亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

We explore a kind of first-order predicate logic with intended semantics in the reals. Compared to other approaches in the literature, we work predominantly in the multiplicative reals $[0,\infty]$, showing they support three generations of connectives, that we call non-linear, linear additive, and linear multiplicative. Means and harmonic means emerge as natural candidates for bounded existential and universal quantifiers, and in fact we see they behave as expected in relation to the other logical connectives. We explain this fact through the well-known fact that min/max and arithmetic mean/harmonic mean sit at opposite ends of a spectrum, that of p-means. We give syntax and semantics for this quantitative predicate logic, and as example applications, we show how softmax is the quantitative semantics of argmax, and R\'enyi entropy/Hill numbers are additive/multiplicative semantics of the same formula. Indeed, the additive reals also fit into the story by exploiting the Napierian duality $-\log \dashv 1/\exp$, which highlights a formal distinction between 'additive' and 'multiplicative' quantities. Finally, we describe two attempts at a categorical semantics via enriched hyperdoctrines. We discuss why hyperdoctrines are in fact probably inadequate for this kind of logic.

Token merging has emerged as a new paradigm that can accelerate the inference of Vision Transformers (ViTs) without any retraining or fine-tuning. To push the frontier of training-free acceleration in ViTs, we improve token merging by adding the perspectives of 1) activation outliers and 2) hierarchical representations. Through a careful analysis of the attention behavior in ViTs, we characterize a delayed onset of the convergent attention phenomenon, which makes token merging undesirable in the bottom blocks of ViTs. Moreover, we augment token merging with a hierarchical processing scheme to capture multi-scale redundancy between visual tokens. Combining these two insights, we build a unified inference framework called DSM: Delayed Spatial Merging. We extensively evaluate DSM on various ViT model scales (Tiny to Huge) and tasks (ImageNet-1k and transfer learning), achieving up to 1.8$\times$ FLOP reduction and 1.6$\times$ throughput speedup at a negligible loss while being two orders of magnitude faster than existing methods.

Given a graph $G=(V,E)$ and an integer $k\in \mathbb{N}$, we investigate the 2-Eigenvalue Vertex Deletion (2-EVD) problem. The objective is to remove at most $k$ vertices such that the adjacency matrix of the resulting graph has at most two eigenvalues. It is established that the adjacency matrix of a graph has at most two eigenvalues if and only if the graph is a collection of equal-sized cliques. Thus, the 2-Eigenvalue Vertex Deletion amounts to removing a set of at most $k$ vertices to transform the graph into a collection of equal-sized cliques. The 2-Eigenvalue Edge Editing (2-EEE), 2-Eigenvalue Edge Deletion (2-EED) and 2-Eigenvalue Edge Addition (2-EEA) problems are defined analogously. We present a kernel of size $\mathcal{O}(k^{3})$ for $2$-EVD, along with an FPT algorithm with a running time of $\mathcal{O}^{*}(2^{k})$. For the problem $2$-EEE, we provide a kernel of size $\mathcal{O}(k^{2})$. Additionally, we present linear kernels of size $5k$ and $6k$ for $2$-EEA and $2$-EED respectively. For the $2$-EED, we also construct an algorithm with running time $\mathcal{O}^{*}(1.47^{k})$ . These results address open questions posed by Misra et al. (ISAAC 2023) regarding the complexity of these problems when parameterized by the solution size.

To analyze the worst-case running time of branching algorithms, the majority of work in exponential time algorithms focuses on designing complicated branching rules over developing better analysis methods for simple algorithms. In the mid-$2000$s, Fomin et al. [2005] introduced measure & conquer, an advanced general analysis method, sparking widespread adoption for obtaining tighter worst-case running time upper bounds for many fundamental NP-complete problems. Yet, much potential in this direction remains untapped, as most subsequent work applied it without further advancement. Motivated by this, we present piecewise analysis, a new general method that analyzes the running time of branching algorithms. Our approach is to define a similarity ratio that divides instances into groups and then analyze the running time within each group separately. The similarity ratio is a scale between two parameters of an instance I. Instead of relying on a single measure and a single analysis for the whole instance space, our method allows to take advantage of different intrinsic properties of instances with different similarity ratios. To showcase its potential, we reanalyze two $17$-year-old algorithms from Fomin et al. [2007] that solve $4$-Coloring and #$3$-Coloring respectively. The original analysis in their paper gave running times of $O(1.7272^n)$ and $O(1.6262^n)$ respectively for these algorithms, our analysis improves these running times to $O(1.7207^n)$ and $O(1.6225^n)$.

