Generalized Relevance Learning Grassmann Quantization

Due to advancements in digital cameras, it is easy to gather multiple images (or videos) from an object under different conditions. Therefore, image-set classification has attracted more attention, and different solutions were proposed to model them. A popular way to model image sets is subspaces, which form a manifold called the Grassmann manifold. In this contribution, we extend the application of Generalized Relevance Learning Vector Quantization to deal with Grassmann manifold. The proposed model returns a set of prototype subspaces and a relevance vector. While prototypes model typical behaviours within classes, the relevance factors specify the most discriminative principal vectors (or images) for the classification task. They both provide insights into the model's decisions by highlighting influential images and pixels for predictions. Moreover, due to learning prototypes, the model complexity of the new method during inference is independent of dataset size, unlike previous works. We applied it to several recognition tasks including handwritten digit recognition, face recognition, activity recognition, and object recognition. Experiments demonstrate that it outperforms previous works with lower complexity and can successfully model the variation, such as handwritten style or lighting conditions. Moreover, the presence of relevances makes the model robust to the selection of subspaces' dimensionality

Dynamical Scaling Behavior of Percolation Clusters in Scale-free Networks

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of tree-like networks. Close to the percolation threshold we find characteristic scaling functions for the density of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic power laws rho(lambda) ~ lambda^alpha_1 or rho(lambda) ~ lambda^alpha_2 for small lambda, where alpha_1 holds below and alpha_2 at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.Comment: 9 pages, 6 figure

A family of complex potentials with real spectrum

We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other PT symmetric models, which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.Comment: Six figures and four table

Band Distributions for Quantum Chaos on the Torus

Band distributions (BDs) are introduced describing quantization in a toral phase space. A BD is the uniform average of an eigenstate phase-space probability distribution over a band of toral boundary conditions. A general explicit expression for the Wigner BD is obtained. It is shown that the Wigner functions for {\em all} of the band eigenstates can be reproduced from the Wigner BD. Also, BDs are shown to be closer to classical distributions than eigenstate distributions. Generalized BDs, associated with sets of adjacent bands, are used to extend in a natural way the Chern-index characterization of the classical-quantum correspondence on the torus to arbitrary rational values of the scaled Planck constant.Comment: 12 REVTEX page

Gegenbauer-solvable quantum chain model

In an innovative inverse-problem construction the measured, experimental energies $E_1$, $E_2$, ...$E_N$ of a quantum bound-state system are assumed fitted by an N-plet of zeros of a classical orthogonal polynomial $f_N(E)$. We reconstruct the underlying Hamiltonian $H$ (in the most elementary nearest-neighbor-interaction form) and the underlying Hilbert space ${\cal H}$ of states (the rich menu of non-equivalent inner products is offered). The Gegenbauer's ultraspherical polynomials $f_n(x)=C_n^\alpha(x)$ are chosen for the detailed illustration of technicalities.Comment: 29 pp., 1 fi

Signal-background separation and energy reconstruction of gamma rays using pattern spectra and convolutional neural networks for the Small-Sized Telescopes of the Cherenkov Telescope Array

Imaging Atmospheric Cherenkov Telescopes (IACTs) detect very high-energy gamma rays from ground level by capturing the Cherenkov light of the induced particle showers. Convolutional neural networks (CNNs) can be trained on IACT camera images of such events to differentiate the signal from the background and to reconstruct the energy of the initial gamma ray. Pattern spectra provide a 2-dimensional histogram of the sizes and shapes of features comprising an image and they can be used as an input for a CNN to significantly reduce the computational power required to train it. In this work, we generate pattern spectra from simulated gamma-ray and proton images to train a CNN for signal-background separation and energy reconstruction for the Small-Sized Telescopes (SSTs) of the Cherenkov Telescope Array (CTA). A comparison of our results with a CNN directly trained on CTA images shows that the pattern spectra-based analysis is about a factor of three less computationally expensive but not able to compete with the performance of the CTA images-based analysis. Thus, we conclude that the CTA images must be comprised of additional information not represented by the pattern spectra.Comment: 10 pages, 9 figures, submitted to Nuclear Instruments and Methods in Physics Research - section

Effects of Electron Correlations on Hofstadter Spectrum

By allowing interactions between electrons, a new Harper's equation is derived to examine the effects of electron correlations on the Hofstadter energy spectra. It is shown that the structure of the Hofstadter butterfly ofr the system of correlated electrons is modified only in the band gaps and the band widths, but not in the characteristics of self-similarity and the Cantor set.Comment: 13 pages, 5 Postscript figure

The kink Casimir energy in a lattice sine-Gordon model

The Casimir energy of quantum fluctuations about the classical kink configuration is computed numerically for a recently proposed lattice sine-Gordon model. This energy depends periodically on the kink position and is found to be approximately sinusoidal.Comment: 10 pages, 4 postscript figure

An importance sampling algorithm for generating exact eigenstates of the nuclear Hamiltonian

We endow a recently devised algorithm for generating exact eigensolutions of large matrices with an importance sampling, which is in control of the extent and accuracy of the truncation of their dimensions. We made several tests on typical nuclei using a correlated basis obtained from partitioning the shell model space. The sampling so implemented allows not only for a substantial reduction of the shell model space but also for an extrapolation to exact eigenvalues and E2 strengths.Comment: A compressed file composed of a text in latex of 19 pages and 9 figures in p

Monte Carlo Study of the Separation of Energy Scales in Quantum Spin 1/2 Chains with Bond Disorder

One-dimensional Heisenberg spin 1/2 chains with random ferro- and antiferromagnetic bonds are realized in systems such as $Sr_3 CuPt_{1-x} Ir_x O_6$. We have investigated numerically the thermodynamic properties of a generic random bond model and of a realistic model of $Sr_3 CuPt_{1-x} Ir_x O_6$ by the quantum Monte Carlo loop algorithm. For the first time we demonstrate the separation into three different temperature regimes for the original Hamiltonian based on an exact treatment, especially we show that the intermediate temperature regime is well-defined and observable in both the specific heat and the magnetic susceptibility. The crossover between the regimes is indicated by peaks in the specific heat. The uniform magnetic susceptibility shows Curie-like behavior in the high-, intermediate- and low-temperature regime, with different values of the Curie constant in each regime. We show that these regimes are overlapping in the realistic model and give numerical data for the analysis of experimental tests.Comment: 7 pages, 5 eps-figures included, typeset using JPSJ.sty, accepted for publication in J. Phys. Soc. Jpn. 68, Vol. 3. (1999