In this paper we prove that the $\ell_0$ isoperimetric coefficient for any axis-aligned cubes, $\psi_{\mathcal{C}}$, is $\Theta(n^{-1/2})$ and that the isoperimetric coefficient for any measurable body $K$, $\psi_K$, is of order $O(n^{-1/2})$. As a corollary we deduce that axis-aligned cubes essentially "maximize" the $\ell_0$ isoperimetric coefficient: There exists a positive constant $q > 0$ such that $\psi_K \leq q \cdot \psi_{\mathcal{C}}$, whenever $\mathcal{C}$ is an axis-aligned cube and $K$ is any measurable set. Lastly, we give immediate applications of our results to the mixing time of Coordinate-Hit-and-Run for sampling points uniformly from convex bodies.
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This paper presents a {\delta}-PI algorithm which is based on damped Newton method for the H{\infty} tracking control problem of unknown continuous-time nonlinear system. A discounted performance function and an augmented system are used to get the tracking Hamilton-Jacobi-Isaac (HJI) equation. Tracking HJI equation is a nonlinear partial differential equation, traditional reinforcement learning methods for solving the tracking HJI equation are mostly based on the Newton method, which usually only satisfies local convergence and needs a good initial guess. Based upon the damped Newton iteration operator equation, a generalized tracking Bellman equation is derived firstly. The {\delta}-PI algorithm can seek the optimal solution of the tracking HJI equation by iteratively solving the generalized tracking Bellman equation. On-policy learning and off-policy learning {\delta}-PI reinforcement learning methods are provided, respectively. Off-policy version {\delta}-PI algorithm is a model-free algorithm which can be performed without making use of a priori knowledge of the system dynamics. NN-based implementation scheme for the off-policy {\delta}-PI algorithms is shown. The suitability of the model-free {\delta}-PI algorithm is illustrated with a nonlinear system simulation.
This paper proposes to develop a new variant of the two-time-scale stochastic approximation to find the roots of two coupled nonlinear operators, assuming only noisy samples of these operators can be observed. Our key idea is to leverage the classic Ruppert-Polyak averaging technique to dynamically estimate the operators through their samples. The estimated values of these averaging steps will then be used in the two-time-scale stochastic approximation updates to find the desired solution. Our main theoretical result is to show that under the strongly monotone condition of the underlying nonlinear operators the mean-squared errors of the iterates generated by the proposed method converge to zero at an optimal rate $\mathcal{O}(1/k)$, where $k$ is the number of iterations. Our result significantly improves the existing result of two-time-scale stochastic approximation, where the best known finite-time convergence rate is $\mathcal{O}(1/k^{2/3})$.
Adhesive and quasiadhesive categories provide a general framework for the study of algebraic graph rewriting systems. In a quasiadhesive category any two regular subobjects have a join which is again a regular subobject. Vice versa, if regular monos are adhesive, then the existence of a regular join for any pair of regular subobjects entails quasiadhesivity. It is also known (quasi)adhesive categories can be embedded in a Grothendieck topos via a functor preserving pullbacks and pushouts along (regular) monomorphisms. In this paper we extend these results to $\mathcal{M}, \mathcal{N}$-adhesive categories, a concept recently introduced to generalize the notion of (quasi)adhesivity. We introduce the notion of $\mathcal{N}$-adhesive morphism, which allows us to express $\mathcal{M}, \mathcal{N}$-adhesivity as a condition on the subobjects's posets. Moreover, $\mathcal{N}$-adhesive morphisms allows us to show how an $\mathcal{M},\mathcal{N}$-adhesive category can be embedded into a Grothendieck topos, preserving pullbacks and $\mathcal{M}, \mathcal{N}$-pushouts.
We design an algorithm for computing the $L$-series associated to an Anderson $t$-motives, exhibiting quasilinear complexity with respect to the target precision. Based on experiments, we conjecture that the order of vanishing at $T=1$ of the $v$-adic $L$-series of a given Anderson $t$-motive with good reduction does not depend on the finite place $v$.
Diffusion models have emerged as powerful generative tools, rivaling GANs in sample quality and mirroring the likelihood scores of autoregressive models. A subset of these models, exemplified by DDIMs, exhibit an inherent asymmetry: they are trained over $T$ steps but only sample from a subset of $T$ during generation. This selective sampling approach, though optimized for speed, inadvertently misses out on vital information from the unsampled steps, leading to potential compromises in sample quality. To address this issue, we present the S$^{2}$-DMs, which is a new training method by using an innovative $L_{skip}$, meticulously designed to reintegrate the information omitted during the selective sampling phase. The benefits of this approach are manifold: it notably enhances sample quality, is exceptionally simple to implement, requires minimal code modifications, and is flexible enough to be compatible with various sampling algorithms. On the CIFAR10 dataset, models trained using our algorithm showed an improvement of 3.27% to 14.06% over models trained with traditional methods across various sampling algorithms (DDIMs, PNDMs, DEIS) and different numbers of sampling steps (10, 20, ..., 1000). On the CELEBA dataset, the improvement ranged from 8.97% to 27.08%. Access to the code and additional resources is provided in the github.
