In this paper, we describe an algorithm for approximating functions of the form $f(x)=\int_{a}^{b} x^{\mu} \sigma(\mu) \, d \mu$ over $[0,1] \subset \mathbb{R}$, where $0<a<b<\infty$ and $\sigma(\mu)$ is some signed Radon measure over $[a,b]$ or some distribution supported on $[a,b]$. Given the desired accuracy $\epsilon$ and the values of $a$ and $b$, our method determines a priori a collection of non-integer powers $\{t_j\}_{j=1}^N$, so that the functions are approximated by series of the form $f(x)\approx \sum_{j=1}^N c_j x^{t_j}$, where the expansion coefficients can be found by solving a square, low-dimensional Vandermonde-like linear system using the collocation points $\{x_j\}_{j=1}^N$, also determined a priori by $\epsilon$ and the values of $a$ and $b$. We prove that our method has a small uniform approximation error which is proportional to $\epsilon$ multiplied by some small constants. We demonstrate the performance of our algorithm with several numerical experiments, and show that the number of singular powers and collocation points grows as $N=O(\log{\frac{1}{\epsilon}})$.
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In generative compressed sensing (GCS), we want to recover a signal $\mathbf{x}^* \in \mathbb{R}^n$ from $m$ measurements ($m\ll n$) using a generative prior $\mathbf{x}^*\in G(\mathbb{B}_2^k(r))$, where $G$ is typically an $L$-Lipschitz continuous generative model and $\mathbb{B}_2^k(r)$ represents the radius-$r$ $\ell_2$-ball in $\mathbb{R}^k$. Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $\mathbf{x}^*$ rather than for all $\mathbf{x}^*$ simultaneously. In this paper, we build a unified framework to derive uniform recovery guarantees for nonlinear GCS where the observation model is nonlinear and possibly discontinuous or unknown. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. Specifically, using a single realization of the sensing ensemble and generalized Lasso, {\em all} $\mathbf{x}^*\in G(\mathbb{B}_2^k(r))$ can be recovered up to an $\ell_2$-error at most $\epsilon$ using roughly $\tilde{O}({k}/{\epsilon^2})$ samples, with omitted logarithmic factors typically being dominated by $\log L$. Notably, this almost coincides with existing non-uniform guarantees up to logarithmic factors, hence the uniformity costs very little. As part of our technical contributions, we introduce the Lipschitz approximation to handle discontinuous observation models. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy. Experimental results are presented to corroborate our theory.
Consider the {$\ell_{\alpha}$} regularized linear regression, also termed Bridge regression. For $\alpha\in (0,1)$, Bridge regression enjoys several statistical properties of interest such as sparsity and near-unbiasedness of the estimates (Fan and Li, 2001). However, the main difficulty lies in the non-convex nature of the penalty for these values of $\alpha$, which makes an optimization procedure challenging and usually it is only possible to find a local optimum. To address this issue, Polson et al. (2013) took a sampling based fully Bayesian approach to this problem, using the correspondence between the Bridge penalty and a power exponential prior on the regression coefficients. However, their sampling procedure relies on Markov chain Monte Carlo (MCMC) techniques, which are inherently sequential and not scalable to large problem dimensions. Cross validation approaches are similarly computation-intensive. To this end, our contribution is a novel \emph{non-iterative} method to fit a Bridge regression model. The main contribution lies in an explicit formula for Stein's unbiased risk estimate for the out of sample prediction risk of Bridge regression, which can then be optimized to select the desired tuning parameters, allowing us to completely bypass MCMC as well as computation-intensive cross validation approaches. Our procedure yields results in a fraction of computational times compared to iterative schemes, without any appreciable loss in statistical performance. An R implementation is publicly available online at: //github.com/loriaJ/Sure-tuned_BridgeRegression .
The multiple scattering theory (MST) is a Green's function method that has been widely used in electronic structure calculations for crystalline disordered systems. The key property of the MST method is the scattering path matrix (SPM) that characterizes the Green's function within a local solution representation. This paper studies various approximations of the SPM, under the condition that an appropriate reference is used for perturbation. In particular, we justify the convergence of the SPM approximations with respect to the size of scattering region and scattering length of reference, which are the central numerical parameters to achieve a linear scaling method with MST. We also present some numerical experiments on several typical systems to support the theory.
This paper studies the prediction of a target $\mathbf{z}$ from a pair of random variables $(\mathbf{x},\mathbf{y})$, where the ground-truth predictor is additive $\mathbb{E}[\mathbf{z} \mid \mathbf{x},\mathbf{y}] = f_\star(\mathbf{x}) +g_{\star}(\mathbf{y})$. We study the performance of empirical risk minimization (ERM) over functions $f+g$, $f \in F$ and $g \in G$, fit on a given training distribution, but evaluated on a test distribution which exhibits covariate shift. We show that, when the class $F$ is "simpler" than $G$ (measured, e.g., in terms of its metric entropy), our predictor is more resilient to $\textbf{heterogenous covariate shifts}$ in which the shift in $\mathbf{x}$ is much greater than that in $\mathbf{y}$. Our analysis proceeds by demonstrating that ERM behaves $\textbf{qualitatively similarly to orthogonal machine learning}$: the rate at which ERM recovers the $f$-component of the predictor has only a lower-order dependence on the complexity of the class $G$, adjusted for partial non-indentifiability introduced by the additive structure. These results rely on a novel H\"older style inequality for the Dudley integral which may be of independent interest. Moreover, we corroborate our theoretical findings with experiments demonstrating improved resilience to shifts in "simpler" features across numerous domains.
We make an experimental comparison of methods for computing the numerical radius of an $n\times n$ complex matrix, based on two well-known characterizations, the first a nonconvex optimization problem in one real variable and the second a convex optimization problem in $n^{2}+1$ real variables. We make comparisons with respect to both accuracy and computation time using publicly available software.
