the aim of this article is to study the existence and give a consistent approximation procedure of the boundary value problem for the conservative dynamical system where is a smooth potential on .we assume that is an open subset of as this is the relevant case for the applications we have in mind ; extensions to a more general setting are possible but not discussed here .  for the boundary conditions , we write with and . here , is part of the problem and has to be determined ( however , the total energy , defined as the sum of kinetic and potential energy , is fixed ) .the focus on the boundary - value problem is motivated by applications , as discussed below .equation  ( furnished with various initial or boundary conditions ) can be reformulated as the classic hamiltonian problem for , where is the _ hamiltonian _   mathematically , the existence of solutions to , often more succinctly written as with , is a classical problem .periodic solutions have been a particular focus , and existence results obtained until the early 1980s are discussed in the beautiful survey article  .already for periodic solutions , a clear distinction has to be made for local results ( that is , short time solutions ) and global solutions describing solutions in the large . apparently the first global global result was obtained by seifert  for a hamiltonian which is slightly more general than the one in .the key idea of his proof is based on differential geometry , using an equivalent reformulation of  in which solutions can be found as a geodesic in a ( degenerate ) riemannian metric , the so - called jacobi metric .a curve shortening procedure proposed by g. d. birkhoff ( * ? ? ?* section v.7 ) can then be applied to show the existence of a geodesic .this result has later been extended by weinstein , and a more general result based on a different variational approach was given by rabinowitz  .  in the sciences , the interest in non - periodic long time solutionshas recently been rejuvenated by various applications .namely , complex systems in physics , chemistry or biology can often be described by a potential energy landscape with many wells , separated by barriers .a common problem is then to find a trajectory joining a given initial point ( configuration ) with a given final point .we study this problem in the situation where the dynamics is determined by , and the points given in  are potentially far apart .in particular , the two configurations will generically be located in different wells of the energy landscape .rare events are an example of these transitions between two wells .typically , thermally activated reactions have many deep wells separated by large energy barriers .reactants will then spend most of the time jostling around in one well before a rare spontaneous fluctuation occurs that lifts the atoms of the reactant over the barrier into the next ( product ) valley .information on rare events is crucial since they represent important changes in the system , such as chemical reactions or conformational modifications of molecules .a major challenge in molecular dynamics ( md ) is that these hopping events take place so rarely that the computational limits of md simulations can be easily exceeded .since the problem  is central in md , a number of solution strategies have been proposed ; see  for a brief review of some methods .further , for practitioners of md , the question arises whether any numerical approximation shadows a physical one  ( and if so , whether it shadows a generic physical trajectory ) .the lack of hyperbolicity rules out standard tools to prove shadowing ( e.g. , ( * ? ? ? * theorem 18.1.3 ) ) .thus , for md , computations are `` based on trust '' ( * ? ? ?* section 4.3 ) .birkhoff s algorithm , for the toy example of the euclidean metric in and for .the initial curve is plotted as a dashed line .points with odd index are marked by black dots , points with even index by grey dots . in the first step ,the points with even indices are kept fixed , and joined by a geodesic .new positions for the points with odd indices on the new curve are determined ( solid curve ) . in a next step , these points are joined by geodesics , which determines new positions for the points with even indices ( dotted curve ) .the curves ( slowly ) converge to the geodesic line segment connecting and .,height=226 ]  one further difficulty for the computation of hamiltonian trajectories , in particular for md , is that these trajectories are often chaotic , and one has to restrict oneself to averaged statistical information .however , of particular interest in the analysis of rare events are trajectories going directly from one well of a potential to another one ; such transition paths can be used to define so - called reaction coordinates . often the most efficient algorithms using for example path sampling assume a knowledge of these coordinates and thus these non - chaotic trajectories are of significant practical importance .the method presented in this article is concerned with the calculation of such non - chaotic transition trajectories .even these relatively `` simple '' trajectories are in practise very hard to compute , since they correspond to rare events and take place on very long time scales .   rather working with the hamiltonian boundary value problem  directly, we use an equivalent variational formulation , namely the maupertuis principle , according to which trajectories to  with total energy are suitably re - parametrised geodesics with respect to the _ jacobi metric _ ( more generally , if is equipped with a riemannian metric , then ) .so hamiltonian trajectories are critical points of the length functional associated with .while the equivalence has been known for centuries , it seems that little advantage has been taken of the fact that this variational formulation has a very convenient mathematical structure .note that hamiltonian problems such as  are commonly indefinite , while a geodesic problem is elliptic and thus bounded from below .so the problem reduces to that of finding geodesics in the jacobi metric .this problem has been addressed by birkhoff , who described a curve shortening procedure to find global geodesics under the assumption that local ( sufficiently short ) geodesics can be found explicitly .this assumption is not met here since local geodesics have to be approximated ; the main task is to show that nevertheless global convergence can be obtained for a suitably devised local approximation scheme . since the local scheme we propose also relies on a birkhoff curve shortening idea , we present his idea in the global setting first to keep the presentation self - contained . initially , one joins the given initial and final point and by an arbitrary curve .then sufficiently many points are marked on the curve so that a local geodesic can be computed between next to nearest neighbours ( see figure [ fig : birkhoffeuclid ] ) .note that the argument assumes that local geodesics can be computed with sufficient accuracy .in the first step , the points with even indices are kept fixed , and joined by a geodesic .new positions for the points with odd indices on the new curve are determined ( solid curve in figure [ fig : birkhoffeuclid ] ) .the procedure of joining next to nearest neighbouring points by local geodesics is then repeated for the points with odd indices ( dotted curve in figure [ fig : birkhoffeuclid ] ) .it is not hard to show that this iterative procedure decreases the length . under suitable assumptions ( e.g. ,  ), this method can be shown to converge to a global geodesic ; however , there are situations such as for degenerating metrics where convergence will not take place .the central result of this article is an analogous local result , introducing a sequence of approximating sequences converging to a sufficiently short geodesic .the trade - off is that the result is local ( applicable only to sufficient short geodesics ) , but does not assume that the approximating sequences consist of exact geodesic segments .we rely on the observation that the global birkhoff argument localises the geodesic problem in a geometrically tractable way .that is , one can restrict the local analysis to points which are sufficiently close .we then show that they can be joined by a geodesic which in addition can be represented as a graph and prove consistency and convergence of the proposed local approximation .   