Recently, Hegerfeld and Kratsch [ESA 2023] obtained the first tight algorithmic results for hard connectivity problems parameterized by clique-width. Concretely, they gave one-sided error Monte-Carlo algorithms that given a $k$-clique-expression solve Connected Vertex Cover in time $6^kn^{O(1)}$ and Connected Dominating Set in time $5^kn^{O(1)}$. Moreover, under the Strong Exponential-Time Hypothesis (SETH) these results were showed to be tight. However, they leave open several important benchmark problems, whose complexity relative to treewidth had been settled by Cygan et al. [SODA 2011 & TALG 2018]. Among which is the Steiner Tree problem. As a key obstruction they point out the exponential gap between the rank of certain compatibility matrices, which is often used for algorithms, and the largest triangular submatrix therein, which is essential for current lower bound methods. Concretely, for Steiner Tree the $GF(2)$-rank is $4^k$, while no triangular submatrix larger than $3^k$ was known. This yields time $4^kn^{O(1)}$, while the obtainable impossibility of time $(3-\varepsilon)^kn^{O(1)}$ under SETH was already known relative to pathwidth. We close this gap by showing that Steiner Tree can be solved in time $3^kn^{O(1)}$ given a $k$-clique-expression. Hence, for all parameters between cutwidth and clique-width it has the same tight complexity. We first show that there is a ``representative submatrix'' of GF(2)-rank $3^k$ (ruling out larger triangular submatrices). At first glance, this only allows to count (modulo 2) the number of representations of valid solutions, but not the number of solutions (even if a unique solution exists). We show how to overcome this problem by isolating a unique representative of a unique solution, if one exists. We believe that our approach will be instrumental for settling further open problems in this research program.
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Regular physics-informed neural networks (PINNs) predict the solution of partial differential equations using sparse labeled data but only over a single domain. On the other hand, fully supervised learning models are first trained usually over a few thousand domains with known solutions (i.e., labeled data) and then predict the solution over a few hundred unseen domains. Physics-informed PointNet (PIPN) is primarily designed to fill this gap between PINNs (as weakly supervised learning models) and fully supervised learning models. In this article, we demonstrate that PIPN predicts the solution of desired partial differential equations over a few hundred domains simultaneously, while it only uses sparse labeled data. This framework benefits fast geometric designs in the industry when only sparse labeled data are available. Particularly, we show that PIPN predicts the solution of a plane stress problem over more than 500 domains with different geometries, simultaneously. Moreover, we pioneer implementing the concept of remarkable batch size (i.e., the number of geometries fed into PIPN at each sub-epoch) into PIPN. Specifically, we try batch sizes of 7, 14, 19, 38, 76, and 133. Additionally, the effect of the PIPN size, symmetric function in the PIPN architecture, and static and dynamic weights for the component of the sparse labeled data in the loss function are investigated.
Recent advancements in evaluating matrix-exponential functions have opened the doors to the practical use of exponential time-integration methods in numerical weather prediction (NWP). The success of exponential methods in shallow water simulations has led to the question of whether they can be beneficial in a 3D atmospheric model. In this paper, we take the first step forward by evaluating the behavior of exponential time-integration methods in the Navy's compressible deep-atmosphere nonhydrostatic global model (NEPTUNE-Navy Environmental Prediction sysTem Utilizing a Nonhydrostatic Engine). Simulations are conducted on a set of idealized test cases designed to assess key features of a nonhydrostatic model and demonstrate that exponential integrators capture the desired large and small-scale traits, yielding results comparable to those found in the literature. We propose a new upper boundary absorbing layer independent of reference state and shown to be effective in both idealized and real-data simulations. A real-data forecast using an exponential method with full physics is presented, providing a positive outlook for using exponential integrators for NWP.
The Sinkhorn algorithm is a numerical method for the solution of optimal transport problems. Here, I give a brief survey of this algorithm, with a strong emphasis on its geometric origin: it is natural to view it as a discretization, by standard methods, of a non-linear integral equation. In the appendix, I also provide a short summary of an early result of Beurling on product measures, directly related to the Sinkhorn algorithm.
