marginal stability may be responsible for the slow relaxation dynamics and the trapping of the system in such trajectories

on the minimization of convex functionals of probability distributions under band constraints the problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below a system of sufficient and necessary first order optimality conditions as well as a bound on the optimality gap of feasible candidate solutions are derived based on these results two numerical algorithms are proposed that iteratively solve the system of optimality conditions on a grid of discrete points both algorithms use a block coordinate descent strategy and terminate once the optimality gap falls below the desired tolerance while the first algorithm is conceptually simpler and more efficient it is not guaranteed to converge for objective functions that are not strictly convex this shortcoming is overcome in the second algorithm which uses an additional outer proximal iteration and which is proven to converge under mild assumptions two examples are given to demonstrate the theoretical usefulness of the optimality conditions as well as the high efficiency and accuracy of the proposed numerical algorithms

higher order nonlinear degenerate parabolic equations this paper is concerned with nonlinear degenerate parabolic equations of the form ut 1 m 1 d f u d2m 1u 0 with f u u n n 1 near u 0 and d x under appropriate boundary conditions it is shown that there exists a weak solution u some of the main results of the paper are that u 0 if u0 0 and that the support of u t when u0 0 increases with t for the last property we require that n 2 and m 1

collisions and phase vortex interactions in dissipative ginzburg landau dynamics in this paper we describe a natural framework for the vortex dynamics in the parabolic complex ginzburg landau equation in r this general setting does not rely on any assumption of well preparedness and has the advantage to be valid even after collision times we analyze carefully collisions leading to annihilation a new phenomenon is identified the phase vortex interaction related to persistence of low frequency oscillations and leading to an unexpected drift in the motion of vortices

on uniqueness for the harmonic map heat flow in supercritical dimensions we examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension d 3 it is shown that generically singular data can give rise to two distinct solutions that are both stable and satisfy the local energy inequality we also discuss how uniqueness can be retrieved 2017 wiley periodicals inc

greedy algorithms for classification consistency convergence rates and adaptivity many regression and classification algorithms proposed over the years can be described as greedy procedures for the stagewise minimization of an appropriate cost function some examples include additive models matching pursuit and boosting in this work we focus on the classification problem for which many recent algorithms have been proposed and applied successfully for a specific regularized form of greedy stagewise optimization we prove consistency of the approach under rather general conditions focusing on specific classes of problems we provide conditions under which our greedy procedure achieves the nearly minimax rate of convergence implying that the procedure cannot be improved in a worst case setting we also construct a fully adaptive procedure which without knowing the smoothness parameter of the decision boundary converges at the same rate as if the smoothness parameter were known

primal explicit max margin feature selection for nonlinear support vector machines embedding feature selection in nonlinear support vector machines svms leads to a challenging non convex minimization problem which can be prone to suboptimal solutions this paper develops an effective algorithm to directly solve the embedded feature selection primal problem we use a trust region method which is better suited for non convex optimization compared to line search methods and guarantees convergence to a minimizer we devise an alternating optimization approach to tackle the problem efficiently breaking it down into a convex subproblem corresponding to standard svm optimization and a non convex subproblem for feature selection importantly we show that a straightforward alternating optimization approach can be susceptible to saddle point solutions we propose a novel technique which shares an explicit margin variable to overcome saddle point convergence and improve solution quality experiment results show our method outperforms the state of the art embedded svm feature selection method as well as other leading filter and wrapper approaches highlightswe solve a primal embedded feature selection problem for nonlinear svms use smooth hinge loss and trust region algorithm for bound constrained minimization propose alternating optimization approach to break problem into two smaller ones propose explicit margin maximization to solve feature selection subproblem show our approach improves state of art and other algorithms on various data

