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

Consider $n$ iid real-valued random vectors of size $k$ having iid coordinates with a general distribution function $F$. A vector is a maximum if and only if there is no other vector in the sample which weakly dominates it in all coordinates. Let $p_{k,n}$ be the probability that the first vector is a maximum. The main result of the present paper is that if $k\equiv k_n$ is growing at a slower (faster) rate than a certain factor of $\log(n)$, then $p_{k,n} \rightarrow 0$ (resp. $p_{k,n}\rightarrow1$) as $n\to\infty$. Furthermore, the factor is fully characterized as a functional of $F$. We also study the effect of $F$ on $p_{k,n}$, showing that while $p_{k,n}$ may be highly affected by the choice of $F$, the phase transition is the same for all distribution functions up to a constant factor.

Hyperdimensional Computing (HDC), also known as Vector-Symbolic Architectures (VSA), is a promising framework for the development of cognitive architectures and artificial intelligence systems, as well as for technical applications and emerging neuromorphic and nanoscale hardware. HDC/VSA operate with hypervectors, i.e., distributed vector representations of large fixed dimension (usually > 1000). One of the key ingredients of HDC/VSA are the methods for encoding data of various types (from numeric scalars and vectors to graphs) into hypervectors. In this paper, we propose an approach for the formation of hypervectors of sequences that provides both an equivariance with respect to the shift of sequences and preserves the similarity of sequences with identical elements at nearby positions. Our methods represent the sequence elements by compositional hypervectors and exploit permutations of hypervectors for representing the order of sequence elements. We experimentally explored the proposed representations using a diverse set of tasks with data in the form of symbolic strings. Although our approach is feature-free as it forms the hypervector of a sequence from the hypervectors of its symbols at their positions, it demonstrated the performance on a par with the methods that apply various features, such as subsequences. The proposed techniques were designed for the HDC/VSA model known as Sparse Binary Distributed Representations. However, they can be adapted to hypervectors in formats of other HDC/VSA models, as well as for representing sequences of types other than symbolic strings.

Blades manufactured through flank and point milling will likely exhibit geometric variability. Gauging the aerodynamic repercussions of such variability, prior to manufacturing a component, is challenging enough, let alone trying to predict what the amplified impact of any in-service degradation will be. While rules of thumb that govern the tolerance band can be devised based on expected boundary layer characteristics at known regions and levels of degradation, it remains a challenge to translate these insights into quantitative bounds for manufacturing. In this work, we tackle this challenge by leveraging ideas from dimension reduction to construct low-dimensional representations of aerodynamic performance metrics. These low-dimensional models can identify a subspace which contains designs that are invariant in performance -- the inactive subspace. By sampling within this subspace, we design techniques for drafting manufacturing tolerances and for quantifying whether a scanned component should be used or scrapped. We introduce the blade envelope as a computational manufacturing guide for a blade that is also amenable to qualitative visualizations. In this paper, the first of two parts, we discuss its underlying concept and detail its computational methodology, assuming one is interested only in the single objective of ensuring that the loss of all manufactured blades remains constant. To demonstrate the utility of our ideas we devise a series of computational experiments with the Von Karman Institute's LS89 turbine blade.

We study the phase synchronization problem with noisy measurements $Y=z^*z^{*H}+\sigma W\in\mathbb{C}^{n\times n}$, where $z^*$ is an $n$-dimensional complex unit-modulus vector and $W$ is a complex-valued Gaussian random matrix. It is assumed that each entry $Y_{jk}$ is observed with probability $p$. We prove that an SDP relaxation of the MLE achieves the error bound $(1+o(1))\frac{\sigma^2}{2np}$ under a normalized squared $\ell_2$ loss. This result matches the minimax lower bound of the problem, and even the leading constant is sharp. The analysis of the SDP is based on an equivalent non-convex programming whose solution can be characterized as a fixed point of the generalized power iteration lifted to a higher dimensional space. This viewpoint unifies the proofs of the statistical optimality of three different methods: MLE, SDP, and generalized power method. The technique is also applied to the analysis of the SDP for $\mathbb{Z}_2$ synchronization, and we achieve the minimax optimal error $\exp\left(-(1-o(1))\frac{np}{2\sigma^2}\right)$ with a sharp constant in the exponent.

Let $Q_{n}^{r}$ be the graph with vertex set $\{-1,1\}^{n}$ in which two vertices are joined if their Hamming distance is at most $r$. The edge-isoperimetric problem for $Q_{n}^{r}$ is that: For every $(n,r,M)$ such that $1\le r\le n$ and $1\le M\le2^{n}$, determine the minimum edge-boundary size of a subset of vertices of $Q_{n}^{r}$ with a given size $M$. In this paper, we apply two different approaches to prove bounds for this problem. The first approach is a linear programming approach and the second is a probabilistic approach. Our bound derived by the first approach generalizes the tight bound for $M=2^{n-1}$ derived by Kahn, Kalai, and Linial in 1989. Moreover, our bound is also tight for $M=2^{n-2}$ and $r\le\frac{n}{2}-1$. Our bounds derived by the second approach are expressed in terms of the \emph{noise stability}, and they are shown to be asymptotically tight as $n\to\infty$ when $r=2\lfloor\frac{\beta n}{2}\rfloor+1$ and $M=\lfloor\alpha2^{n}\rfloor$ for fixed $\alpha,\beta\in(0,1)$, and is tight up to a factor $2$ when $r=2\lfloor\frac{\beta n}{2}\rfloor$ and $M=\lfloor\alpha2^{n}\rfloor$. In fact, the edge-isoperimetric problem is equivalent to a ball-noise stability problem which is a variant of the traditional (i.i.d.-) noise stability problem. Our results can be interpreted as bounds for the ball-noise stability problem.