the main result of this article is a consistent approximation of what we call local geodesics ; important aspects are that the proof is constructive and yields bounds on the allowed distance between points to be joined by local geodesics .the bounds are not in terms of the usual estimates from differential geometry ( such as the injectivity radius , that is , the radius for which there is a unique geodesic starting at the centre , with arbitrary velocity ) , but are expressed explicitly in terms of the total and potential energy of the molecular hamiltonian system .we point out that locality of the geodesics does not necessarily require that the two end points are very close ; an important aspect of the proof is that we may consider local geodesics which can be represented as graphs .global geodesics can violate this assumption , while suitably small segments of geodesics remain geodesics and can be represented as graphs .birkhoff s idea to segment an original connecting curve allows us to confine our algorithm to such local geodesics .the efficiency and applicability of a global birkhoff method as in  will depend on the chosen parametrisation ( number and location of points in figure [ fig : birkhoffeuclid ] ) .for example , it is possible that refinements in a numerical implementation are required .however , the proof shows that in the setting studied in this article , no refinement or reparametrisation is required .  as a by - product, we show the existence of a continuous ( physical ) trajectory for suitable points , , using the jacobi formulation as seifert  , but replacing the periodic boundary conditions considered there by equation .there are some related existence results  , which also rely on the jacobi formulation .the novelty of the results presented here are twofold : ( i ) while a formulation using the jacobi metric is natural , a difficulty is that the metric degenerates at the boundary of the configuration space , where kinetic energy and potential energy agree .we provide _ a priori _ bounds to ensure that the geodesic stays away from ; the bounds depend on the location of the boundary points and or on the total energy .bounds could be obtained along the lines of thought presented in this paper , but in a simpler fashion .the reason why we give a more complicated argument is that the approximation we give here is constructive , giving existence of a solution and at the same time a consistent approximation procedure .thus , we obtain an approximation procedure which may not necessarily be the most efficient but one for which can dispense with the need for trust .since the algorithm we develop is consistent , the issue of shadowing is answered in an affirmative way for the procedure we propose , under the assumptions made on the potential energy and the bounds in term of total energy made on the end points .the existence of geodesics joining a given initial and final point in open domains , which is trivial within the radius of injectivity , is not obvious if the two points are further away from each other .we point out that the argument of this paper automatically proves the existence of such a geodesics for the jacobi metric with sufficiently large total energy .other existence results , proceeding along quite different lines , can be found elsewhere  .we remark that we will not address the question of how to choose ; this choice typically requires insight in the physics , chemistry or biology of the problem in question and thus can not be answered in the general mathematical framework considered here .however , the existence result given here can be interpreted in two ways : given the initial point and the total energy , the arguments provide estimates on possible locations of the final point so that and can be joined by a trajectory with total energy .alternatively , given and , the analysis provides lower bounds on such that and can be joined with this total energy .it is easy to see that no general existence theorem can hold if , and are unrestricted ( determines the configuration manifold , and in particular for small the configuration manifold may be disconnected , so and could be in different components ) .approximations which are proven to be consistent , such as godunov s scheme for hyperbolic equations , are often less efficient than algorithms for which consistency can not be shown .the approximation introduced in this article is no exception .yet , it is often possible to take inspiration from a consistent approximation and deduce efficient ( but not provably consistent ) formulations .this is the case for the formulation introduced here ; a related flow model approximation has been shown to be able to detect different trajectories joining points in different wells of the energy landscape of the mller potential and the collinear reaction  . while our formulation relies both on the choice of maupertuis formulation and birkhoff s curve shortening , which seems to be new to the field of numerical methods for molecular dynamics , the curve - shortening procedures resembles other rubber - band algorithms  .for the isothermal case , a nudged elastic band method has been proposed by e , ren and vanden - eijnden  . there , the aim is to find minimal energy paths , which are defined as paths along which the orthogonal component of the deterministic vector field vanishes .the approach in  to reduce the orthogonal contributions iteratively bears many similarities with the method presented here . as examples of string methods with temperature, we refer to the pioneering work by e , ren and vanden - eijnden  .maupertuis principle has been used before in md  , but without the connection to birkhoff s curve shortening method .we also refer the reader to the recent work by cameron , kohn and vanden - eijnden which gives an analysis and in particular convergence results for a steepest descent string method  .the usual numerical approach for a boundary value problem is a shooting method ; there , the existence of a solution is assumed as well as the closeness of an initial guess to the solution .( there are abstract existence results available , see for example  ; however , as noted by stoer and bulirsch  , the abstract formulation of the boundary conditions to be imposed rules out the condition , for the first - order system considered there , even for the case . )the method discussed in this paper provides both an explicit existence proof and estimates on the closeness required .we point out that the birkhoff algorithm also provides a strategy for gluing together local geodesics to obtain global paths , which converge to a global geodesic .note that this latter aspect of the birkhoff approach defines a natural tool to localise the computation of geodesics which can be exploited for parallelisation .this aspect is discussed in more detail in  .an advantage of birkhoff s method is that these local steps are intrinsically parallelisable .finally , we mention connections to the onsager - machlup / freidlin - wentzell theory . there , an action functional is derived , with the minimal action path describing the most likely trajectory .the connection between that theory , hamilton - jacobi theory and the maupertuis principle as discussed here is a topic with many open questions ; we refer to  for results in this direction together with applications to rare events .   throughout the presentation, is the _ configuration manifold _ of a system and thus describes all possible states the system can occupy .the coordinates of the _ phase space _ ( cotangent bundle ) are , position and momentum .analogously , the coordinates of the tangent bundle are , where denotes the velocity .we assume that the system dynamics is conservative with degrees of freedom .then , the _ hamiltonian _ is defined as . here , the kinetic energy is a function of the momenta only and is the potential energy , depending on the coordinates alone . the _lagrangian _ of the system is a function , namely . for a wide class of applications , it is sufficient to consider , the inner product for a system with particles with mass , and to assume that the lagrangian is of the standard form   this paper is organised as follows : in section [ sec : continuous - setting ] , we review the maupertuis principle and the birkhoff curve shortening algorithm , both for the continuous setting .section [ sec : discr - sett - birkh ] describes the analogous discrete setting , contains the relevant _ a priori _ estimates and introduces the discrete birkhoff procedure for a fixed discretisation .section [ sec : birkhoff - refinement ] describes the birkhoff refinement and convergence to the continuous limit .numerical simulations and numerical convergence rates for a model problem are the content of section [ sec : numer - investigations ] .our construction will be guided by a variational formulation , equivalent to , where convergent approximations can be obtained with relative ease .this continuous setting is sketched in the present section .it is a well known fact that solutions to  with pre - assigned total energy are re - parametrised geodesics in _jacobi s metric _ .this formulation is sometimes denoted _maupertuis principle _ , or _jacobi s least action principle_. for the special lagrangian , the _routhian _ associated with jacobi s principle is obviously , is a metric in those regions of where .the action functional { : = } \int_a^b r{\left(}\gamma,\gamma'{\right)}{\ , \mathrm{d}}\tau\ ] ] is the measure of the length of in this metric . for a given curve , the value  ] , and , .note that is restricted to the values by the definitions of and . by assumption, however , has left the set , hence .proposition [ prop : rhombus ] then implies .this contradicts , so case 2 is in fact impossible .  