direct estimation of density functionals using a polynomial basis a number of fundamental quantities in statistical signal processing and information theory can be expressed as integral functions of two probability density functions such quantities are called density functionals as they map density functions onto the real line for example information divergence functions measure the dissimilarity between two probability density functions and are useful in a number of applications typically estimating these quantities requires complete knowledge of the underlying distribution followed by multidimensional integration existing methods make parametric assumptions about the data distribution or use nonparametric density estimation followed by high dimensional integration in this paper we propose a new alternative we introduce the concept of data driven basis functions functions of distributions whose value we can estimate given only samples from the underlying distributions without requiring distribution fitting or direct integration we derive a new data driven complete basis that is similar to the deterministic bernstein polynomial basis and develop two methods for performing basis expansions of functionals of two distributions we also show that the new basis set allows us to approximate functions of distributions as closely as desired finally we evaluate the methodology by developing data driven estimators for the kullback leibler divergences and the hellinger distance and by constructing empirical estimates of tight bounds on the bayes error rate

role of saddles in topologically driven phase transitions the case of the d dimensional spherical model analyzing the d dimensional spherical model we show that underlying saddles defined through a map in the configuration space play a central role in driving the phase transition at the phase transition point the underlying saddle energy reaches its lowest value corresponding to the trivial boundary topological singularity

linear stability of steady states for thin film and cahn hilliard type equations we study the linear stability of smooth steady states of the evolution equation h t f h xxx x g h h x x ah under both periodic and neumann boundary conditions if a 0 we assume f 1 in particular we consider positive periodic steady states of thin film equations where a 0 and f g might have degeneracies such as f 0 0 as well as singularities like g 0

some inequalities for information divergence and related measures of discrimination inequalities which connect information divergence with other measures of discrimination or distance between probability distributions are used in information theory and its applications to mathematical statistics ergodic theory and other scientific fields we suggest new inequalities of this type often based on underlying identities as a consequence we obtain certain improvements of the well known pinsker inequality our study depends on two measures of discrimination called capacitory discrimination and triangular discrimination the discussion contains references to related research and comparison with other measures of discrimination e g ali silvey csiszar 1996 1966 divergences and in particular the hellinger distance

energy levels of steady states for thin film type equations abstract we study the phase space of the evolution equationht f h hxxx x g h hx x by means of a dissipated energy here h x t 0 and at h 0 the coefficient functions f 0 and g can either degenerate to 0 or blow up to or tend to a nonzero constant we first show that all positive periodic steady states are saddles in the energy landscape with respect to zero mean perturbations if g f 0 or if the perturbations are allowed to have period longer than that of the steady state for power law coefficients f y yn and g y b ym for some b 0 we analytically determine the relative energy levels of distinct steady states for example with m n 1 2 and for suitable choices of the period and mean value we find three fundamentally different steady states the first is a constant steady state that is stable and is a local minimum of the energy the second is a positive periodic steady state that is linearly unstable and has higher energy than the constant steady state it is a saddle point for the energy the third is a periodic collection of droplet compactly supported steady states having lower energy than either the positive steady state or the constant one since the energy must decrease along every orbit these results significantly constrain the dynamics of the evolution equation

information divergence and risk for binary experiments we unify f divergences bregman divergences surrogate regret bounds proper scoring rules cost curves roc curves and statistical information we do this by systematically studying integral and variational representations of these objects and in so doing identify their representation primitives which all are related to cost sensitive binary classification as well as developing relationships between generative and discriminative views of learning the new machinery leads to tight and more general surrogate regret bounds and generalised pinsker inequalities relating f divergences to variational divergence the new viewpoint also illuminates existing algorithms it provides a new derivation of support vector machines in terms of divergences and relates maximum mean discrepancy to fisher linear discriminants

