constructing descriptive and discriminative nonlinear features rayleigh coefficients in kernel feature spaces we incorporate prior knowledge to construct nonlinear algorithms for invariant feature extraction and discrimination employing a unified framework in terms of a nonlinearized variant of the rayleigh coefficient we propose nonlinear generalizations of fisher s discriminant and oriented pca using support vector kernel functions extensive simulations show the utility of our approach

a sequential algorithm for sparse support vector classifiers support vector machines svms though accurate are not preferred in applications requiring high classification speed or when deployed in systems of limited computational resources due to the large number of support vectors involved in the model to overcome this problem we have devised a primal svm method with the following properties 1 it solves for the svm representation without the need to invoke the representer theorem 2 forward and backward selections are combined to approach the final globally optimal solution and 3 a criterion is introduced for identification of support vectors leading to a much reduced support vector set in addition to introducing this method the paper analyzes the complexity of the algorithm and presents test results on three public benchmark problems and a human activity recognition application these applications demonstrate the effectiveness and efficiency of the proposed algorithm

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

existence regularity and properties of generalized apparent horizons we prove a conjecture of tom ilmanen and hubert bray on the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing

on stable complete minimal hypersurfaces in r n 1 abstract by using the interior curvature estimate and gromov s compactness theorem we prove that a stable complete minimal hypersurface in rn l with finite total scalar curvature must be a hyperplane this extends the result obtained by p b rard in remarques sur l quation de j simons pitman monographs surveys pure appl math vol 52 1991 pp 47 57

hierarchical mixing linear support vector machines for nonlinear classification support vector machines svms play a very dominant role in data classification because of their good generalization performance however they suffer from the high computational complexity in the classification stage when there are a considerable number of support vectors svs it is desirable to design efficient algorithms in the classification stage to deal with datasets obtained from real time pattern recognition systems to this end we propose a novel classifier called hmlsvms hierarchical mixing linear support vector machines which has a hierarchical structure with a mixing linear svm classifier at each node it predicts the label of a sample using only a few hyperplanes we also give a generalization error bound for the class of locally linear svms llsvms based on the rademacher theory which ensures that overfitting can be effectively avoided experimental evaluations show that the proposed classifier achieves a high efficiency in the classification stage while the classification performance approaches that of kernel svms highlightsh mlsvms does not require a large memory to store kernel values in the training process because h mlsvms is composed of lsvms and there are very efficient algorithms for training lsvms 48 49 the hierarchical structure makes h mlsvms predict the labels of new arrived samples via a few of lsvms and it is much faster than nonlinear svms classifiers we quantify the generalization error bound for the class of llsvms based on the rademacher complexity and the stop criterion based on minimizing this bound ensures that h mlsvms can effectively avoid overfitting and have a good classification performance

partial regularity of stable solutions to the supercritical equations and its applications abstract we prove the partial regularity of stable solutions of supercritical elliptic equations as an application we prove that any smooth stable entire solution to supercritical equations with p in a suitable range is radially symmetric

min max minimal hypersurface in manifolds with convex boundary and hbox ric ge 0 let m n 1 partial m g be a compact manifold with non negative ricci curvature convex boundary and 2 le n le 6 we show that the min max minimal hypersurface with respect to one parameter families of hypersurfaces in m partial m is orientable of index one and multiplicity one

minimal hypersurfaces of least area in this paper we study closed embedded minimal hypersurfaces in a riemannian n 1 manifold 2 n 6 that minimize area among such hypersurfaces we show they exist and arise either by minimization techniques or by min max methods they have index at most 1 we apply this to obtain a lower area bound for such minimal surfaces in some hyperbolic 3 manifolds

sparseness of support vector machines support vector machines svms construct decision functions that are linear combinations of kernel evaluations on the training set the samples with non vanishing coefficients are called support vectors in this work we establish lower asymptotical bounds on the number of support vectors on our way we prove several results which are of great importance for the understanding of svms in particular we describe to which limit svm decision functions tend discuss the corresponding notion of convergence and provide some results on the stability of svms using subdifferential calculus in the associated reproducing kernel hilbert space