using the inclusion , we conclude that for a finite discretisation length , the number of distinguished polygons represented on the discrete grid is finite .each affirmative birkhoff step is strictly reducing the length , thus does not allow us to visit the same polygon twice .hence the method terminates after finitely many affirmative birkhoff steps . as already argued in ( i ), the final polygon has to lie in as otherwise one could show the existence of a further affirmative birkhoff step .this section consists of three parts . in the first part, we define a sequence of polygons , which are the the final polygons of the birkhoff method for the discretisation length . as , we will show that the curves converge to a curve . in the second part, we recall a weak formulation of the geodesic equation . in the third part , we show that the limit satisfies this weak geodesic equation , hence is smooth and a stationary curve for the jacobi length functional .[ def : refine ] let be large enough such that and .let us define the starting polygon as the of the straight line segment with endpoints , .  for , we want to halve the discretisation length , that is , for  .we embed the polygon into the finer grid by introducing new vertices at the midpoints of the connecting line segments . in the notation of definition [ def : poly ] , this means the embedded polygon has the vertices we define . for this choice , the vertices of lie on the finer grid .then is the birkhoff map of for the discretisation length .  to simplify the notation, we write on each level   we now want to prove that the birkhoff refinement laid out above will lead to a converging sequence of polygons if we start the iteration with properly chosen initial points .specifically , let us first assume that the total energy , and thus , is fixed , and let us choose a nonempty compact set .then let be as in the proof of proposition [ prop : rhombus ] , with the norm taken over the set .[ def;admis ] any pair of endpoints is _ admissible _ if they meet the following two conditions , which depend on :  1 . , with as in lemma [ lem : par - nbhd ] ; 2 .  the set defined in lemma [ lem : par - nbhd ] is contained in .[ theo : dq - est ] for an admissible pair , we consider the sequence of graphs with representing the polygons obtained by the birkhoff refinement . as ,the centred differences satisfy for all the combined estimates   observe that the initial polygon is contained in , since all its finite differences vanish .  by inductionlet .according to definition [ def : refine ] , we embed into the finer grid of size , which is half the size of .the recursive definition of implies , with .  with respect to ,the embedded polygons are in ; this follows since all newly introduced vertices have a vanishing second difference quotient at odd nodes , whereas all quotients with even are doubled in size , from proposition [ prop : par0inv ] , it follows that the final polygon obtained from the birkhoff map is in fact in the smaller set .this is crucial for our argument , since it shows that one application of the birkhoff map after a refinement retains the same discrete curvature bound as before .this shows that every polygon with step - size belongs to . by lemma [ lem : par - nbhd ]it obeys also a bound on the first difference quotients , which is uniform in .[ cor : conv - aa ] for an admissible pair , we consider a sequence of graphs with representing the polygons obtained by the birkhoff refinement .then the functions converge in to a limit .we write for the limit graph .  as ;s) ] for any by arzel - ascoli .let us remark that the graphs of all polygons and hence the limit belong to the subset of defined in lemma [ lem : par - nbhd ] .let us consider a geodesic which can be represented as a graph . in this subsection, we derive a weak formulation for its governing equation .thus , we consider a curve represented as a graph , .the length of this curve in the jacobi metric is given by =\int_{-\ell}^ {       \ell}\sqrt{g^{\mathrm{jac}}_{ij}({\gamma}(x))\dot{\gamma}^i\dot{\gamma}^j}{\ , \mathrm{d}}x  = \int_{-\ell}^{\ell}e^{h({\gamma}(x))}\sqrt{1+{{\left\lvert{\tfrac{\mathrm{d}}{\mathrm{d } x}}f\right\rvert}}^2}{\ , \mathrm{d}}x.\ ] ] for a _ orthogonal _ perturbation , where has compact support in , we deduce via integration by parts    & = \int_{-\ell}^{\ell }   \left[y(x)\cdot\nabla_ne^{h({\gamma}(x))}\sqrt{1+{{\left\lvert{\tfrac{\mathrm{d}}{\mathrm{d } x}}f(x)\right\rvert}}^2}+e^{h({\gamma}(x))}\frac{{\tfrac{\mathrm{d}}{\mathrm{d } x}}f(x)\cdot { \tfrac{\mathrm{d}}{\mathrm{d } x}}y(x)}{\sqrt{1+{{\left\lvert{\tfrac{\mathrm{d}}{\mathrm{d } x}}f(x)\right\rvert}}^2}}\right ] { \ , \mathrm{d}}x  .\end{aligned}\ ] ] thus , a graph with is stationary for the length functional if the variational derivative vanishes for all functions .we will use this weak formulation , rather than the more common one obtained by a further integration by parts .   using a smoother interpolation of the grid points representing the polygons of the birkhoff refinementwe obtain a better convergence and a smoother characterisation of its limit than the previous result of corollary [ cor : conv - aa ] .[ theo : char - lim ] for an admissible pair , consider the sequence of polygons obtained by the birkhoff refinement .the functions converge in to a limit .furthermore , and the limit graph satisfies {\ , \mathrm{d}}x    .\label{eq : weak - geo }  \end{aligned}\ ] ] for every function .analogously to the definition of in , we define it is convenient to write for in analogy to  in further analogy to , let we want to examine of  evaluated at a node for and , where and . theorem [ theo : dq - est ] provides the necessary estimates for the difference quotients , so that since and by . further , by definition [ def : refine ] , thus , we can write we identify the right hand side as a product of a function evaluated on the piecewise constant interpolation times a function evaluated on the piecewise linear interpolation , that is , for all there holds   , \end{gathered}\ ] ] where is the centred difference quotient and we remark that the right - hand of  side is globally defined for , and piecewise constant .given an arbitrary fixed function , we can multiply  with and integrate by parts to obtain { \ , \mathrm{d}}x  \\   & = e_i   \cdot\int_{-\ell+\frac { { \varepsilon}_k}2}^{\ell-\frac { { \varepsilon}_k}2}y(x)\left [    d_{{\varepsilon}_k}{\left(}e^h{\left(}x,{{{\gamma}_k^\mathrm{pc}}}(x){\right)}{{f}}{\left(}{\tfrac{\mathrm{d}}{\mathrm{d } x}}{{{\gamma}_k^\mathrm{pl}}}(x){\right)}{\right)}+{{\nabla_n e^h}}{\left(}x,{{{\gamma}_k^\mathrm{pc}}}(x){\right)}{{p_f}}{\left(}{\tfrac{\mathrm{d}}{\mathrm{d } x}}{{{\gamma}_k^\mathrm{pl}}}(x){\right)}\right ] { \ , \mathrm{d}}x  \\  & = e_i  \cdot\int_{-\ell+\frac { { \varepsilon}_k}2}^{\ell-\frac { { \varepsilon}_k}2}\left [   - { \left(}d_{{\varepsilon}_k}y(x){\right)}e^h{\left(}x,{{{\gamma}_k^\mathrm{pc}}}(x){\right)}{{f}}{\left(}{\tfrac{\mathrm{d}}{\mathrm{d } x}}{{{\gamma}_k^\mathrm{pl}}}(x){\right)}+y(x){{\nabla_n e^h}}{\left(}x,{{{\gamma}_k^\mathrm{pc}}}(x){\right)}{{p_f}}{\left(}{\tfrac{\mathrm{d}}{\mathrm{d } x}}{{{\gamma}_k^\mathrm{pl}}}(x){\right)}\right ] { \ , \mathrm{d}}x  .\end{split }  \label{eq : int - est}\end{aligned}\ ] ]  we observe that the right - hand side does not depend on .now recall that stoppage of the algorithm on level implies that for both choices of the sign of .let us assume for the moment that piecewise constant interpolation and the piecewise linear interpolation converge strongly to the same limit as in the sense that then the argument can be finished as follows .observe that is non - negative for both choices of , whereas the last line of  does not depend on the chosen anymore . on each interval , we first choose to have the same sign as .then the sum on the left - hand side is non - negative .we pass to the limit in  and find { \ , \mathrm{d}}x  ; \end{gathered}\ ] ] here the convergence of the difference quotient follows from ( * ? ? ?* lemma 7.24 ) .similarly , we then choose to have always the opposite sign and obtain the reversed inequality .together this yields { \ , \mathrm{d}}x  .\ ] ] as we can test all normal directions , with , in this fashion , we obtain the vectorial identity { \ , \mathrm{d}}x\ ] ] in . by substituting the definitions of and , we recover the claim .the previously assumed convergence of the interpolants is established in the following arguments .[ lem : fin - el ] given a function , let be the piecewise constant interpolation on an equidistant grid of size , and similarly the piecewise linear interpolation .then the following error estimates hold   this is a standard argument in finite elements , see for example ( * ? ? ?* theorem 0.8.7 and section 4 ) .  