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

Neural networks with the Rectified Linear Unit (ReLU) nonlinearity are described by a vector of parameters $\theta$, and realized as a piecewise linear continuous function $R_{\theta}: x \in \mathbb R^{d} \mapsto R_{\theta}(x) \in \mathbb R^{k}$. Natural scalings and permutations operations on the parameters $\theta$ leave the realization unchanged, leading to equivalence classes of parameters that yield the same realization. These considerations in turn lead to the notion of identifiability -- the ability to recover (the equivalence class of) $\theta$ from the sole knowledge of its realization $R_{\theta}$. The overall objective of this paper is to introduce an embedding for ReLU neural networks of any depth, $\Phi(\theta)$, that is invariant to scalings and that provides a locally linear parameterization of the realization of the network. Leveraging these two key properties, we derive some conditions under which a deep ReLU network is indeed locally identifiable from the knowledge of the realization on a finite set of samples $x_{i} \in \mathbb R^{d}$. We study the shallow case in more depth, establishing necessary and sufficient conditions for the network to be identifiable from a bounded subset $\mathcal X \subseteq \mathbb R^{d}$.

Matrix sparsification is a well-known approach in the design of efficient algorithms, where one approximates a matrix $A$ with a sparse matrix $A'$. Achlioptas and McSherry [2007] initiated a long line of work on spectral-norm sparsification, which aims to guarantee that $\|A'-A\|\leq \epsilon \|A\|$ for error parameter $\epsilon>0$. Various forms of matrix approximation motivate considering this problem with a guarantee according to the Schatten $p$-norm for general $p$, which includes the spectral norm as the special case $p=\infty$. We investigate the relation between fixed but different $p\neq q$, that is, whether sparsification in Schatten $p$-norm implies (existentially and/or algorithmically) sparsification in Schatten $q$-norm with similar sparsity. An affirmative answer could be tremendously useful, as it will identify which value of $p$ to focus on. Our main finding is a surprising contrast between this question and the analogous case of $\ell_p$-norm sparsification for vectors: For vectors, the answer is affirmative for $p<q$ and negative for $p>q$, but for matrices we answer negatively for almost all $p\neq q$.

We present an algorithm for the maximum matching problem in dynamic (insertion-deletions) streams with *asymptotically optimal* space complexity: for any $n$-vertex graph, our algorithm with high probability outputs an $\alpha$-approximate matching in a single pass using $O(n^2/\alpha^3)$ bits of space. A long line of work on the dynamic streaming matching problem has reduced the gap between space upper and lower bounds first to $n^{o(1)}$ factors [Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to $\text{polylog}{(n)}$ factors [Dark-Konrad; CCC 2020]. Our upper bound now matches the Dark-Konrad lower bound up to $O(1)$ factors, thus completing this research direction. Our approach consists of two main steps: we first (provably) identify a family of graphs, similar to the instances used in prior work to establish the lower bounds for this problem, as the only "hard" instances to focus on. These graphs include an induced subgraph which is both sparse and contains a large matching. We then design a dynamic streaming algorithm for this family of graphs which is more efficient than prior work. The key to this efficiency is a novel sketching method, which bypasses the typical loss of $\text{polylog}{(n)}$-factors in space compared to standard $L_0$-sampling primitives, and can be of independent interest in designing optimal algorithms for other streaming problems.

Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedeness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defined over the intersections of FE mesh elements with a ball of radius $\delta$, where $\delta$ is the range of nonlocal interaction. Identifying and parameterizing these intersections is a nontrivial computational geometry problem. In this work, we propose a quadrature technique where the inner integration is performed using quadrature points distributed over the full ball, without regard for how it intersects elements, and weights are computed based on the generalized moving least squares method. Thus, as opposed to all previously employed methods, our technique does not require element-by-element integration and fully circumvents the computation of element-ball intersections. This paper considers one- and two-dimensional implementations of piecewise linear continuous FE approximations, focusing on the case where the element size h and the nonlocal radius $\delta$ are proportional, as is typical of practical computations. When boundary conditions are treated carefully and the outer integral of the variational form is computed accurately, the proposed method is asymptotically compatible in the limit of $h \sim \delta \to 0$, featuring at least first-order convergence in L^2 for all dimensions, using both uniform and nonuniform grids.

We prove a bound of $O( k (n+m)\log^{d-1})$ on the number of incidences between $n$ points and $m$ axis parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points. That is, the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work by a factor of $\log^d n$, for $d >2$. We also study other variants of the problem. For halfspaces, using shallow cuttings, we get a near linear bound in two and three dimensions. Finally, we present near linear bound for the case of shapes in the plane with low union complexity (e.g. fat triangles).

The low-rank matrix approximation problem is ubiquitous in computational mathematics. Traditionally, this problem is solved in spectral or Frobenius norms, where the accuracy of the approximation is related to the rate of decrease of the singular values of the matrix. However, recent results indicate that this requirement is not necessary for other norms. In this paper, we propose a method for solving the low-rank approximation problem in the Chebyshev norm, which is capable of efficiently constructing accurate approximations for matrices, whose singular values do not decrease or decrease slowly.

Kernel matrices, which arise from discretizing a kernel function $k(x,x')$, have a variety of applications in mathematics and engineering. Classically, the celebrated fast multipole method was designed to perform matrix multiplication on kernel matrices of dimension $N$ in time almost linear in $N$ by using techniques later generalized into the linear algebraic framework of hierarchical matrices. In light of this success, we propose a quantum algorithm for efficiently performing matrix operations on hierarchical matrices by implementing a quantum block-encoding of the hierarchical matrix structure. When applied to many kernel matrices, our quantum algorithm can solve quantum linear systems of dimension $N$ in time $O(\kappa \operatorname{polylog}(\frac{N}{\varepsilon}))$, where $\kappa$ and $\varepsilon$ are the condition number and error bound of the matrix operation. This runtime is exponentially faster than any existing quantum algorithms for implementing dense kernel matrices. Finally, we discuss possible applications of our methodology in solving integral equations or accelerating computations in N-body problems.

Starting from the local structures to study hierarchical trees is a common research method. However, the cumbersome analysis and description make the naive method challenging to adapt to the increasingly complex hierarchical tree problems. To improve the efficiency of hierarchical tree research, we propose an embeddable matrix representation for hierarchical trees, called Generation Matrix. It can transform the abstract hierarchical tree into a concrete matrix representation and then take the hierarchical tree as a whole to study, which dramatically reduces the complexity of research. Mathematical analysis shows that Generation Matrix can simulate various recursive algorithms without accessing local structures and provides a variety of interpretable matrix operations to support the research of hierarchical trees. Applying Generation Matrix to differential privacy hierarchical tree release, we propose a Generation Matrix-based optimally consistent release algorithm (GMC). It provides an exceptionally concise process description so that we can describe its core steps as a simple matrix expression rather than multiple complicated recursive processes like existing algorithms. Our experiments show that GMC takes only a few seconds to complete a release for large-scale datasets with more than 10 million nodes. The calculation efficiency is increased by up to 100 times compared with the state-of-the-art schemes.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.