the lubrication approximation for thin viscous films regularity and long time behavior of weak solutions we consider the fourth order degenerate diffusion equation in one space dimension this equation derived from a lubrication approximation models the surface tension dominated motion of thin viscous films and spreading droplets 15 the equation with f h h also models a thin neck of fluid in the hele shaw cell 10 11 23 in such problems h x t is the local thickness of the the film or neck this paper considers the properties of weak solutions that are more relevant to the droplet problem than to hele shaw for simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets we consider two problems the first has initial data h0 0 and f h h n 0 n 3 we show that there exists a weak nonnegative solution for all time also we show that this solution becomes a strong positive solution after some finite time t and asymptotically approaches its means as t the weak solution is in the classical sense of distributions for 3 8 n 3 and in a weaker sense introduced in 1 for the remaining 0 n 3 8 furthermore the solutions have high enough regularity to just include the unique source type solutions 2 with zero slope at the edge of the support they do not include any of the less regular solutions with positive slope at the edge of the support second we consider strictly positive initial data h0 m 0 and f h h n 0 n for this problem we show that even if a finite time singularity of the form h 0 does occur there exists a weak nonnegative solution for all time t this weak solution becomes strong and positive again after some critical time t as in the first problem we show that the solution approaches its mean as t the main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity we show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result 1996 john wiley sons inc

computational topology for configuration spaces of hard disks we explore the topology of configuration spaces of hard disks experimentally and show that several changes in the topology can already be observed with a small number of particles the results illustrate a theorem of baryshnikov bubenik and kahle 2 that critical points correspond to configurations of disks with balanced mechanical stresses and suggest conjectures about the asymptotic topology as the number of disks tends to infinity

nonparametric decentralized detection using kernel methods we consider the problem of decentralized detection under constraints on the number of bits that can be transmitted by each sensor in contrast to most previous work in which the joint distribution of sensor observations is assumed to be known we address the problem when only a set of empirical samples is available we propose a novel algorithm using the framework of empirical risk minimization and marginalized kernels and analyze its computational and statistical properties both theoretically and empirically we provide an efficient implementation of the algorithm and demonstrate its performance on both simulated and real data sets

estimating divergence functionals and the likelihood ratio by convex risk minimization we develop and analyze m estimation methods for divergence functionals and the likelihood ratios of two probability distributions our method is based on a nonasymptotic variational characterization of f divergences which allows the problem of estimating divergences to be tackled via convex empirical risk optimization the resulting estimators are simple to implement requiring only the solution of standard convex programs we present an analysis of consistency and convergence for these estimators given conditions only on the ratios of densities we show that our estimators can achieve optimal minimax rates for the likelihood ratio and the divergence functionals in certain regimes we derive an efficient optimization algorithm for computing our estimates and illustrate their convergence behavior and practical viability by simulations

long wave instabilities and saturation in thin film equations hocherman and rosenau conjectured that long wave unstable cahn hilliard type interface models develop nite time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term phys d 67 113 125 1993 we consider this conjecture for a class of equations often used to model thin lms in a lubrication context by showing that if the solutions are uniformly bounded above or below e g are nonnegative then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded for example they conjecture that for the long wave unstable equation ht h n hxxx x h m hx x m n leads to blow up using a conservation of volume constraint for the case m n 0 we conjecture a dierent critical exponent for possible singularities of nonnegative solutions we prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions specically for the above equation solutions are bounded if m n 2 the bound is proved using energy methods and is then used to prove the existence of non negative weak solutions in the sense of distributions we present preliminary numerical evidence suggesting that m n 2 can allow blow up

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

statistical behavior and consistency of classification methods based on convex risk minimization we study how closely the optimal bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function the measurement of closeness is characterized by the loss function used in the estimation we show that such a classification scheme can be generally regarded as a nonmaximum likelihood conditional in class probability estimate and we use this analysis to compare various convex loss functions that have appeared in the literature furthermore the theoretical insight allows us to design good loss functions with desirable properties another aspect of our analysis is to demonstrate the consistency of certain classification methods using convex risk minimization this study sheds light on the good performance of some recently proposed linear classification methods including boosting and support vector machines it also shows their limitations and suggests possible improvements

