It is shown that every $n$-vertex graph that admits a 2-bend RAC drawing in the plane, where the edges are polylines with two bends per edge and any pair of edges can only cross at a right angle, has at most $24n-26$ edges for $n\geq 3$. This improves upon the previous upper bound of $74.2n$; this is the first improvement in more than 12 years. A crucial ingredient of the proof is an upper bound on the size of plane multigraphs with polyline edges in which the first and last segments are either parallel or orthogonal.
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In this short piece, I delved into the connections of Nobel laureates by applying Network Science methods to and public data collected from Wikipedia. I uncovered the existence of a central "giant component" in the Nobel laureate network, highlighting the core-periphery structure and the disparity in visibility among laureates. I explored the dominance of laureates in the fields of science and humanities, revealing a polarization that contradicts the trend of interdisciplinary research. Furthermore, it the finding sheds light on the underrepresentation of female laureates in certain Nobel Prize categories.
Imagine a polygon-shaped platform $P$ and only one static spotlight outside $P$; which direction should the spotlight face to light most of $P$? This problem occurs in maximising the visibility, as well as in limiting the uncertainty in localisation problems. More formally, we define the following maximum cover problem: "Given a convex polygon $P$ and a Field Of View (FOV) with a given centre and inner angle $\phi$; find the direction (an angle of rotation $\theta$) of the FOV such that the intersection between the FOV and $P$ has the maximum area". In this paper, we provide the theoretical foundation for the analysis of the maximum cover with a rotating field of view. The main challenge is that the function of the area $A_{\phi}(\theta)$, with the angle of rotation $\theta$ and the fixed inner angle $\phi$, cannot be approximated directly. We found an alternative way to express it by various compositions of a function $A_{\theta}(\phi)$ (with a restricted inner angle $\phi$ and a fixed direction $\theta$). We show that $A_{\theta}(\phi)$ has an analytical solution in the special case of a two-sector intersection and later provides a constrictive solution for the original problem. Since the optimal solution is a real number, we develop an algorithm that approximates the direction of the field of view, with precision $\varepsilon$, and complexity $\mathcal{O}(n(\log{n}+(\log{\varepsilon})/\phi))$.
In the pooled data problem, the goal is to identify the categories associated with a large collection of items via a sequence of pooled tests. Each pooled test reveals the number of items of each category within the pool. We study an approximate message passing (AMP) algorithm for estimating the categories and rigorously characterize its performance, in both the noiseless and noisy settings. For the noiseless setting, we show that the AMP algorithm is equivalent to one recently proposed by El Alaoui et al. Our results provide a rigorous version of their performance guarantees, previously obtained via non-rigorous techniques. For the case of pooled data with two categories, known as quantitative group testing (QGT), we use the AMP guarantees to compute precise limiting values of the false positive rate and the false negative rate. Though the pooled data problem and QGT are both instances of estimation in a linear model, existing AMP theory cannot be directly applied since the design matrices are binary valued. The key technical ingredient in our result is a rigorous analysis of AMP for generalized linear models defined via generalized white noise design matrices. This result, established using a recent universality result of Wang et al., is of independent interest. Our theoretical results are validated by numerical simulations. For comparison, we propose estimators based on convex relaxation and iterative thresholding, without providing theoretical guarantees. Our simulations indicate that AMP outperforms the convex programming estimator for a range of QGT scenarios, but the convex program performs better for pooled data with three categories.
Text-guided diffusion models such as DALLE-2, Imagen, eDiff-I, and Stable Diffusion are able to generate an effectively endless variety of images given only a short text prompt describing the desired image content. In many cases the images are of very high quality. However, these models often struggle to compose scenes containing several key objects such as characters in specified positional relationships. The missing capability to ``direct'' the placement of characters and objects both within and across images is crucial in storytelling, as recognized in the literature on film and animation theory. In this work, we take a particularly straightforward approach to providing the needed direction. Drawing on the observation that the cross-attention maps for prompt words reflect the spatial layout of objects denoted by those words, we introduce an optimization objective that produces ``activation'' at desired positions in these cross-attention maps. The resulting approach is a step toward generalizing the applicability of text-guided diffusion models beyond single images to collections of related images, as in storybooks. Directed Diffusion provides easy high-level positional control over multiple objects, while making use of an existing pre-trained model and maintaining a coherent blend between the positioned objects and the background. Moreover, it requires only a few lines to implement.
