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

Dependence is undoubtedly a central concept in statistics. Though, it proves difficult to locate in the literature a formal definition which goes beyond the self-evident 'dependence = non-independence'. This absence has allowed the term 'dependence' and its declination to be used vaguely and indiscriminately for qualifying a variety of disparate notions, leading to numerous incongruities. For example, the classical Pearson's, Spearman's or Kendall's correlations are widely regarded as 'dependence measures' of major interest, in spite of returning 0 in some cases of deterministic relationships between the variables at play, evidently not measuring dependence at all. Arguing that research on such a fundamental topic would benefit from a slightly more rigid framework, this paper suggests a general definition of the dependence between two random variables defined on the same probability space. Natural enough for aligning with intuition, that definition is still sufficiently precise for allowing unequivocal identification of a 'universal' representation of the dependence structure of any bivariate distribution. Links between this representation and familiar concepts are highlighted, and ultimately, the idea of a dependence measure based on that universal representation is explored and shown to satisfy Renyi's postulates.

Microring resonators (MRRs) are promising devices for time-delay photonic reservoir computing, but the impact of the different physical effects taking place in the MRRs on the reservoir computing performance is yet to be fully understood. We numerically analyze the impact of linear losses as well as thermo-optic and free-carrier effects relaxation times on the prediction error of the time-series task NARMA-10. We demonstrate the existence of three regions, defined by the input power and the frequency detuning between the optical source and the microring resonance, that reveal the cavity transition from linear to nonlinear regimes. One of these regions offers very low error in time-series prediction under relatively low input power and number of nodes while the other regions either lack nonlinearity or become unstable. This study provides insight into the design of the MRR and the optimization of its physical properties for improving the prediction performance of time-delay reservoir computing.

Subspace codes are the $q$-analog of binary block codes in the Hamming metric. Here the codewords are vector spaces over a finite field. They have e.g. applications in random linear network coding, distributed storage, and cryptography. In this chapter we survey known constructions and upper bounds for subspace codes.

We provide a non-unit disk framework to solve combinatorial optimization problems such as Maximum Cut (Max-Cut) and Maximum Independent Set (MIS) on a Rydberg quantum annealer. Our setup consists of a many-body interacting Rydberg system where locally controllable light shifts are applied to individual qubits in order to map the graph problem onto the Ising spin model. Exploiting the flexibility that optical tweezers offer in terms of spatial arrangement, our numerical simulations implement the local-detuning protocol while globally driving the Rydberg annealer to the desired many-body ground state, which is also the solution to the optimization problem. Using optimal control methods, these solutions are obtained for prototype graphs with varying sizes at time scales well within the system lifetime and with approximation ratios close to one. The non-blockade approach facilitates the encoding of graph problems with specific topologies that can be realized in two-dimensional Rydberg configurations and is applicable to both unweighted as well as weighted graphs. A comparative analysis with fast simulated annealing is provided which highlights the advantages of our scheme in terms of system size, hardness of the graph, and the number of iterations required to converge to the solution.

Deep neural networks (DNNs) often fail silently with over-confident predictions on out-of-distribution (OOD) samples, posing risks in real-world deployments. Existing techniques predominantly emphasize either the feature representation space or the gradient norms computed with respect to DNN parameters, yet they overlook the intricate gradient distribution and the topology of classification regions. To address this gap, we introduce GRadient-aware Out-Of-Distribution detection in interpolated manifolds (GROOD), a novel framework that relies on the discriminative power of gradient space to distinguish between in-distribution (ID) and OOD samples. To build this space, GROOD relies on class prototypes together with a prototype that specifically captures OOD characteristics. Uniquely, our approach incorporates a targeted mix-up operation at an early intermediate layer of the DNN to refine the separation of gradient spaces between ID and OOD samples. We quantify OOD detection efficacy using the distance to the nearest neighbor gradients derived from the training set, yielding a robust OOD score. Experimental evaluations substantiate that the introduction of targeted input mix-upamplifies the separation between ID and OOD in the gradient space, yielding impressive results across diverse datasets. Notably, when benchmarked against ImageNet-1k, GROOD surpasses the established robustness of state-of-the-art baselines. Through this work, we establish the utility of leveraging gradient spaces and class prototypes for enhanced OOD detection for DNN in image classification.

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Rational best approximations (in a Chebyshev sense) to real functions are characterized by an equioscillating approximation error. Similar results do not hold true for rational best approximations to complex functions in general. In the present work, we consider unitary rational approximations to the exponential function on the imaginary axis, which map the imaginary axis to the unit circle. In the class of unitary rational functions, best approximations are shown to exist, to be uniquely characterized by equioscillation of a phase error, and to possess a super-linear convergence rate. Furthermore, the best approximations have full degree (i.e., non-degenerate), achieve their maximum approximation error at points of equioscillation, and interpolate at intermediate points. Asymptotic properties of poles, interpolation nodes, and equioscillation points of these approximants are studied. Three algorithms, which are found very effective to compute unitary rational approximations including candidates for best approximations, are sketched briefly. Some consequences to numerical time-integration are discussed. In particular, time propagators based on unitary best approximants are unitary, symmetric and A-stable.

Classification of $N$ points becomes a simultaneous control problem when viewed through the lens of neural ordinary differential equations (neural ODEs), which represent the time-continuous limit of residual networks. For the narrow model, with one neuron per hidden layer, it has been shown that the task can be achieved using $O(N)$ neurons. In this study, we focus on estimating the number of neurons required for efficient cluster-based classification, particularly in the worst-case scenario where points are independently and uniformly distributed in $[0,1]^d$. Our analysis provides a novel method for quantifying the probability of requiring fewer than $O(N)$ neurons, emphasizing the asymptotic behavior as both $d$ and $N$ increase. Additionally, under the sole assumption that the data are in general position, we propose a new constructive algorithm that simultaneously classifies clusters of $d$ points from any initial configuration, effectively reducing the maximal complexity to $O(N/d)$ neurons.

In this note we prove almost sure unisolvence of RBF interpolation on randomly distributed sequences by a wide class of polyharmonic splines (including Thin-Plate Splines), without polynomial addition.

Interactions between genes and environmental factors may play a key role in the etiology of many common disorders. Several regularized generalized linear models (GLMs) have been proposed for hierarchical selection of gene by environment interaction (GEI) effects, where a GEI effect is selected only if the corresponding genetic main effect is also selected in the model. However, none of these methods allow to include random effects to account for population structure, subject relatedness and shared environmental exposure. In this paper, we develop a unified approach based on regularized penalized quasi-likelihood (PQL) estimation to perform hierarchical selection of GEI effects in sparse regularized mixed models. We compare the selection and prediction accuracy of our proposed model with existing methods through simulations under the presence of population structure and shared environmental exposure. We show that for all simulation scenarios, compared to other penalized methods, our proposed method enforced sparsity by controlling the number of false positives in the model while having the best predictive performance. Finally, we apply our method to a real data application using the Orofacial Pain: Prospective Evaluation and Risk Assessment (OPPERA) study, and found that our method retrieves previously reported significant loci.