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Christoffel polynomials are classical tools from approximation theory. They can be used to estimate the (compact) support of a measure $\mu$ on $\mathbb{R}^d$ based on its low-degree moments. Recently, they have been applied to problems in data science, including outlier detection and support inference. A major downside of Christoffel polynomials in such applications is the fact that, in order to compute their coefficients, one must invert a matrix whose size grows rapidly with the dimension $d$. In this paper, we propose a modification of the Christoffel polynomial which is significantly cheaper to compute, but retains many of its desirable properties. Our approach relies on sparsity of the underlying measure $\mu$, described by a graphical model. The complexity of our modification depends on the treewidth of this model., Comment: 21 pages, 1 figure

Mode-locked lasers are of interest for applications such as biological imaging, non-linear frequency conversion, and single-photon generation. In the infrared, chip-integrated mode-locked lasers have been demonstrated through integration of laser diodes with low-loss photonic circuits. However additional challenges, such as a higher propagation loss and smaller alignment tolerances have prevented the realization of such lasers in the visible range. Here, we demonstrate the first chip-integrated mode-locked diode laser in the visible using an integrated photonic circuit for cavity extension. Based on a gallium arsenide gain chip and a low-loss silicon nitride feedback circuit, the laser is passively mode-locked using a saturable absorber implemented by focused ion beam milling. At a center wavelength of 642 nm, the laser shows an average output power of 3.4 mW, with a spectral bandwidth of 1.5 nm at a repetition rate of 7.84 GHz., Comment: 7 pages, 4 figures

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex) optimization problems. Various hierarchies of (lower and upper) bounds have been introduced, having the remarkable property that they converge asymptotically to the global minimum. These bounds exploit algebraic representations of positive polynomials in terms of sums of squares and can be computed using semidefinite optimization. Our focus lies in the performance analysis of these hierarchies of bounds, namely, in how far the bounds are from the global minimum as the degrees of the sums of squares they involve tend to infinity. We present the main state-of-the-art results and offer a gentle introductory overview over the various techniques that have been recently developed to establish them, stemming from the theory of orthogonal polynomials, approximation theory, Fourier analysis, and more., Comment: 21 pages, 1 figure

Rubinfeld & Vasilyan recently introduced the framework of testable learning as an extension of the classical agnostic model. It relaxes distributional assumptions which are difficult to verify by conditions that can be checked efficiently by a tester. The tester has to accept whenever the data truly satisfies the original assumptions, and the learner has to succeed whenever the tester accepts. We focus on the setting where the tester has to accept standard Gaussian data. There, it is known that basic concept classes such as halfspaces can be learned testably with the same time complexity as in the (distribution-specific) agnostic model. In this work, we ask whether there is a price to pay for testably learning more complex concept classes. In particular, we consider polynomial threshold functions (PTFs), which naturally generalize halfspaces. We show that PTFs of arbitrary constant degree can be testably learned up to excess error $\varepsilon > 0$ in time $n^{\mathrm{poly}(1/\varepsilon)}$. This qualitatively matches the best known guarantees in the agnostic model. Our results build on a connection between testable learning and fooling. In particular, we show that distributions that approximately match at least $\mathrm{poly}(1/\varepsilon)$ moments of the standard Gaussian fool constant-degree PTFs (up to error $\varepsilon$). As a secondary result, we prove that a direct approach to show testable learning (without fooling), which was successfully used for halfspaces, cannot work for PTFs., Comment: 53 pages

We demonstrate absorber-free passive and hybrid mode-locking at sub-GHz repetition rates using a hybrid integrated extended cavity diode laser around 1550 nm. The laser is based on InP as gain medium and a long Si$_3$N$_4$ feedback circuit, with three highly frequency selective microring resonators. The feedback resonators not only increases the cavity length up to 0.6 m to achieve sub-GHz repetition rates but also serve as a dispersive narrowband mirror for sharp spectral filtering, which enables Fourier domain mode-locking. We observe passive mode-locking with repetition rates below 500 MHz, with $\approx$ 15 comb lines at around 0.2 mW total power. To stabilize the repetition rate, hybrid mode-locking is demonstrated by weak RF modulation of the diode current at frequencies around 500 MHz. The RF injection reduces the Lorentzian linewidth component from 8.9 kHz to a detection limited value around 300 mHz. To measure the locking range of the repetition rate, the injected RF frequency is tuned with regard to the passive mode-locking frequency and the injected RF power is varied. The locking range increases approximately as a square-root function of the injected RF power. At 1 mW injection a wide locking range of about 80 MHz is obtained. We observe the laser maintaining stable mode-locking also when the DC diode pump current is increased from 40 mA to 190 mA, provided that the cavity length is maintained constant with thermo-refractive tuning., Comment: This manuscript is under review for publication with optics express