on the bayes risk consistency of regularized boosting methods the probability of error of classification methods based on convex combinations of simple base classifiers by boosting algorithms is investigated the main result of the paper is that certain regularized boosting algorithms provide bayes risk consistent classifiers under the sole assumption that the bayes classifier may be approximated by a convex combination of the base classifiers nonasymptotic distribution free bounds are also developed which offer interesting new insight into how boosting works and help explain its success in practical classification problems

strong data processing inequalities and phi sobolev inequalities for discrete channels the noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions for instance the worst case ratio of output relative entropy to input relative entropy for all possible pairs of input distributions is bounded from above by unity by the data processing theorem however for a fixed reference input distribution this quantity may be strictly smaller than one giving the so called strong data processing inequalities sdpis the same considerations apply to an arbitrary phi divergence this paper presents a systematic study of optimal constants in the sdpis for discrete channels including their variational characterizations upper and lower bounds structural results for channels on product probability spaces and the relationship between the sdpis and the so called phi sobolev inequalities another class of inequalities that can be used to quantify the noisiness of a channel by controlling entropy like functionals of the input distribution by suitable measures of input output correlation several applications to information theory discrete probability and statistical physics are discussed

existence and partial regularity results for the heat flow for harmonic maps on considere le probleme d evolution t u m u n u 0 dans m r u 0 u 0 sur m ou m et n sont des varietes de riemann lisses compactes et u 0 l application harmonique de m dans n on demontre l existence d une solution globale faible

convexity classification and risk bounds many of the classification algorithms developed in the machine learning literature including the support vector machine and boosting can be viewed as minimum contrast methods that minimize a convex surrogate of the 0 1 loss function the convexity makes these algorithms computationally efficient the use of a surrogate however has statistical consequences that must be balanced against the computational virtues of convexity to study these issues we provide a general quantitative relationship between the risk as assessed using the 0 1 loss and the risk as assessed using any nonnegative surrogate loss function we show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function that it satisfies a pointwise form of fisher consistency for classification the relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications we also present a refined version of this result in the

empirically estimable classification bounds based on a nonparametric divergence measure information divergence functions play a critical role in statistics and information theory in this paper we show that a nonparametric f divergence measure can be used to provide improved bounds on the minimum binary classification probability of error for the case when the training and test data are drawn from the same distribution and for the case where there exists some mismatch between training and test distributions we confirm these theoretical results by designing feature selection algorithms using the criteria from these bounds and by evaluating the algorithms on a series of pathological speech classification tasks

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

relationship between phase transitions and topological changes in one dimensional models we address the question of the quantitative relationship between thermodynamic phase transitions and topological changes in the potential energy manifold analyzing two classes of one dimensional models the burkhardt solid on solid model and the peyrard bishop model for dna thermal denaturation both in the confining and nonconfining version these models apparently do not fit m kastner phys rev lett 93 150601 2004 in the general idea that the phase transition is signaled by a topological discontinuity we show that in both models the phase transition energy v c is actually noncoincident with and always higher than the energy v ensuremath theta at which a topological change appears however applying a procedure already successfully employed in other cases as the mean field ensuremath phi 4 model i e introducing a map mathcal m v ensuremath rightarrow v s from levels of the energy hypersurface v to the level of the stationary points visited at temperature t we find that mathcal m v c v ensuremath theta this result enhances the relevance of the underlying stationary points in determining the thermodynamics of a system and extends the validity of the topological approach to the study of phase transition to the elusive one dimensional systems considered here

