A Measure Version of Gaussian Heuristic

Most applicable lattice reduction algorithms used in practice are BKZ (Block-Korkine-Zolotarev) type algorithms as the blockwise generalizations of the LLL algorithm (Lenstra-Lenstra-Lovasz). Its original version was proposed by Schnorr and Euchner in 1991. The quality of reduced lattice bases is measured by the Hermitian factor $\frac{||{\bf b}_1||}{vol({\bf L})^{1/d}}$ and the $d$-th root of this factor which is called root Hermitian factor. In Asiacrypt 2011 paper Y. Chen and Phong Q. Nguyen used BKZ with extreme pruning enumeration subroutine to handle the large block size lattice reduction with the purpose that the better root Hermitian factors can be expected. This BKZ 2.0 algorithm has been served as a base stone for the security evaluation of recent lattice-based cryptosystems such as fully homomorphic encryption and cryptographic multilinear mappings. In this paper we propose a measure version of Gaussian heuristic. This is a strict mathematical proven theorem. It can be used to give a strict mathematical proof for conjectured or simulated root Hermitian factors in BKZ 2.0 type algorithms and BKZ or slide reduction with large block-sizes. The theoretical analysis of these heuristic assumptions in the simulator of BKZ 2.0 type algorithms are also given

A Statistical Approach to Topological Data Analysis

Until very recently, topological data analysis and topological inference methods mostlyrelied on deterministic approaches. The major part of this habilitation thesis presents astatistical approach to such topological methods. We first develop model selection toolsfor selecting simplicial complexes in a given filtration. Next, we study the estimationof persistent homology on metric spaces. We also study a robust version of topologicaldata analysis. Related to this last topic, we also investigate the problem of Wassersteindeconvolution. The second part of the habilitation thesis gathers our contributions inother fields of statistics, including a model selection method for Gaussian mixtures, animplementation of the slope heuristic for calibrating penalties, and a study of Breiman’spermutation importance measure in the context of random forests

Non-perturbative \lambda\Phi^4 in D=1+1: an example of the constructive quantum field theory approach in a schematic way

During the '70, several relativistic quantum field theory models in $D=1+1$ and also in $D=2+1$ have been constructed in a non-perturbative way. That was done in the so-called {\it constructive quantum field theory} approach, whose main results have been obtained by a clever use of Euclidean functional methods. Although in the construction of a single model there are several technical steps, some of them involving long proofs, the constructive quantum field theory approach contains conceptual insights about relativistic quantum field theory that deserved to be known and which are accessible without entering in technical details. The purpose of this note is to illustrate such insights by providing an oversimplified schematic exposition of the simple case of $\lambda\Phi^4$ (with $m>0$) in $D=1+1$. Because of the absence of ultraviolet divergences in its perturbative version, this simple example -although does not capture all the difficulties in the constructive quantum field theory approach- allows to stress those difficulties inherent to the non-perturbative definition. We have made an effort in order to avoid several of the long technical intermediate steps without missing the main ideas and making contact with the usual language of the perturbative approach.Comment: 63 pages. Typos correcte

Heuristic Spike Sorting Tuner (HSST), a framework to determine optimal parameter selection for a generic spike sorting algorithm

Extracellular microelectrodes frequently record neural activity from more than one neuron in the vicinity of the electrode. The process of labeling each recorded spike waveform with the identity of its source neuron is called spike sorting and is often approached from an abstracted statistical perspective. However, these approaches do not consider neurophysiological realities and may ignore important features that could improve the accuracy of these methods. Further, standard algorithms typically require selection of at least one free parameter, which can have significant effects on the quality of the output. We describe a Heuristic Spike Sorting Tuner (HSST) that determines the optimal choice of the free parameters for a given spike sorting algorithm based on the neurophysiological qualification of unit isolation and signal discrimination. A set of heuristic metrics are used to score the output of a spike sorting algorithm over a range of free parameters resulting in optimal sorting quality. We demonstrate that these metrics can be used to tune parameters in several spike sorting algorithms. The HSST algorithm shows robustness to variations in signal to noise ratio, number and relative size of units per channel. Moreover, the HSST algorithm is computationally efficient, operates unsupervised, and is parallelizable for batch processing

Block-diagonal covariance selection for high-dimensional Gaussian graphical models

Gaussian graphical models are widely utilized to infer and visualize networks of dependencies between continuous variables. However, inferring the graph is difficult when the sample size is small compared to the number of variables. To reduce the number of parameters to estimate in the model, we propose a non-asymptotic model selection procedure supported by strong theoretical guarantees based on an oracle inequality and a minimax lower bound. The covariance matrix of the model is approximated by a block-diagonal matrix. The structure of this matrix is detected by thresholding the sample covariance matrix, where the threshold is selected using the slope heuristic. Based on the block-diagonal structure of the covariance matrix, the estimation problem is divided into several independent problems: subsequently, the network of dependencies between variables is inferred using the graphical lasso algorithm in each block. The performance of the procedure is illustrated on simulated data. An application to a real gene expression dataset with a limited sample size is also presented: the dimension reduction allows attention to be objectively focused on interactions among smaller subsets of genes, leading to a more parsimonious and interpretable modular network.Comment: Accepted in JAS

A note on detecting statistical outliers in psychophysical data

This paper considers how to identify statistical outliers in psychophysical datasets, where the underlying sampling distributions are unknown. Eight methods are described, and each is evaluated using Monte Carlo simulations of a typical psychophysical experiment. The best method is shown to be one based on a measure of spread known as S n . This is shown to be more sensitive than popular heuristics based on standard deviations from the mean, and more robust than non-parametric methods based on percentiles or interquartile range. MATLAB code for computing S n is included

Ab initio lifetime correction to scattering states for time-dependent electronic-structure calculations with incomplete basis sets

We propose a method for obtaining effective lifetimes of scattering electronic states for avoiding the artificially confinement of the wave function due to the use of incomplete basis sets in time-dependent electronic-structure calculations of atoms and molecules. In this method, using a fitting procedure, the lifetimes are extracted from the spatial asymptotic decay of the approximate scattering wave functions obtained with a given basis set. The method is based on a rigorous analysis of the complex-energy solutions of the Schr{\"o}dinger equation. It gives lifetimes adapted to any given basis set without using any empirical parameters. The method can be considered as an ab initio version of the heuristic lifetime model of Klinkusch et al. [J. Chem. Phys. 131, 114304 (2009)]. The method is validated on the H and He atoms using Gaussian-type basis sets for calculation of high-harmonic-generation spectra

Info-Greedy sequential adaptive compressed sensing

We present an information-theoretic framework for sequential adaptive compressed sensing, Info-Greedy Sensing, where measurements are chosen to maximize the extracted information conditioned on the previous measurements. We show that the widely used bisection approach is Info-Greedy for a family of $k$-sparse signals by connecting compressed sensing and blackbox complexity of sequential query algorithms, and present Info-Greedy algorithms for Gaussian and Gaussian Mixture Model (GMM) signals, as well as ways to design sparse Info-Greedy measurements. Numerical examples demonstrate the good performance of the proposed algorithms using simulated and real data: Info-Greedy Sensing shows significant improvement over random projection for signals with sparse and low-rank covariance matrices, and adaptivity brings robustness when there is a mismatch between the assumed and the true distributions.Comment: Preliminary results presented at Allerton Conference 2014. To appear in IEEE Journal Selected Topics on Signal Processin