Let $\mathbf{X} \subseteq \mathbb{R}^n$ be a closed set, and consider the problem of computing the minimum $f_{\min}$ of a polynomial $f$ on $\mathbf{X}$. Given a measure $\mu$ supported on $\mathbf{X}$, Lasserre (SIAM J. Optim. 21(3), 2011) proposes a decreasing sequence of upper bounds on $f_{\min}$, each of which may be computed by solving a semidefinite program. When $\mathbf{X}$ is compact, these bounds converge to $f_{\min}$ under minor assumptions on $\mu$. Later, Lasserre (Math. Program. 190, 2020) introduces a related, but far more economical sequence of upper bounds which rely on the push-forward measure of $\mu$ by $f$. While these new bounds are weaker a priori, they actually achieve similar asymptotic convergence rates on compact sets. In this work, we show that no such free lunch exists in the non-compact setting. While convergence of the standard bounds to $f_{\min}$ is guaranteed when $\mathbf{X} = \mathbb{R}^n$ and $\mu$ is a Gaussian distribution, we prove that the bounds relying on the push-forward measure fail to converge to $f_{\min}$ in that setting already for polynomials of degree $6$., Comment: v2: Made a change/fix to Definition 4 in the case n > 1, alpha != 2. Added explicit proof of Theorem 5 (Appendix A). Added explicit extension to case n >= 2 (Appendix B). Extended statement and proof of Theorem 7 to all alpha > 0. Implemented reviewer comments

We present hybrid-integrated extended cavity diode lasers tunable around 637 nm, with a gain-wide spectral coverage of 8 nm. This tuning range allows addressing the zero-phonon line of nitrogen vacancy centers and includes the wavelength of HeNe lasers (633 nm). The lasers provide wide mode-hop free tuning up to 97 GHz and a narrow intrinsic linewidth down to 10 kHz. The maximum output power is 2.5 mW in a single-mode fiber, corresponding to an on-chip power of 4.0 mW. Full integration and packaging in a standard housing with fiber pigtails provide high intrinsic stability and will enable integration into complex optical systems., Comment: 11 pages, 7 figures

The Fermilab Accelerator Science and Technology (FAST) facility is currently in operation with its linac based on TESLA-type superconducting rf cavities. Using a 3-MHz micropulse repetition rate with a long macropulse composed of up to 3000 micropulses, and with beam energies demonstrated at 300 MeV and projected to reach 800 MeV with two additional cryomodules, the feasibilities for a vacuum ultraviolet (VUV) and an extreme ultraviolet (EUV) free-electron laser oscillator (FELO) with the two energies are evaluated. We have used both the GINGER code with an oscillator module and the MINERVA/OPC code to assess FELO saturation prospects at 120 nm with a 5.0-cm-period undulator of 4.5-m length and the MINERVA/OPC code to assess the FELO at 13.4 nm with adjusted parameters. The simulation results support saturation at both of these wavelengths which are much shorter than the demonstrated shortest wavelength record of 168.6 nm from a storage-ring-based FELO. This indicates superconducting rf linac-driven FELOs can be extended into this VUV-EUV wavelength regime previously only reached with single-pass FEL configurations. In addition, emittance-dilution effects due to wakefields in the cavities and the resulting sub-macropulse centroid slew effects on FELO performance are addressed using MINERVA/OPC simulations for the first time., Comment: Journal submitted Phys.Rev.Accel.Beams (OA)

We investigate on-chip spectral control of supercontinuum generation, taking advantage of the additional spatial degree of freedom in strongly-coupled dual-core waveguides. Using numerical integration of the multi-mode generalized nonlinear Schr\"odinger equation, we show that, with proper waveguide cross-section design, selective excitation of supermodes can vary the dispersion to its extremes, i.e., all-normal or anomalous dispersion can be selected via phase shifting in a Mach-Zehnder input circuit. The resulting control allows to provide vastly different supercontinuum spectra with the same waveguide circuit.