Despite having the same basic prophet inequality setup and model of loss aversion, conclusions in our multi-dimensional model differs considerably from the one-dimensional model of Kleinberg et al. For example, Kleinberg et al. gives a tight closed-form on the competitive ratio that an online decision-maker can achieve as a function of $\lambda$, for any $\lambda \geq 0$. In our multi-dimensional model, there is a sharp phase transition: if $k$ denotes the number of dimensions, then when $\lambda \cdot (k-1) \geq 1$, no non-trivial competitive ratio is possible. On the other hand, when $\lambda \cdot (k-1) < 1$, we give a tight bound on the achievable competitive ratio (similar to Kleinberg et al.). As another example, Kleinberg et al. uncovers an exponential improvement in their competitive ratio for the random-order vs. worst-case prophet inequality problem. In our model with $k\geq 2$ dimensions, the gap is at most a constant-factor. We uncover several additional key differences in the multi- and single-dimensional models.
Panoptic segmentation assigns semantic and instance ID labels to every pixel of an image. As permutations of instance IDs are also valid solutions, the task requires learning of high-dimensional one-to-many mapping. As a result, state-of-the-art approaches use customized architectures and task-specific loss functions. We formulate panoptic segmentation as a discrete data generation problem, without relying on inductive bias of the task. A diffusion model is proposed to model panoptic masks, with a simple architecture and generic loss function. By simply adding past predictions as a conditioning signal, our method is capable of modeling video (in a streaming setting) and thereby learns to track object instances automatically. With extensive experiments, we demonstrate that our simple approach can perform competitively to state-of-the-art specialist methods in similar settings.
We explore the space of matrix-generated (0, m, 2)-nets and (0, 2)-sequences in base 2, also known as digital dyadic nets and sequences. In computer graphics, they are arguably leading the competition for use in rendering. We provide a complete characterization of the design space and count the possible number of constructions with and without considering possible reorderings of the point set. Based on this analysis, we then show that every digital dyadic net can be reordered into a sequence, together with a corresponding algorithm. Finally, we present a novel family of self-similar digital dyadic sequences, to be named $\xi$-sequences, that spans a subspace with fewer degrees of freedom. Those $\xi$-sequences are extremely efficient to sample and compute, and we demonstrate their advantages over the classic Sobol (0, 2)-sequence.
Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractional analogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paper is on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previous conditions under which the size of the support of fractional edge covers is bounded independently of the size of the hypergraph itself. This allows us to extend previous tractability results for checking if the fractional hypertree width of a given hypergraph is $\leq k$ for some constant $k$. We also show how our results translate to fractional vertex covers.
AI is undergoing a paradigm shift with the rise of models (e.g., BERT, DALL-E, GPT-3) that are trained on broad data at scale and are adaptable to a wide range of downstream tasks. We call these models foundation models to underscore their critically central yet incomplete character. This report provides a thorough account of the opportunities and risks of foundation models, ranging from their capabilities (e.g., language, vision, robotics, reasoning, human interaction) and technical principles(e.g., model architectures, training procedures, data, systems, security, evaluation, theory) to their applications (e.g., law, healthcare, education) and societal impact (e.g., inequity, misuse, economic and environmental impact, legal and ethical considerations). Though foundation models are based on standard deep learning and transfer learning, their scale results in new emergent capabilities,and their effectiveness across so many tasks incentivizes homogenization. Homogenization provides powerful leverage but demands caution, as the defects of the foundation model are inherited by all the adapted models downstream. Despite the impending widespread deployment of foundation models, we currently lack a clear understanding of how they work, when they fail, and what they are even capable of due to their emergent properties. To tackle these questions, we believe much of the critical research on foundation models will require deep interdisciplinary collaboration commensurate with their fundamentally sociotechnical nature.
Video captioning is a challenging task that requires a deep understanding of visual scenes. State-of-the-art methods generate captions using either scene-level or object-level information but without explicitly modeling object interactions. Thus, they often fail to make visually grounded predictions, and are sensitive to spurious correlations. In this paper, we propose a novel spatio-temporal graph model for video captioning that exploits object interactions in space and time. Our model builds interpretable links and is able to provide explicit visual grounding. To avoid unstable performance caused by the variable number of objects, we further propose an object-aware knowledge distillation mechanism, in which local object information is used to regularize global scene features. We demonstrate the efficacy of our approach through extensive experiments on two benchmarks, showing our approach yields competitive performance with interpretable predictions.
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