A study of the classification of low-dimensional data with supervised manifold learning

Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of supervised manifold learning for classification. We consider nonlinear dimensionality reduction algorithms that yield linearly separable embeddings of training data and present generalization bounds for this type of algorithms. A necessary condition for satisfactory generalization performance is that the embedding allow the construction of a sufficiently regular interpolation function in relation with the separation margin of the embedding. We show that for supervised embeddings satisfying this condition, the classification error decays at an exponential rate with the number of training samples. Finally, we examine the separability of supervised nonlinear embeddings that aim to preserve the low-dimensional geometric structure of data based on graph representations. The proposed analysis is supported by experiments on several real data sets

Fractionally-addressed delay lines

While traditional implementations of variable-length digital delay lines are based on a circular buffer accessed by two pointers, we propose an implementation where a single fractional pointer is used both for read and write operations. On modern general-purpose architectures, the proposed method is nearly as efficient as the popularinterpolated circular buffer, and it behaves well for delay-length modulations commonly found in digital audio effects. The physical interpretation of the new implementation shows that it is suitable for simulating tension or density modulations in wave-propagating media.Comment: 11 pages, 19 figures, to be published in IEEE Transactions on Speech and Audio Processing Corrected ACM-clas

Nonlinear Supervised Dimensionality Reduction via Smooth Regular Embeddings

The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing methods primarily focus on the embedding of the training data, and the generalization of the embedding to initially unseen test data is rather ignored. In this work, we build on recent theoretical results on the generalization performance of supervised manifold learning algorithms. Motivated by these performance bounds, we propose a supervised manifold learning method that computes a nonlinear embedding while constructing a smooth and regular interpolation function that extends the embedding to the whole data space in order to achieve satisfactory generalization. The embedding and the interpolator are jointly learnt such that the Lipschitz regularity of the interpolator is imposed while ensuring the separation between different classes. Experimental results on several image data sets show that the proposed method outperforms traditional classifiers and the supervised dimensionality reduction algorithms in comparison in terms of classification accuracy in most settings

Restoration of chiral and U(1)AU(1)_A symmetries in excited hadrons

The effective restoration of $SU(2)_L \times SU(2)_R$ and $U(1)_A$ chiral symmetries of QCD in excited hadrons is reviewed. While the low-lying hadron spectrum is mostly shaped by the spontaneous breaking of chiral symmetry, in the high-lying hadrons the role of the quark condensate of the vacuum becomes negligible and the chiral symmetry is effectively restored. This implies that the mass generation mechanisms in the low- and high-lying hadrons are essentially different. The fundamental origin of this phenomenon is a suppression of quark quantum loop effects in high-lying hadrons relative to the classical contributions that preserve both chiral and $U(1)_A$ symmetries. Microscopically the chiral symmetry breaking is induced by the dynamical Lorentz-scalar mass of quarks due to their coupling with the quark condensate of the vacuum. This mass is strongly momentum-dependent, however, and vanishes in the high-lying hadrons where the typical momentum of valence quarks is large. This physics is illustrated within the solvable chirally-symmetric and confining model. Effective Lagrangians for the approximate chiral multiplets at the hadron level are constructed which can be used as phenomenological effective field theories in the effective chiral restoration regime. Different ramifications and implications of the effective chiral restoration for the string description of excited hadrons, the decoupling of excited hadrons from the Goldstone bosons, the glueball - quark-antiquark mixing and the OZI rule violations are discussed.Comment: 64 pages. To appear in Physics Report

Some non monotone schemes for Hamilton-Jacobi-Bellman equations

We extend the theory of Barles Jakobsen to develop numerical schemes for Hamilton Jacobi Bellman equations. We show that the monotonicity of the schemes can be relaxed still leading to the convergence to the viscosity solution of the equation. We give some examples of such numerical schemes and show that the bounds obtained by the framework developed are not tight. At last we test some numerical schemes.Comment: 24 page

Chiral multiplets of excited mesons

It is shown that experimental meson states with spins J=0,1,2,3 in the energy range 1.9 - 2.4 GeV obtained in recent partial wave analysis of proton-antiproton annihilation at LEAR remarkably confirm all predictions of chiral symmetry restoration. Classification of excited $\bar q q$ mesons according to the representations of chiral $U(2)_L \times U(2)_R$ group is performed. There are two important predictions of chiral symmetry restoration in highly excited mesons: (i) physical states must fill out approximately degenerate parity-chiral multiplets; (ii) some of the physical states with the given $I,J^{PC}$ are members of one parity-chiral multiplet, while the other states with the same $I,J^{PC}$ are members of the other parity-chiral multiplet. For example, while some of the excited $\rho(1,1^{--})$ states are systematically degenerate with $a_1(1,1^{++})$ states forming (0,1)+(1,0) chiral multiplets, the other excited $\rho(1,1^{--})$ states are degenerate with $h_1(0,1^{+-})$ states ((1/2,1/2) chiral multiplets). Hence, one of the predictions of chiral symmetry restoration is that the combined amount of $a_1(1,1^{++})$ and $h_1(0,1^{+-})$ states must coincide with the amount of $\rho(1,1^{--})$ states in the chirally restored regime. It is shown that the same rule applies (and experimentally confirmed) to many other meson states.Comment: 14 pages, discussion and conclusion section is largely extende