for a fixed refinement level ,let be the point set associated to , the output of the birkhoff map at level .now , we first construct a quadratic interpolation of with the special property that its piecewise constant and piecewise linear interpolation coincide with the interpolations and introduced before . specifically , in our situation the quadratic interpolation can be chosen in such a way that is bounded independently of .explicitly , to construct , any two neighbouring nodes and are connected via two quadratic splines on and , with matching conditions   we recall that the polygons obtained by the birkhoff refinement satisfy the finite difference estimates . for the chosen assignments of the first derivatives at the nodes , the piecewise quadratic interpolation is continuously differentiable throughout .further , it can be computed that the bound on the second difference quotients implies a bound which holds uniformly in .hence the sequence is uniformly bounded in , and we infer convergence in to by arzel - ascoli .this implies that as .choosing as the function in lemma [ lem : fin - el ] , we infer for all that all expressions tend to by . this proves the previously assumed claims  and .here we present experimental convergence rates and estimates for the computational effort for a simple benchmark problem .first , we obtain an explicit solution for a special metric , which will serve as comparison for the numerical approximations on different discretisation levels .   from the weak formulation of the geodesic equation for graphs , we obtain by integration by parts the following strong version which has to satisfied by a geodesic connecting the two points being a graph , .if we assume further that does not depend on , the last equation simplifies to that is , we need to solve the simpler equation let us restrict ourselves to linear , not depending on the ` horizontal ' co - ordinate , that is , with and a unit vector , .then , hence , . with the ansatz , the equation can be rewritten as hence we are left with the scalar equation for alone .its solution is .thus , integrating once again we deduce . matching the boundary conditions , we find that the geodesic connecting the two points is with we remark that this representation requires , or equivalently .note that for , the metric approaches the constant euclidean metric , and we obtain the straight line segment in the limit .  for the other extreme , we remark that if , the geodesic converges to the set of two parallel lines connected at infinity .in fact , this set of constant and finite length will be the minimising configuration for all values of .we now present computations in with , so connecting .let us consider three different metrics by choosing three values , , and , all less than .larger values of emphasise the difference of compared to the flat euclidean metric , hence one expects a larger curvature in the geodesic , which bends further away from the horizontal straight connection between the boundary points .we also explore how the performance of the algorithm depends on this geometrical feature .we begin by presenting numerical statistics of the geodesic computation . the plots in figure [ fig : exp - metric - log ]show the computational error of the polygonal approximation compared to the explicit geodesic ( measured in the norm ) in the left panel and the computational effort , measured in the number of affirmative birkhoff steps , in the right panel .the error decreases as decreases for .in fact , the calculation shows the error is proportional to , where the exponent varies from 0.40 to 0.53 , depending on the choice of in the metric .the effort increases as decreases , and the simulation shows that the effort turns out to be ( inversely ) proportional to , where ranges from 1.71 to 2.2 , as is taken from , , .  hence , for geodesics of larger curvature variation , the birkhoff procedure starts closer but converges with smaller rate in , whereas the effort increases with largely uniform rate in of about .figure [ fig : exp - metric ] shows how the polygons on the discrete level approach the exact solution we show the approximation for the three different values for simultaneously in one graph . in each case, the solid line is the exact geodesic , the dotted line is the polygonal approximation for , with the thicker dots indicating the stencil points .the intermediate polygon with densely distributed stencils is the approximation for .j. z. gratefully acknowledges the financial support of the epsrc through an advanced research fellowship ( gr / s99037/1 ) , grant ep / k027743/1 and the leverhulme trust ( rpg-2013 - 261 ) .both authors are grateful to michael ortiz for pointing out the relevance of the jacobi principle .they benefited from helpful discussions during the first annual meeting of the epsrc network `` mathematical challenges of molecular dynamics : a chemo - mathematical forum '' ( ep / f03685x/1 ) and thank the anonymous reviewers for helpful comments .10 ajit banerjee and noah p. adams .dynamics of classical systems based on the principle of stationary action . , 92(12):73307339 , 1990 .v. benci , d. fortunato , and f. giannoni . on the existence of geodesics in static lorentz manifolds with singular boundary ., 19(2):255289 , 1992 .george d. birkhoff . .with an addendum by jurgen moser .american mathematical society colloquium publications , vol .american mathematical society , providence , r.i .susanne c. brenner and l. ridgway scott. , volume 15 of _ texts in applied mathematics_. springer , new york , third edition , 2008 .maria cameron , robert v. kohn , and eric vanden - eijnden .the string method as a dynamical system ., 21(2):193230 , 2011 .weinan e , weiqing ren , and eric vanden - 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lehrbuch .[ springer textbook ] .springer - verlag , berlin , third edition , 1990 .eine einfhrung unter bercksichtigung von vorlesungen von f. l. bauer .[ an introduction , with reference to lectures by f. l. bauer ] .

Abstract: this article studies the existence of long - time solutions to the hamiltonian boundary value problem , and their consistent numerical approximation . such a boundary value problem is , for example , common in molecular dynamics , where one aims at finding a dynamic trajectory that joins a given initial state with a final one , with the evolution being governed by classical ( hamiltonian ) dynamics . the setting considered here is sufficiently general so that long time transition trajectories connecting two configurations can be included , provided the total energy is chosen suitably . in particular , the formulation presented here can be used to detect transition paths between two stable basins and thus to prove the existence of long - time trajectories . the starting point is the formulation of the equation of motion of classical mechanics in the framework of jacobi s principle ; a curve shortening procedure inspired by birkhoff s method is then developed to find geodesic solutions . this approach can be viewed as a string method .    dedicated to professor armin leutbecher on the occasion of his 80th birthday


computing a symmetric positive definite solution of an overdetermined linear system of equations arises in a number of physical problems such as estimating the mass inertia matrix in the design of controllers for solid structures and robots ; see , e.g. , , , . modeling a deformable structure also leads to such a mathematical problem ; e.g. , see .the problem turns into finding an optimal solution of the system where , with , are given and a symmetric positive definite matrix is to be computed as a solution of ( [ 1 ] ) .in some special applications , the data matrix has a simple structure , which may be taken into consideration for efficiently organized computations .estimation of the covariance matrix and computation of the correlation matrix in finance are two such examples where the data matrices are respectively block diagonal and the identity matrix ; e.g. , see . + a number of least squares formulations have been proposed for physical problems , which may be classified as ordinary and error in variables ( eiv ) models .+ also , single or multiple right hand side least squares may arise . with a single right hand side, we have an overdetermined linear system of equations , where , , with , are known and the vector is to be computed . in an ordinary least squares formulation , the error is only attributed to .so , to minimize the corresponding error , the following mathematical problem is devised : there are a number of methods for solving ( [ 2 ] ) , identified as direct and iterative methods .a well known direct method is based on using the qr factorization of the matrix  .an iterative method has also been introduced in for solving ( [ 2 ] ) using the gmres algorithm .+ in an eiv model , however , errors in both and are considered ; e.g. , see .total least squares formulation is a well - known eiv model , where the goal is to solve the following mathematical problem ( e.g. , see and ) : \| \\ & s. t. &  ( d+\delta d)x = t+\delta t. \label{3}\end{aligned}\ ] ]  we note that in ( [ 2 ] ) and ( [ 3 ] ) respectively denote the vector 2-norm and the matrix frobenius norm. both direct and iterative methods have been presented for solving ( [ 3 ] ) . moreover ,the scaled total least squares formulation has been considered to unify both ordinary and total leats squares formulation ; e.g. , see . in ascaled toal least squares formulation , the mathematical problem \| \nonumber\\ & s. t. ( d+\delta d)x=\lambda t+\delta t\end{aligned}\ ] ] is to be solved for an arbitrary scalar .zhou has studied the effect of perturbation and gave an error analysis of such a formulation .