Optimal Rates for Random Fourier Features

Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show good versatility, they are computationally intensive and have poor scalability to large data as they require operations on Gram matrices. In order to mitigate this serious computational limitation, recently randomized constructions have been proposed in the literature, which allow the application of fast linear algorithms. Random Fourier features (RFF) are among the most popular and widely applied constructions: they provide an easily computable, low-dimensional feature representation for shift-invariant kernels. Despite the popularity of RFFs, very little is understood theoretically about their approximation quality. In this paper, we provide a detailed finite-sample theoretical analysis about the approximation quality of RFFs by (i) establishing optimal (in terms of the RFF dimension, and growing set size) performance guarantees in uniform norm, and (ii) presenting guarantees in $L^r$ ($1\le r<\infty$) norms. We also propose an RFF approximation to derivatives of a kernel with a theoretical study on its approximation quality.Comment: To appear at NIPS-201

The equation of state at high temperatures from lattice QCD

We present results for the equation of state upto previously unreachable, high temperatures. Since the temperature range is quite large, a comparison with perturbation theory can be done directly.Comment: 7 pages, 5 figures, Lattice 200

The nature of the finite temperature QCD transition as a function of the quark masses

The finite temperature QCD transition for physical quark masses is a crossover. For smaller quark masses a first-order phase transition is expected. Using Symanzik improved gauge and stout improved fermion action for 2+1 flavour staggered QCD we give estimates/bounds for the phase line separating the first-order region from the crossover one. The calculations are carried out on two different lattice spacings. Our conclusion for the critical mass is $m_0 \lesssim 0.07 \cdot m_{phys}$ for $N_T=4$ and $m_0 \lesssim 0.12 \cdot m_{phys}$ for $N_T=6$ lattices.Comment: Talk presented at the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, Germany. 7 pages, 6 figure

CCD photometry and new models of 5 minor planets

We present new R filtered CCD observations of 5 faint and moderately faint asteroids carried out between October, 1998 and January, 1999. The achieved accuracy is between 0.01-0.03 mag, depending mainly on the target brightness. The obtained sinodic periods and amplitudes: 683 Lanzia - 4.6+/-0.2 h, 0.13 mag; 725 Amanda - >3.0 h, >=0.40 mag; 852 Wladilena - 4.62+/-0.01 h, 0.32 mag (December, 1998) and 0.27 mag (January, 1999); 1627 Ivar - 4.80+/-0.01, 0.77 mag (December, 1998) and 0.92 mag (January, 1999). The Near Earth Object 1998 PG unambiguously showed doubly-periodic lightcurve, suggesting the possibility of a relatively fast precession (P_1=1.3 h, P_2=5.3 h). Collecting all data from the literature, we determined new models for 3 minor planets. The resulting spin vectors and triaxial ellipsoids have been calculated by an amplitude-method. Sidereal periods and senses of rotation were calculated for two asteroids (683 and 1627) by a modified epoch-method. The results are: 683 - lambda_p=15/195+/-25 deg, beta_p=52+/-15 deg, a/b=1.15+/-0.05, b/c=1.05+/-0.05, P_sid=0.1964156+/-0.0000001 d, retrograde; 852 - lambda_p=30/210+/-20 deg, beta_p=30+/-10 deg, a/b=2.3+/-0.3, b/c=1.2+/-0.2; 1627 - lambda_p=145/325+/-8 deg, beta_p=34+/-6 deg, a/b=2.0+/-0.1, b/c=1.09+/-0.05, P_sid=0.1999154+/-0.0000003 d, retrograde. The obtained shape of 1627 is in good agreement with radar images by Ostro et al. (1990).Comment: 8 pages, accepted for publication in Astronomy and Astrophysics Suppl. Serie

Exact Solution of Noncommutative Field Theory in Background Magnetic Fields

We obtain the exact non-perturbative solution of a scalar field theory defined on a space with noncommuting position and momentum coordinates. The model describes non-locally interacting charged particles in a background magnetic field. It is an exactly solvable quantum field theory which has non-trivial interactions only when it is defined with a finite ultraviolet cutoff. We propose that small perturbations of this theory can produce solvable models with renormalizable interactions.Comment: 9 Pages AMSTeX; Typos correcte

Quiver Gauge Theory of Nonabelian Vortices and Noncommutative Instantons in Higher Dimensions

We construct explicit BPS and non-BPS solutions of the Yang-Mills equations on the noncommutative space R^{2n}_\theta x S^2 which have manifest spherical symmetry. Using SU(2)-equivariant dimensional reduction techniques, we show that the solutions imply an equivalence between instantons on R^{2n}_\theta x S^2 and nonabelian vortices on R^{2n}_\theta, which can be interpreted as a blowing-up of a chain of D0-branes on R^{2n}_\theta into a chain of spherical D2-branes on R^{2n} x S^2. The low-energy dynamics of these configurations is described by a quiver gauge theory which can be formulated in terms of new geometrical objects generalizing superconnections. This formalism enables the explicit assignment of D0-brane charges in equivariant K-theory to the instanton solutions.Comment: 45 pages, 4 figures; v2: minor correction

Lattice SU(3) thermodynamics and the onset of perturbative behaviour

We present the equation of state (pressure, trace anomaly, energy density and entropy density) of the SU(3) gauge theory from lattice field theory in an unprecedented precision and temperature range. We control both finite size and cut-off effects. The studied temperature window ($0.7... 1000 T_c$) stretches from the glueball dominated system into the perturbative regime, which allows us to discuss the range of validity of these approaches. From the critical couplings on fine lattices we get T_c/\Lambdamsbar=1.26(7) and use this ratio to express the perturbative free energy in $T_c$ units. We also determine the preferred renormalization scale of the Hard Thermal Loop scheme and we fit the unknown $g^6$ order perturbative coefficient at extreme high temperatures $T>100T_c$. We furthermore quantify the nonperturbative contribution to the trace anomaly using two simple functional forms.Comment: 7 pages, Contribution to the The XXVIII International Symposium on Lattice Field Theory; June 14 - 19, 2010, Villasimius, Sardinia, Ital