harmonic maps and constant mean curvature surfaces in h 2 x r we introduce a it hyperbolic gauss map into the poincar e disk for any surface in bbb h 2 times bbb r with regular vertical projection and prove that if the surface has constant mean curvature h 1 2 this hyperbolic gauss map is harmonic conversely we show that every nowhere conformal harmonic map from an open simply connected riemann surface sigma into the poincar e disk is the hyperbolic gauss map of a two parameter family of such surfaces as an application we obtain that any holomorphic quadratic differential on sigma can be realized as the abresch rosenberg holomorphic differential of some and generically infinitely many complete surfaces with h 1 2 in bbb h 2 times bbb r a similar result applies to minimal surfaces in the heisenberg group rm nil 3 finally we classify all complete minimal vertical graphs in bbb h 2 times bbb r

large isoperimetric surfaces in initial data sets we study the isoperimetric structure of asymptotically flat riemannian 3 manifolds m g that are mathcal c 0 asymptotic to schwarzschild of mass m gt 0 refining an argument due to h bray we obtain an effective volume comparison theorem in schwarzschild we use it to show that isoperimetric regions exist in m g for all sufficiently large volumes and that they are close to centered coordinate spheres this implies that the volume preserving stable constant mean curvature spheres constructed by g huisken and s t yau as well as r ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume this confirms a conjecture of h bray our results are consistent with the uniqueness results for volume preserving stable constant mean curvature surfaces in initial data sets obtained by g huisken and s t yau and strengthened by j qing and g tian the additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work

isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space the classical isoperimetric inequality for a domain sigma subset mathbb r k with smooth boundary partial sigma is displaystyle k k omega k mathrm vol sigma k 1 leq mathrm vol partial sigma k where equality holds if and only if sigma is a ball in mathbb r k

min max theory for free boundary minimal hypersurfaces ii general morse index bounds and applications for any smooth riemannian metric on an n 1 dimensional compact manifold with boundary m partial m where 3 le n 1 le 7 we establish general upper bounds for the morse index of free boundary minimal hypersurfaces produced by min max theory in the almgren pitts setting we apply our morse index estimates to prove that for almost every in the c infty baire sense riemannan metric the union of all compact properly embedded free boundary minimal hypersurfaces is dense in m if partial m is further assumed to have a strictly mean convex point we show the existence of infinitely many compact properly embedded free boundary minimal hypersurfaces whose boundaries are non empty our results prove a conjecture of yau for generic metrics in the free boundary setting

a study on reduced support vector machines recently the reduced support vector machine rsvm was proposed as an alternate of the standard svm motivated by resolving the difficulty on handling large data sets using svm with nonlinear kernels it preselects a subset of data as support vectors and solves a smaller optimization problem however several issues of its practical use have not been fully discussed yet for example we do not know if it possesses comparable generalization ability as the standard svm in addition we would like to see for how large problems rsvm outperforms svm on training time in this paper we show that the rsvm formulation is already in a form of linear svm and discuss four rsvm implementations experiments indicate that in general the test accuracy of rsvm are a little lower than that of the standard svm in addition for problems with up to tens of thousands of data if the percentage of support vectors is not high existing implementations for svm is quite competitive on the training time thus from this empirical study rsvm will be mainly useful for either larger problems or those with many support vectors experiments in this paper also serve as comparisons of 1 different implementations for linear svm and 2 standard svm using linear and quadratic cost functions

inequivalent complexity criteria for free boundary minimal surfaces abstract we obtain a series of results in the global theory of free boundary minimal surfaces which in particular provide a rather complete picture for the way different complexity criteria such as area topology and morse index compare beyond the regime where effective estimates are at disposal

curvature estimates for minimal hypersurfaces in singular spaces we here prove pointwise curvature estimates for minimal hypersurfaces in singular spaces using the integral curvature estimate and a generalized simons inequality which were established recently a further basic ingredient is a new sobolev type inequality for stationary hypersurfaces

three circles theorems for schrodinger operators on cylindrical ends and geometric applications we show that for a schrodinger operator with bounded potential on a manifold with cylindrical ends the space of solutions that grows at most exponentially at infinity is finite dimensional and for a dense set of potentials or equivalently for a surface for a fixed potential and a dense set of metrics the constant function 0 is the only solution that vanishes at infinity clearly for general potentials there can be many solutions that vanish at infinity one of the key ingredients in these results is a three circles inequality or log convexity inequality for the sobolev norm of a solution u to a schrodinger equation on a product n 0 t w heren is a closed manifold with a certain spectral gap examples of such n s are all round spheres s n for n 1 and all zoll surfaces finally we discuss some examples arising in geometry of such manifolds and schrodinger operators c

