Spectrum Bandit Optimization

We consider the problem of allocating radio channels to links in a wireless network. Links interact through interference, modelled as a conflict graph (i.e., two interfering links cannot be simultaneously active on the same channel). We aim at identifying the channel allocation maximizing the total network throughput over a finite time horizon. Should we know the average radio conditions on each channel and on each link, an optimal allocation would be obtained by solving an Integer Linear Program (ILP). When radio conditions are unknown a priori, we look for a sequential channel allocation policy that converges to the optimal allocation while minimizing on the way the throughput loss or {\it regret} due to the need for exploring sub-optimal allocations. We formulate this problem as a generic linear bandit problem, and analyze it first in a stochastic setting where radio conditions are driven by a stationary stochastic process, and then in an adversarial setting where radio conditions can evolve arbitrarily. We provide new algorithms in both settings and derive upper bounds on their regrets.Comment: 21 page

Spectral alignment of correlated Gaussian random matrices

In this paper we analyze a simple method ($EIG1$) for the problem of matrix alignment, consisting in aligning their leading eigenvectors: given $A$ and $B$, we compute $v_1$ and $v'_1$ two leading eigenvectors of $A$ and $B$. The algorithm returns a permutation $\hat{\Pi}$ such that the rank of the coordinate $\hat{\Pi}(i)$ in $v_1$ is the rank of the coordinate $i$ in $v'_1$ (up to the sign of $v'_1$). We consider a model where $A$ belongs to the Gaussian Orthogonal Ensemble (GOE), and $B= \Pi^T (A+\sigma H) \Pi$, where $\Pi$ is a permutation matrix and $H$ is an independent copy of $A$. We show the following 0-1 law: under the condition $\sigma N^{7/6+\epsilon} \to 0$, the $EIG1$ method recovers all but a vanishing part of the underlying permutation $\Pi$. When $\sigma N^{7/6-\epsilon} \to \infty$, this algorithm cannot recover more than $o(N)$ correct matches. This result gives an understanding of the simplest and fastest spectral method for matrix alignment (or complete weighted graph alignment), and involves proof methods and techniques which could be of independent interest.Comment: 29 pages, 4 figure

Low-threshold heterogeneously integrated InP/SOI lasers with a double adiabatic taper coupler

We report on a heterogeneously integrated InP/silicon-on-insulator (SOI) laser source realized through divinylsiloxane-bis-benzocyclobutene (DVS-BCB) wafer bonding. The hybrid lasers present several new features. The III-V waveguide has a width of only 1.7 mu m, reducing the power consumption of the device. The silicon waveguide thickness is 400 nm, compatible with high-performance modulator designs and allowing efficient coupling to a standard 220-nm high index contrast silicon waveguide layer. In order to make the mode coupling efficient, both the III-V waveguide and silicon waveguide are tapered, with a tip width for the III-V waveguide of around 800 nm. These new features lead to good laser performance: a lasing threshold as low as 30 mA and an output power of more than 4 mW at room temperature in continuous-wave operation regime. Continuous wave lasing up to 70 degrees C is obtained

Demonstration of a heterogeneously integrated III-V/SOI single wavelength tunable laser

A heterogeneously integrated III-V-on-silicon laser is reported, integrating a III-V gain section, a silicon ring resonator for wavelength selection and two silicon Bragg grating reflectors as back and front mirrors. Single wavelength operation with a side mode suppression ratio higher than 45 dB is obtained. An output power up to 10 mW at 20 °C and a thermo-optic wavelength tuning range of 8 nm are achieved. The laser linewidth is found to be 1.7 MHz

On rigidity, orientability and cores of random graphs with sliders

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph G ∈ G(n, c/n) the threshold value of c for the appearance of a linear-sized rigid component as a function of q, generalizing results of [7]. We show that this appearance of a giant component undergoes a continuous transition for q ≤ 1/2 and a discontinuous transition for q > 1/2. In our proofs, we introduce a generalized notion of orientability interpolating between 1-and 2-orientability, of cores interpolating between 2-core and 3-core, and of extended cores interpolating between 2 + 1-core and 3 + 2-core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of [7] about the size of the 3 + 2-core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest

Matchings on infinite graphs

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page

Spectral density of random graphs with topological constraints

The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case an exact solution is found for the spectral density in the form of consistency equations depending on the statistical properties of the graph ensemble in question. We highlight the effect of these topological constraints on the resulting spectral density.Comment: 24 pages, 6 figure