In this paper we approximate high-dimensional functions $f\colon\mathbb T^d\to\mathbb C$ by sparse trigonometric polynomials based on function evaluations. Recently it was shown that a dimension-incremental sparse Fourier transform (SFT) approach does not require the signal to be exactly sparse and is applicable in this setting. We combine this approach with subsampling techniques for rank-1 lattices. This way our approach benefits from the underlying structure in the sampling points making fast Fourier algorithms applicable whilst achieving the good sampling complexity of random points (logarithmic oversampling). In our analysis we show detection guarantees of the frequencies corresponding to the Fourier coefficients of largest magnitude. In numerical experiments we make a comparison to full rank-1 lattices and uniformly random points to confirm our findings.
相關內容
Weighted low rank approximation is a fundamental problem in numerical linear algebra, and it has many applications in machine learning. Given a matrix $M \in \mathbb{R}^{n \times n}$, a weight matrix $W \in \mathbb{R}_{\geq 0}^{n \times n}$, a parameter $k$, the goal is to output two matrices $U, V \in \mathbb{R}^{n \times k}$ such that $\| W \circ (M - U V^\top) \|_F$ is minimized, where $\circ$ denotes the Hadamard product. Such a problem is known to be NP-hard and even hard to approximate assuming Exponential Time Hypothesis [GG11, RSW16]. Meanwhile, alternating minimization is a good heuristic solution for approximating weighted low rank approximation. The work [LLR16] shows that, under mild assumptions, alternating minimization does provide provable guarantees. In this work, we develop an efficient and robust framework for alternating minimization. For weighted low rank approximation, this improves the runtime of [LLR16] from $n^2 k^2$ to $n^2k$. At the heart of our work framework is a high-accuracy multiple response regression solver together with a robust analysis of alternating minimization.
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a $2$-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a simple rounding algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair $(G, S)$, consisting of a graph with an odd cycle transversal. If $S$ is a stable set, we prove a tight approximation ratio of $1 + 1/\rho$, where $2\rho -1$ denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph $\tilde{G} := G /S$ and satisfies $\rho \in [2,\infty]$. If $S$ is an arbitrary set, we prove a tight approximation ratio of $\left(1+1/\rho \right) (1 - \alpha) + 2 \alpha$, where $\alpha \in [0,1]$ is a natural parameter measuring the quality of the set $S$. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph $\tilde{G}$. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals and show optimality of the analysis.
A grid-overlay finite difference method is proposed for the numerical approximation of the fractional Laplacian on arbitrary bounded domains. The method uses an unstructured simplicial mesh and an overlay uniform grid for the underlying domain and constructs the approximation based on a uniform-grid finite difference approximation and a data transfer from the unstructured mesh to the uniform grid. The method takes full advantage of both uniform-grid finite difference approximation in efficient matrix-vector multiplication via the fast Fourier transform and unstructured meshes for complex geometries. It is shown that its stiffness matrix is similar to a symmetric and positive definite matrix and thus invertible if the data transfer has full column rank and positive column sums. Piecewise linear interpolation is studied as a special example for the data transfer. It is proved that the full column rank and positive column sums of linear interpolation is guaranteed if the spacing of the uniform grid is smaller than or equal to a positive bound proportional to the minimum element height of the unstructured mesh. Moreover, a sparse preconditioner is proposed for the iterative solution of the resulting linear system for the homogeneous Dirichlet problem of the fractional Laplacian. Numerical examples demonstrate that the new method has similar convergence behavior as existing finite difference and finite element methods and that the sparse preconditioning is effective. Furthermore, the new method can readily be incorporated with existing mesh adaptation strategies. Numerical results obtained by combining with the so-called MMPDE moving mesh method are also presented.
A growing number of central authorities use assignment mechanisms to allocate students to schools in a way that reflects student preferences and school priorities. However, most real-world mechanisms give students an incentive to be strategic and misreport their preferences. In this paper, we provide an identification approach for causal effects of school assignment on future outcomes that accounts for strategic misreporting. Misreporting may invalidate existing point-identification approaches, and we derive sharp bounds for causal effects that are robust to strategic behavior. Our approach applies to any mechanism as long as there exist placement scores and cutoffs that characterize that mechanism's allocation rule. We use data from a deferred acceptance mechanism that assigns students to more than 1,000 university-major combinations in Chile. Students behave strategically because the mechanism in Chile constrains the number of majors that students submit in their preferences to eight options. Our methodology takes that into account and partially identifies the effect of changes in school assignment on various graduation outcomes.
Sequences with low aperiodic autocorrelation are used in communications and remote sensing for synchronization and ranging. The autocorrelation demerit factor of a sequence is the sum of the squared magnitudes of its autocorrelation values at every nonzero shift when we normalize the sequence to have unit Euclidean length. The merit factor, introduced by Golay, is the reciprocal of the demerit factor. We consider the uniform probability measure on the $2^\ell$ binary sequences of length $\ell$ and investigate the distribution of the demerit factors of these sequences. Previous researchers have calculated the mean and variance of this distribution. We develop new combinatorial techniques to calculate the $p$th central moment of the demerit factor for binary sequences of length $\ell$. These techniques prove that for $p\geq 2$ and $\ell \geq 4$, all the central moments are strictly positive. For any given $p$, one may use the technique to obtain an exact formula for the $p$th central moment of the demerit factor as a function of the length $\ell$. The previously obtained formula for variance is confirmed by our technique with a short calculation, and we demonstrate that our techniques go beyond this by also deriving an exact formula for the skewness.
