High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation

The ratio between two probability density functions is an important component of various tasks, including selection bias correction, novelty detection and classification. Recently, several estimators of this ratio have been proposed. Most of these methods fail if the sample space is high-dimensional, and hence require a dimension reduction step, the result of which can be a significant loss of information. Here we propose a simple-to-implement, fully nonparametric density ratio estimator that expands the ratio in terms of the eigenfunctions of a kernel-based operator; these functions reflect the underlying geometry of the data (e.g., submanifold structure), often leading to better estimates without an explicit dimension reduction step. We show how our general framework can be extended to address another important problem, the estimation of a likelihood function in situations where that function cannot be well-approximated by an analytical form. One is often faced with this situation when performing statistical inference with data from the sciences, due the complexity of the data and of the processes that generated those data. We emphasize applications where using existing likelihood-free methods of inference would be challenging due to the high dimensionality of the sample space, but where our spectral series method yields a reasonable estimate of the likelihood function. We provide theoretical guarantees and illustrate the effectiveness of our proposed method with numerical experiments.Comment: With supplementary materia

On the structure of framed vertex operator algebras and their pointwise frame stabilizers

In this paper, we study the structure of a general framed vertex operator algebra. We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA V_C. This result would give a prospect on the classification of framed vertex operator algebras. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in this pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous Moonshine VOA are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of the moonshine VOA is isomorphic to the moonshine VOA itself.Comment: Version 3: 59 pages. Corrected version. 54 pages on my LaTeX system version 2: We add Theorem 5.16 in which we give a necessary and sufficient condtion for a code to be a structure code of a holomorphic framed VOA. "hyperref" style is also introduce

Quantum interference in attosecond transient absorption of laser-dressed helium atoms

We calculate the transient absorption of an isolated attosecond pulse by helium atoms subject to a delayed infrared (\ir) laser pulse. With the central frequency of the broad attosecond spectrum near the ionization threshold, the absorption spectrum is strongly modulated at the sub-\ir-cycle level. Given that the absorption spectrum results from a time-integrated measurement, we investigate the extent to which the delay-dependence of the absorption yields information about the attosecond dynamics of the atom-field energy exchange. We find two configurations in which this is possible. The first involves multi photon transitions between bound states that result in interference between different excitation pathways. The other involves the modification of the bound state absorption lines by the IR field, which we find can result in a sub-cycle time dependence only when ionization limits the duration of the strong field interaction

Oblique and curved D-branes in IIB plane-wave string theory

Oblique Dp-branes in the maximally supersymmetric type IIB plane-wave background are constructed in terms of boundary states, as well as from the open string point of view. These Dp-branes, whose existence was anticipated by Hikida and Yamaguchi from general supersymmetry arguments, have an isometry that is a subgroup of the diagonal SO(4) symmetry of the background. The oblique D3-brane is found to preserve four dynamical and four kinematical supersymmetries while the oblique D5-brane preserves one half of both the dynamical and kinematical supersymmetries. We also discuss the open-string boundary conditions for curved D7- and D5-branes, and analyze their supersymmetry.Comment: 27 page