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

the local picture theorem on the scale of topology we prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular riemannian threemanifold in a certain small intrinsic neighborhood of a point of almost minimal injectivity radius this structure theorem includes a limit object which we call a minimal parking garage structure on r 3 whose theory we also develop

building sparse multiple kernel svm classifiers the support vector machines svms have been very successful in many machine learning problems however they can be slow during testing because of the possibly large number of support vectors obtained recently wu 2005 proposed a sparse formulation that restricts the svm to use a small number of expansion vectors in this paper we further extend this idea by integrating with techniques from multiple kernel learning mkl the kernel function in this sparse svm formulation no longer needs to be fixed but can be automatically learned as a linear combination of kernels two formulations of such sparse multiple kernel classifiers are proposed the first one is based on a convex combination of the given base kernels while the second one uses a convex combination of the so called ldquoequivalentrdquo kernels empirically the second formulation is particularly competitive experiments on a large number of toy and real world data sets show that the resultant classifier is compact and accurate and can also be easily trained by simply alternating linear program and standard svm solver

helicoidal minimal surfaces of prescribed genus for every genus g we prove that mathbf s 2 times mathbf r contains complete properly embedded genus g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch we also show that as the radius of the mathbf s 2 tends to infinity these examples converge smoothly to complete properly embedded minimal surfaces in mathbf r 3 that are helicoidal at infinity we prove that helicoidal surfaces in mathbf r 3 of every prescribed genus occur as such limits of examples in mathbf s 2 times mathbf r

a sparsity driven kernel machine based on minimizing a generalization error bound a new sparsity driven kernel classifier is presented based on the minimization of a recently derived data dependent generalization error bound the objective function consists of the usual hinge loss function penalizing training errors and a concave penalty function of the expansion coefficients the problem of minimizing the non convex bound is addressed by a successive linearization approach whereby the problem is transformed into a sequence of linear programs the algorithm produced comparable error rates to the standard support vector machine but significantly reduced the number of support vectors and the concomitant classification time

minimal hypersurfaces and bordism of positive scalar curvature metrics let y g be a compact riemannian manifold of positive scalar curvature psc it is well known due to schoen yau that any closed stable minimal hypersurface of y also admits a psc metric we establish an analogous result for stable minimal hypersurfaces with free boundary furthermore we combine this result with tools from geometric measure theory and conformal geometry to study psc bordism for instance assume y 0 g 0 and y 1 g 1 are closed psc manifolds equipped with stable minimal hypersurfaces x 0 subset y 0 and x 1 subset y 1 under natural topological conditions we show that a psc bordism z bar g y 0 g 0 rightsquigarrow y 1 g 1 gives rise to a psc bordism between x 0 and x 1 equipped with the psc metrics given by the schoen yau construction

the size of the singular set in mean curvature flow of mean convex sets in this paper we study the singularities that form when a hypersurface of positive mean curvature moves with a velocity that is equal at each point to the mean curvature of the surface at that point it is most convenient to describe the results in terms of the level set flow also called biggest flow i2 of chen giga goto cgg and evans spruck es under the level set flow any closed set k in r generates a one parameter family of closed sets ft k t 0 with f0 k k if the boundary of k is a smooth compact hypersurface then so is the boundary of ft k for t in some interval 0 and for such t s the evolution coincides with motion by mean curvature as defined classically by partial differential equations and differential geometry as in h1 however if k is compact then the boundary of ft k will necessarily become singular at some finite time our goal is to show that the singular sets are necessarily quite small this we do provided the initial set k is compact and mean convex in the sense that ft k interior k for all t 0 in case m k is a smooth hypersurface we have the following equivalent characterization of mean convexity k is mean convex if and only if the mean curvature of m with respect to the inward unit normal is everywhere non negative given a compact set k let k be the region in spacetime swept out by the ft k k x t r r t 0 x ft k a point x x t in the boundary of k is called a regular point if 1 x has a neighborhood in which k is a smooth manifold with boundary and 2 the tangent plane to k at x is not horizontal i e is not r 0 note that if x x t is a regular point then in a neighborhood of x ft k is a smooth manifold with boundary in r a point x x t in k with t 0 that is not a regular point is called a singular point

