Abstract: An efficient method for optimizing single-determinant wave functions of medium and large systems is presented. It is based on a minimization of the energy functional using a new set of variables to perform orbital transformations. With this method convergence of the wave function is guaranteed. Preconditioners with different computational cost and efficiency have been constructed. Depending on the preconditioner, the method needs a number of iterations that is very similar to the established diagonalization–DIIS approach, in cases where the latter converges well. Diagonalization of the Kohn–Sham matrix can be avoided and the sparsity of the overlap and Kohn–Sham matrix can be exploited. If sparsity is taken into account, the method scales as O(MN2), where M is the total number of basis functions and N is the number of occupied orbitals. The relative performance of the method is optimal for large systems that are described with high quality basis sets, and for which the density matrices are not yet sparse....

Abstract: Let f : Rs → R be a real-valued function, and let x = (x1,...,xs)T ∈ Rs be partitioned into t subsets of non-overlapping variables as x = (X1,...,Xt)T, with Xi ∈ Rpi for i = 1,...,t, Σi=1tpi = s. Alternating optimization (AO) is an iterative procedure for minimizing f(x) = f(X1, X2,..., Xt) jointly over all variables by alternating restricted minimizations over the individual subsets of variables X1,...., Xt. Alternating optimization has been (more or less) studied and used in a wide variety of areas. Here a self-contained and general convergence theory is presented that is applicable to all partitionings of x. Under reasonable assumptions, the general AO approach is shown to be locally, q-linearly convergent, and to also exhibit a type of global convergence.

Abstract: A side effect occurs when a procedure or function changes the value of a global variable. This is one reason why the use of global variables is deplored in programming; they allow the possibility of side effects, which are usually unplanned and, therefore, undesirable. But this is not always so; sometimes, as in database systems, the database itself is global to all procedures, and modifying it is just what many of the procedures in the system are supposed to do. Another example would be a procedure to generate random numbers where the ith random number, Xi, is generated as f(Xi-1) for some function f. A global variable X would contain the value of Xi-1 and a side effect would replace this with Xi. A third example of benign side effects is in functions, such as input and output functions, that both produce results and return a value giving information about them. For example, the scanf function in the CI/O library reads values into its arguments (a side effect) and returns a count of the number of items read.

Abstract: We formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution. This allows us to define a good notion of integral with respect to irregular paths with Hoelder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons' theory of rough paths in Hoelder topology.

Abstract: We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u t −Δu+ɛ−2 f(u)=0 arising from phase transition in materials science, where ɛ is a small parameter known as an ``interaction length''. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ɛ. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u 0 . In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow.

Abstract: A computer-implemented method generates a computer model of one or more teeth by receiving as input a digital data set of meshes representing the teeth; selecting a curved coordinate system with mappings to and from a 3D space; and generating a function in the curved coordinate system to represent each tooth.

Abstract: f ,w hereP is the transition kernel of the Markov chain and α ∈ C is a constant. The function ˇ f is an eigenfunction, with corresponding eigenvalue λ, for the kernel (e αF P) = e αF (x) P( x, dy). A “multiplicative” mean ergodic theorem. For all complex α in a neighborhood of the origin, the normalized mean of exp(αSt ) (and not the logarithm of the mean) converges to ˇ f exponentially fast, where ˇ f is a solution of the multiplicative Poisson equation. Edgeworth expansions. Rates are obtained for the convergence of the distribution function of the normalized partial sums St to the standard Gaussian distribution. The first term in this expansion is of order (1/ √ t) and it depends on the initial condition of the Markov chain through the solution � F of the associated Poisson equation (and not the solution ˇ f of the multiplicative Poisson equation). Large deviations. The partial sums are shown to satisfy a large deviations principle in a neighborhood of the mean. This result, proved under geometric ergodicity alone, cannot in general be extended to the whole real line. Exact large deviations asymptotics. Rates of convergence are obtained for the large deviations estimates above. The polynomial preexponent is of order (1/ √ t) and its coefficient depends on the initial condition of the Markov chain through the solution ˇ f of the multiplicative Poisson equation. Extensions of these results to continuous-time Markov processes are also given.