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

exploring the energy landscape of xy models we investigate the energy landscape of two and three dimensional xy models with nearest neighbor interactions by analytically constructing several classes of stationary points of the hamiltonian these classes are analyzed in particular with respect to possible signatures of the thermodynamic phase transitions of the models we find that even after explicitly breaking the global o 2 symmetry of the xy spins an exponentially large class of stationary points are singular and occur in continuous one parameter families this property may complicate the use of theoretical tools developed for the investigation of phase transitions based on stationary points of the energy landscape and we discuss strategies to avoid these difficulties

process consistency for adaboost recent experiments and theoretical studies show that adaboost can overfit in the limit of large time if running the algorithm forever is suboptimal a natural question is how low can the prediction error be during the process of adaboost we show under general regularity conditions that during the process of adaboost a consistent prediction is generated which has the prediction error approximating the optimal bayes error as the sample size increases this result suggests that while running the algorithm forever can be suboptimal it is reasonable to expect that some regularization method via truncation of the process may lead to a near optimal performance for sufficiently large sample size

topological hypothesis on phase transitions the simplest case we critically analyze the possibility of finding signatures of a phase transition by looking exclusively at static quantities of statistical systems like e g the topology of potential energy submanifolds pes s this topological hypothesis has been successfully tested in a few statistical models but up to now there has been no rigorous proof of its general validity we make a new test of it analyzing the probably simplest example of a nontrivial system undergoing a continuous phase transition the completely connected version of the spherical model going through the topological properties of its pes it is shown that as expected the phase transition is correlated with a change in their topology nevertheless this change as reflected in the behavior of a particular topological invariant the euler characteristic is small at variance with the strong singularity observed in other systems furthermore it is shown that in the presence of an external field when the phase transition is destroyed a similar topology change in the submanifolds is still observed at the maximum value of the potential energy manifold a level which nevertheless is thermodynamically inaccessible this suggests that static properties of the pes s are not enough in order to decide whether a phase transition will take place some input from dynamics seems necessary

consistency of support vector machines and other regularized kernel classifiers it is shown that various classifiers that are based on minimization of a regularized risk are universally consistent i e they can asymptotically learn in every classification task the role of the loss functions used in these algorithms is considered in detail as an application of our general framework several types of support vector machines svms as well as regularization networks are treated our methods combine techniques from stochastics approximation theory and functional analysis

on the mean field spherical model exact solutions are obtained for the mean field spherical model with or without an external magnetic field for any finite or infinite number n of degrees of freedom both in the microcanonical and in the canonical ensemble the canonical result allows for an exact discussion of the loci of the fisher zeros of the canonical partition function the microcanonical entropy is found to be nonanalytic for arbitrary finite n the mean field spherical model of finite size n is shown to be equivalent to a mixed isovector isotensor model on a lattice of two sites partial equivalence of statistical ensembles is observed for the mean field spherical model in the thermodynamic limit a discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence

topology and phase transitions the case of the short range spherical model we characterize the topology of the phase space of the berlin kac spherical model in the context of the so called topological hypothesis for spins lying in hypercubic lattices of dimension d for zero external field we are able to characterize the topology exactly up to homology we find that even though there is a continuum of changes in the topology of the corresponding manifolds for d 3 there are abrupt discontinuities in some topological functions that could be good candidates to associate with the phase transitions that occur at the thermodynamic level we show however that these changes do not coincide with the phase transitions and conversely that no topological discontinuity can be associated to the points where the phase transitions take place at variance with what happens in the mean field version of this same model we show that these abrupt topological changes are accessible thermodynamically we conclude that even in short range systems the topological mechanism does not seem to be responsible for the triggering of a phase transition we also analyze the case of spins connected to a macroscopic number of but not all neighbors and find that similar to the results found for the fully connected version in this case the topological hypothesis seems to hold the phase transition coincides with an accumulation point of the topological changes present in configuration space the question of the ensemble equivalence in the short range spherical model is also considered

