we consider the minimization problem of an unknown risk function , where is the dimension of the statistical model .we assume the existence of a risk minimizer where the risk function corresponds to the expectation of an appropriate loss function w.r.t . an unknown distribution . in empirical risk minimization , this quantity is usually estimated by its empirical version from an i.i.dhowever , in many problems such as local -estimation or errors - in - variables models , a nuisance parameter can be involved in the empirical version .this parameter most often coincides with some bandwidth related to a kernel that gives rise to `` kernel empirical risk minimization . ''one typically deals with this issue in pointwise estimation , as , for example , in polzehl and spokoiny with localized likelihoods or in chichignoud and lederer with local -estimators . in learning theory , many authors have recently investigated supervised and unsupervised learning with errors in variables . as a rule ,such matters require one to plug deconvolution kernels into the empirical risk , as loustau and marteau in noisy discriminant analysis or hall and lahiri in quantile and moment estimation ; see also dattner , reiss and trabs .  in the above papers, the authors studied the theoretical properties of kernel empirical risk minimizers and proposed deterministic choices of bandwidths to deduce optimal minimax results .as usual , these optimal bandwidths are related to the smoothness of the target function or the underlying density and are not achievable in practice .adaptivity is therefore one of the biggest challenges . in this respect ,data - driven bandwidth selections have been already proposed in , which are all based on lepski - type procedures .lepski - type procedures are rather appropriate to construct data - driven bandwidths involved in kernels ; for further details , see , for example , .it is well known that they suffer from the restriction to isotropic bandwidths with multidimensional data , which is the consideration of nested neighborhoods ( hyper - cube ) .many improvements were made by kerkyacharian , lepski and picard and more recently by goldenshluger and lepski to select anisotropic bandwidths ( hyper - rectangle ) .nevertheless , their approach still does not provide anisotropic bandwidth selection for nonlinear estimators , which is the scope of this paper .the only work we can mention is in a restrictive case , which is pointwise estimation in nonparametric regression .therefore , the study of data - driven selection of anisotropic bandwidths is still an open issue .moreover , this field is of great interest in practice , especially in image denoising ; see , for example , .the main contribution of our paper is to bring new insights to the problem of bandwidth selection in kernel empirical risk minimization in a possible anisotropic framework .to this end , we first introduce a new criterion called _ gradient excess risk _ , which makes the anisotropic bandwidth selection possible .we then provide a novel data - driven selection based on the comparison of `` gradient empirical risks . ''that can be viewed as an extension of the so - called goldenshluger lepski method ( gl method ; see ) and of the empirical risk comparison method ( erc method ; see ) . eventually , we derive an upper bound for the gradient excess risk ( called gradient inequality ) and optimal results in many settings , such as pointwise and global estimation in nonparametric regression and clustering with errors in variables .note that we consider the risk minimization over the finite dimensional set . in statistical learning or nonparametricestimation , one usually aims at estimating a functional object belonging to some hilbert space .however , in many examples , the target function can be approximated by a finite object , thanks , for instance , to a suitable decomposition in a basis of the hilbert space .this is typically the case in local -estimation , where the target function is assumed to be locally polynomial ( and even constant in many cases ) .moreover , in statistical learning , one is often interested in the estimation of a finite number of parameters , as in clustering .the extension to the infinite - dimensional case is discussed in section [ sdiscussion ] .the structure of this paper is as follows : the main ideas behind the gradient excess risk are introduced in the remainder of this section .an upper bound for the gradient excess risk of the data - driven procedure is presented in section [ gradientinequality ] .this procedure is applied to clustering in section [ sectionkmeans ] and to robust nonparametric regression in section [ sectionlocalglobal ] .additionally , a discussion of our assumptions and an outlook are given in section [ sdiscussion ] , and section [ ssimu ] illustrates the behavior of the method with numerical results .the proofs are finally conducted in the .   in the literature , such as in statistical learning ,the excess risk is the main criterion to measure the performance of some estimator .originally , vapnik and chervonenkis proposed to control this quantity via the empirical process theory , which gives rise to slow rates for the excess risk ; see also . in the last decade, many authors have improved such a bound by giving fast rates using the so - called localization technique ; see and boucheron , bousquet and lugosi for an overview in classification .this technique consists of studying the increments of an empirical process in the neighborhood of the target .in particular , it requires a variance - risk correspondence , equivalent to the eminent margin assumption .as far as we know , this complicated modus operandi is the major obstacle to the anisotropic bandwidth selection issue . inwhat follows , we introduce an alternative criterion to solve this issue , namely the gradient excess risk ( -excess risk , for short , in the sequel ) .this quantity is defined as whereas denotes the euclidean norm on and denotes the gradient of the risk . with a slight abuse of notation, denotes the gradient , whereas denotes the -excess risk . under regularity assumptions on , the -excess risk is linked with the excess risk , thanks to the following lemma .[ lemmadmargin ] let , defined as in ( [ oracle ] ) , and be the euclidean ball of centered at , with radius .assume is , each second partial derivative of is bounded on by a constant and the hessian matrix is positive definite at .then , for small enough , we have where is the smallest eigenvalue of .  