Abstract: We explore the learnability of two-valued functions from samples using the paradigm of Data Compression. A first algorithm (compression) choses a small subset of the sample which is called the kernel. A second algorithm predicts future values of the function from the kernel, i.e. the algorithm acts as an hypothesis for the function to be learned. The second algorithm must be able to reconstruct the correct function values when given a point of the original sample. We demonstrate that the existence of a suitable data compression scheme is sufficient to ensure learnability. We express the probability that the hypothesis predicts the function correctly on a random sample point as a function of the sample and kernel sizes. No assumptions are made on the probability distributions according to which the sample points are generated. This approach provides an alternative to that of [BEHW86], which uses the Vapnik-Chervonenkis dimension to classify learnable geometric concepts. Our bounds are derived directly from the kernel size of the algorithms rather than from the Vapnik-Chervonenkis dimension of the hypothesis class. The proofs are simpler and the introduced compression scheme provides a rigorous model for studying data compression in connection with machine learning.

Abstract: In Part I of this article we propose a general cross-validation criterian for selecting among a collection of estimators of a particular parameter of interest based on n i.i.d. observations. It is assumed that the parameter of interest minimizes the expectation (w.r.t. to the distribution of the observed data structure) of a particular loss function of a candidate parameter value and the observed data structure, possibly indexed by a nuisance parameter. The proposed cross-validation criterian is defined as the empirical mean over the validation sample of the loss function at the parameter estimate based on the training sample, averaged over random splits of the observed sample. The cross-validation selector is now the estimator which minimizes this cross-validation criterion. We illustrate that this general methodology covers, in particular, the selection problems in the current literature, but results in a wide range of new selection methods. We prove a finite sample oracle inequality, and asymptotic optimality of the cross-validated selector under general conditions. The asymptotic optimality states that the cross-validation selector performs asymptotically exactly as well as the selector which for each given data set makes the best choice (knowing the true data generating distribution). Our general framework allows, in particular, the situation in which the observed data structure is a censored version of the full data structure of interest, and where the parameter of interest is a parameter of the full data structure distribution. As examples of the parameter of the full data distribution we consider a density of (a part of) the full data structure, a conditional expectation of an outcome, given explanatory variables, a marginal survival function of a failure time, and multivariate conditional expectation of an outcome vector, given covariates. In part II of this article we show that the general estimating function methodology for censored data structures as provided in van der Laan, Robins (2002) yields the wished loss functions for the selection among estimators of a full-data distribution parameter of interest based on censored data. The corresponding cross-validation selector generalizes any of the existing selection methods in regression and density estimation (including model selection) to the censored data case. Under general conditions, our optimality results now show that the corresponing cross-validation selector performs asymptotically exactly as well as the selector which for each given data set makes the best choice (knowing the true full data distribution). In Part III of this article we propose a general estimator which is defined as follows. For a collection of subspaces and the complete parameter space, one defines an epsilon-net (i.e., a finite set of points whose epsilon-spheres cover the complete parameter space). For each epsilon and subspace one defines now a corresponding minimum cross-valided empirical risk estimator as the minimizer of cross-validated risk over the subspace-specific epsilon-net. In the special case that the loss function has no nuisance parameter, which thus covers the classical regression and density estimation cases, this epsilon and subspace specific minimum risk estimator reduces to the minimizer of the empirical risk over the corresponding epsilon-net. Finally, one selects epsilon and the subspace with the cross-validation selector. We refer to the resulting estimator as the cross-validated adaptive epsilon-net estimator. We prove an oracle inequality for this estimator which implies that the estimator minimax adaptive in the sense that it achieves the minimax optimal rate of convergence for the smallest of the guessed subspaces containing the true parameter value. Cross-Validation for Estimator Selection 1 Stating the Selection Problem. Let O1, . . . , On be n i.i.d. observations of O ∼ P0, where P0 is known to be an element of a statistical model M. Let ψ0(·) = ψ(· | P0) be a parameter (function) of P0 of interest. Let the parameter set for this parameter be Ψ = {ψ(· | P ) : P ∈ M}. Let (O,ψ) → L(O,ψ | η0) ∈ IR be a “loss function”, possibly depending on a nuisance parameter η0 = η(P0), which maps a candidate parameter value ψ and observation O into a real number, whose expectation is minimized at ψ0: ψ0 = argminψ∈Ψ ∫ L(o, ψ | η0)dP0(o) (1) = argminψ∈ΨE0L(O,ψ | η0). Let Pn be the empirical distribution of O1, . . . , On. Let ψk(·) = ψk(· | Pn) ∈ Ψ, k = 1, . . . , K(n), be a collection of estimators (i.e., algorithms one can apply to data) of ψ0(·). The choice of loss function. Different choices of loss functions can satisfy (1). In fact, (1) can define a class of possible loss functions. Different choices of loss functions result in estimators of ψ0 with different behavior. Consequently, the choice of loss function is an interesting issue to be addressed. We suggest the following reasonable strategy for selecting a loss function. Firstly, among the loss functions identifying ψ0 as the minimizer of its risk (i.e., satisfying (1), one wishes to choose a loss function which identifies the wished measure of performance/Risk θ(ψ | P0) ≡ ∫ L(O,ψ | η0)dP0(O) for a candidate ψ ∈ Ψ. Identifying such a function θ(ψ | P0) on the parameter set Ψ does still not uniquely identify the loss function L(O,ψ | η0). Secondly, given this function θ(ψ | P0), we now wish to choose the loss function so that for a locally consistent estimator ηn of η0, 1/n ∑ i L(Oi, ψ | ηn) is a locally efficient estimator of θ(ψ | P0). That is, let L(O,ψ | η0) be a parametrization of