a general regularity theory for stable codimension 1 integral varifolds we give a necessary and sufficient geometric structural condition which we call the a structural hypothesis for a stable codimension 1 integral varifold on a smooth riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities the a structural hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded c 1 a hypersurfaces with boundary meeting only along their common boundary we establish that whenever a stable integral n varifold on a smooth n 1 dimensional riemannian manifold satisfies the a structural hypothesis for some a 0 1 2 its singular set is empty if n 6 discrete if n 7 and has hausdorff dimension n 7 if n 8 in view of well known examples this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses we also establish compactness of mass bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a structural hypothesis for some a 0 1 2 the a structural hypothesis on an n varifold for any a 0 1 2 is readily implied by either of the following two hypotheses i the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary ii the singular set of the varifold has vanishing n 1 dimensional hausdorff measure thus our theory subsumes the well known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions an optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems

notes on submanifolds in warped products in this paper we study submanifolds in the warped product m fr where m is a riemannian manifold f is the warping function we investigate the graphs with positive gaussian curvature in m2 r the minimal graphs in m fr and the totally geodesic submanifolds in m fr

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

extended polynomial growth transforms for design and training of generalized support vector machines growth transformations constitute a class of fixed point multiplicative update algorithms that were originally proposed for optimizing polynomial and rational functions over a domain of probability measures in this paper we extend this framework to the domain of bounded real variables which can be applied towards optimizing the dual cost function of a generic support vector machine svm the approach can therefore not only be used to train traditional soft margin binary svms one class svms and probabilistic svms but can also be used to design novel variants of svms with different types of convex and quasi convex loss functions in this paper we propose an efficient training algorithm based on polynomial growth transforms and compare and contrast the properties of different svm variants using several synthetic and benchmark data sets the preliminary experiments show that the proposed multiplicative update algorithm is more scalable and yields better convergence compared to standard quadratic and nonlinear programming solvers while the formulation and the underlying algorithms have been validated in this paper only for svm based learning the proposed approach is general and can be applied to a wide variety of optimization problems and statistical learning models

volume of minimal hypersurfaces in manifolds with nonnegative ricci curvature we establish a min max estimate on the volume width of a closed riemannian manifold with nonnegative ricci curvature more precisely we show that every closed riemannian manifold with nonnegative ricci curvature admits a pl morse function whose level set volume is bounded in terms of the volume of the manifold as a consequence of this sweep out estimate there exists an embedded closed possibly singular minimal hypersurface whose volume is bounded in terms of the volume of the manifold

non compactness of the space of minimal hypersurfaces we show that the space of min max minimal hypersurfaces is non compact when the manifold has an analytic metric of positive ricci curvature and dimension 3 le n 1 le 7 furthermore we show that bumpy metrics with positive ricci curvature admit minimal hypersurfaces with unbounded mathrm index mathrm area when combined with the recent work fo f c marques and a neves we then deduce some new properties regarding the infinitely many minimal hypersurfaces they found