Letter-to-letter transducers are a standard formalism for modeling reactive systems. Often, two transducers that model similar systems differ locally from one another, by behaving similarly, up to permutations of the input and output letters within "rounds". In this work, we introduce and study notions of simulation by rounds and equivalence by rounds of transducers. In our setting, words are partitioned to consecutive subwords of a fixed length $k$, called rounds. Then, a transducer $\mathcal{T}_1$ is $k$-round simulated by transducer $\mathcal{T}_2$ if, intuitively, for every input word $x$, we can permute the letters within each round in $x$, such that the output of $\mathcal{T}_2$ on the permuted word is itself a permutation of the output of $\mathcal{T}_1$ on $x$. Finally, two transducers are $k$-round equivalent if they simulate each other. We solve two main decision problems, namely whether $\mathcal{T}_2$ $k$-round simulates $\mathcal{T}_1$ (1) when $k$ is given as input, and (2) for an existentially quantified $k$. We demonstrate the usefulness of the definitions by applying them to process symmetry: a setting in which a permutation in the identities of processes in a multi-process system naturally gives rise to two transducers, whose $k$-round equivalence corresponds to stability against such permutations.
Consider that there are $k\le n$ agents in a simple, connected, and undirected graph $G=(V,E)$ with $n$ nodes and $m$ edges. The goal of the dispersion problem is to move these $k$ agents to distinct nodes. Agents can communicate only when they are at the same node, and no other means of communication such as whiteboards are available. We assume that the agents operate synchronously. We consider two scenarios: when all agents are initially located at any single node (rooted setting) and when they are initially distributed over any one or more nodes (general setting). Kshemkalyani and Sharma presented a dispersion algorithm for the general setting, which uses $O(m_k)$ time and $\log(k+\delta)$ bits of memory per agent [OPODIS 2021]. Here, $m_k$ is the maximum number of edges in any induced subgraph of $G$ with $k$ nodes, and $\delta$ is the maximum degree of $G$. This algorithm is the fastest in the literature, as no algorithm with $o(m_k)$ time has been discovered even for the rooted setting. In this paper, we present faster algorithms for both the rooted and general settings. First, we present an algorithm for the rooted setting that solves the dispersion problem in $O(k\log \min(k,\delta))=O(k\log k)$ time using $O(\log \delta)$ bits of memory per agent. Next, we propose an algorithm for the general setting that achieves dispersion in $O(k (\log k)\cdot (\log \min(k,\delta))=O(k \log^2 k)$ time using $O(\log (k+\delta))$ bits.
A kernelization for a parameterized decision problem $\mathcal{Q}$ is a polynomial-time preprocessing algorithm that reduces any parameterized instance $(x,k)$ into an instance $(x',k')$ whose size is bounded by a function of $k$ alone and which has the same yes/no answer for $\mathcal{Q}$. Such preprocessing algorithms cannot exist in the context of counting problems, when the answer to be preserved is the number of solutions, since this number can be arbitrarily large compared to $k$. However, we show that for counting minimum feedback vertex sets of size at most $k$, and for counting minimum dominating sets of size at most $k$ in a planar graph, there is a polynomial-time algorithm that either outputs the answer or reduces to an instance $(G',k')$ of size polynomial in $k$ with the same number of minimum solutions. This shows that a meaningful theory of kernelization for counting problems is possible and opens the door for future developments. Our algorithms exploit that if the number of solutions exceeds $2^{\mathsf{poly}(k)}$, the size of the input is exponential in terms of $k$ so that the running time of a parameterized counting algorithm can be bounded by $\mathsf{poly}(n)$. Otherwise, we can use gadgets that slightly increase $k$ to represent choices among $2^{O(k)}$ options by only $\mathsf{poly}(k)$ vertices.
In this paper, we prove the following non-linear generalization of the classical Sylvester-Gallai theorem. Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, and $\mathcal{F}=\{F_1,\cdots,F_m\} \subset \mathbb{K}[x_1,\cdots,x_N]$ be a set of irreducible homogeneous polynomials of degree at most $d$ such that $F_i$ is not a scalar multiple of $F_j$ for $i\neq j$. Suppose that for any two distinct $F_i,F_j\in \mathcal{F}$, there is $k\neq i,j$ such that $F_k\in \mathrm{rad}(F_i,F_j)$. We prove that such radical SG configurations must be low dimensional. More precisely, we show that there exists a function $\lambda : \mathbb{N} \to \mathbb{N}$, independent of $\mathbb{K},N$ and $m$, such that any such configuration $\mathcal{F}$ must satisfy $$ \dim (\mathrm{span}_{\mathbb{K}}{\mathcal{F}}) \leq \lambda(d). $$ Our result confirms a conjecture of Gupta [Gup14, Conjecture 2] and generalizes the quadratic and cubic Sylvester-Gallai theorems of [S20,OS22]. Our result takes us one step closer towards the first deterministic polynomial time algorithm for the Polynomial Identity Testing (PIT) problem for depth-4 circuits of bounded top and bottom fanins. Our result, when combined with the Stillman uniformity type results of [AH20a,DLL19,ESS21], yields uniform bounds for several algebraic invariants such as projective dimension, Betti numbers and Castelnuovo-Mumford regularity of ideals generated by radical SG configurations.
The property that the velocity $\boldsymbol{u}$ belongs to $L^\infty(0,T;L^2(\Omega)^d)$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretisation methods are uniformly stable in the $L^\infty(0,T;L^2(\Omega)^d)$-norm. In this work, we establish that this is indeed the case for Discontinuous Galerkin (DG) discretisations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretisation is equivalent to the RadauIIA Implicit Runge-Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest.
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