the proof is based on the inverse function theorem and a taylor expansion of the function .let us explain how this lemma , together with standard probabilistic tools , leads to fast rates for the excess risk . in this section, denotes the usual empirical risk with associated gradient and associated erm for ease of exposition . under the assumptions of lemma [ lemmadmargin ] , , and we have the following heuristic : \\[-8pt]\nonumber & \leq & \sup_{{\theta}\in{{\mathbb r}}^m}\bigl { \vert}{g}({\theta})-{\widehat{g}}({\theta})\bigr{\vert}_2\lesssim n^{-1/2},\end{aligned}\ ] ] where denotes the inequality up to some positive constantthe last bound only requires a concentration inequality applied to the empirical process .therefore , this heuristic provides fast rates for the excess risk without any localization technique .furthermore , similar bounds can be obtained for the -norm using the same path .indeed , under the same assumptions , the assertion of lemma [ lemmadmargin ] holds , replacing the square root of the excess risk by ( see the proof of lemma [ lemmadmargin ] ) , and then optimal rates are deduced .  from the model selection point of view ,standard penalization techniques based on localization suffer from the dependency on parameters involved in the margin assumption .more precisely , in the strong margin assumption framework , the construction of the penalty requires the knowledge of , related to the hessian matrix of the risk .although many authors have recently investigated the adaptivity w.r.t .these parameters , by proposing `` margin - adaptive '' procedures ( see for the propagation method , for aggregation and for the slope heuristic ) , the theory is not completed and remains a hard issue ; see the related discussion in section [ sdiscussion ] .as an alternative , our data - driven procedure does not suffer from the dependency on since we focus on a gradient inequality in section [ gradientinequality ] .   in this section ,the kernel empirical risk minimization is properly defined and illustrated with two examples : local estimators and deconvolution -means .for some , consider a random variable distributed according to , absolutely continuous w.r.t . the lebesgue measure . inwhat follows , we observe a sample of independent and identically distributed ( i.i.d . )random variables according to .moreover , we call a kernel of order a symmetric function , , which satisfies the following properties :  * , *  , * , .  for any , the dilation is defined as where . for a given kernel , we define the kernel empirical risk indexed by an anisotropic bandwidth ^d ] , assuming that ^d ] , where stands for the usual fourier transform .as introduced in section [ sectionkerm ] , we have at our disposal the family of kernel erm defined as where ^d}f(x)g(\cdot - x)\,dx ] for simplicity ) . from an adaptive point of view , chichignoud and loustau recently investigated the problem of choosing the bandwidth in ( [ noisykmeans ] ) .they established fast rates of convergence up to a logarithmic term for a data - driven selection of , based on a comparison of kernel empirical risks .however , their approach is restricted to isotropic bandwidth selection and depends on the parameters involved in the margin assumption ( in particular on in lemma [ lemmadmargin ] ) .  in the following , adaptive fast rates of convergence for the excess riskare obtained via the gradient approach . for this purpose, we assume that the hessian matrix is positive definite . this assumption was considered for the first time in pollard and is often referred as pollard s regularity assumptions ; see also  . under these assumptions , we can state the same kind of result as lemma [ lemmadmargin ] in the framework of clustering with -means .  [ dclustering ]let be a minimizer of ( [ distortion ] ) , and assume is continuous and is positive definite .let be the euclidean ball center at with radius .then , for sufficiently small , where is a constant which depends on , and .we have at our disposal a family of kernel erm with associated kernel empirical risk defined in ( [ noisykmeans ] ) .we propose to apply the selection rule ( [ defrule ] ) to choose the bandwidth . in this problemas well , we first consider the -excess risk approach to establish adaptive fast rates of convergence for the excess risk . for any ,the -empirical risk vector of is given by ^d}w ( { \mathbf{c}},x)\widetilde k_h(z_i - x)\,dx \biggr)_{(u , j)\in\{1 , \ldots , d\}\times\{1,\ldots , k\ } } \\ & = & \biggl(-\frac{1}{n}\sum_{i=1}^n2 \int_{v_j({\mathbf{c}})}\bigl(x^u - c_j^u \bigr)\widetilde k_h(z_i - x)\,dx \biggr)_{(u , j)\in\{1 , \ldots , d\}\times\{1,\ldots , k\}},\end{aligned}\ ] ] where denotes the coordinate of and , are open vorono cells associated to , defined as ^d\dvtx \forall u\neq j , { \vert}x - c_j{\vert}_2<{\vert}x - c_u{\vert}_2\} ] .this assumption is standard in deconvolution estimation and is satisfied by many standard kernels , such as the _ sinc _ kernel .we also consider a kernel of order , according to the definition of section [ sectionkerm ] .kernels of order satisfying ( * k1 * ) could be constructed by using the so - called meyer wavelet ; see .additionally , we need an assumption on the noise distribution :  there exist a vector and a positive constant such that for all , (t)\bigr{\vert}\geq\rho\prod_{j=1}^d \biggl(\frac { t_j^2 + 1}{2 } \biggr)^{-\beta_j/2}. \ ] ]  * na( ) * deals with a polynomial behavior of the fourier transform of the noise density .an exponential decreasing of the characteristic function of is not considered in this paper for simplicity ; see in multivariate deconvolution for such a study .we are now ready to compute some majorant functions in our context . forsome , let ^d ] , such that .for any integer , let us introduce the function defined as where is linear in and independent of ; see the for details .  