Abstract: It is shown that a function u satisfying |∂t+Δu|≦M(|u|+|∇u|), |u(x, t)|≦MeM|x|2 in (ℝn \ (BR) × [0, T] and u(x, 0) = 0 for xℝn \ BR must vanish identically in ℝn \ BR×[0, T].

Abstract: In 1986, Fiat and Shamir proposed a general method for transforming secure 3-round public-coin identiflcation schemes into digital signature schemes. The idea of the transformation was to replace the random message of the verifler in the identiflcation scheme, with the value of some deterministic\hash" function evaluated on various quantities in the protocol and on the message to be signed. The Fiat-Shamir methodology for producing digital signature schemes quickly gained popularity as it yields e‐cient and easy to implement digital signature schemes. The most important question however remained open: are the digital signatures produced by the Fiat-Shamir methodology secure? In this paper, we answer this question negatively. We show that there exist secure 3round public-coin identiflcation schemes for which the Fiat-Shamir transformation yields insecure digital signature schemes for any \hash" function used by the transformation. This is in contrast to the work of Pointcheval and Stern which proved that the FiatShamir methodology always produces digital signatures secure against chosen message attack in the \Random Oracle Model" { when the hash function is modelled by a random oracle. Among other things, we make new usage of Barak’s technique for taking advantage of non black-box access to a program, this time in the context of digital signatures.

Abstract: An improved apparatus and method for creating high quality virtual reality panoramas is disclosed that yields dramatic improvements during the authoring and projecting cycles, with speeds up to several orders of magnitude faster than prior systems. In a preferred embodiment, a series of rectilinear images taken from a plurality of rows are pairwise registered with one another, and locally optimized using a pairwise objective function (local error function) that minimizes certain parameters in a projective transformation, using an improved iterative procedure. The local error function values for the pairwise registrations are then saved and used to construct a quadratic surface to approximate a global optimization function (global error function). The chain rule is used to avoid the direct evaluation of the global objective function, saving computation. In one embodiment concerning the blending aspect of the present invention, an improved procedure is described that relies on Laplacian and Gaussian pyramids, using a blend mask whose boundaries are determined by the grassfire transform. An improved iterative procedure is disclosed for the blending that also determines at what level of the pyramid to perform blending, and results in low frequency image components being blended over a wider region and high frequency components being blended over a narrower region. Human interaction and input is also provided to allow manual projective registration, initial calibration and feedback in the selection of photos and convergence of the system.

Abstract: The problem of estimating the state of a discrete-time linear system can be addressed by minimizing an estimation cost function dependent on a batch of recent measure and input vectors. This problem has been solved by introducing a receding-horizon objective function that includes also a weighted penalty term related to the prediction of the state. For such an estimator, convergence results and unbiasedness properties have been proved. The issues concerning the design of this filter are discussed in terms of the choice of the free parameters in the cost function. The performance of the proposed receding-horizon filter is evaluated and compared with other techniques by means of a numerical example.