+ a least squares problem with multiple right hand side vectors can also be formulated as an overdetermined system of equations , where , , with , are given and the matrix is to be computed . with ordinary and total least squares formulations , the respective mathematical problems are : and \| \\ & s. t. &  ( d+\delta d)x = t+\delta t \nonumber\\ & & x \in { \mathbb{r}}^{n \times k}.\label{5}\end{aligned}\ ] ]  common methods for solving ( [ 4 ] ) are similar to the ones for ( [ 2 ] ) ; see , e.g. , , . solving ( [ 5 ] ) is possible by using the method described in , based on the svd factorization of the matrix $ ] .connections between ordinary least squares and total least squares formulations have been discussed in . + here, we consider a newly defined eiv model for solving a positive definite linear problem . our goal is to compute a symmetric positive definite solution to the overdetermined system of equations , where both matrices and may contain errors .we refer to this problem as positive definite linear system of equations later .no eiv model , even the well - known total least squares formulation , is considered for solving the positive definite linear system of equations in the literature .several approaches have been proposed for this problem , commonly considering the ordinary least squares formulation and minimizing the error over all symmetric positive definite matrices , where is the frobenious norm ; see e.g. .larson discussed a method for solving a positive definite least squares problem considering the corresponding normal system of equations .he considered both symmetric and positive definite least squares problems .krislock proposed an interior point method for solving a variety of least squares problems with positive semi - definite constraints .woodgate described a new algorithm for solving a similar problem in which a symmetric positive semi - definite matrix is computed to minimize , with known and .hu presented a quadratic programming approach to handle the positive definite constraint . in her method , the upper and lower bounds for the entries of the target matrix can be given as extra constraints .in real measurements , however , both the data and target matrices may contain errors ; hence , the total least squares formulation appears to be appropriate .+ the rest of our work is organized as follows . in section 2 , we define a new error function and discuss some of its characteristics . a method for solving the resulting optimization problem with the assumption that has full column rank is presented in section 3 . in section 4 ,we generalize the method to the case of data matrix having an arbitrary rank . in section 5, a detailed discussion is made on computational complexity of both methods .computational results and comparisons with available methods are given in section 6 .section 7 gives our concluding remarks .consider a single equation , where and .errors in the entry of and are respectively equal to and ; e.g. , see . + in , was considered as a value to represent errors in both and . as shown in figure [ f1 ], is the height of the triangle abc which turns to be equal to . here , to represent the errors in both and , we define the area error to be which is equal to for .+ considering the problem of finding a symmetric and positive definite solution to the overdetermined system of linear equations , in which both and include error , the values and are predicted values for and from the model ; hence , vectors and are the entries of errors in the column of and , respectively . extending the error formulation ( [ 77 ] ) , the value seems to be an appropriate measure of error .we also have with standing for trace of a matrix . therefore, the problem can be formulated as where is symmetric and by , we mean is positive definite .problem ( 9 ) poses a newly defined eiv model for solving the positive definite linear system of equations .+ in lemma [ 9 ] , we represent an equivalent formulation for the error , .first , consider to a well - known property of positive definite matrices .* note * a matrix is positive definite if and only if there exists a nonsingular matrix such that .+  + the following results about the trace operator are also well - known ; e.g. , see .+  [ 8 ] for an nonsingulartible matrix and arbitrary matrices , and we have +  + + +  +  [ 9 ] the error , defined by ( [ 88 ] ) , is equal to where and denotes the frobenius norm of a matrix .substituting in ( [ 88 ] ) and using lemma [ 8 ] , we get   considering this new formulation for , it can be concluded that by use of our newly defined eiv model , computing a symmetric and positive definite solution to the over - determined system of equations is equivalent to computing a nonsingular matrix to be the solution of and letting .a similar result is obtained by considering the over - determined system with and multiplying both sides by .we have , or equivalently , now , to assign a solution to ( [ 11 ] ) , it makes sense to minimize the norm of residual .thus , to compute , it is sufficient to let to be the solution of   * note * an appropriate characteristic of the error formulation proposed by ( [ 88 ] ) is that for a symmetric and positive definite matrix , the value of is nonnegative and it is equal to zero if and only if .here , we are to develop an algorithm for solving ( [ 6 ] ) with the assumption that has full column rank .+ using lemma [ 8 ] , with being symmetric , we have + so , ( [ 6 ] ) can be written as where and and the symmetric and positive definite matrix is to be computed .+  for each satisfying the first order necessary conditions of ( [ 12 ] ) , the sufficient optimality conditions described in theorem [ 14 ] are satisfied and since is convex on the cone of symmetric positive definite matrices , we can confirm that the symmetric positive definite matrix satisfying the kkt necessary conditions mentioned in theorem [ 13 ] is the unique global solution of ( [ 12 ] ) .+   as mentioned in theorem [ 13 ] , the kkt conditions lead to the nonlinear matrix equation note that ( [ 116 ] ) is an special case of the continuous time riccati equation ( care ) ,  with , and . there is a matlab routine to solve care for arbitrary values of , , , , and . to use the routine , it is sufficient to type the command  .... x = care(a , b , q , r , s , e ) , ....  for the input arguments as in ( [ 115 ] ) .higham developed an effective method for computing the positive definite solution to this special care when and are symmetric and positive definite using well - known decompositions .lancaster and rodman ( ) also discussed solving different types of algebraic riccati equations .moreover , they derived a perturbation analysis for these matrix equations .+ * note * ( qr decomposition ) the qr decomposition of a matrix with , is a decomposition of the form , where is an upper triangular matrix and satisfies .moreover , if has full column rank , then also has full column rank .+  + * note * ( cholesky decomposition ) a cholesky decomposition of a symmetric positive definite matrix is a decomposition of the form , where , known as the cholesky factor of , is an nonsingular upper triangular matrix .+  + * note * ( spectral decomposition ) all eigenvalues of a symmetric matrix , , are real and there exists an orthonormal matrix with columns representing the corresponding eigenvectors .thus , there exist an orthonormal matrix with columns equal to the eigenvectors of and a diagonal matrix containing the eigenvalues such that . also , if is positive definite , then all of its eigenvalues are positive , and so we can set .thus , spectral decomposition for a symmetric positive definite matrix is a decomposition of the form , where and is a diagonal matrix .+  +  [70 ] assume with are known and .let be the qr factorization of .let and .define the matrix and compute its spectral decomposition , that is , .then , ( [ 12 ] ) has a unique solution , given by   based on theorem [ 14 ] and the afterwards discussion , it is sufficient to show that satisfies the necessary optimality conditions , .note that from , we have substituting , we have +   * note * to compute , it is also possible to first compute and then calculate the cholesky decomposition for . however , because of more stability , in theorem [ 70 ] the qr decomposition of is used . +  + we are now ready to outline the steps of our proposed algorithm .+  + * solving the eiv model for positive definite linear system using qr decomposition . *+ compute the qr decomposition for d and let .+ let , where and compute the spectral decomposition of , that is , + set + set .+  + note that algorithm 1 computes the solution of ( [ 6 ] ) directly .+ the following theorem shows that by use of spectral decomposition of a method similar to the one introduced in is in hand for solving the continuous time riccati equation .