The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O'Donnell (2017) and later Raghavendra & Weitz (2017) show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz (2017). In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible., Comment: 10 pages

Hybrid integrated diode lasers based on combining semiconductor optical amplifiers with low-loss Si3N4-based feedback circuits enable great laser performance for advanced photonic circuits. In particular, using high-Q Si3N4 ring resonators for frequency-selective feedback provides wide spectral coverage, mode-hop free tuning, and high frequency stability on short timescales, showing as ultra-narrow intrinsic linewidths. However, many applications also require long-term stability, which can be provided by locking the laser frequency to a suitable reference. We present the stabilization of a hybrid-integrated laser, which is widely tunable around the central wavelength of 1550 nm, to a fiber-based optical frequency discriminator (OFD) and to an acetylene absorption line. By locking the laser to the OFD, the laser's fractional frequency stability is improved down to 2.1$\cdot$10$^{-12}$ over an averaging time of 0.5 ms. For absolute stability over longer times of several days, we successfully lock the laser frequency to an acetylene absorption line. This limits the frequency deviations of the laser to a range of less than 12 MHz over 5 days., Comment: 12 pages, 6 figures

The Positivstellens\"atze of Putinar and Schm\"udgen show that any polynomial $f$ positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of these results, namely to show bounds on the required degree of the sums of squares in such representations. These effective Positivstellens\"atze have direct implications for the convergence rate of the celebrated moment-SOS hierarchy in polynomial optimization. In this paper, we restrict to the fundamental case of the hypercube $\mathrm{B}^{n} = [-1, 1]^n$. We show an upper degree bound for Putinar-type representations on $\mathrm{B}^{n}$ of the order $O(f_{\max}/f_{\min})$, where $f_{\max}$, $f_{\min}$ are the maximum and minimum of $f$ on $\mathrm{B}^{n}$, respectively. Previously, specialized results of this kind were available only for Schm\"udgen-type representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound in $\Omega(\sqrt[8]{f_{\max}/f_{\min}})$. This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities., Comment: Final version. Improved presentation and included a more detailed comparison with Bach and Rudi, Exponential Convergence of Sum-of-Squares Hierarchies for Trigonometric Polynomials, SIAM Opt., 2023

Hybrid integrated diode lasers have so far been realized using silicon, polymer, and silicon nitride (Si3N4) waveguide platforms for extending on-chip tunable light engines from the infrared throughout the visible range. Here we demonstrate the first hybrid integrated laser using the aluminum oxide (Al2O3) deep-UV capable waveguide platform. By permanently coupling low-loss Al2O3 frequency-tunable Vernier feedback circuits with GaN double-pass amplifiers in a hermetically sealed housing, we demonstrate the first extended cavity diode laser (ECDL) in the near UV. The laser shows a maximum fiber-coupled output power of 0.74 mW, corresponding to about 3.5 mW on chip, and tunes more than 4.4 nm in wavelength from 408.1 nm to 403.7 nm. Integrating stable, single-mode and tunable lasers into a deep-UV platform opens a new path for chip-integrated photonic applications., Comment: 14 pages, 3 figures

Persistent homology has been widely used to study the topology of point clouds in $\mathbb{R}^n$. Standard approaches are very sensitive to outliers, and their computational complexity depends badly on the number of data points. In this paper we introduce a novel persistence module for a point cloud using the theory of Christoffel-Darboux kernels. This module is robust to (statistical) outliers in the data, and can be computed in time linear in the number of data points. We illustrate the benefits and limitations of our new module with various numerical examples in $\mathbb{R}^n$, for $n=1, 2, 3$. Our work expands upon recent applications of Christoffel-Darboux kernels in the context of statistical data analysis and geometric inference (Lasserre, Pauwels and Putinar, 2022). There, these kernels are used to construct a polynomial whose level sets capture the geometry of a point cloud in a precise sense. We show that the persistent homology associated to the sublevel set filtration of this polynomial is stable with respect to the Wasserstein distance. Moreover, we show that the persistent homology of this filtration can be computed in singly exponential time in the ambient dimension $n$, using a recent algorithm of Basu & Karisani (2022)., Comment: 22 pages, 11 figures, 1 table

The effects on beam dynamics from long-range and short-range wakefields from TESLA-type cavities are considered in regard to a proposed FEL oscillator (FE-LO) operating at 120 nm. This would be driven by the Fermilab Accelerator Science and Technology (FAST) linac at 300 MeV with a 3-MHz micropulse repetition rate. Our wakefield studies showed measurable effects on submacropulse centroid stability and on submicropulse head-tail kicks that can lead to emittance degradation. In the case of the former, we use MINERVA/OPC to simulate the ~100-$\mu$m centroid slew effects on the saturated output power levels of the FELO.