Learning a graph topology to reveal the underlying relationship between data entities plays an important role in various machine learning and data analysis tasks. Under the assumption that structured data vary smoothly over a graph, the problem can be formulated as a regularised convex optimisation over a positive semidefinite cone and solved by iterative algorithms. Classic methods require an explicit convex function to reflect generic topological priors, e.g. the $\ell_1$ penalty for enforcing sparsity, which limits the flexibility and expressiveness in learning rich topological structures. We propose to learn a mapping from node data to the graph structure based on the idea of learning to optimise (L2O). Specifically, our model first unrolls an iterative primal-dual splitting algorithm into a neural network. The key structural proximal projection is replaced with a variational autoencoder that refines the estimated graph with enhanced topological properties. The model is trained in an end-to-end fashion with pairs of node data and graph samples. Experiments on both synthetic and real-world data demonstrate that our model is more efficient than classic iterative algorithms in learning a graph with specific topological properties.

We address the question of characterizing and finding optimal representations for supervised learning. Traditionally, this question has been tackled using the Information Bottleneck, which compresses the inputs while retaining information about the targets, in a decoder-agnostic fashion. In machine learning, however, our goal is not compression but rather generalization, which is intimately linked to the predictive family or decoder of interest (e.g. linear classifier). We propose the Decodable Information Bottleneck (DIB) that considers information retention and compression from the perspective of the desired predictive family. As a result, DIB gives rise to representations that are optimal in terms of expected test performance and can be estimated with guarantees. Empirically, we show that the framework can be used to enforce a small generalization gap on downstream classifiers and to predict the generalization ability of neural networks.

Unsupervised (or self-supervised) graph representation learning is essential to facilitate various graph data mining tasks when external supervision is unavailable. The challenge is to encode the information about the graph structure and the attributes associated with the nodes and edges into a low dimensional space. Most existing unsupervised methods promote similar representations across nodes that are topologically close. Recently, it was shown that leveraging additional graph-level information, e.g., information that is shared among all nodes, encourages the representations to be mindful of the global properties of the graph, which greatly improves their quality. However, in most graphs, there is significantly more structure that can be captured, e.g., nodes tend to belong to (multiple) clusters that represent structurally similar nodes. Motivated by this observation, we propose a graph representation learning method called Graph InfoClust (GIC), that seeks to additionally capture cluster-level information content. These clusters are computed by a differentiable K-means method and are jointly optimized by maximizing the mutual information between nodes of the same clusters. This optimization leads the node representations to capture richer information and nodal interactions, which improves their quality. Experiments show that GIC outperforms state-of-art methods in various downstream tasks (node classification, link prediction, and node clustering) with a 0.9% to 6.1% gain over the best competing approach, on average.

Knowledge distillation is typically conducted by training a small model (the student) to mimic a large and cumbersome model (the teacher). The idea is to compress the knowledge from the teacher by using its output probabilities as soft-labels to optimize the student. However, when the teacher is considerably large, there is no guarantee that the internal knowledge of the teacher will be transferred into the student; even if the student closely matches the soft-labels, its internal representations may be considerably different. This internal mismatch can undermine the generalization capabilities originally intended to be transferred from the teacher to the student. In this paper, we propose to distill the internal representations of a large model such as BERT into a simplified version of it. We formulate two ways to distill such representations and various algorithms to conduct the distillation. We experiment with datasets from the GLUE benchmark and consistently show that adding knowledge distillation from internal representations is a more powerful method than only using soft-label distillation.

We present the problem of selecting relevant premises for a proof of a given statement. When stated as a binary classification task for pairs (conjecture, axiom), it can be efficiently solved using artificial neural networks. The key difference between our advance to solve this problem and previous approaches is the use of just functional signatures of premises. To further improve the performance of the model, we use dimensionality reduction technique, to replace long and sparse signature vectors with their compact and dense embedded versions. These are obtained by firstly defining the concept of a context for each functor symbol, and then training a simple neural network to predict the distribution of other functor symbols in the context of this functor. After training the network, the output of its hidden layer is used to construct a lower dimensional embedding of a functional signature (for each premise) with a distributed representation of features. This allows us to use 512-dimensional embeddings for conjecture-axiom pairs, containing enough information about the original statements to reach the accuracy of 76.45% in premise selection task, only with simple two-layer densely connected neural networks.