This paper introduces a novel class of fair and interpolatory curves called $p\kappa$-curves. These curves are comprised of smoothly stitched B\'ezier curve segments, where the curvature distribution of each segment is made to closely resemble a parabola, resulting in an aesthetically pleasing shape. Moreover, each segment passes through an interpolated point at a parameter where the parabola has an extremum, encouraging the alignment of interpolated points with curvature extrema. To achieve these properties, we tailor an energy function that guides the optimization process to obtain the desired curve characteristics. Additionally, we develop an efficient algorithm and an initialization method, enabling interactive modeling of the $p\kappa$-curves without the need for global optimization. We provide various examples and comparisons with existing state-of-the-art methods to demonstrate the curve modeling capabilities and visually pleasing appearance of $p\kappa$-curves.
相關內容
Ordinary differential equations (ODEs) can provide mechanistic models of temporally local changes of processes, where parameters are often informed by external knowledge. While ODEs are popular in systems modeling, they are less established for statistical modeling of longitudinal cohort data, e.g., in a clinical setting. Yet, modeling of local changes could also be attractive for assessing the trajectory of an individual in a cohort in the immediate future given its current status, where ODE parameters could be informed by further characteristics of the individual. However, several hurdles so far limit such use of ODEs, as compared to regression-based function fitting approaches. The potentially higher level of noise in cohort data might be detrimental to ODEs, as the shape of the ODE solution heavily depends on the initial value. In addition, larger numbers of variables multiply such problems and might be difficult to handle for ODEs. To address this, we propose to use each observation in the course of time as the initial value to obtain multiple local ODE solutions and build a combined estimator of the underlying dynamics. Neural networks are used for obtaining a low-dimensional latent space for dynamic modeling from a potentially large number of variables, and for obtaining patient-specific ODE parameters from baseline variables. Simultaneous identification of dynamic models and of a latent space is enabled by recently developed differentiable programming techniques. We illustrate the proposed approach in an application with spinal muscular atrophy patients and a corresponding simulation study. In particular, modeling of local changes in health status at any point in time is contrasted to the interpretation of functions obtained from a global regression. This more generally highlights how different application settings might demand different modeling strategies.
Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality.
The purpose of this paper is to develop the anti-Gauss cubature rule for approximating integrals defined on the square whose integrand function may have algebraic singularities at the boundaries. An application of such a rule to the numerical solution of second-kind Fredholm integral equations is also explored. The stability, convergence, and conditioning of the proposed Nystr\"om-type method are studied. The numerical solution of the resulting dense linear system is also investigated and several numerical tests are presented.
This paper introduces a novel approach to approximate a broad range of reaction-convection-diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves accuracy of $O(h^k)$ in the energy norm, where $k$ represents the underlying polynomial degree. To validate the approach, a series of numerical experiments is conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favourable performance of the current approach.
This paper develops a two-stage stochastic model to investigate evolution of random fields on the unit sphere $\bS^2$ in $\R^3$. The model is defined by a time-fractional stochastic diffusion equation on $\bS^2$ governed by a diffusion operator with the time-fractional derivative defined in the Riemann-Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on $\bS^2$ as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on $\bS^2$. The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree $L\geq1$. The rate of convergence of the truncation errors as a function of $L$ and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic H\"{o}lder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.
Linear principal component analysis (PCA), nonlinear PCA, and linear independent component analysis (ICA) -- those are three methods with single-layer autoencoder formulations for learning linear transformations from data. Linear PCA learns orthogonal transformations (rotations) that orient axes to maximise variance, but it suffers from a subspace rotational indeterminacy: it fails to find a unique rotation for axes that share the same variance. Both nonlinear PCA and linear ICA reduce the subspace indeterminacy from rotational to permutational by maximising statistical independence under the assumption of unit variance. The relationship between all three can be understood by the singular value decomposition of the linear ICA transformation into a sequence of rotation, scale, rotation. Linear PCA learns the first rotation; nonlinear PCA learns the second. The scale is simply the inverse of the standard deviations. The problem is that, in contrast to linear PCA, conventional nonlinear PCA cannot be used directly on the data to learn the first rotation, the first being special as it reduces dimensionality and orders by variances. In this paper, we have identified the cause, and as a solution we propose $\sigma$-PCA: a unified neural model for linear and nonlinear PCA as single-layer autoencoders. One of its key ingredients: modelling not just the rotation but also the scale -- the variances. This model bridges the disparity between linear and nonlinear PCA. And so, like linear PCA, it can learn a semi-orthogonal transformation that reduces dimensionality and orders by variances, but, unlike linear PCA, it does not suffer from rotational indeterminacy.
We propose a new randomized method for solving systems of nonlinear equations, which can find sparse solutions or solutions under certain simple constraints. The scheme only takes gradients of component functions and uses Bregman projections onto the solution space of a Newton equation. In the special case of euclidean projections, the method is known as nonlinear Kaczmarz method. Furthermore, if the component functions are nonnegative, we are in the setting of optimization under the interpolation assumption and the method reduces to SGD with the recently proposed stochastic Polyak step size. For general Bregman projections, our method is a stochastic mirror descent with a novel adaptive step size. We prove that in the convex setting each iteration of our method results in a smaller Bregman distance to exact solutions as compared to the standard Polyak step. Our generalization to Bregman projections comes with the price that a convex one-dimensional optimization problem needs to be solved in each iteration. This can typically be done with globalized Newton iterations. Convergence is proved in two classical settings of nonlinearity: for convex nonnegative functions and locally for functions which fulfill the tangential cone condition. Finally, we show examples in which the proposed method outperforms similar methods with the same memory requirements.
We introduce numerical solvers for the steady-state Boltzmann equation based on the symmetric Gauss-Seidel (SGS) method. Due to the quadratic collision operator in the Boltzmann equation, the SGS method requires solving a nonlinear system on each grid cell, and we consider two methods, namely Newton's method and the fixed-point iteration, in our numerical tests. For small Knudsen numbers, our method has an efficiency between the classical source iteration and the modern generalized synthetic iterative scheme, and the complexity of its implementation is closer to the source iteration. A variety of numerical tests are carried out to demonstrate its performance, and it is concluded that the proposed method is suitable for applications with moderate to large Knudsen numbers.
The main reason for query model's prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values can't be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. Finally, we study how large certificate complexity and block sensitivity can be as compared to sensitivity for symmetric functions (even up to constant factors). We show tight separations, i.e., give upper bounds on possible separations and construct functions achieving the same.
The classical Zarankiewicz's problem asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{t,t}$. In one of the cornerstones of extremal graph theory, K\H{o}v\'ari S\'os and Tur\'an proved an upper bound of $O(n^{2-\frac{1}{t}})$. In a celebrated result, Fox et al. obtained an improved bound of $O(n^{2-\frac{1}{d}})$ for graphs of VC-dimension $d$ (where $d<t$). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. At SODA'23, Chan and Har-Peled further improved Basit et al.'s bounds and presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of $O(n \log \log n)$ for the incidence graph of points and pseudo-discs in the plane. In this paper we present a new approach to Zarankiewicz's problem, via $\epsilon$-t-nets - a recently introduced generalization of the classical notion of $\epsilon$-nets. We show that the existence of `small'-sized $\epsilon$-t-nets implies upper bounds for Zarankiewicz's problem. Using the new approach, we obtain a sharp bound of $O(n)$ for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain a short proof of the $O(n^{2-\frac{1}{d}})$ bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp $O(n\frac{\log n}{\log \log n})$ bound for the intersection graph of two families of axis-parallel rectangles.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191