then , for sufficiently large , the function is a majorant , that is , where denotes the expectation w.r.t . to the sample and ^{dk}}{\vert}t({\mathbf{c}}){\vert}_2 ] belongs to the anisotropic nikolskii class if for all , and for all ^d ] , any , we have < \infty,\end{aligned}\ ] ] where is driven in ( [ defrulekmeans ] ) .this theorem is a direct application of theorem [ thmainresult ] , lemma [ dclustering ] and the majorant construction .it gives adaptive fast rates of convergence for the excess risk of  and significantly improves the result stated in for two reasons : first , the selection rule allows the extension to the anisotropic case ; besides , there is no logarithmic term in the adaptive rate . in our opinion ,the localization technique used in seems to be the major obstacle to avoid the extra term .in this section , we apply the gradient inequality to the framework of local -estimation in nonparametric robust regression. it will give adaptive minimax results for nonlinear estimators for both pointwise and global estimation .let us specify the model beforehand .for some , we observe a training set of i.i.d .pairs , distributed according to the probability measure on ^d\times{{\mathbb r}} ] , .the noise variables are assumed to be i.i.d . with symmetric density w.r.t .the lebesgue measure .we also assume is continuous at and . for simplicity ,the design points are assumed to be i.i.d . according to the uniform law on ^d ] to avoid discussion on boundary effects .we will consider a point for pointwise estimation and the -risk for global estimation , for .next , we introduce an estimate of at any with the local constant approach ( lca ) . the key idea of lca , as described , for example , in , chapter 1 , is to approximate the target function by a constant in a neighborhood of size of a given point , which corresponds to a model of dimension . to deal with heavy - tailed noises ,we especially employ the huber loss ( see ) defined as follows . for any scale and , the parameter selects the level of robustness of the huber loss between the square loss ( large value of ) and the absolute loss ( small value of ) .let ^d ] , such that , for any , the function is a majorant according to definition [ defmajorant ] .eventually , we introduce the data - driven bandwidth following the schema of the selection rule in section [ gradientinequality ] , where .to derive minimax adaptive rates for local estimation , we start with the definition of the anisotropic hlder class .  [ defholderanisot ] let and be fixed .we say that ^d\rightarrow[-l , l ] ] , and ^d}\biggl{\vert}\frac{\partial^{l}}{\partial x_j^{l}}f(x)\biggr{\vert}\leq l\qquad \forall l=0 , \ldots , \lfloor s_j\rfloor , \ ] ] where is the largest integer strictly less than .[ thholderadapation ] for any ^d ] , and therefore their lepski - type procedure depends on this quantity . thanks to the gradient approach , we obtain the same result without the dependency on the parameter , which corresponds to in the general setting .the selection rule is therefore robust w.r.t . to the fluctuations of this parameter , in particular when is small ( median estimator ) .the aim of this section is to derive adaptive minimax results for for the -risk . to this end, we need to modify the selection rule ( [ defruleloc ] ) including a global ( -norm ) comparison of -empirical risks . for this purpose , for all , we denote the -empirical risks at a given point as and where the dependence in is explicitly written. then we define , for and for any function , the -norm and -semi - norm }\bigl{\vert}\omega(t,\cdot)\bigr { \vert}_q . \ ] ]  the construction of majorants is based on uniform bounds for -norms of empirical processes .recently , goldenshluger and lepski investigated this topic , theorem 2 . for any integer ,let us introduce the function defined as where is an absolute constant which does not depend on .then , for any , the function is a majorant according to definition [ defmajorant ] .we finally select the bandwidth according to   the above choice of the bandwidth leads to the estimator with the following adaptive minimax properties for the -risk over anisotropic nikolskii classes ; see definition [ defnikolskiianisot ] .[ thnikolskiiadapation ] for any ^d ] denotes the multi - dimensional bracket between and , and denotes the operator norm associated to the euclidean norm .since and is continuous , we now have }{\big|\!\big|\!\big|}j_{{g}^{-1}}(u){\big|\!\big|\!\big|}_2= { \big|\!\big|\!\big|}j_{{g}^{-1 } } \bigl(g \bigl({\theta}^\star\bigr)\bigr){\big|\!\big|\!\big|}_2 . \ ] ] then , for small enough , we have with ( [ mvt2 ] ) where is the hessian matrix of .( [ secondstep ] ) follows easily , and the proof is complete .proof of lemma [ dclustering ] the hessian matrix of involves integrals over faces of the vorono diagram . for ,let us denote the face ( possibly empty ) common to and as .moreover , denote the -dimensional lebesgue measure .then , since is continuous and ^d ] .then , for any , we have   the proof of lemma [ lemmatal ] is omitted ; see .we hence have to compile the quantities and associated with the random variable the compilation of uses the same path as , lemma 3 .more precisely , we can apply a chaining argument to the function , for any .then we have , together with a maximum inequality due to , chapter 6 , where for provided that ( thanks to the definition of and sufficiently large ) .the constant can be explicitly computed .this calculation is omitted for simplicity . besides , using , lemma 1 , with , we have \leq\frac { b_6}{\pi_h(2 \beta)}:=v(h),\ ] ] whereas , lemma 2 , allows us to write where are absolute constants .hence , lemma [ lemmatal ] , together with ( [ eqlemma1])([mbound ] ) , gives us , for all , moreover , note that from the previous calculations , we have and , where depend on and , respectively . provided that ( thanks to the definition of and sufficiently large ) , we come up with this gives us the first part of the majorant of lemma [ lemmamajkmeans ] .the last step is to show a similar bound for the auxiliary empirical process .this can be easily done by using lemma [ lemmatal ] together with the previous results .then we have for any , where with a slight abuse of notation , the maximum is understood coordinatewise . using the union bound ,the definition of allows us to write where we choose with .proof of theorem [ thmkmeans ] the proof of theorem [ thmkmeans ] is a direct application of theorem [ thmainresult ] and lemma [ lemmamajkmeans ] .indeed , for any , for large enough , we have with probability , where is defined as the control of the bias function is as follows : ^{dk}}\sum_{i , j } \biggl\ { \int _ { v_j}2\bigl(x^i - c_j^i \bigr ) \bigl({\mathbb e}_{p^z } \widetilde k_{h,\eta}(z - x)- { \mathbb e}_{p^z}\widetilde k_\eta(z - x ) \bigr)\,dx \biggr \}^2 \\ & & \qquad = \sup_{{\mathbf{c}}\in[0,1]^{dk}}\sum _ { i , j } \biggl\{\int_{v_j}2 \bigl(x^i - c_j^i\bigr ) \bigl ( { \mathbb e}_{p^x } k_{h,\eta}(x - x)-{\mathbb e}_{p^x } k_\eta(x - x ) \bigr)\,dx \biggr\}^2 \\ & & \qquad \leq 4\sup_{{\mathbf{c}}\in[0,1]^{dk}}\sum_{i , j}\int _ { v_j}\bigl(x^i - c_j^i \bigr)^2\,dx\bigl{\vert}k_{\eta}*(k_h*f - f)\bigr { \vert}_2 ^ 2 \\ & & \qquad \leq 4k\bigl{\vert}\mathcal{f}[k]\bigr{\vert}_{\infty}{\vert}f_h - f{\vert}_2 ^ 2,\end{aligned}\ ] ] where is the usual nonparametric bias term in deconvolution estimation . besides, note that ^{dk}}\sum_{i , j } \biggl\{\int _ { v_j}2\bigl(x^i - c_j^i \bigr ) \bigl({\mathbb e}_{p^x } k_{h}(x - x)-f(x ) \bigr)\,dx \biggr \}^2 \\ & & \qquad \leq 4\sup_{{\mathbf{c}}\in[0,1]^{dk}}\sum_{i , j}\int _ { v_j}\bigl(x^i - c_j^i \bigr)^2\,dx{\vert}k_{h}*f - f{\vert}_2 ^ 2.