We consider the construction of a polyhedral Delaunay partition as a limit of the sequence of power diagrams (radical partitions). The dual Voronoi diagram is obtained as a limit of the sequence of weighted Delaunay partitions. The problem is reduced to the construction of two dual convex polyhedra, inscribed and superscribed around a circular paraboloid, as a limit of the sequence of pairs of general dual convex polyhedra. The sequence of primal polyhedra should converge to the superscribed polyhedron and the sequence of the dual polyhedra converges to the inscribed polyhedron. We are interested in the case when the vertices of primal polyhedra can move or merge together, i.e., no new faces are allowed for dual polyhedra. These rules define the transformation of the set of initial spheres into the set of Delaunay spheres using radius variation and sphere movement and elimination. Existence theorems are still unavailable but we suggest a functional measuring the deviation of the convex polyhedron from the one inscribed into the paraboloid. It is the discrete Dirichlet functional for the power function which is a linear interpolant of the vertical distance of the dual vertices from the paraboloid. The functional's absolute minimizer is attained on the constant power field, meaning that the inscribed polyhedron can be obtained by a simple translation. This formulation of the functional for the dual surface is not quadratic since the unknowns are the vertices of the primal polyhedron, hence, the transformation of the set of spheres into Delaunay spheres is not unique. We concentrate on the experimental confirmation of the approach viability and put aside mesh quality problems. The zero value of the gradient of the proposed functional defines a manifold describing the evolution of Delaunay spheres. Hence, Delaunay-Voronoi meshes can be optimized using the manifold as a constraint.
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In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with $m$ edges and $n$ vertices in $O(mn)$ time with space complexity $O(n)$.
Certain simplicial complexes are used to construct a subset $D$ of $\mathbb{F}_{2^n}^m$ and $D$, in turn, defines the linear code $C_{D}$ over $\mathbb{F}_{2^n}$ that consists of $(v\cdot d)_{d\in D}$ for $v\in \mathbb{F}_{2^n}^m$. Here we deal with the case $n=3$, that is, when $C_{D}$ is an octanary code. We establish a relation between $C_{D}$ and its binary subfield code $C_{D}^{(2)}$ with the help of a generator matrix. For a given length and dimension, a code is called distance optimal if it has the highest possible distance. With respect to the Griesmer bound, five infinite families of distance optimal codes are obtained, and sufficient conditions for certain linear codes to be minimal are established.
We investigate the feature compression of high-dimensional ridge regression using the optimal subsampling technique. Specifically, based on the basic framework of random sampling algorithm on feature for ridge regression and the A-optimal design criterion, we first obtain a set of optimal subsampling probabilities. Considering that the obtained probabilities are uneconomical, we then propose the nearly optimal ones. With these probabilities, a two step iterative algorithm is established which has lower computational cost and higher accuracy. We provide theoretical analysis and numerical experiments to support the proposed methods. Numerical results demonstrate the decent performance of our methods.
Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.
Motivated by problems from neuroimaging in which existing approaches make use of "mass univariate" analysis which neglects spatial structure entirely, but the full joint modelling of all quantities of interest is computationally infeasible, a novel method for incorporating spatial dependence within a (potentially large) family of model-selection problems is presented. Spatial dependence is encoded via a Markov random field model for which a variant of the pseudo-marginal Markov chain Monte Carlo algorithm is developed and extended by a further augmentation of the underlying state space. This approach allows the exploitation of existing unbiased marginal likelihood estimators used in settings in which spatial independence is normally assumed thereby facilitating the incorporation of spatial dependence using non-spatial estimates with minimal additional development effort. The proposed algorithm can be realistically used for analysis of %smaller subsets of large image moderately sized data sets such as $2$D slices of whole $3$D dynamic PET brain images or other regions of interest. Principled approximations of the proposed method, together with simple extensions based on the augmented spaces, are investigated and shown to provide similar results to the full pseudo-marginal method. Such approximations and extensions allow the improved performance obtained by incorporating spatial dependence to be obtained at negligible additional cost. An application to measured PET image data shows notable improvements in revealing underlying spatial structure when compared to current methods that assume spatial independence.
In the storied Colonel Blotto game, two colonels allocate $a$ and $b$ troops, respectively, to $k$ distinct battlefields. A colonel wins a battle if they assign more troops to that particular battle, and each colonel seeks to maximize their total number of victories. Despite the problem's formulation in 1921, the first polynomial-time algorithm to compute Nash equilibrium (NE) strategies for this game was discovered only quite recently. In 2016, \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} formulated a breakthrough algorithm to compute NE strategies for the Colonel Blotto game\footnote{To the best of our knowledge, the algorithm from \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} has computational complexity $O(k^{14}\max\{a,b\}^{13})$}, receiving substantial media coverage (e.g. \citep{Insider}, \citep{NSF}, \citep{ScienceDaily}). In this work, we present the first known $\epsilon$-approximation algorithm to compute NE strategies in the two-player Colonel Blotto game in runtime $\widetilde{O}(\epsilon^{-4} k^8 \max\{a,b\}^2)$ for arbitrary settings of these parameters. Moreover, this algorithm computes approximate coarse correlated equilibrium strategies in the multiplayer (continuous and discrete) Colonel Blotto game (when there are $\ell > 2$ colonels) with runtime $\widetilde{O}(\ell \epsilon^{-4} k^8 n^2 + \ell^2 \epsilon^{-2} k^3 n (n+k))$, where $n$ is the maximum troop count. Before this work, no polynomial-time algorithm was known to compute exact or approximate equilibrium (in any sense) strategies for multiplayer Colonel Blotto with arbitrary parameters. Our algorithm computes these approximate equilibria by a novel (to the author's knowledge) sampling technique with which we implicitly perform multiplicative weights update over the exponentially many strategies available to each player.
The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.
The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.
The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.
We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting directions. Then we give a new information-theoretic proof of a finite version of de Finetti's classical representation theorem for finite-valued random variables. We derive an upper bound on the relative entropy between the distribution of the first $k$ in a sequence of $n$ exchangeable random variables, and an appropriate mixture over product distributions. The mixing measure is characterised as the law of the empirical measure of the original sequence, and de Finetti's result is recovered as a corollary. The proof is nicely motivated by the Gibbs conditioning principle in connection with statistical mechanics, and it follows along an appealing sequence of steps. The technical estimates required for these steps are obtained via the use of a collection of combinatorial tools known within information theory as `the method of types.'
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