Abstract: In this paper we present an approach to approximate reachability computation for nonlinear continuous systems. Rather than studying a complex nonlinear system x = g(x), we study an approximating system x = f(x) which is easier to handle. The class of approximating systems we consider in this paper is piecewise linear, obtained by interpolating g over a mesh. In order to be conservative, we add a bounded input in the approximating system to account for the interpolation error. We thus develop a reachability method for systems with input, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets. This method is then extended to the approximate piecewise linear systems arising in our construction. The final result is a reachability algorithm for nonlinear continuous systems which allows to compute conservative approximations with as great degree of accuracy as desired, and more importantly, it has good convergence rate. If g is a C2 function, our method is of order 2. Furthermore, the method can be straightforwardly extended to hybrid systems.

Abstract: We have established and implemented a fully ab initio method which allows one to calculate optical absorption spectra, including excitonic effects, without solving the cumbersome Bethe-Salpeter equation, but obtaining results of the same precision. This breakthrough has been achieved in the framework of time-dependent density-functional theory, using new exchange-correlation kernels f x e that are free of any empirical parameter. We show that the same excitonic effects in the optical spectra can be reproduced through different f x c 's ranging from frequency-dependent ones to a static one, by varying the kernel's spatial degrees of freedom. This indicates that the key quantity is not f x c , but f x c . combined with a response function. We present results for the optical absorption of bulk Si and SiC in good agreement with experiment, almost indistinguishable from those of the Bethe-Salpeter approach.

Abstract: This paper studies the problem of characterizing the simplest aperiodic discrete point sets, using invariants based on topological dynamics A Delone set of finite type is a Delone set X such that X - X is locally finite Such sets are characterized by their patch-counting function N(T) of radius T being finite for all T We formulate conjectures relating slow growth of the patch-counting function N(T) to the set X having a nontrivial translation symmetry A Delone set X of finite type is repetitive if there is a function M(T) such that every closed ball of radius M(T) + T contains a complete copy of each kind of patch of radius T that occurs in X This is equivalent to the minimality of an associated topological dynamical system with ℝ-action There is a lower bound for M(T) in terms of N(T), namely M(T) ≥ c(N(T) for some positive constant c depending on the Delone set constants r, R, but there is no general upper bound for M(T) purely in terms of N(T) The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function M(T) For example, the function M(T) is bounded if and only if X is a periodic crystal A set X is linearly repetitive if M(T) = O(T) as T → ∞ and is densely repetitive if M(T) = O(N(T)) as T → ∞ We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, ie the associated topological dynamical system is strictly ergodic It follows that such sets are diffractive, in the sense of having a well-defined diffraction measure In the reverse direction, we construct a repetitive Delone set X in ℝ which has M(T) = O (T (log T) (log log log T)), but does not have uniform patch frequencies Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets and we propose considering them as a notion of 'perfectly ordered quasicrystals'

Abstract: A novel solution to the inter-camera color calibration problem, which is very important for multicamera systems is presented. We propose a distance metric and a model function to evaluate the inter-camera radiometric properties. Instead of depending on the shape assumptions of brightness transfer function to find separate radiometric responses, we derive a nonparametric function to model color distortion for pair-wise camera combinations. Our method is based on correlation matrix analysis and dynamic programming. The correlation matrix is computed from three 1-D color histograms, and the model function is obtained from a minimum cost path traced within the matrix. The model function enables accurate compensation of color mismatches, which cannot be done with conventional distance metrics. Furthermore, we show that our metric can be reduced to other commonly used metrics with suitable simplification. Our simulations prove the effectiveness of the proposed method even for severe color distortions.

Abstract: A heuristic method to construct uniform approximations to analytic transcendental functions is developed as a generalization of the Hermite-Pade interpolation to infinite intervals. The resulting uniform approximants are built from elementary functions using known series and asymptotic expansions of the given transcendental function. In one case (Lambert's W function) we obtained a uniform approximation valid in the entire complex plane. Several examples of the application of this method to selected transcendental functions are given.