\end{aligned}\ ] ] then we need a control of the bias function , \bigr{\vert}_{\infty } \bigr){\vert}k_h*f - f{\vert}_2\qquad \forall h\in{{\mathcal h}}.\ ] ] by using comte and lacour , proposition 3 , we directly have for all , \bigr{\vert}_{\infty } \bigr)l\sum_{j=1}^dh_j^{s_j}\qquad \forall h\in{{\mathcal h}}.\ ] ] now , we have to use a result such as lemma [ dclustering ] , for our family of estimators .in other words , we need to check that this family of estimators is consistent with respect to the euclidean norm in .[ consistencykmeans ] assume is continuous , ^d ] such that since is a 2-lipschitz function , it yields the proof is complete .proof of theorem [ thholderadapation ] from , theorem 1 , we notice that all estimators are consistent , and thus , for sufficiently large , the assumption of lemma [ lemlocalmargincondition ] holds for all . using theorem [ thmainresult ] with and lemma [ lemlocalmargincondition ] , we get with .the control of over hlder classes is based on the same schema as in , applied to the function . for any and any , we then want to show }\sup _ { y\in{{\mathcal t}}}\biggl{\vert}\int k_{h}(x - y ) \bigl[f_t(x)-f_t(y ) \bigr]\,dx\biggr{\vert}\nonumber\\[-8pt]\\[-8pt]\nonumber & \leq & l { \vert}k{\vert}_\infty\sum_{j=1}^dh_j^ { s_j}.\end{aligned}\ ] ] by definition , we see that }{\vert}{\mathbb e}k_{h}(w - x_0 ) [ f_t(w)-f_t(x_0 ) ] { \vert} ] , \leq\bigl{\vert}f(x)-f(y)\bigr { \vert}.\end{aligned}\ ] ] therefore , ( [ eqglobalbiascontrol ] ) holds for all .then , using theorem [ thmainresult ] with , lemma [ lemlocalmargincondition ] and ( [ eqglobalbiascontrol ] ) , there exists an absolute constant such that computing these infimums , we obtain the assertion of the theorem .

Abstract: in this paper , we deal with the data - driven selection of multidimensional and possibly anisotropic bandwidths in the general framework of kernel empirical risk minimization . we propose a universal selection rule , which leads to optimal adaptive results in a large variety of statistical models such as nonparametric robust regression and statistical learning with errors in variables . these results are stated in the context of smooth loss functions , where the gradient of the risk appears as a good criterion to measure the performance of our estimators . the selection rule consists of a comparison of gradient empirical risks . it can be viewed as a nontrivial improvement of the goldenshluger  lepski method to nonlinear estimators . furthermore , one main advantage of our selection rule is the nondependency on the hessian matrix of the risk , usually involved in standard adaptive procedures .  ./style / arxiv - general.cfg


in recent years the manufacturing of freeform surfaces has become increasingly feasible .these freeform surfaces offer an elegant way of simultaneous irradiance and phase control .therefore , the development of numerical algorithms for the calculation of continuous freeform surfaces for control of the irradiance and/or the phase of a beam is of great interest .+ in this work the problem of designing continuous double freeform surfaces in a geometrical optics approximation is considered , in which two collimated beams of arbitrary irradiance are mapped onto each other .several methods for phase and irradiance control with double freeform surfaces have been proposed in literature .+ one of the first design methods for the mapping of two wavefronts by coupled freeform surfaces is the simultaneous multiple surface ( sms ) method , which was developed by benitez and miano .the surfaces are thereby constructed from generalized cartesian ovals and by applying constant optical path length ( opl ) conditions .the design method can be utilized for numerous applications in imaging and nonimaging optics . +zhang et al . and shengqian et al . solve the design problem by describing it in the form of a monge - ampre type pde , discretizing the equation by finite differences and then solving the resulting nonlinear equation system by the newton method .the design method can be applied to a variety of wavefront shapes .+ an alternative approach to construct freeform surfaces for irradiance and phase control is from ray mapping methods .these methods are based on the seperation of the design process into two separate steps : the calculation of an _ integrable _ ray mapping between the source and the target irradiance and the subsequent construction of the _ continuous _ freeform surfacesfrom the mapping .the integrability thereby ensures the continuity of the freeform surfaces and the mapping of the input irradiance onto the ouptut irradiance . since the integrability depends on the physical properties of the optical system it is in general a nontrivial task to find such a mapping . +as shown in several publications , there is a strong relation between the inverse problem of nonimaging optics and optimal mass transport ( omt) .the cost function , which has to be applied to a certain optical configuration , is thereby problem - specific .for example , the mapping of two collimated beams with arbitrary irradiance onto each other with double freeform _ mirrors _ is described by a quadratic cost function and can be solved by corresponding numerical schemes . the same problem statement with double freeform _ lenses _ , which is considered in this work ,is described by a different cost function , which depends on the opl between the freeform surfaces as it was shown by rubinstein and wolansky .+ the investigations presented here are inspired by several publications , in which the authors applied the quadratic omt cost function to calculate a ray mapping to deal with the lens design problem . with this ray mapping, designs have been demonstrated of both single freeform surfaces for irradiance control for collimated input beams and point sources , and that of double freeform surfaces for irradiance and phase control . as demonstrated for illumination control with single freeform surfaces in ref . and for collimated beam shaping with double freeform surfaces in ref . the design problems are thereby restricted to a paraxial approximation .+ here we first investigate the design by ray - mapping methods of double freeform surfaces which map two collimated beams with arbitrary irradiance onto each other _ beyond _ the paraxial approximation . to overcome the restriction to the paraxial regime , which is necessary for the construction of compact systems, the design problem will be modeled by two coupled pde s .this involves on one hand the jacobian equation , expressing the local energy conservation , and on the other hand a ray mapping pde , enforcing the surface continuity and the constant opl .the pdes will then be solved by an optimization scheme with the omt mapping from the quadratic cost function as the intial iterate , leading to a construction approach for the freeform surfaces .