+  [ 18 ] let and with , and let the spectral decomposition of be .define the matrix and compute its spectral decomposition , .then , the unique minimizer of ( [ 12 ] ) is   similar to the proof of theorem [ 70 ] , it is sufficient to show that the mentioned satisfies . substituting , we have    next , based on theorem [ 18 ] , we outline an algorithm for solving ( [ 6 ] ) .+  + * solving the eiv model for positive definite linear system using spectral decomposition . *+ let and compute its spectral decomposition : + let , where and compute the spectral decomposition of , that is , + set + set .+  + in section 4 we generalize our proposed method for solving positive definite linear system of equations when the data matrix is rank deficient .since the data matrix is usually produced from experimental measurements , we may have . here , we are to generalize algorithm 1 for solving ( [ 6 ] ) , assuming that . in section 4.1we outline two algorithms to compute the general solution of ( [ 6 ] ) .it will be shown that , in general , ( [ 6 ] ) may not have a unique solution .hence , in section 4.2 we discuss how to find a particular solution of ( [ 6 ] ) having desirable characteristics for control problems .+   based on theorems [ 13 ] and [ 14 ] , a symmetric positive definite matrix is a solution of ( [ 6 ] ) if and only if therefore , in the following , we discuss how to find a symmetric positive definite matrix satisfying ( [ 19 ] ) .+ first we note that in case and are rank deficient , there might be no solution for ( [ 19 ] ) , and if there is any , it is not necessarily a unique solution ; see , e.g. , .higham considered to as a solution of ( [ 19 ] ) , which is symmetric and positive semidefinite. however , we are interested in finding a symmetric positive definite solution to ( [ 19 ] ) .hence , in the following , first the necessary and sufficient conditions on and to guarantee the existence of positive definite solution to ( [ 19 ] ) are discussed .we then outline two algrithms to compute such a solution .let the spectral decomposition of be , where is a diagonal matrix having the positive eigenvalues of as its diagonal entries . substituting the decomposition in ( [ 19 ] ) , we get + since is orthonormal , ( [ 20 ] ) can be written as   then , letting and , we have thus , the matrix is a solution of ( [ 6 ] ) if and only if is symmetric positive definite and satisfies ( [ 21 ] ) .substituting the block form , where , and , in ( [ 21 ] ) leads to which is satisfied if and only if +     before discussing how to compute , we show that if ( ) has a symmetric and positive definite solution , then must be nonsingular .the matrix as a main minor of the positive definite matrix is nonsingular . is also supposed to be nonsingular .hence , it can be concluded from ( [ 22 ] ) that is nonsingular .let and suppose satisfies .consider problem ( [ 6 ] ) corresponding to the data and target matrices and as follows : we know from theorems [ 13 ] and [ 14 ] that the necessary and sufficient optimality conditions for the unique solution of problem ( [ 25 ] ) implies ( [ 22 ] ) .thus , can be computed using algorithm 1 for the input arguments and . substituting the computed in ( [ 23 ] ) , the linear system of equations arises , where are known and is to be computed . since is positive definite and is nonsingular ,the coefficient matrix of the linear system ( [ 26 ] ) is nonsingular and can be uniquely computed .+ it is clear that since is symmetric , is the same as .now , we check whether the computed and satisfy ( [ 24 ] ) .inconsistency of ( [ 25 ] ) means that there is no symmetric positive definite matrix satisfying ( [ 22])-([24 ] ) , and if so , ( [ 6 ] ) has no solution .thus , in solving an specific positive definite system with rank deficient data and target matrices using the presented eiv model , a straightforward method to investigate the existence of solution is to check whether ( [ 24 ] ) holds for the given data and target matrices . on the other hand , for numerical results, it is necessary to generate meaningful test problems .hence , in the following two lemmas , we investigate the necessary and sufficient conditions for satisfaction of ( [ 24 ] ) .+   from ( [ 22 ] ) , we have and from ( [ 24 ] ) , we get  manipulating ( [ 24 ] ) with ( [ 29 ] ) and ( [ 30 ] ) , we get considering the block form , where and , we have rewriting ( [ 31 ] ) results in which is equivalent to ( e.g. , see ) where is in the null space of .thus , ( [ 19 ] ) has a positive definite solution if and only if   * note * for real problems with arbitrary values of and , the necessary and sufficient condition given in lemma may not be satisfied , in general .hence , we are to propose a threshold to determine if is close enough to zero . in the following ,we show that if , for a sufficiently small scalar , then computed from ( [ 23 ] ) is a proper approximation for the solution of ( [ 24 ] ) . substituting in ( [ 332 ] ) , we have and let satisfy ( [ 24 ] ) , that is , then , we have letting and , ( [ 331 ] ) , we get   and where and are the column of and respectively . now , since the 2 norm of each column of is equal to one , every entry of is less than or equal to one .moreover , under the assumption , none of the entries of are greater than .hence , we have and are the row of and the column of respectively .now , ( [ 337 ] ) together with ( [ 4444 ] ) gives hence , there is a constant such that and are the entry of and respectively . letting , from ( [ 555 ] ) we get for and and hence , assuming we have which means that if is close enough to zero , then the computed solution from the approximate satisfaction of ( [ 24 ] ) would be close enough to the exact solution .+  + in the following lemma , we give a sufficient condition which guarantees the existence of a solution for .we later use this result to generate consistent test problems in section 6 .  [ 117 ] let the spectral decomposition of be where and . a sufficient condition for satisfaction of ([ 24 ] ) is that where and satisfy and .a possible choice for in lemma [ 117 ] is zero , for which ( [ 32 ] ) is equivalent to with in the null space of . to obtain a simplified sufficient condition for existence of a positive definite solution to ( [ 19 ] ) , we let . multiplying ( [ 33 ] ) by and respectively on the left and right , and substituting the spectral decomposition of , we get letting , we get since has full rank , we get now , since is nonsingular , ( [ 35 ] ) holds if and only if this leads to + since is orthonormal , we have .hence , we get + substituting ( [ 45 ] ) in ( [ 44 ] ) , we get which is satisfied if and only if .since the columns of form an orthogonal basis for the null space of  , it can be concluded that each column of is a linear combination of the columns of .thus , is a necessary condition for ( [ 36 ] ) to be satisfied , and since both and have orthogonal columns , satisfies . on the other hand , we know from the definition of the spectral decomposition that .thus , manipulating ( [ 46 ] ) with ( [ 47 ] ) , we get which holds if and only if there exists a matrix such that and it can be concluded from ( [ 46 ] ) and ( [ 49 ] ) that , where and .  [ 89 ] the matrices and defined in lemma [ 27 ] can set to be rotation matrices to satisfy thus , to compute a target matrix , , satisfying lemma [ 27 ] , it is sufficient to first compute from ( [ 28 ] ) with and arbitrary rotation matrices and as defined in lemma [ 27 ] and then set , where and are arbitrary orthonormal and diagonal matrices .thus , problem ( [ 6 ] ) has a solution if and only if the data and target matrices satisfy ( [ 3344 ] ) . in this case , , and its transpose , , are respectively computed from ( [ 22 ] ) and ( [ 23 ] ) .hence , the only remaining step is to compute so that is symmetric and positive definite .+ we know that is symmetric positive definite if and only if there exists a nonsingular lower triangular matrix so that where is lower triangular and nonsingular. considering the block forms and , where is an matrix and and are nonsingular lower triangular matrices , we get thus ,      therefore , to compute a symmetric positive definite , ( [ 40])([43 ] ) must be satisfied .let be the cholesky decomposition of . satisfies ( [ 40 ] ) . substituting in ( [ 41 ] ) , is computed uniquely by solving the resulting linear system . since ( [ 42 ] ) is transpose of ( [ 41 ] ) , it does not give any additional information . finally , to compute a matrix to satisfy ( [ 43 ] ) , it is sufficient to choose an arbitrary lower triangular nonsingular matrix and substitute it in ( [ 43 ] ) .the resulting gives a symmetric positive definite as follows :    * solving the eiv model for positive definite linear system with rank deficient data and target matrices using spectral decomposition . *+ as the upper bounds for absolute error is taken to be close to the machine ( or user s ) zero .+ let and compute its spectral decomposition :   let and .+ compute and let let , assume satisfies .+ perform * algorithm 1 * with input parameters and , and let .