The theta body of a graph, introduced by Gr\"otschel, Lov\'asz, and Schrijver in 1986, is a tractable relaxation of the independent-set polytope derived from the Lov\'asz theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the high-dimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: triangle-free graphs and Mantel's theorem, and bounds on the density of triangle-avoiding sets in the Hamming cube., Comment: 23 pages, 2 figures

We consider the unlabeled motion-planning problem of $m$ unit-disc robots moving in a simple polygonal workspace of $n$ edges. The goal is to find a motion plan that moves the robots to a given set of $m$ target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end. If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time $O(n \log n + mn + m^2)$ if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance $4$ apart, any two target positions are at least distance $4$ apart, and any pair of a start and a target positions is at least distance $3$ apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary., Comment: A shorter version of this paper appeared in the Proceedings of the 38th International Symposium on Computational Geometry (SoCG 2022)

A general theory has been developed for a polydisperse semi-flexible multi-block copolymer melt. Using the Bawendi-Freed approach to model semi-flexible chains an expression for the Landau free energy is derived in the weak segregation regime which includes the density and orientation order-parameters. The orientation order-parameter is described in the smectic phase and in more complicated structures such as the hexagonal phase. The Landau free energy contains contributions of two kinds of interactions. The first kind is the Flory- Huggins interaction which describes the incompatibility of chemically different blocks and may induce microphase separation. The second kind is the Maier- Saupe interaction which may induce nematic ordening. In the framework of the weak segragation limit the Landau theory allows to predict phase structures in the melt as a function of the composition, persistence length and the strength of the Flory-Huggins and Maier-Saupe interaction. The general theory is applied to a simple system of monodisperse semi-flexible diblock copolymers. In several phase diagrams a number of possible phase structures is predicted such as the bcc, hexagonal, smectic-A, smectic-C and nematic phase. The infuence of the Maier-Saupe interaction on the microphase structure is thoroughly discussed., Comment: 63 pages, 6 figures, to be published in The Journal of Chemical Physics

Coherent optomechanical interaction between acoustic and optical waves known as stimulated Brillouin scattering (SBS) can enable ultra-high resolution signal processing and narrow linewidth lasers important for next generation wireless communications, precision sensing, and quantum information processing. While SBS has recently been studied extensively in integrated waveguides, many implementations rely on complicated fabrication schemes, using suspended waveguides, or non-standard materials such as As$_2$S$_3$. The absence of SBS in standard and mature fabrication platforms prevents large-scale circuit integration and severely limits the potential of this technology. Notably, SBS in standard silicon nitride integration platform is currently rendered out of reach due to the lack of acoustic guiding and the infinitesimal photo-elastic response of the material. In this paper, we experimentally demonstrate advanced control of backward SBS in multilayer silicon nitride waveguides. By optimizing the separation between two silicon nitride layers, we unlock gigahertz acoustic waveguiding in this platform for the first time, leading up to 15 $\times$ higher SBS gain coefficient than previously possible in silicon nitride waveguides. Using the same principle, we experimentally demonstrate on-demand inhibition of SBS by preventing acoustic guiding in the waveguide platform. We utilize the enhanced SBS gain to demonstrate a microwave photonic notch filter with high rejection (30 dB). We accomplish this in a low-loss, standard, and versatile silicon nitride integration platform without the need of suspending the SBS-active waveguide or hybrid integration with other materials., Comment: 21 pages, 5 figures

Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set ${\mathbf{X} \subseteq \mathbb{R}^n}$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schm\"udgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schm\"udgen-type certificates converge to the global minimum of $f$ at a rate in $O(1/r^2)$, matching recently obtained convergence rates for the hypersphere and hypercube $[-1,1]^n$. For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel-Darboux kernel, and make use of closed form expressions for this kernel derived by Xu., Comment: v3: Fixed a technical error in the (proof of) Lemma 18. This has a (minor) impact on the constants appearing in Theorems 3, 4