+ to do so , the work is structured as follows . in section [ sec:2 ] , by using the law of refraction , the constant opl condition and a surface continuity condition , a pde for an integrable ray mapping , is derived . together with the jacobian equation it builds a system of pdes for the determination of the mapping components .it is argued that the pde system is fulfilled within the paraxial approximation by the quadratic cost function omt map . in section [ sec:3 ] , a method for solving the pde system for general lens lens distances is presented .it is based on discretizing the pdes with finite differences and solving the resulting system of nonlinear equations by a standard optimization scheme with the quadratic cost function omt map as the initial iterate .a summary of the design algorithm and a detailed discussion of the implemenation is presented in section [ sec:4 ] , followed by the application of the presented method to the design of a single - lens and a two - lens system in section [ sec:5 ] .finally , in section [ sec:6 ] , a short discussion of the results is presented .in ref . design method was presented for the construction of a single freeform surface for a collimated input beam with irradiance and an arbitrary illumination pattern on a target plane .it was shown that in the paraxial approximation the design process can be decoupled into two separate steps . in the first stepa raymapping is calculated from the theory of optimal mass transport , and in the second step the freeform surface is constructed from the mapping .+ there are several basic physical principles that a ray mapping needs to fulfill .firstly , to map the source irradiance onto the target irradiance , the ray mapping should be energy conserving .the local energy conservation is expressed through the jacobian eq . secondly , in case of freeform illumination optics , the mapping should allow the calculation of _ continuous _ freeform surfaces .as shown in several publications , these so called _ integrable _ ray mappings are related to problem - specific cost functions representing different optical settings , where one has to distinguish between point sources and/or collimated beams , mirrors and/or lenses and so on .the cost function defines a metric between the source distribution and the target illumination pattern and therefore represents an additional constraint to the underdetermined eq .( [ eq:1 ] ) . in the case of a single freeform surface for the redistribution of collimated input beams , the quadratic cost function which is valid in the paraxial approximation , was studied by the authors . a key property there was the vanishing curl characterizing the quadratic cost function in eq . ( [ eq:2 ] ) .+ as shown by rubinstein and wolansky , the cost function for collimated beam shaping with double freeform _ lenses _ takes a different form than eq .( [ eq:2]) .the authors propose to minimize the corresponding cost function by a steepest descent algorithm to get the ray mapping , but unfortunatly a numerically stable implementation is a nontrivial problem .+ due to its applicability in the paraxial approximation ( see below ) and the availability of numerous published stable numerical schemes for its calculation , the quadratic cost function omt mapping serves as an initial iterate for the optimization scheme presented below . it will therefore build the basis of the design approach presented in sections [ sec:3 ] and [ sec:4 ] .   [ cols="^ " , ]   the freeform calculation with the initial maps shows strong deviations from the predefined specifications , whereas the optimized maps show high quality illumination patterns and a wavefront uniformity beyond the diffraction limit .remaining deviations from the ideal wavefront result ( besides fundamental numerical limitiations ) are from eq .( [ eq:14b ] ) , which is not fulfilled exactly and therefore leads to an error accumulation along the integration path of eqs .( [ eq:9 ] ) and ( [ eq:12 ] ) .the precision of the illumination pattern on the other hand is mainly limited by the precision of .the main deviations from the predefined distribution are resulting from steep gradients , which can be seen especially for the example iap " in fig .[ fig:6]d ) .this is in agreement with fig . [fig:3]b ) and observations which were made in ref . .a design method for the calculation of compact continuous double freeform surfaces for collimated beam shaping with complex irradiance patterns was presented .the method is based on the ray mapping condition ( [ eq:14b ] ) , which was derived from the law of refraction and the surface continuity condition in section [ sec:2 ] and builds together with the jacobian eq .( [ eq:14a ] ) a system of nonlinear pdes for the unknown ray mapping .+ due to the satisfaction of eq .( [ eq:14 ] ) for infinite lens distances by the mapping from omt with the quadratic cost function ( [ eq:2 ] ) , this mapping serves as an ideal initial iterate for an optimization approach for solving the system of eq .( [ eq:14 ] ) . as it was shown by approximating the eqs .( [ eq:14 ] ) by finite differences and using a standard optimization scheme from matlab s optimization toolbox one can ensure a fast convergence to the solution of the eqs .( [ eq:14 ] ) .this was demonstrated by applying the presented method to two design examples with complex target distributions and validating the results by ray tracing .the double freeform surfaces showed thereby high accuracy for the irradiance patterns and the wavefront , which was assessed by the calculation of the corresponding values of the normalized differences .+ further improvements can be made by using omt methods for more complex boundaries of the source and target distributions , which requires the replacement of sulman s method ( with e.g. ) and the generalization of eq .( [ eq:16 ] ) to more complex boundary shapes .the scalability of the distance of the initial map by to the solution of the eqs .( [ eq:14 ] ) suggests the application of e.g. the newton algorithm for a faster optimization .