In the Steiner Tree problem we are given an undirected edge-weighted graph as input, along with a set $K$ of vertices called terminals. The task is to output a minimum-weight connected subgraph that spans all the terminals. The famous Dreyfus-Wagner algorithm running in $3^{|K|} \mathsf{poly}(n)$ time shows that the problem is fixed-parameter tractable parameterized by the number of terminals. We present fixed-parameter tractable algorithms for Steiner Tree using structurally smaller parameterizations. Our first result concerns the parameterization by a multiway cut $S$ of the terminals, which is a vertex set $S$ (possibly containing terminals) such that each connected component of $G-S$ contains at most one terminal. We show that Steiner Tree can be solved in $2^{O(|S|\log|S|)}\mathsf{poly}(n)$ time and polynomial space, where $S$ is a minimum multiway cut for $K$. The algorithm is based on the insight that, after guessing how an optimal Steiner tree interacts with a multiway cut $S$, computing a minimum-cost solution of this type can be formulated as minimum-cost bipartite matching. Our second result concerns a new hybrid parameterization called $K$-free treewidth that simultaneously refines the number of terminals $|K|$ and the treewidth of the input graph. By utilizing recent work on $\mathcal{H}$-Treewidth in order to find a corresponding decomposition of the graph, we give an algorithm that solves Steiner Tree in time $2^{O(k)} \mathsf{poly}(n)$, where $k$ denotes the $K$-free treewidth of the input graph. To obtain this running time, we show how the rank-based approach for solving Steiner Tree parameterized by treewidth can be extended to work in the setting of $K$-free treewidth, by exploiting existing algorithms parameterized by $|K|$ to compute the table entries of leaf bags of a tree $K$-free decomposition.

We prove impossibility results for adaptivity in non-smooth stochastic convex optimization. Given a set of problem parameters we wish to adapt to, we define a "price of adaptivity" (PoA) that, roughly speaking, measures the multiplicative increase in suboptimality due to uncertainty in these parameters. When the initial distance to the optimum is unknown but a gradient norm bound is known, we show that the PoA is at least logarithmic for expected suboptimality, and double-logarithmic for median suboptimality. When there is uncertainty in both distance and gradient norm, we show that the PoA must be polynomial in the level of uncertainty. Our lower bounds nearly match existing upper bounds, and establish that there is no parameter-free lunch. En route, we also establish tight upper and lower bounds for (known-parameter) high-probability stochastic convex optimization with heavy-tailed and bounded noise, respectively.

A subset of points in a metric space is said to resolve it if each point in the space is uniquely characterized by its distance to each point in the subset. In particular, resolving sets can be used to represent points in abstract metric spaces as Euclidean vectors. Importantly, due to the triangle inequality, points close by in the space are represented as vectors with similar coordinates, which may find applications in classification problems of symbolic objects under suitably chosen metrics. In this manuscript, we address the resolvability of Jaccard spaces, i.e., metric spaces of the form $(2^X,\text{Jac})$, where $2^X$ is the power set of a finite set $X$, and $\text{Jac}$ is the Jaccard distance between subsets of $X$. Specifically, for different $a,b\in 2^X$, $\text{Jac}(a,b)=|a\Delta b|/|a\cup b|$, where $|\cdot|$ denotes size (i.e., cardinality) and $\Delta$ denotes the symmetric difference of sets. We combine probabilistic and linear algebra arguments to construct highly likely but nearly optimal (i.e., of minimal size) resolving sets of $(2^X,\text{Jac})$. In particular, we show that the metric dimension of $(2^X,\text{Jac})$, i.e., the minimum size of a resolving set of this space, is $\Theta(|X|/\ln|X|)$. In addition, we show that a much smaller subset of $2^X$ suffices to resolve, with high probability, all different pairs of subsets of $X$ of cardinality at most $\sqrt{|X|}/\ln|X|$, up to a factor.

Graph Convolutional Networks (GCNs) have been widely applied in various fields due to their significant power on processing graph-structured data. Typical GCN and its variants work under a homophily assumption (i.e., nodes with same class are prone to connect to each other), while ignoring the heterophily which exists in many real-world networks (i.e., nodes with different classes tend to form edges). Existing methods deal with heterophily by mainly aggregating higher-order neighborhoods or combing the immediate representations, which leads to noise and irrelevant information in the result. But these methods did not change the propagation mechanism which works under homophily assumption (that is a fundamental part of GCNs). This makes it difficult to distinguish the representation of nodes from different classes. To address this problem, in this paper we design a novel propagation mechanism, which can automatically change the propagation and aggregation process according to homophily or heterophily between node pairs. To adaptively learn the propagation process, we introduce two measurements of homophily degree between node pairs, which is learned based on topological and attribute information, respectively. Then we incorporate the learnable homophily degree into the graph convolution framework, which is trained in an end-to-end schema, enabling it to go beyond the assumption of homophily. More importantly, we theoretically prove that our model can constrain the similarity of representations between nodes according to their homophily degree. Experiments on seven real-world datasets demonstrate that this new approach outperforms the state-of-the-art methods under heterophily or low homophily, and gains competitive performance under homophily.

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.

Humans perceive the world by concurrently processing and fusing high-dimensional inputs from multiple modalities such as vision and audio. Machine perception models, in stark contrast, are typically modality-specific and optimised for unimodal benchmarks, and hence late-stage fusion of final representations or predictions from each modality (`late-fusion') is still a dominant paradigm for multimodal video classification. Instead, we introduce a novel transformer based architecture that uses `fusion bottlenecks' for modality fusion at multiple layers. Compared to traditional pairwise self-attention, our model forces information between different modalities to pass through a small number of bottleneck latents, requiring the model to collate and condense the most relevant information in each modality and only share what is necessary. We find that such a strategy improves fusion performance, at the same time reducing computational cost. We conduct thorough ablation studies, and achieve state-of-the-art results on multiple audio-visual classification benchmarks including Audioset, Epic-Kitchens and VGGSound. All code and models will be released.