Convergence bounds for empirical nonlinear least-squares

We consider best approximation problems in a nonlinear subset [[EQUATION]] of a Banach space of functions [[EQUATION]] . The norm is assumed to be a generalization of the [[EQUATION]] -norm for which only a weighted Monte Carlo estimate [[EQUATION]] can be computed. The objective is to obtain an approximation [[EQUATION]] of an unknown function [[EQUATION]] by minimizing the empirical norm [[EQUATION]] . We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the nonlinear least squares setting. Several model classes are examined where analytical statements can be made about the RIP and the results are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.

Bounds on the Lattice Point Enumerator via Slices and Projections

Gardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

On 2-rainbow domination number of functigraph and its complement

Let \(G\) be a graph and \(f:V (G)\rightarrow P(\{1,2\})\) be a function where for every vertex \(v\in V(G)\), with \(f(v)=\emptyset\) we have \(\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}\). Then \(f\) is a \(2\)-rainbow dominating function or a \(2RDF\) of \(G\). The weight of \(f\) is \(\omega(f)=\sum_{v\in V(G)} |f(v)|\). The minimum weight of all \(2\)-rainbow dominating functions is \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{r2}(G)\). Let \(G_1\) and \(G_2\) be two copies of a graph G with disjoint vertex sets \(V(G_1)\) and \(V(G_2)\), and let \(\sigma\) be a function from \(V(G_1)\) to \(V(G_2)\). We define the functigraph \(C(G,\sigma)\) to be the graph that has the vertex set \(V(C(G,\sigma)) = V(G_1)\cup V(G_2)\), and the edge set \(E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}\). In this paper, \(2\)-rainbow domination number of the functigraph of \(C(G,\sigma)\) and its complement are investigated. We obtain a general bound for \(\gamma_{r2}(C(G,\sigma))\) and we show that this bound is sharp.

On Perimeters and Volumes of Fattened Sets

In this paper we analyze the shape of fattened sets; given a compact set C⊂RN let Cr be its r-fattened set; we prove a general bound rP(Cr)≤NL({Cr∖C}) between the perimeter of Cr and the Lebesgue measure of Cr∖C. We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, P(Cr) is integrable for r∈(0,a). We further show that for any integrable continuous decreasing function ψ:(0,1/2)→(0,∞) there exists a compact set C⊂RN such that P(Cr)≥ψ(r). These results solve a conjecture left open in (Mennucci and Duci, 2015) and provide new insight in applications where the fattened set plays an important role.

Universal bound on sampling bosons in linear optics and its computational implications

ABSTRACT In linear optics, photons are scattered in a network through passive optical elements including beam splitters and phase shifters, leading to many intriguing applications in physics, such as Mach–Zehnder interferometry, the Hong–Ou–Mandel effect, and tests of fundamental quantum mechanics. Here we present the fundamental limit in the transition amplitudes of bosons, applicable to all physical linear optical networks. Apart from boson sampling, this transition bound results in many other interesting applications, including behaviors of Bose–Einstein condensates (BEC) in optical networks, counterparts of Hong–Ou–Mandel effects for multiple photons, and approximating permanents of matrices. In addition, this general bound implies the existence of a polynomial-time randomized algorithm for estimating the transition amplitudes of bosons, which represents a solution to an open problem raised by Aaronson and Hance (Quantum Inf Comput 2012; 14: 541–59). Consequently, this bound implies that computational decision problems encoded in linear optics, prepared and detected in the Fock basis, can be solved efficiently by classical computers within additive errors. Furthermore, our result also leads to a classical sampling algorithm that can be applied to calculate the many-body wave functions and the S-matrix of bosonic particles.

GOLDFELD’S CONJECTURE AND CONGRUENCES BETWEEN HEEGNER POINTS

Given an elliptic curve$E$over$\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (respectively 1). We show that this conjecture holds whenever$E$has a rational 3-isogeny. We also prove the analogous result for the sextic twists of$j$-invariant 0 curves. For a more general elliptic curve$E$, we show that the number of quadratic twists of$E$up to twisting discriminant$X$of analytic rank 0 (respectively 1) is$\gg X/\log ^{5/6}X$, improving the current best general bound toward Goldfeld’s conjecture due to Ono–Skinner (respectively Perelli–Pomykala). To prove these results, we establish a congruence formula between$p$-adic logarithms of Heegner points and apply it in the special cases$p=3$and$p=2$to construct the desired twists explicitly. As a by-product, we also prove the corresponding$p$-part of the Birch and Swinnerton–Dyer conjecture for these explicit twists.

THE DOMINATION GAME ON SPLIT GRAPHS

We investigate the domination game and the game domination number $\unicode[STIX]{x1D6FE}_{g}$ in the class of split graphs. We prove that $\unicode[STIX]{x1D6FE}_{g}(G)\leq n/2$ for any isolate-free $n$-vertex split graph $G$, thus strengthening the conjectured $3n/5$ general bound and supporting Rall’s $\lceil n/2\rceil$-conjecture. We also characterise split graphs of even order with $\unicode[STIX]{x1D6FE}_{g}(G)=n/2$.

On the Strong Chromatic Index of Sparse Graphs

The strong chromatic index of a graph $G$, denoted $\chi'_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi'_{s,\ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors.We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with ${\rm girth}(G) \geq 41$ then $\chi'_{s,\ell}(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and ${\rm girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper.Finally, if $G$ is a planar graph with maximum degree at most four and ${\rm girth}(G) \geq 28$, then $\chi'_s(G)N \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its applications to the strong edge coloring, Applied Mathematics and Computation, 325 (2018), 246-251] in this case.