thin films with high surface tension this paper is a review of work on thin fluid films where surface tension is a driving mechanism its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations in general the introduction of surface tension into standard lubrication theory leads to a fourth order nonlinear parabolic equation pad h t pad x left c frac h 3 3 frac partial 3 h partial x 3 f h h x h xx right 0 label abeq1 where h h x t is the fluid film height for steady situations this equation may be integrated once and a third order ordinary differential equation is obtained appropriate forms of this equation have been used to model fluid flows in physical situations such as coating draining of foams and the movement of contact lenses in the introduction a form of the above equation is derived for flow driven by surface tension surface tension gradients gravity and long range molecular forces modifications to the equation due to slip the effect of two free surfaces two phase fluids and higher dimensional forms are also discussed the second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given the third section reviews analytical work on the model equations as well as the generalized lubrication equation pad h t pad x left h n h xxx right 0 in particular the discussion focusses on asymptotic results travelling waves stability and similarity solutions numerical work is also discussed while for analytical results the reader is directed to existing literature

on surrogate loss functions and f divergences the goal of binary classification is to estimate a discriminant function y from observations of covariate vectors and corresponding binary labels we consider an elaboration of this problem in which the covariates are not available directly but are transformed by a dimensionality reducing quantizer q we present conditions on loss functions such that empirical risk minimization yields bayes consistency when both the discriminant function and the quantizer are estimated these conditions are stated in terms of a general correspondence between loss functions and a class of functionals known as ali silvey or divergence functionals whereas this correspondence was established by blackwell proc 2nd berkeley symp probab statist 1 1951 93 102 univ california press berkeley for the 0 1 loss we extend the correspondence to the broader class of surrogate loss functions that play a key role in the general theory of bayes consistency for binary classification our result makes it possible to pick out the strict subset of surrogate loss functions that yield bayes consistency for joint estimation of the discriminant function and the quantizer

on the evolution of harmonic maps in higher dimensions on etablit des resultats de regularite partielle et l existence de solutions regulieres globales pour le probleme d evolution pour des applications harmoniques avec des donnees petites

semi supervised nearest mean classification through a constrained log likelihood we cast a semi supervised nearest mean classifier previously introduced by the first author in a more principled log likelihood formulation that is subject to constraints this in turn leads us to make the important suggestion to not only investigate error rates of semi supervised learners but also consider the risk they originally aim to optimize we demonstrate empirically that in terms of classification error mixed results are obtained when comparing supervised to semi supervised nearest mean classification while in terms of log likelihood on the test set the semi supervised method consistently outperforms its supervised counterpart comparisons to self learning a standard approach in semi supervised learning are included to further clarify the way in which our constrained nearest mean classifier improves over regular supervised nearest mean classification

the mean field φ 4 model entropy analyticity and configuration space topology a large deviation technique is applied to the mean field model phi4 providing an exact expression for the configurational entropy s v m as a function of the potential energy v and the magnetization m although a continuous phase transition occurs at some critical energy vc the entropy is found to be a real analytic function in both arguments and it is only the maximization over m which gives rise to a nonanalyticity in s v supm s v m this mechanism of nonanalyticity generation by maximization over one variable of a real analytic entropy function is restricted to systems with long range interactions and has for continuous phase transitions the generic occurrence of classical critical exponents as an immediate consequence furthermore this mechanism can provide an explanation why contradictory to the so called topological hypothesis the phase transition in the mean field model need not be accompanied by a topology change in the family of constant energy submanifolds

quantization for decentralized hypothesis testing under communication constraints in a decentralized hypothesis testing network several peripheral nodes observe an environment and communicate their observations to a central node for the final decision the presence of capacity constraints introduces theoretical and practical problems the following problem is addressed given that the peripheral encoders that satisfy these constraints are scalar quantizers how should they be designed in order that the central test to be performed on their output indices is most powerful the scheme is called cooperative design separate encoding since the quantizers process separate observations but have a common goal they seek to maximize a system wide performance measure the bhattacharyya distance of the joint index space as such a criterion is suggested and a design algorithm to optimize arbitrarily many quantizers cyclically is proposed a simplified version of the algorithm namely an independent design separate encoding scheme where the correlation is either absent or neglected for the sake of simplicity is outlined performances are compared through worked examples