+ in our future research , we want to generalize the presented approach to double freeform surfaces for wavefronts different from the plane case , like e.g. spherical wavefronts .the authors want to thank ralf hambach and mateusz oleszko for valuable discussions , johannes stock for comments on the manuscript and david musick for a grammar and spelling check. we also want to acknowledge the federal ministry of education and research germany for financial support through the project fo+ ( fkz : o3wkck1d ) and kosimo ( fkz:031pt609x ) .   p. benitez and j. c. miano , `` simultaneous multiple surface optical design method in three dimensions , '' opt . eng . *43*(7 ) , 14891502 ( 2004 ) .j. chaves , `` introduction to nonimaging optics , second edition '' , crc press , 2015 , 321406 .    c. shengqian , r. wu , a. li , and z. zheng , `` design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a monge - ampre type equation , '' j. opt . *18*(12 ) , 125602 ( 2016 ) . t. glimm and v.i .oliker , `` optical design of single reflector systems and the monge-kantorovich mass transfer problem , '' j. o. math . sc . * 117*(3 ) , 40964018 ( 2003 ) .t. glimm and v.i .oliker , `` optical design of two - reflector systems , the monge - kantorovich mass transfer problem and fermat s principle , '' indiana university math .j. * 53*(5 ) , 12551277 ( 2004 ) .x .- j . wang , `` on the design of a reflector antenna ii , '' calc . var . * 20*(3 ) , 329 - 341 ( 2004 ) .j. rubinstein and g. wolansky , `` intensity control with a free - form lens , '' j. opt .a * 24*(2 ) , 463469 ( 2007 ) .t. glimm , `` a rigorous analysis using optimal transport theory for a two - reflector design problem with a point source , '' inverse problems * 26*(4 ) , p. 16oliker , `` designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport , '' arch .rational mech .201*(3 ) , 1013 - 1045 ( 2011 ) .oliker , j. rubinstein , and g. wolansky , `` ray mapping and illumination control , '' j. photon . energy *3*(1 ) , 035599 ( 2013 ) .         l.l .doskolovich , e.a .bezus , m.a .moiseev , d.a .bykov , and n.l .kazanskiy , `` analytical source - target mapping method for the design of freeform mirrors generating prescribed 2d intensity distributions , '' opt .express * 24*(10 ) , 1096210971 ( 2016 )

Abstract: numerous applications require the simultaneous redistribution of the irradiance and phase of a laser beam . the beam shape is thereby determined by the respective application . an elegant way to control the irradiance and phase at the same time is from double freeform surfaces . in this work the numerical design of continuous double freeform surfaces from ray mapping methods for collimated beam shaping with arbitrary irradiances is considered . these methods consist of the calculation of a proper ray mapping between the source and the target irradiance and the subsequent construction of the freeform surfaces . by combining the law of refraction , the constant optical path length and the surface continuity condition , a partial differential equation ( pde ) for the ray mapping is derived . it is shown that the pde can be fulfilled in a small - angle approximation by a mapping derived from optimal mass transport with a quadratic cost function . to overcome the restriction to the paraxial regime we use this mapping as an initial iterate for the simultaneous solution of the jacobian equation and the ray mapping pde by an optimization . the presented approach enables the efficient calculation of compact double freeform surfaces for complex target irradiances . this is demonstrated by applying it to the design of a single - lens and a two - lens system .  * * *  2


in recent years , advances in information and communication technologies have been demanding high data - rate wireless communications . in order to realize high data - rate wireless communications , bandwidth of systems should be as wide as possible , which means that such systems should have a large degree of freedom .those systems are also required to be able to operate efficiently even in bad and uncertain environments .for example , in urban areas , there are many obstacles , such as buildings , cars , and people , which interact with wireless communication systems as reflecting / scattering bodies , thereby making communication environment very complex . in analyzing wireless communication systems ,therefore , one has to regard them as systems with very high dimensionality and randomness .this is why statistical mechanics approach is expected to be useful in studying wireless communication systems .  in this paper, we consider a problem arising from considerations of multiple - access channels . typically , a wireless communication system has to accommodate multiple users simultaneously .in such a system , signals coming from different users interfere with each other .how to mitigate such multiple - access interference ( mai ) is one of the most important problems in wireless communications .code - division multiple - access ( cdma )  provides an effective scheme to mitigate mai , and is widely used in various commercial systems . in cdma ,an information symbol of a user is modulated with a spreading sequence assigned to the user .receiver has to estimate information symbols based on received sequences by utilizing knowledge of spreading sequences of the users .a conventional choice to mitigate mai is to use pseudorandom sequences as the spreading sequences .although analysis of such randomly - spread cdma systems was thought to be difficult , it has turned out that _ replica method _ , which is an analytical tool developed in the research field of statistical physics of disordered systems ( spin glasses ) , is very useful for the analysis  .more precisely , these studies have revealed that replica method allows us to evaluate `` macroscopic '' performance of cdma systems with independent and identically - distributed ( i.i.d . )random spreading sequences in the large - system limit , such as mutual information between inputs and outputs , bit error rate , and so on .  