A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semi-linear problems with trilinear non-linearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space, and more importantly, modifies the trilinear term in the stream-function vorticity formulation of the incompressible 2D Navier-Stokes and the von K\'{a}rm\'{a}n equations. This enables the first efficient and reliable a posteriori error estimates for the 2D Navier-Stokes equations in the stream-function vorticity formulation for Morley, two discontinuous Galerkin, $C^0$ interior penalty, and WOPSIP discretizations with piecewise quadratic polynomials.
Establishing the correspondences between newly acquired points and historically accumulated data (i.e., map) through nearest neighbors search is crucial in numerous robotic applications.However, static tree data structures are inadequate to handle large and dynamically growing maps in real-time.To address this issue, we present the i-Octree, a dynamic octree data structure that supports both fast nearest neighbor search and real-time dynamic updates, such as point insertion, deletion, and on-tree down-sampling. The i-Octree is built upon a leaf-based octree and has two key features: a local spatially continuous storing strategy that allows for fast access to points while minimizing memory usage, and local on-tree updates that significantly reduce computation time compared to existing static or dynamic tree structures.The experiments show that i-Octree surpasses state-of-the-art methods by reducing run-time by over 50% on real-world open datasets.
The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit residual-based error estimator as well as a separate marking strategy based on the alternative error estimator and an optimal data approximation algorithm. This paper reviews and discusses available convergence results. In addition, all three strategies are investigated empirically for a set of benchmarks examples of second-order elliptic partial differential equations in two spatial dimensions. Particular interest is on the choice of the marking and refinement parameters and the approximation of the given data. The numerical experiments are reproducible using the author's software package octAFEM available on the platform Code Ocean.
Speech emotion conversion is the task of converting the expressed emotion of a spoken utterance to a target emotion while preserving the lexical content and speaker identity. While most existing works in speech emotion conversion rely on acted-out datasets and parallel data samples, in this work we specifically focus on more challenging in-the-wild scenarios and do not rely on parallel data. To this end, we propose a diffusion-based generative model for speech emotion conversion, the EmoConv-Diff, that is trained to reconstruct an input utterance while also conditioning on its emotion. Subsequently, at inference, a target emotion embedding is employed to convert the emotion of the input utterance to the given target emotion. As opposed to performing emotion conversion on categorical representations, we use a continuous arousal dimension to represent emotions while also achieving intensity control. We validate the proposed methodology on a large in-the-wild dataset, the MSP-Podcast v1.10. Our results show that the proposed diffusion model is indeed capable of synthesizing speech with a controllable target emotion. Crucially, the proposed approach shows improved performance along the extreme values of arousal and thereby addresses a common challenge in the speech emotion conversion literature.
We revisit the Pseudo-Bayesian approach to the problem of estimating density matrix in quantum state tomography in this paper. Pseudo-Bayesian inference has been shown to offer a powerful paradign for quantum tomography with attractive theoretical and empirical results. However, the computation of (Pseudo-)Bayesian estimators, due to sampling from complex and high-dimensional distribution, pose significant challenges that hampers their usages in practical settings. To overcome this problem, we present an efficient adaptive MCMC sampling method for the Pseudo-Bayesian estimator. We show in simulations that our approach is substantially faster than the previous implementation by at least two orders of magnitude which is significant for practical quantum tomography.
The myriad complex systems with multiway interactions motivate the extension of graph-based pairwise connections to higher-order relations. In particular, the simplicial complex has inspired generalizations of graph neural networks (GNNs) to simplicial complex-based models. Learning on such systems requires large amounts of data, which can be expensive or impossible to obtain. We propose data augmentation of simplicial complexes through both linear and nonlinear mixup mechanisms that return mixtures of existing labeled samples. In addition to traditional pairwise mixup, we present a convex clustering mixup approach for a data-driven relationship among several simplicial complexes. We theoretically demonstrate that the resultant synthetic simplicial complexes interpolate among existing data with respect to homomorphism densities. Our method is demonstrated on both synthetic and real-world datasets for simplicial complex classification.
We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^\infty(0,T;H^1(\Omega))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of $\gamma$.
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