Given a positive integer $d$, the class $d$-DIR is defined as all those intersection graphs formed from a finite collection of line segments in ${\mathbb R}^2$ having at most $d$ slopes. Since each slope induces an interval graph, it easily follows for every $G$ in $d$-DIR with clique number at most $\omega$ that the chromatic number $\chi(G)$ of $G$ is at most $d\omega$. We show for every even value of $\omega$ how to construct a graph in $d$-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvo\v{r}\'ak and Noorizadeh. Furthermore, we show that the $\chi$-binding function of $d$-DIR is $\omega \mapsto d\omega$ for $\omega$ even and $\omega \mapsto d(\omega-1)+1$ for $\omega$ odd. This refutes said conjecture of Bhattacharya, Dvo\v{r}\'ak and Noorizadeh.
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This paper analyzes a $\theta$-method and 3-point time filter. This approach adds one additional line of code to the existing source code of $\theta$-method. We prove the method's $0$-stability, accuracy, and $A$-stability for both constant time step and variable time step. Some numerical tests are performed to validate the theoretical results.
Given a graph $G=(V,E)$ and an integer $k$, the Cluster Editing problem asks whether we can transform $G$ into a union of vertex-disjoint cliques by at most $k$ modifications (edge deletions or insertions). In this paper, we study the following variant of Cluster Editing. We are given a graph $G=(V,E)$, a packing $\cal H$ of modification-disjoint induced $P_3$s (no pair of $P_3$s in $\cal H$ share an edge or non-edge) and an integer $\ell$. The task is to decide whether $G$ can be transformed into a union of vertex-disjoint cliques by at most $\ell+|\cal H|$ modifications (edge deletions or insertions). We show that this problem is NP-hard even when $\ell=0$ (in which case the problem asks to turn $G$ into a disjoint union of cliques by performing exactly one edge deletion or insertion per element of $\cal H$) and when each vertex is in at most 23 $P_3$s of the packing. This answers negatively a question of van Bevern, Froese, and Komusiewicz (CSR 2016, ToCS 2018), repeated by C. Komusiewicz at Shonan meeting no. 144 in March 2019. We then initiate the study to find the largest integer $c$ such that the problem remains tractable when restricting to packings such that each vertex is in at most $c$ packed $P_3$s. Here packed $P_3$s are those belonging to the packing $\cal H$. Van Bevern et al. showed that the case $c = 1$ is fixed-parameter tractable with respect to $\ell$ and we show that the case $c = 2$ is solvable in $|V|^{2\ell + O(1)}$ time.
We address speech enhancement based on variational autoencoders, which involves learning a speech prior distribution in the time-frequency (TF) domain. A zero-mean complex-valued Gaussian distribution is usually assumed for the generative model, where the speech information is encoded in the variance as a function of a latent variable. In contrast to this commonly used approach, we propose a weighted variance generative model, where the contribution of each spectrogram time-frame in parameter learning is weighted. We impose a Gamma prior distribution on the weights, which would effectively lead to a Student's t-distribution instead of Gaussian for speech generative modeling. We develop efficient training and speech enhancement algorithms based on the proposed generative model. Our experimental results on spectrogram auto-encoding and speech enhancement demonstrate the effectiveness and robustness of the proposed approach compared to the standard unweighted variance model.
We introduce and analyse a family of hash and predicate functions that are more likely to produce collisions for small reducible configurations of vectors. These may offer practical improvements to lattice sieving for short vectors. In particular, in one asymptotic regime the family exhibits significantly different convergent behaviour than existing hash functions and predicates.
It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into $\ell_1$ with constant distortion. However, given an $n$-vertex weighted planar graph, the best upper bound on the distortion is only $O(\sqrt{\log n})$, by Rao [SoCG99]. In this paper we study the case where there is a set $K$ of terminals, and the goal is to embed only the terminals into $\ell_1$ with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into $\ell_1$. The more general case, where the set of terminals can be covered by $\gamma$ faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of $O(\log \gamma)$ by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to $O(\sqrt{\log\gamma})$. Since every planar graph has at most $O(n)$ faces, any further improvement on this result, will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into $\ell_1$. Therefore, our result provides a polynomial time $O(\sqrt{\log \gamma})$-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by $\gamma$ faces.
A common approach to evaluating the significance of a collection of $p$-values combines them with a pooling function, in particular when the original data are not available. These pooled $p$-values convert a sample of $p$-values into a single number which behaves like a univariate $p$-value. To clarify discussion of these functions, a telescoping series of alternative hypotheses are introduced that communicate the strength and prevalence of non-null evidence in the $p$-values before general pooling formulae are discussed. A pattern noticed in the UMP pooled $p$-value for a particular alternative motivates the definition and discussion of central and marginal rejection levels at $\alpha$. It is proven that central rejection is always greater than or equal to marginal rejection, motivating a quotient to measure the balance between the two for pooled $p$-values. A combining function based on the $\chi^2_{\kappa}$ quantile transformation is proposed to control this quotient and shown to be robust to mis-specified parameters relative to the UMP. Different powers for different parameter settings motivate a map of plausible alternatives based on where this pooled $p$-value is minimized.
We propose $\mathbb{VD}$-$\mathbb{GR}$ - a novel visual dialog model that combines pre-trained language models (LMs) with graph neural networks (GNNs). Prior works mainly focused on one class of models at the expense of the other, thus missing out on the opportunity of combining their respective benefits. At the core of $\mathbb{VD}$-$\mathbb{GR}$ is a novel integration mechanism that alternates between spatial-temporal multi-modal GNNs and BERT layers, and that covers three distinct contributions: First, we use multi-modal GNNs to process the features of each modality (image, question, and dialog history) and exploit their local structures before performing BERT global attention. Second, we propose hub-nodes that link to all other nodes within one modality graph, allowing the model to propagate information from one GNN (modality) to the other in a cascaded manner. Third, we augment the BERT hidden states with fine-grained multi-modal GNN features before passing them to the next $\mathbb{VD}$-$\mathbb{GR}$ layer. Evaluations on VisDial v1.0, VisDial v0.9, VisDialConv, and VisPro show that $\mathbb{VD}$-$\mathbb{GR}$ achieves new state-of-the-art results across all four datasets.
The paper presents a spectral representation for general type two-sided discrete time signals from $\ell_\infty$, i.e for all bounded discrete time signals, including signals that do not vanish at $\pm\infty$. This representation allows to extend on the general type signals from $\ell_\infty$ the notions of transfer functions, spectrum gaps, and filters, and to obtain some frequency conditions of predictability and data recoverability.
We construct a monotone continuous $Q^1$ finite element method on the uniform mesh for the anisotropic diffusion problem with a diagonally dominant diffusion coefficient matrix. The monotonicity implies the discrete maximum principle. Convergence of the new scheme is rigorously proven. On quadrilateral meshes, the matrix coefficient conditions translate into specific a mesh constraint.
We describe Bayes factors functions based on z, t, $\chi^2$, and F statistics and the prior distributions used to define alternative hypotheses. The non-local alternative prior distributions are centered on standardized effects, which index the Bayes factor function. The prior densities include a dispersion parameter that models the variation of effect sizes across replicated experiments. We examine the convergence rates of Bayes factor functions under true null and true alternative hypotheses. Several examples illustrate the application of the Bayes factor functions to replicated experimental designs and compare the conclusions from these analyses to other default Bayes factor methods.
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