+ solve the linear system ( [ 23 ] ) to compute and let .+ compute the spectral decomposition for , that is , compute .+ if * stop * ( ( [ 6 ] ) has no solution ) + else + let the cholesky decomposition of be and set .+ solve the lower triangular system ( [ 41 ] ) to compute .+ let be an arbitrary nonsingular lower triangular matrix and compute using ( [ 43 ] ) .+ let and . + compute + endif .+  + next , we show how to use the complete orthogonal decomposition of the data matrix instead of the spectral decomposition of . + * note * ( complete orthogonal decomposition ) let be an arbitrary matrix with .there exist , and so that is upper triangular , , and +  + next , algorithm 4 is presented using the complete orthogonal decomposition of . +  * solving the eiv model for positive definite linear system with rank deficient data and target matrices using complete orthogonal decomposition . * + as the upper bounds for absolute error is taken to be close to the machine ( or user s ) zero .+ compute the complete orthogonal decomposition of , that is , let , and , where consists of the first columns of .+ compute and let let , assume satisfies .+ perform * algorithm 1 * with input parameters and , and let .+ solve the linear system ( [ 23 ] ) to compute and let .+ compute the spectral decomposition for , that is , compute .+ if * stop * ( ( [ 6 ] ) has no solution ) + else + let the cholesky decomposition of be and set .+ solve the lower triangular system ( [ 41 ] ) to compute .+ let be an arbitrary nonsingular lower triangular matrix and compute using ( [ 43 ] ) .+ let and . + compute + endif .+  + thus , based on the above study , the computational complexity of pdeiv - qr is lower than that of pdeiv - spec , for all matrix sizes .but , for the case of rank deficient data matrix , depending on the matrix size and rank , one of the algorithms pdeiv - rd - spec and pdeiv - rd - cod may have a lower computational complexity .+  10 alizadeh f. , pierre j. , heaberly a. , overton m. l. : primal - dual interior point methods for semidefinite programming : convergence rates , stability and numerical result , siam j. optim ., 8 , 746 - 768 ( 1998 ) aubry a. , maio a. d. , pallotta l. , farina a. : maximum likelihood estimation of a structured covariance matrix with a condition number constraint , ieee trans . on signal processing , 60(6 ) , 3004 - 3021 ( 2012 ) cheng c. l. , kukush a. , mastronardi n. , paige c. , van huffel s. : total least squares and errors - in - variables modeling , comput stat data an , 52 , 1076 - 1079 ( 2007 ) deng y. , boley d. : on the optimal approximation for the symmetric procrustes problems of the matrix equation axb = c , proceedings of the international conference on computational and mathematical methods in science and engineering , chicago , 159 - 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Abstract: the need to estimate a positive definite solution to an overdetermined linear system of equations with multiple right hand side vectors arises in several process control contexts . the coefficient and the right hand side matrices are respectively named data and target matrices . a number of optimization methods were proposed for solving such problems , in which the data matrix is unrealistically assumed to be error free . here , considering error in measured data and target matrices , we present an approach to solve a positive definite constrained linear system of equations based on the use of a newly defined error function . to minimize the defined error function , we derive necessary and sufficient optimality conditions and outline a direct algorithm to compute the solution . we provide a comparison of our proposed approach and two existing methods , the interior point method and a method based on quadratic programming . two important characteristics of our proposed method as compared to the existing methods are computing the solution directly and considering error both in data and target matrices . moreover , numerical test results show that the new approach leads to smaller standard deviations of error entries and smaller effective rank as desired by control problems . furthermore , in a comparative study , using the dolan - mor performance profiles , we show the approach to be more efficient . +  error in variables models , positive definiteness constraints , overdetermined linear system of equations , multiple right hand side vectors  65f05 , 65f20 , 49m05


generative adversarial nets were recently introduced as an alternative framework for training generative models in order to sidestep the difficulty of approximating many intractable probabilistic computations .adversarial nets have the advantages that markov chains are never needed , only backpropagation is used to obtain gradients , no inference is required during learning , and a wide variety of factors and interactions can easily be incorporated into the model .furthermore , as demonstrated in , it can produce state of the art log - likelihood estimates and realistic samples .  in an unconditioned generative model, there is no control on modes of the data being generated .however , by conditioning the model on additional information it is possible to direct the data generation process .such conditioning could be based on class labels , on some part of data for inpainting like , or even on data from different modality .  in this workwe show how can we construct the conditional adversarial net . andfor empirical results we demonstrate two set of experiment .one on mnist digit data set conditioned on class labels and one on mir flickr 25,000 dataset for multi - modal learning .despite the many recent successes of supervised neural networks ( and convolutional networks in particular ) , it remains challenging to scale such models to accommodate an extremely large number of predicted output categories .a second issue is that much of the work to date has focused on learning one - to - one mappings from input to output .however , many interesting problems are more naturally thought of as a probabilistic one - to - many mapping .for instance in the case of image labeling there may be many different tags that could appropriately applied to a given image , and different ( human ) annotators may use different ( but typically synonymous or related ) terms to describe the same image .  one way to help address the first issue is to leverage additional information from other modalities : for instance , by using natural language corpora to learn a vector representation for labels in which geometric relations are semantically meaningful .when making predictions in such spaces , we benefit from the fact that when prediction errors we are still often ` close ' to the truth ( e.g. predicting table instead of chair ) , and also from the fact that we can naturally make predictive generalizations to labels that were not seen during training time . works such as shown that even a simple linear mapping from image feature - space to word - representation - space can yield improved classification performance .  one way to addressthe second problem is to use a conditional probabilistic generative model , the input is taken to be the conditioning variable and the one - to - many mapping is instantiated as a conditional predictive distribution . take a similar approach to this problem , and train a multi - modal deep boltzmann machine on the mir flickr 25,000 dataset as we do in this work .additionally , in the authors show how to train a supervised multi - modal neural language model , and they are able to generate descriptive sentence for images .generative adversarial nets were recently introduced as a novel way to train a generative model .they consists of two ` adversarial ' models : a generative model that captures the data distribution , and a discriminative model that estimates the probability that a sample came from the training data rather than .both and could be a non - linear mapping function , such as a multi - layer perceptron .  to learn a generator distribution over data data ,the generator builds a mapping function from a prior noise distribution to data space as .and the discriminator , , outputs a single scalar representing the probability that came form training data rather than . and are both trained simultaneously : we adjust parameters for to minimize and adjust parameters for to minimize , as if they are following the two - player min - max game with value function :   + \mathbb{e}_{\bm{z } \sim p_z(\bm{z})}[\log ( 1 - d(g(\bm{z})))].\ ] ]   generative adversarial nets can be extended to a conditional model if both the generator and discriminator are conditioned on some extra information . could be any kind of auxiliary information , such as class labels or data from other modalities .we can perform the conditioning by feeding into the both the discriminator and generator as additional input layer .  in the generatorthe prior input noise , and are combined in joint hidden representation , and the adversarial training framework allows for considerable flexibility in how this hidden representation is composed .  