since theoretical performance of cdma systems is affected by choices of spreading sequences , design of spreading sequences is an important problem in cdma .there have been several researches in which the problem of designing spreading sequences is formulated in terms of an optimization problem . for example , rupf and massey  discussed optimization of spreading sequences with the channel capacity of cdma systems as a measure of optimization .they showed that so - called welch bound equality ( wbe ) spreading sequences , which minimize the total squared correlation ( tsc ) of spreading sequences , achieve the channel capacity .we restrict ourselves to considering the optimization problem of spreading sequences of cdma systems with the mutual information between inputs and outputs as the measure of optimization .there have been several researches in which such optimization problems are discussed  .they have assumed inputs of the system to be gaussian distributed and discussed theoretical upper bound ( i.e. , channel capacity ) of mutual information between inputs and outputs . if one wishes to consider realistic wireless communication systems , however , it is important to study the optimization problem under the assumption of non - gaussian inputs .the objective of this paper is therefore to discuss the optimization problem of spreading sequences of cdma systems when one allows non - gaussian inputs .we would like to emphasize that , unlike previous statistical - mechanics studies of cdma systems  whose objectives are basically to analyze macroscopic system performance , we show in this paper that the statistical - mechanics approach is also useful in dealing with design problems in wireless communication , with the optimization problem of spreading sequences of cdma systems as a demonstrative example .a digest version of this paper has been presented as a conference paper  .we consider the following real - valued -user cdma channel model , where is an information symbol of user .we assume that are i.i.d .random variables , and let be the prior probability of , whose mean and variance are assumed to be zero and one , respectively . is the spreading sequence of user in the symbol interval of interest , and denotes the spreading factor of the cdma channel model .we assume that the power of the spreading sequences is normalized to one , so that holds for .we assume additive white gaussian noise ( awgn ) : so that is the variance of awgn .let us introduce the following notations : ^t ] , ^t ] , in the large - system limit . under the assumption of random spreading , we consider the average conditional mutual information in the large - system limit , where denotes expectation with respect to , and where is conditional mutual information between and given . we discuss maximization of with respect to characteristics of the random matrix .the average mutual information  is decomposed into two terms , \\  & = \mathcal{f } - \frac{1}{2 \beta } \left ( 1 + \log ( 2\pi \sigma^2)\right ) , \label{eq : capa2}\end{aligned}\ ] ] with where denotes expectation with respect to and .direct calculation of the right - hand side of  is in general computationally intractable . in order to evaluate ,we invoke the replica method .substituting the identity to the right - hand side of , we obtain we assume that the limit with respect to and the limit and the differentiation with respect to are interchangeable without altering the final result , obtaining the limit allows us to apply the saddle - point method to evaluate a relevant quantity .we apply replica trick in order to evaluate , in which we first evaluate assuming that is a non - negative integer , and then perform the limit and the differentiation with respect to , assuming that is real .evaluation of  basically goes in a similar manner as  .detailed analysis is described in the appendix . here, we only show the result .the average mutual information in the large - system limit is given by where are parameters whose values are to be determined later , and where is a probability density function of output of a scalar awgn channel with the noise variance , when the channel input is generated from the distribution .the function is defined as where is the r - transform  of the limiting eigenvalue distribution of the correlation matrix , which is defined on the basis of the hilbert transform in the upper half of complex plane .] of , as the parameters are to be determined from the following saddle - point equations : where denotes posterior mean estimate of the channel input of the scalar awgn channel introduced above , defined as and where in  denotes expectation with respect to the channel input and output of the scalar awgn channel .( left figure ) and (right figure ) with .both distributions have probability weight at .,title="fig:",scaledwidth=45.0% ] ( left figure ) and (right figure ) with .both distributions have probability weight at .,title="fig:",scaledwidth=45.0% ]  the average mutual information  depends on the limiting eigenvalue distribution of as well as the prior distribution .when we assume that are i.i.d .random variables whose mean and variance are zero and one , respectively , our result is reduced to that obtained by guo and verd  . in this case , the limiting eigenvalue distribution is given by the so - called marcenko - pastur law  , where , and , ( see figure [ fig : eig ] ) , whose r - transform is given by substituting  to , one can confirm the above - mentioned fact .our analysis also includes the case of the system with wbe spreading sequences .since the characteristic of wbe spreading sequences is expressed as , the correlation matrix of wbe spreading sequences has trivial zero eigenvalue and the eigenvalue , with multiplicities and , respectively ( see figure [ fig : eig ] ) , and therefore whose r - transform is given by one can evaluate , via , the mutual information when orthogonally - invariant random wbe spreading sequences are employed . in figure[ fig : random - wbe ] , we show a comparison of the mutual information when the above two spreading sequences are employed , and when probability distribution of is given by , . one can confirm that wbe spreading sequences achieve higher mutual information than i.i.d .random spreading sequences do .   when ( wbe ) and ( random ) are specified , in case of .]   choices of spreading sequences affect the average mutual information through the limiting eigenvalue distribution of the correlation matrix .then , we regard the average mutual information as a functional of , and seek the eigenvalue distribution which maximizes the average mutual information . hereafter, we consider the case of since optimal spreading sequences in the case of are obviously orthogonal spreading sequences . in optimizing with respect to , the following two constraints should be imposed on : first , since , the matrix has trivial zero eigenvalues with multiplicity .second , since we have normalized the power of spreading sequences as , the matrix should satisfy where are the eigenvalues of matrix . in terms of , the constraintis expressed as we rewrite in view of these constraints as where satisfies as the normalization as a probability distribution , and which corresponds to the normalization of the power of spreading sequences .  