reduced support vector machines a statistical theory in dealing with large data sets the reduced support vector machine rsvm was proposed for the practical objective to overcome some computational difficulties as well as to reduce the model complexity in this paper we study the rsvm from the viewpoint of sampling design its robustness and the spectral analysis of the reduced kernel we consider the nonlinear separating surface as a mixture of kernels instead of a full model the rsvm uses a reduced mixture with kernels sampled from certain candidate set our main results center on two major themes one is the robustness of the random subset mixture model the other is the spectral analysis of the reduced kernel the robustness is judged by a few criteria as follows 1 model variation measure 2 model bias deviation between the reduced model and the full model and 3 test power in distinguishing the reduced model from the full one for the spectral analysis we compare the eigenstructures of the full kernel matrix and the approximation kernel matrix the approximation kernels are generated by uniform random subsets the small discrepancies between them indicate that the approximation kernels can retain most of the relevant information for learning tasks in the full kernel we focus on some statistical theory of the reduced set method mainly in the context of the rsvm the use of a uniform random subset is not limited to the rsvm this approach can act as a supplemental algorithm on top of a basic optimization algorithm wherein the actual optimization takes place on the subset approximated data the statistical properties discussed in this paper are still valid

equidistribution of minimal hypersurfaces for generic metrics for almost all riemannian metrics in the c infty baire sense on a closed manifold m n 1 3 le n 1 le 7 we prove that there is a sequence of closed smooth embedded connected minimal hypersurfaces that is equidistributed in m this gives a quantitative version of the main result of irie et al ann math 187 3 963 972 2018 that established density of minimal hypersurfaces for generic metrics as in irie et al 2018 the main tool is the weyl law for the volume spectrum proven by liokumovich et al ann math 187 3 933 961 2018

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

existence of infinitely many minimal hypersurfaces in positive ricci curvature in the early 1980s s t yau conjectured that any compact riemannian three manifold admits an infinite number of closed immersed minimal surfaces we use min max theory for the area functional to prove this conjecture in the positive ricci curvature setting more precisely we show that every compact riemannian manifold with positive ricci curvature and dimension at most seven contains infinitely many smooth closed embedded minimal hypersurfaces in the last section we mention some open problems related with the geometry of these minimal hypersurfaces

jenkins serrin type results for the jang equation let m g k be an initial data set for the einstein equations of general relativity we prove that there exist solutions of the plateau problem for marginally outer trapped surfaces motss that are stable in the sense of motss this answers a question of g galloway and n o murchadha raised in some remarks on the size of bodies and black holes classical quantum gravity 25 2008 no 10 105009 9 mr 2416045 and is an ingredient in the proof of the spacetime positive mass theorem given by l h huang d lee r schoen and the first named author we show that a canonical solution of the jang equation exists in the complement of the union of all weakly future outer trapped regions in the initial data set with respect to a given end provided that this complement contains no weakly past outer trapped regions the graph of this solution relates the area of the horizon to the global geometry of the initial data set in a non trivial way we prove the existence of a scherk type solution of the jang equation outside the union of all weakly future or past outer trapped regions in the initial data set this result is a natural exterior analogue for the jang equation of the classical jenkins serrin theory we extend and complement existence theorems for scherk type constant mean curvature graphs over polygonal domains in m g where m g is a complete riemannian surface we can dispense with the a priori assumptions that a sub solution exists and that m g has particular symmetries also our method generalizes to higher dimensions

a robust minimax approach to classification when constructing a classifier the probability of correct classification of future data points should be maximized we consider a binary classification problem where the mean and covariance matrix of each class are assumed to be known no further assumptions are made with respect to the class conditional distributions misclassification probabilities are then controlled in a worst case setting that is under all possible choices of class conditional densities with given mean and covariance matrix we minimize the worst case maximum probability of misclassification of future data points for a linear decision boundary this desideratum is translated in a very direct way into a convex second order cone optimization problem with complexity similar to a support vector machine problem the minimax problem can be interpreted geometrically as minimizing the maximum of the mahalanobis distances to the two classes we address the issue of robustness with respect to estimation errors in the means and covariances of the classes via a simple modification of the input data we also show how to exploit mercer kernels in this setting to obtain nonlinear decision boundaries yielding a classifier which proves to be competitive with current methods including support vector machines an important feature of this method is that a worst case bound on the probability of misclassification of future data is always obtained explicitly