in the discriminator and are presented as inputs and to a discriminative function ( embodied again by a mlp in this case ) .the objective function of a two - player minimax game would be as eq [ eq : minimaxgame - definition - conditioned ] + \mathbb{e}_{\bm{z } \sim p_z(\bm{z})}[\log ( 1 - d(g(\bm{z } | \bm{y})))].\ ] ]  fig [ fig : diagram ] illustrates the structure of a simple conditional adversarial net .we trained a conditional adversarial net on mnist images conditioned on their class labels , encoded as one - hot vectors .  in the generator net ,a noise prior with dimensionality 100 was drawn from a uniform distribution within the unit hypercube .both and are mapped to hidden layers with rectified linear unit ( relu ) activation , with layer sizes 200 and 1000 respectively , before both being mapped to second , combined hidden relu layer of dimensionality 1200 .we then have a final sigmoid unit layer as our output for generating the 784-dimensional mnist samples .the discriminator maps to a maxout layer with 240 units and 5 pieces , and to a maxout layer with 50 units and 5 pieces . both of the hidden layers mapped to a joint maxout layer with 240 units and 4 pieces before being fed to the sigmoid layer .( the precise architecture of the discriminator is not critical as long as it has sufficient power ; we have found that maxout units are typically well suited to the task . )the model was trained using stochastic gradient decent with mini - batches of size 100 and initial learning rate of which was exponentially decreased down to with decay factor of .also momentum was used with initial value of which was increased up to .dropout with probability of 0.5 was applied to both the generator and discriminator . andbest estimate of log - likelihood on the validation set was used as stopping point .table [ table : parzen ] shows gaussian parzen window log - likelihood estimate for the mnist dataset test data .1000 samples were drawn from each 10 class and a gaussian parzen window was fitted to these samples .we then estimate the log - likelihood of the test set using the parzen window distribution .( see for more details of how this estimate is constructed . )the conditional adversarial net results that we present are comparable with some other network based , but are outperformed by several other approaches including non - conditional adversarial nets .we present these results more as a proof - of - concept than as demonstration of efficacy , and believe that with further exploration of hyper - parameter space and architecture that the conditional model should match or exceed the non - conditional results .fig [ fig : mnist ] shows some of the generated samples .each row is conditioned on one label and each column is a different generated sample .  .parzen window - based log - likelihood estimates for mnist .we followed the same procedure as for computing these values . [ cols="^,^",options="header " , ]photo sites such as flickr are a rich source of labeled data in the form of images and their associated user - generated metadata ( ugm ) in particular user - tags .user - generated metadata differ from more ` canonical ' image labelling schems in that they are typically more descriptive , and are semantically much closer to how humans describe images with natural language rather than just identifying the objects present in an image .another aspect of ugm is that synoymy is prevalent and different users may use different vocabulary to describe the same concepts consequently , having an efficient way to normalize these labels becomes important .conceptual word embeddings can be very useful here since related concepts end up being represented by similar vectors .  in this sectionwe demonstrate automated tagging of images , with multi - label predictions , using conditional adversarial nets to generate a ( possibly multi - modal ) distribution of tag - vectors conditional on image features .  for image features we pre - train a convolutional model similar to the one from on the full imagenet dataset with 21,000 labels .we use the output of the last fully connected layer with 4096 units as image representations .  for the world representationwe first gather a corpus of text from concatenation of user - tags , titles and descriptions from yfcc100 m dataset metadata .after pre - processing and cleaning of the text we trained a skip - gram model with word vector size of 200 .and we omitted any word appearing less than 200 times from the vocabulary , thereby ending up with a dictionary of size 247465 .we keep the convolutional model and the language model fixed during training of the adversarial net . and leave the experiments when we even backpropagate through these models as future work .  for our experiments we use mir flickr 25,000 dataset , and extract the image and tags features using the convolutional model and language model we described above .images without any tag were omitted from our experiments and annotations were treated as extra tags .the first 150,000 examples were used as training set .images with multiple tags were repeated inside the training set once for each associated tag .  for evaluation , we generate 100 samples for each image and find top 20 closest words using cosine similarity of vector representation of the words in the vocabulary to each sample .then we select the top 10 most common words among all 100 samples .table [ table : samples ] shows some samples of the user assigned tags and annotations along with the generated tags .the best working model s generator receives gaussian noise of size 100 as noise prior and maps it to 500 dimension relu layer .and maps 4096 dimension image feature vector to 2000 dimension relu hidden layer .both of these layers are mapped to a joint representation of 200 dimension linear layer which would output the generated word vectors .the discriminator is consisted of 500 and 1200 dimension relu hidden layers for word vectors and image features respectively and maxout layer with 1000 units and 3 pieces as the join layer which is finally fed to the one single sigmoid unit .the model was trained using stochastic gradient decent with mini - batches of size 100 and initial learning rate of which was exponentially decreased down to with decay factor of .also momentum was used with initial value of which was increased up to .dropout with probability of 0.5 was applied to both the generator and discriminator .the hyper - parameters and architectural choices were obtained by cross - validation and a mix of random grid search and manual selection ( albeit over a somewhat limited search space . )l | b3.7 cm | b3.7 cm & user tags + annotations & generated tags +  & montanha , trem , inverno , frio , people , male , plant life , tree , structures , transport , car & taxi , passenger , line , transportation , railway station , passengers , railways , signals , rail , rails +  & food , raspberry , delicious , homemade & chicken , fattening , cooked , peanut , cream , cookie , house made , bread , biscuit , bakes +  & water , river & creek , lake , along , near , river , rocky , treeline , valley , woods , waters +  & people , portrait , female , baby , indoor & love , people , posing , girl , young , strangers , pretty , women , happy , life +  [ table : samples ]the results shown in this paper are extremely preliminary , but they demonstrate the potential of conditional adversarial nets and show promise for interesting and useful applications .   also , in the current experiments we only use each tag individually . but by using multiple tags at the same time ( effectively posing generative problem as one of ` set generation ' ) we hope to achieve better results .  another obvious direction left for futurework is to construct a joint training scheme to learn the language model .works such as has shown that we can learn a language model for suited for the specific task .this project was developed in pylearn2  framework , and we would like to thank pylearn2 developers . we also like to thank ian goodfellow for helpful discussion during his affiliation at university of montreal .the authors gratefully acknowledge the support from the vision & machine learning , and production engineering teams at flickr ( in alphabetical order : andrew stadlen , arel cordero , clayton mellina , cyprien noel , frank liu , gerry pesavento , huy nguyen , jack culpepper , john ko , pierre garrigues , rob hess , stacey svetlichnaya , tobi baumgartner , and ye lu ) .

Abstract: * generative adversarial nets * were recently introduced as a novel way to train generative models . in this work we introduce the conditional version of generative adversarial nets , which can be constructed by simply feeding the data , , we wish to condition on to both the generator and discriminator . we show that this model can generate mnist digits conditioned on class labels . we also illustrate how this model could be used to learn a multi - modal model , and provide preliminary examples of an application to image tagging in which we demonstrate how this approach can generate descriptive tags which are not part of training labels .