in order to discuss the extremum of with respect to ,we consider first - order perturbations of .since the parameters affected by the perturbation of are , the functional derivative of with respect to is expressed as since should satisfy the saddle - point equations  and , the derivatives of with respect to the parameters should be zero at the saddle point . therefore , one can safely ignore the effects of perturbations via and .our next observation is that , if one can find an eigenvalue distribution which maximizes , which is the only term having the first - order effect in , it also maximizes the average mutual information .we rewrite as since and , one can make the following statement : if there is a distribution whose r - transform satisfies for any r - transform of the distribution which satisfies the constraints , also maximizes . we summarize the above arguments in the next proposition .if one can find an eigenvalue distribution which maximizes r - transform for , also maximizes the average mutual information .it should be noted that the existence of is not guaranteed at this stage .however , in the following , we show that there is a distribution which satisfies the condition of proposition 1 .  as a next step ,we convert the optimization problem in terms of r - transform into the one in terms of hilbert transform . since hilbert transform is a monotonically decreasing function of , andsince hilbert transform has the relation  with r - transform , it follows that also decreases monotonically with respect to .this fact leads us to the following statement : if one can find an eigenvalue distribution which maximizes for , and which satisfies , also maximizes for , with summarizing the arguments so far , we can state the next proposition .if one can find an eigenvalue distribution whose hilbert transform satisfies for any eigenvalue distribution with hilbert transform , which satisfies  , also maximizes the average mutual information .  following the above proposition, we consider the maximization problem of the hilbert transform for .substituting  to , the hilbert transform is rewritten as since the first term , which is derived from the trivial zero eigenvalues , has no room for optimization , we maximize the second term under the constraints  and .let us consider the following integral : where is a linear function tangential to at . since the function is a linear function , the expectation of with respect to is constant under the constraints  and .thus , the quantities  and  are maximized by the same eigenvalue distribution .we here consider a maximization problem of the objective function  for with the constraint , but _ without _ the constraint .since we have only to consider  ] , which are drawn from the same probability distribution as , we rewrite as performing the integral with respect to , we obtain \bigg\}.\end{gathered}\ ] ] where matrix is given by the expectation with respect to can be performed via the so - called itzykson - zuber integral  ( see also  ) , since is assumed orthogonally invariant and rank of is at most , as \right\ } = \frac{1}{2}\mathrm{tr}g(v),\ ] ] where is the function defined in .thus , we obtain the following equation , ignoring vanishing terms in the large - system limit , \right\}. \label{eq : moment}\end{gathered}\ ] ]  we next take expectation of  with respect to .since eigenvalues of the matrix are functions of only through their inner products , we rewrite the expectation with respect to into the one with respect to the matrix as \mu_k(q)\mathrm{d}q , \label{eq : pre - saddle}\ ] ] where is the exponent of , (q)-\frac{1}{2\beta}\log(n+1)-\frac{n}{2\beta}\log(2\pi\sigma^2),\ ] ] and where is the following measure utilizing the saddle - point method  , we evaluate  in the limit as where is the rate function of the empirical means , defined via a legendre transform as ,\ ] ] where is an symmetric matrix . the cumulant generating function of defined as \right\}.\ ] ]  in order to proceed further , we assume the so - called _ replica symmetry _ :we assume that the extremum of  is invariant under exchanges of the replica indexes . under the assumption of replica symmetry ,we introduce new parameters , using these parameters , the eigenvalues of are expressed as  whose multiplicities are and , respectively . since is an analytic function and , we can express as we also apply the assumption of replica symmetry to as and rewrite the rate function as \bigg\ } \bigg ] .\label{eq : rate - result}\end{gathered}\ ] ] deriving the extremum condition of  with respect to and in the limit , we obtain the following saddle - point equations ,  similarly , as the extremum condition of , we obtain the following saddle - point equation , where follows the probability distribution , and where is the posterior mean estimate of channel input given output in a single - user awgn channel whose variance is and whose input and output are and , respectively . introducing the posterior variance , we obtain the saddle - point equations  and . in order to obtain , we differentiate , which is obtained by substituting  and  to , with respect to , and then take the limit .finally , we obtain the representation of as substituting  to , we obtain the average mutual information .support from the grant - in - aid for scientific research on priority areas , the ministry of education , culture , sports , science and technology , japan ( no .18079010 ) is acknowledged .j. l. massey , t. mittelholzer , welch s bound and sequence sets for code - division multiple - access systems , in : r. capocelli , a. d. santis , u. vaccaro ( eds . ) , sequences ii , methods in communication , security , and computer science , springer - verlag , 1993 , pp . 5378 .p. viswanath , v. anantharam , d. n. c. tse , optimal sequences , power control , and user capacity of synchronous cdma systems with linear mmse multiuser receivers , ieee trans .theory 45 ( 1999 ) 19681983 .

Abstract: statistical mechanics approach is useful not only in analyzing macroscopic system performance of wireless communication systems , but also in discussing design problems of wireless communication systems . in this paper , we discuss a design problem of spreading sequences in code - division multiple - access ( cdma ) systems , as an example demonstrating the usefulness of statistical mechanics approach . we analyze , via replica method , the average mutual information between inputs and outputs of a randomly - spread cdma channel , and discuss the optimization problem with the average mutual information as a measure of optimization . it has been shown that the average mutual information is maximized by orthogonally - invariant random welch bound equality ( wbe ) spreading sequences .   code - division multiple - access ( cdma ) , replica method , average mutual information , large system limit