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

regularized vector field learning with sparse approximation for mismatch removal in vector field learning regularized kernel methods such as regularized least squares require the number of basis functions to be equivalent to the training sample size n the learning process thus has o n 3 and o n 2 in the time and space complexity respectively this poses significant burden on the vector learning problem for large datasets in this paper we propose a sparse approximation to a robust vector field learning method sparse vector field consensus sparsevfc and derive a statistical learning bound on the speed of the convergence we apply sparsevfc to the mismatch removal problem the quantitative results on benchmark datasets demonstrate the significant speed advantage of sparsevfc over the original vfc algorithm two orders of magnitude faster without much performance degradation we also demonstrate the large improvement by sparsevfc over traditional methods like ransac moreover the proposed method is general and it can be applied to other applications in vector field learning

a sharp lower bound for the entropy of closed hypersurfaces up to dimension six the entropy is a natural geometric quantity which measures the complexity of a hypersurface in mathbb r n 1 it is non increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow in colding et al j differ geom 95 1 53 69 2013 colding ilmanen minicozzi white showed that within the class of closed smooth self shrinking solutions of the mean curvature flow in mathbb r n 1 the entropy is uniquely minimized at the round sphere they conjectured that for 2 le n le 6 the round sphere minimizes the entropy among all closed hypersurfaces using an appropriate weak mean curvature flow we prove their conjecture for these dimensions our approach also gives a new proof of the main result of colding et al j differ geom 95 1 53 69 2013 and extends its conclusions to compact singular self shrinking solutions

the nature of singularities in mean curvature flow of mean convex sets let k be a compact subset of r or more generally of an n 1 dimensional riemannian manifold we suppose that k is mean convex if the boundary of k is smooth and connected this means that the mean curvature of k is everywhere nonnegative with respect to the inward unit normal and is not identically 0 more generally it means that ft k is contained in the interior of k for t 0 where ft k is the set obtained by letting k evolve for time t under the level set mean curvature flow as k evolves it traces out a closed set k of spacetime k x t r r x ft k also there is associated to k a brakke flow m t 7 mt of rectifiable varifolds we call the pair m k a mean convex flow let x x t be a point in spacetime with t 0 suppose xi ti is a sequence of points converging to x and i is a sequence of positive numbers tending to infinity translate the pair m and k in spacetime by y 7 y xi ti and then dilate parabolically by y 7 iy i to get new flows mi and ki the sequence mi ki is called a blow up sequence at x general compactness theorems guarantee that this sequence will have subsequential limits a subsequential limit m k is called a limit flow here m t 7 m t

learning sparse kernel classifiers for multi instance classification we propose a direct approach to learning sparse kernel classifiers for multi instance mi classification to improve efficiency while maintaining predictive accuracy the proposed method builds on a convex formulation for mi classification by considering the average score of individual instances for bag level prediction in contrast existing formulations used the maximum score of individual instances in each bag which leads to nonconvex optimization problems based on the convex mi framework we formulate a sparse kernel learning algorithm by imposing additional constraints on the objective function to enforce the maximum number of expansions allowed in the prediction function the formulated sparse learning problem for the mi classification is convex with respect to the classifier weights therefore we can employ an effective optimization strategy to solve the optimization problem that involves the joint learning of both the classifier and the expansion vectors in addition the proposed formulation can explicitly control the complexity of the prediction model while still maintaining competitive predictive performance experimental results on benchmark data sets demonstrate that our proposed approach is effective in building very sparse kernel classifiers while achieving comparable performance to the state of the art mi classifiers

the existence of embedded minimal hypersurfaces we give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth n 1 dimensional riemannian manifolds a theorem proved first by pitts for 2 n 5 and extended later by schoen and simon to any n