Abstract: This paper presents a new type of nonlinear function for independent component analysis to process complex-valued signals, which is used in frequency-domain blind source separation. The new function is based on the polar coordinates of a complex number, whereas the conventional one is based on the Cartesian coordinates. The new function is derived from the probability density function of frequency-domain signals that are assumed to be independent of the phase. We show that the difference between the two types of functions is in the assumed densities of independent components. Experimental results for separating speech signals show that the new nonlinear function behaves better than the conventional one.

Abstract: This paper describes a novel, fast templatematching technique, referred to as bounded partial correlation (BPC), based on the normalised cross-correlation (NCC) function. The technique consists in checking at each search position a suitable elimination condition relying on the evaluation of an upper-bound for the NCC function. The check allows for rapidly skipping the positions that cannot provide a better degree of match with respect to the current best-matching one. The upper-bounding function incorporates partial information from the actual cross-correlation function and can be calculated very efficiently using a recursive scheme. We show also a simple improvement to the basic BPC formulation that provides additional computational benefits and renders the technique more robust with respect to the parameters choice.

Abstract: In this paper we analyze a widely employed test function for global optimization, the Griewank function. While this function has an exponentially increasing number of local minima as its dimension increases, it turns out that a simple Multistart algorithm is able to detect its global minimum more and more easily as the dimension increases. A justification of this counterintuitive behavior is given. Some modifications of the Griewank function are also proposed in order to make it challenging also for large dimensions.

Abstract: A method for statistically reconstructing images from a plurality of transmission measurements having energy diversity and image reconstructor apparatus utilizing the method are provided. A statistical (maximum-likelihood) method for dual-energy X-ray CT accommodates a wide variety of potential system configurations and measurement noise models. Regularized methods (such as penalized-likelihood or Bayesian estimations) are straightforward extensions. One version of the algorithm monotonically decreases the negative log-likelihood cost function each iteration. An ordered-subsets variation of the algorithm provides a fast and practical version. The method and apparatus provide material characterization and quantitatively accurate CT values in a variety of applications. The method and apparatus provide improved noise/dose properties.

Abstract: We investigate the geometry of that function in the plane or 3-space, which associates to each point the square of the shortest distance to a given curve or surface. Particular emphasis is put on second order Taylor approximants and other local quadratic approximants. Their key role in a variety of geometric optimization algorithms is illustrated at hand of registration in Computer Vision and surface approximation.

Abstract: A function f: R → R is continuous at a point u if, given a sequence x = (x n ), lim x = u implies that lim f(x) = f(u). This definition can be modified by replacing lim with an arbitrary linear functional G. Generalizing several definitions that have appeared in the literature, we say that f: R → R is G-continuous at u if G(x) = u implies that G(f(x)) = f(u). When G(x) = lim n n -1 Σ n k=1 x k , Buck showed that if a function f is G-continuous at a single point then f is linear, that is, f(u) = au+b for fixed a and b. Other authors have replaced convergence in arithmetic mean with A-summability, almost convergence and statistical convergence, The results in this paper include a sufficient condition for G-continuity to imply linearity and a necessary condition for continuous functions to be G-continuous, thereby generalizing several known results in the literature. It is also shown that, in many situations, the G-continuous functions must be either precisely the linear functions or precisely the continuous functions. However, examples are found where this dichotomy fails, which, in particular, leads to a counterexample to a conjecture of Spigel and Krupnik.

Abstract: This paper deals with the approximation behaviour of soft computing techniques. First, we give a survey of the results of universal approximation theorems achieved so far in various soft computing areas, mainly in fuzzy control and neural networks. We point out that these techniques have common approximation behaviour in the sense that an arbitrary function of a certain set of functions (usually the set of continuous function, C) can be approximated with arbitrary accuracy ? on a compact domain. The drawback of these results is that one needs unbounded numbers of “building blocks” (i.e. fuzzy sets or hidden neurons) to achieve the prescribed ? accuracy. If the number of building blocks is restricted, it is proved for some fuzzy systems that the universal approximation property is lost, moreover, the set of controllers with bounded number of rules is nowhere dense in the set of continuous functions. Therefore it is reasonable to make a trade-off between accuracy and the number of the building blocks, by determining the functional relationship between them. We survey this topic by showing the results achieved so far, and its inherent limitations. We point out that approximation rates, or constructive proofs can only be given if some characteristic of smoothness is known about the approximated function.