Gaussian graphical models are nowadays commonly applied to the comparison of groups sharing the same variables, by jointy learning their independence structures. We consider the case where there are exactly two dependent groups and the association structure is represented by a family of coloured Gaussian graphical models suited to deal with paired data problems. To learn the two dependent graphs, together with their across-graph association structure, we implement a fused graphical lasso penalty. We carry out a comprehensive analysis of this approach, with special attention to the role played by some relevant submodel classes. In this way, we provide a broad set of tools for the application of Gaussian graphical models to paired data problems. These include results useful for the specification of penalty values in order to obtain a path of lasso solutions and an ADMM algorithm that solves the fused graphical lasso optimization problem. Finally, we present an application of our method to cancer genomics where it is of interest to compare cancer cells with a control sample from histologically normal tissues adjacent to the tumor. All the methods described in this article are implemented in the $\texttt{R}$ package $\texttt{pdglasso}$ availabe at: //github.com/savranciati/pdglasso.
Physics-informed neural networks (PINNs) offer a novel and efficient approach to solving partial differential equations (PDEs). Their success lies in the physics-informed loss, which trains a neural network to satisfy a given PDE at specific points and to approximate the solution. However, the solutions to PDEs are inherently infinite-dimensional, and the distance between the output and the solution is defined by an integral over the domain. Therefore, the physics-informed loss only provides a finite approximation, and selecting appropriate collocation points becomes crucial to suppress the discretization errors, although this aspect has often been overlooked. In this paper, we propose a new technique called good lattice training (GLT) for PINNs, inspired by number theoretic methods for numerical analysis. GLT offers a set of collocation points that are effective even with a small number of points and for multi-dimensional spaces. Our experiments demonstrate that GLT requires 2--20 times fewer collocation points (resulting in lower computational cost) than uniformly random sampling or Latin hypercube sampling, while achieving competitive performance.
Probabilistic shaping is a pragmatic approach to improve the performance of coherent optical fiber communication systems. In the nonlinear regime, the advantages offered by probabilistic shaping might increase thanks to the opportunity to obtain an additional nonlinear shaping gain. Unfortunately, the optimization of conventional shaping techniques, such as probabilistic amplitude shaping (PAS), yields a relevant nonlinear shaping gain only in scenarios of limited practical interest. In this manuscript we use sequence selection to investigate the potential, opportunities, and challenges offered by nonlinear probabilistic shaping. First, we show that ideal sequence selection is able to provide up to 0.13 bit/s/Hz gain with respect to PAS with an optimized blocklength. However, this additional gain is obtained only if the selection metric accounts for the signs of the symbols: they must be known to compute the selection metric, but there is no need to shape them. Furthermore, we show that the selection depends in a non-critical way on the symbol rate and link length: the sequences selected for a certain scenario still provide a relevant gain if these are modified. Then, we analyze and compare several practical implementations of sequence selection by taking into account interaction with forward error correction (FEC) and complexity. Overall, the single block and the multi block FEC-independent bit scrambling are the best options, with a gain up to 0.08 bit/s/Hz. The main challenge and limitation to their practical implementation remains the evaluation of the metric, whose complexity is currently too high. Finally, we show that the nonlinear shaping gain provided by sequence selection persists when carrier phase recovery is included.
Quasi-Newton algorithms are among the most popular iterative methods for solving unconstrained minimization problems, largely due to their favorable superlinear convergence property. However, existing results for these algorithms are limited as they provide either (i) a global convergence guarantee with an asymptotic superlinear convergence rate, or (ii) a local non-asymptotic superlinear rate for the case that the initial point and the initial Hessian approximation are chosen properly. In particular, no current analysis for quasi-Newton methods guarantees global convergence with an explicit superlinear convergence rate. In this paper, we close this gap and present the first globally convergent quasi-Newton method with an explicit non-asymptotic superlinear convergence rate. Unlike classical quasi-Newton methods, we build our algorithm upon the hybrid proximal extragradient method and propose a novel online learning framework for updating the Hessian approximation matrices. Specifically, guided by the convergence analysis, we formulate the Hessian approximation update as an online convex optimization problem in the space of matrices, and we relate the bounded regret of the online problem to the superlinear convergence of our method.
This paper studies inference for the local average treatment effect in randomized controlled trials with imperfect compliance where treatment status is determined according to "matched pairs." By "matched pairs," we mean that units are sampled i.i.d. from the population of interest, paired according to observed, baseline covariates and finally, within each pair, one unit is selected at random for treatment. Under weak assumptions governing the quality of the pairings, we first derive the limiting behavior of the usual Wald (i.e., two-stage least squares) estimator of the local average treatment effect. We show further that the conventional heteroskedasticity-robust estimator of its limiting variance is generally conservative in that its limit in probability is (typically strictly) larger than the limiting variance. We therefore provide an alternative estimator of the limiting variance that is consistent for the desired quantity. Finally, we consider the use of additional observed, baseline covariates not used in pairing units to increase the precision with which we can estimate the local average treatment effect. To this end, we derive the limiting behavior of a two-stage least squares estimator of the local average treatment effect which includes both the additional covariates in addition to pair fixed effects, and show that the limiting variance is always less than or equal to that of the Wald estimator. To complete our analysis, we provide a consistent estimator of this limiting variance. A simulation study confirms the practical relevance of our theoretical results. We use our results to revisit a prominent experiment studying the effect of macroinsurance on microenterprise in Egypt.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191