the asymptotic behavior of properly embedded minimal surfaces of finite topology while advances have been made in recent years in the study of properly embedded minimal surfaces of finite topology in r3 3 5 8 9 11 12 13 23 progress has depended in an essential manner on the special structure of such surfaces with finite total curvature a properly immersed minimal surface with finite total curvature is conformally a compact riemann surface punctured in a finite number of points and its gauss map extends to the compact surface as a meromorphic function 22 in particular all the topological ends are conformally equivalent to a punctured disk and there is a well defined limit tangent plane at each end outside of a sufficiently large compact set such an end is a multisheeted graph over the limit tangent plane and if the end is embedded it is asymptotic to either a plane or a half catenoid see e g schoen 24 for a survey of results on properly immersed surfaces of finite total curvature see 9 1 without the assumption of finite total curvature these results are not true in general as an indication of the relative lack of information consider that it is unknown whether or not the helicoid is the only simply connected minimal surface that is properly embedded and nonplanar a question raised by osserman the helicoid has infinite total curvature and a single annular end if a surface has finite topology then all of its ends are annular any general theory of properly embedded minimal surfaces of finite topology must include an understanding of the behavior of these ends in this paper we prove

min max theory for constant mean curvature hypersurfaces in this paper we develop a min max theory for the construction of constant mean curvature cmc hypersurfaces of prescribed mean curvature in an arbitrary closed manifold as a corollary we prove the existence of a nontrivial smooth closed almost embedded cmc hypersurface of any given mean curvature c moreover if c is nonzero then our min max solution always has multiplicity one

relevance units latent variable model and nonlinear dimensionality reduction a new dimensionality reduction method called relevance units latent variable model rulvm is proposed in this paper rulvm has a close link with the framework of gaussian process latent variable model gplvm and it originates from a recently developed sparse kernel model called relevance units machine rum rum follows the idea of relevance vector machine rvm under the bayesian framework but releases the constraint that relevance vectors rvs have to be selected from the input vectors rum treats relevance units rus as part of the parameters to be learned from the data as a result a rum maintains all the advantages of rvm and offers superior sparsity rulvm inherits the advantages of sparseness offered by the rum and the experimental result shows that rulvm algorithm possesses considerable computational advantages over gplvm algorithm

isoperimetric regions in cones we consider cones c 0 m n and prove that if the ricci curvature of c is nonnegative then geodesic balls about the vertex minimize perimeter for given volume if strict inequality holds then they are the only stable regions

min max theory and the willmore conjecture in 1965 t j willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 we prove this conjecture using the min max theory of minimal surfaces

qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area we prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed riemannian manifolds in terms of their index and area restricting to the case where the hypersurface has dimension less than seven in particular we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index the total curvature along the sequence is quantised in terms of the total curvature of some limit surface plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in euclidean space all of which are finite thus we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary

non proper helicoid like limits of closed minimal surfaces in 3 manifolds we show that there exists a metric with positive scalar curvature on s 2 s 1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that in a neighborhood of a strictly stable 2 sphere is smooth except at two helicoid like singular ities on the 2 sphere the construction is inspired by a recent example by d hoffman and b white

which ambient spaces admit isoperimetric inequalities for submanifolds we give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities to hold

prime ideals in differential operator rings catenarity let r be a commutative algebra over the commutative ring k and let a 51 5 be a finite set of commuting k linear derivations from r to r let t r 61i 6 al 5 n be the corresponding ring of differential operators we define and study an isomorphism of left r modules between t and its associated graded ring r xl a polynomial ring over r this isomorphism is used to study the prime ideals of t with emphasis on the question of catenarity we prove that t is catenary when r is a commutative noetherian universally catenary k algebra and one of the following cases occurs a k is a field of characteristic zero and a acts locally finitely b k is a field of positive characteristic c k is the ring of integers r is affine over k and a acts locally finitely

catenarity and gelfand kirillov dimension in ore extensions if r is a commutative afline domain for example the coordinate ring of an irreducible algebraic variety the sum of the height of any prime ideal in r and the dimension of the corresponding factor ring is the dimension of r this implies that if q and p are prime ideals of r with q 2 p any saturated chain of prime ideals from p to q has length dim r p dim r q and so r is catenary schelter gabber and others have proven analogous statements for some noncommutative affine rings r in this paper we study the question of when an ore extension of a commutative ring is catenary giving both positive and negative results we show that for locally finitedimensional ore extensions of commutative affine rings the above dimension statements are still true if we use the gelfand kirillov dimension