We introduce a new class of algorithms for finding a short vector in lattices defined by codes of co-dimension $k$ over $\mathbb{Z}_P^d$, where $P$ is prime. The co-dimension $1$ case is solved by exploiting the packing properties of the projections mod $P$ of an initial set of non-lattice vectors onto a single dual codeword. The technical tools we introduce are sorting of the projections followed by single-step pairwise Euclidean reduction of the projections, resulting in monotonic convergence of the positive-valued projections to zero. The length of vectors grows by a geometric factor each iteration. For fixed $P$ and $d$, and large enough user-defined input sets, we show that it is possible to minimize the number of iterations, and thus the overall length expansion factor, to obtain a short lattice vector. Thus we obtain a novel approach for controlling the output length, which resolves an open problem posed by Noah Stephens-Davidowitz (the possibility of an approximation scheme for the shortest-vector problem (SVP) which does not reduce to near-exact SVP). In our approach, one may obtain short vectors even when the lattice dimension is quite large, e.g., 8000. For fixed $P$, the algorithm yields shorter vectors for larger $d$. We additionally present a number of extensions and generalizations of our fundamental co-dimension $1$ method. These include a method for obtaining many different lattice vectors by multiplying the dual codeword by an integer and then modding by $P$; a co-dimension $k$ generalization; a large input set generalization; and finally, a "block" generalization, which involves the replacement of pairwise (Euclidean) reduction by a $k$-party (non-Euclidean) reduction. The $k$-block generalization of our algorithm constitutes a class of polynomial-time algorithms indexed by $k\geq 2$, which yield successively improved approximations for the short vector problem.
This paper addresses the challenges of real-time, large-scale, and near-optimal multi-agent pathfinding (MAPF) through enhancements to the recently proposed LaCAM* algorithm. LaCAM* is a scalable search-based algorithm that guarantees the eventual finding of optimal solutions for cumulative transition costs. While it has demonstrated remarkable planning success rates, surpassing various state-of-the-art MAPF methods, its initial solution quality is far from optimal, and its convergence speed to the optimum is slow. To overcome these limitations, this paper introduces several improvement techniques, partly drawing inspiration from other MAPF methods. We provide empirical evidence that the fusion of these techniques significantly improves the solution quality of LaCAM*, thus further pushing the boundaries of MAPF algorithms.
We study the edge-coloring problem in simple $n$-vertex $m$-edge graphs with maximum degree $\Delta$. This is one of the most classical and fundamental graph-algorithmic problems. Vizing's celebrated theorem provides $(\Delta+1)$-edge-coloring in $O(m\cdot n)$ deterministic time. This running time was improved to $O\left(m\cdot\min\left\{\Delta\cdot\log n, \sqrt{n}\right\}\right)$. It is also well-known that $3\left\lceil\frac{\Delta}{2}\right\rceil$-edge-coloring can be computed in $O(m\cdot\log\Delta)$ time deterministically. Duan et al. devised a randomized $(1+\varepsilon)\Delta$-edge-coloring algorithm with running time $O\left(m\cdot\frac{\log^6 n}{\varepsilon^2}\right)$. It was however open if there exists a deterministic near-linear time algorithm for this basic problem. We devise a simple deterministic $(1+\varepsilon)\Delta$-edge-coloring algorithm with running time $O\left(m\cdot\frac{\log n}{\varepsilon}\right)$. We also devise a randomized $(1+\varepsilon)\Delta$-edge-coloring algorithm with running time $O(m\cdot(\varepsilon^{-18}+\log(\varepsilon\cdot\Delta)))$. For $\varepsilon\geq\frac{1}{\log^{1/18}\Delta}$, this running time is $O(m\cdot\log\Delta)$.
This paper presents exact formulas for the probability distribution function (PDF) and moment generating function (MGF) of the sum-product of statistically independent but not necessarily identically distributed (i.n.i.d.) Nakagami-$m$ random variables (RVs) in terms of Meijer's G-function. Additionally, exact series representations are also derived for the sum of double-Nakagami RVs, providing useful insights on the trade-off between accuracy and computational cost. Simple asymptotic analytical expressions are provided to gain further insight into the derived formula, and the achievable diversity order is obtained. The suggested statistical properties are proved to be a highly useful tool for modeling parallel cascaded Nakagami-$m$ fading channels. The application of these new results is illustrated by deriving exact expressions and simple tight upper bounds for the outage probability (OP) and average symbol error rate (ASER) of several binary and multilevel modulation signals in intelligent reflecting surfaces (IRSs)-assisted communication systems operating over Nakagami-$m$ fading channels. It is demonstrated that the new asymptotic expression is highly accurate and can be extended to encompass a wider range of scenarios. To validate the theoretical frameworks and formulations, Monte-Carlo simulation results are presented. Additionally, supplementary simulations are provided to compare the derived results with two common types of approximations available in the literature, namely the central limit theorem (CLT) and gamma distribution.
We study the algorithmic undecidability of abstract dynamical properties for sofic $\mathbb{Z}^{2}$-subshifts and subshifts of finite type (SFTs) on $\mathbb{Z}^{2}$. Within the class of sofic $\mathbb{Z}^{2}$-subshifts, we prove the undecidability of every nontrivial dynamical property. We show that although this is not the case for $\mathbb{Z}^{2}$-SFTs, it is still possible to establish the undecidability of a large class of dynamical properties. This result is analogous to the Adian-Rabin undecidability theorem for group properties. Besides dynamical properties, we consider dynamical invariants of $\mathbb{Z}^{2}$-SFTs taking values in partially ordered sets. It is well known that the topological entropy of a $\mathbb{Z}^{2}$-SFT can not be effectively computed from an SFT presentation. We prove a generalization of this result to \emph{every} dynamical invariant which is nonincreasing by factor maps, and satisfies a mild additional technical condition. Our results are also valid for $\Z^{d}$, $d\geq2$, and more generally for any group where determining whether a subshift of finite type is empty is undecidable.
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