tunable measures for information leakage and applications to privacy utility tradeoffs we introduce a tunable measure for information leakage called maximal alpha leakage this measure quantifies the maximal gain of an adversary in inferring any potentially random function of a dataset from a release of the data the inferential capability of the adversary is in turn quantified by a class of adversarial loss functions that we introduce as alpha loss alpha in 1 infty cup infty the choice of alpha determines the specific adversarial action and ranges from refining a belief about any function of the data for alpha 1 to guessing the most likely value for alpha infty while refining the alpha text th moment of the belief for alpha in between maximal alpha leakage then quantifies the adversarial gain under alpha loss over all possible functions of the data in particular for the extremal values of alpha 1 and alpha infty maximal alpha leakage simplifies to mutual information and maximal leakage respectively for alpha in 1 infty this measure is shown to be the arimoto channel capacity of order alpha we show that maximal alpha leakage satisfies data processing inequalities and a sub additivity property thereby allowing for a weak composition result building upon these properties we use maximal alpha leakage as the privacy measure and study the problem of data publishing with privacy guarantees wherein the utility of the released data is ensured via a hard distortion constraint unlike average distortion hard distortion provides a deterministic guarantee of fidelity we show that under a hard distortion constraint for alpha 1 the optimal mechanism is independent of alpha and therefore the resulting optimal tradeoff is the same for all values of alpha 1 finally the tunability of maximal alpha leakage as a privacy measure is also illustrated for binary data with average hamming distortion as the utility measure

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

topology and phase transitions from an exactly solvable model to a relation between topology and thermodynamics the elsewhere surmized topological origin of phase transitions is given here important evidence through the analytic study of an exactly solvable model for which both topology of submanifolds of configuration space and thermodynamics are worked out the model is a mean field one with a k body interaction it undergoes a second order phase transition for k 2 and a first order one for kg2 this opens a perspective for the understanding of the deep origin of first and second order phase transitions respectively in particular a remarkable theoretical result consists of a mathematical characterization of first order transitions moreover we show that a reduced configuration space can be defined in terms of collective variables such that the correspondence between phase transitions and topology changes becomes one to one for this model finally an unusual relationship is worked out between the microscopic description of a classical n body system and its macroscopic thermodynamic behavior this consists of a functional dependence of thermodynamic entropy upon the morse indexes of the critical points saddles of the constant energy hypersurfaces of the microscopic 2n dimensional phase space thus phase space and configuration space topology is directly related to thermodynamics

parabolic theory as a high dimensional limit of elliptic theory the aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts this technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows specifically we give new proofs of an l 2 carleman estimate for the heat operator and the monotonicity formulas for the frequency function associated to the heat operator the two phase free boundary problem the flow of harmonic maps and the mean curvature flow the proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory in particular each parabolic theorem is proved by taking a high dimensional limit of the related elliptic result

singularities of first kind in the harmonic map and yang mills heat flows by constructing examples which are explicit up to solving an ode we prove that singularities of first kind exist for the harmonic map and yang mills heat flows as a by product we also get a simplified proof of ratto s ding s theorem about the existence of harmonic hopf constructions

composite multiclass losses we consider loss functions for multiclass prediction problems we show when a multiclass loss can be expressed as a proper composite loss which is the composition of a proper loss and a link function we extend existing results for binary losses to multiclass losses we subsume results on classification calibration by relating it to properness we determine the stationarity condition bregman representation order sensitivity and quasi convexity of multiclass proper losses we then characterise the existence and uniqueness of the composite representation for multiclass losses we show how the composite representation is related to other core properties of a loss mixability admissibility and strong convexity of multiclass losses which we characterise in terms of the hessian of the bayes risk we show that the simple integral representation for binary proper losses can not be extended to multiclass losses but offer concrete guidance regarding how to design different loss functions the conclusion drawn from these results is that the proper composite representation is a natural and convenient tool for the design of multiclass loss functions