embedded self similar shrinkers of genus 0 we conrm a well known conjecture that the round sphere is the only compact embedded self similar shrinking solution of mean curvature ow in r 3 with genus 0 more generally we show that the only properly embedded self similar shrinkers in r 3 with vanishing intersection form are the sphere the cylinder and the plane this answers two questions posed by t ilmanen

online kernel principal component analysis a reduced order model kernel principal component analysis kernel pca is an elegant nonlinear extension of one of the most used data analysis and dimensionality reduction techniques the principal component analysis in this paper we propose an online algorithm for kernel pca to this end we examine a kernel based version of oja s rule initially put forward to extract a linear principal axe as with most kernel based machines the model order equals the number of available observations to provide an online scheme we propose to control the model order we discuss theoretical results such as an upper bound on the error of approximating the principal functions with the reduced order model we derive a recursive algorithm to discover the first principal axis and extend it to multiple axes experimental results demonstrate the effectiveness of the proposed approach both on synthetic data set and on images of handwritten digits with comparison to classical kernel pca and iterative kernel pca

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

the jang equation reduction of the spacetime positive energy theorem in dimensions less than eight we extend the jang equation proof of the positive energy theorem due to schoen and yau commun math phys 79 2 231 260 1981 from dimension n 3 to dimensions 3 n 8 this requires us to address several technical difficulties that are not present when n 3 the regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those in schoen and yau commun math phys 79 2 231 260 1981

asymptotically flat three manifolds contain minimal planes abstract let m g be an asymptotically flat 3 manifold containing no closed embedded minimal surfaces we prove that for every point p m there exists a complete properly embedded minimal plane in m containing p

a costa hoffman meeks type surface in mathbb h 2 times mathbb r we show the existence in the space 2 x of a family of embedded minimal surfaces of genus 1 k and finite total extrinsic curvature with two catenoidal type ends and one middle planar end the proof is based on a gluing procedure

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

stable minimal surfaces in m ℝ in 4 we developed the theory of properly embedded minimal surfaces in m r where m is a compact riemannian surface one of the first results in 4 is that a properly embedded noncompact minimal surface in m r of bounded gaussian curvature is quasiperiodic in the following sense given any sequence of vertical translates n of a subsequence of the n converges on compact subsets of m r to another properly embedded minimal surface by the curvature estimates of schoen 8 every properly embedded stable minimal surface in m r has bounded curvature therefore every properly embedded noncompact stable minimal surface in m r is quasiperiodic this quasiperiodicity property will be essential in proving the next theorem throughout this paper m denotes a compact riemannian surface

a direct method for building sparse kernel learning algorithms many kernel learning algorithms including support vector machines result in a kernel machine such as a kernel classifier whose key component is a weight vector in a feature space implicitly introduced by a positive definite kernel function this weight vector is usually obtained by solving a convex optimization problem based on this fact we present a direct method to build sparse kernel learning algorithms by adding one more constraint to the original convex optimization problem such that the sparseness of the resulting kernel machine is explicitly controlled while at the same time performance is kept as high as possible a gradient based approach is provided to solve this modified optimization problem applying this method to the support vectom machine results in a concrete algorithm for building sparse large margin classifiers these classifiers essentially find a discriminating subspace that can be spanned by a small number of vectors and in this subspace the different classes of data are linearly well separated experimental results over several classification benchmarks demonstrate the effectiveness of our approach

multiple kernel learning algorithms in recent years several methods have been proposed to combine multiple kernels instead of using a single one these different kernels may correspond to using different notions of similarity or may be using information coming from multiple sources different representations or different feature subsets in trying to organize and highlight the similarities and differences between them we give a taxonomy of and review several multiple kernel learning algorithms we perform experiments on real data sets for better illustration and comparison of existing algorithms we see that though there may not be large differences in terms of accuracy there is difference between them in complexity as given by the number of stored support vectors the sparsity of the solution as given by the number of used kernels and training time complexity we see that overall using multiple kernels instead of a single one is useful and believe that combining kernels in a nonlinear or data dependent way seems more promising than linear combination in fusing information provided by simple linear kernels whereas linear methods are more reasonable when combining complex gaussian kernels