Standard Gaussian Research Articles

On Dvoretzky's theorem for subspaces of Lp

We prove that for any 2<p<∞ and for every n-dimensional subspace X of Lp, represented on Rn, whose unit ball BX is in Lewis' position one has the following two-level Gaussian concentration inequality:P(|‖Z‖−E‖Z‖|>εE‖Z‖)≤Cexp⁡(−cmin⁡{αpε2n,(εn)2/p}),0<ε<1, where Z is the standard n-dimensional Gaussian vector, αp>0 is a constant depending only on p and C,c>0 are absolute constants. As a consequence we show optimal lower bound on the dimension of random almost spherical sections for these spaces. In particular, for any 2<p<∞ and every n-dimensional subspace X of Lp, the Euclidean space ℓ2k can be (1+ε)-embedded into X with k≥cpmin⁡{ε2n,(εn)2/p}, where cp>0 is a constant depending only on p. This improves upon the previously known estimate due to Figiel, Lindenstrauss and Milman.

From Mahalanobis to Bregman via Monge and Kantorovich

In his celebrated 1936 paper on “the generalized distance in statistics,” P.C. Mahalanobis pioneered the idea that, when defined over a space equipped with some probability measure P, a meaningful distance should be P-specific, with data-driven empirical counterpart. The so-called Mahalanobis distance and the corresponding Mahalanobis outlyingness achieve this objective in the case of a Gaussian P by mapping P to the spherical standard Gaussian, via a transformation based on second-order moments which appears to be an optimal transport in the Monge-Kantorovich sense. In a non-Gaussian context, though, one may feel that second-order moments are not informative enough, or inappropriate; moreover, they might not exist. We therefore propose a distance that fully takes the underlying P into account—not just its second-order features—by considering the potential that characterizes the optimal transport mapping P to the uniform over the unit ball, along with a symmetrized version of the corresponding Bregman divergence.

Boosted Gaussian Bayes Classifier and its application in bank credit scoring

With the explosion of computer science in the last decade, data banks and networksmanagement present a huge part of tomorrows problems. One of them is the development of the best classication method possible in order to exploit the data bases. In classication problems, a representative successful method of the probabilistic model is a Naïve Bayes classier. However, the Naïve Bayes effectiveness still needs to be upgraded. Indeed, Naïve Bayes ignores misclassied instances instead of using it to become an adaptive algorithm. Different works have presented solutions on using Boosting to improve the Gaussian Naïve Bayes algorithm by combining Naïve Bayes classier and Adaboost methods. But despite these works, the Boosted Gaussian Naïve Bayes algorithm is still neglected in the resolution of classication problems. One of the reasons could be the complexity of the implementation of the algorithm compared to a standard Gaussian Naïve Bayes. We present in this paper, one approach of a suitable solution with a pseudo-algorithm that uses Boosting and Gaussian Naïve Bayes principles having the lowest possible complexity. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Gaussian distributions and phase space Weyl–Heisenberg frames

Gaussian states are at the heart of quantum mechanics and play an essential role in quantum information processing. In this paper we provide approximation formulas for the expansion of a general Gaussian symbol in terms of elementary Gaussian functions. For this purpose we introduce the notion of a "phase space frame" associated with a Weyl-Heisenberg frame. Our results give explicit formulas for approximating general Gaussian symbols in phase space by phase space shifted standard Gaussians as well as explicit error estimates and the asymptotic behavior of the approximation.

Dichotomies, structure, and concentration in normed spaces

We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space X=(Rn,‖⋅‖) there exists an invertible linear map T:Rn→Rn withP(|‖TG‖−E‖TG‖|>εE‖TG‖)≤Cexp⁡(−cmax⁡{ε2,ε}log⁡n),ε>0, where G is the standard n-dimensional Gaussian vector and C,c>0 are universal constants. It follows that for every ε∈(0,1) and for every normed space X=(Rn,‖⋅‖) there exists a k-dimensional subspace of X which is (1+ε)-Euclidean and k≥cεlog⁡n/log⁡1ε. This improves by a logarithmic on ε term the best previously known result due to G. Schechtman.

Invariance Principle on the Slice

The non-linear invariance principle of Mossel, O’Donnell, and Oleszkiewicz establishes that if f ( x 1 ,… , x n ) is a multilinear low-degree polynomial with low influences, then the distribution of if f ( b 1 ,…, b n ) is close (in various senses) to the distribution of f ( G 1 ,…, G n ), where B i ∈ R {-1,1} are independent Bernoulli random variables and G i ∼ N(0,1) are independent standard Gaussians. The invariance principle has seen many applications in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans–Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO’s invariance principle works for any two vectors of hypercontractive random variables ( X 1 ,… , X n ),( Y 1 ,… , Y n ) such that (i) Matching moments : X i and Y i have matching first and second moments and (ii) Independence : the variables X 1 ,… , X n are independent, as are Y 1 ,…, Y n . The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions X 1 ,… , X n in which the individual coordinates are not independent. A common example is the uniform distribution on the slice ( [ n ] k ) which consists of all vectors ( x 1 ,…, x n )∈{0,1} n with Hamming weight k . The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdős–Ko–Rado theorems), and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which ( X 1 ,…, X n ) is the uniform distribution on a slice ( [ n ] pn and ( Y 1 ,… , Y n ) consists either of n independent Ber( p ) random variables, or of n independent N( p , p (1- p )) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain’s tail theorem, a version of the Kindler–Safra structural theorem, and a stability version of the t -intersecting Erdős–Ko–Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra, and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

Capacity of the Energy Harvesting Gaussian MAC

We consider an energy harvesting multiple access channel (MAC) where the transmitters are powered by an exogenous stochastic energy harvesting process and equipped with finite batteries. We characterize the capacity region of this channel as $n$ -letter mutual information rate and develop inner and outer bounds that differ by a constant gap. An interesting conclusion that emerges from our results is that the sum-capacity approaches that of a standard additive white Gaussian noise MAC (with only an average constraint on the transmitted power), as the number of users in the MAC becomes large.

Generalizing I-Vector Estimation for Rapid Speaker Recognition

An i-vector is a compact representation that captures both the speaker and session variabilities rendered in a spoken utterance. Over the past years, it has prevailed over other techniques and is now the de facto representation for text-independent speaker recognition. Standard i-vector extraction requires intense computation at run-time. Reducing the computation will allow effective use of i-vector in more applications. Such intense computation arises from the posterior covariance matrix, when estimating the i-vector. There have been studies on how to simplify the computation of posterior covariance matrix with modest success. In this paper, we propose a novel approach to i-vector extraction without the need to evaluate the full posterior covariance thereby speeding up the run-time extraction process. This is achieved by generalizing the i-vector estimation in two ways. First, we introduce the use of occupancy reweighting in conjunction with whitening over the Baum-Welch statistics as part of the preprocessing step. Second, we introduce the so-called subspace-orthogonalizing prior (SOP) to replace the standard Gaussian prior in i-vector formulation. Experiments conducted on the extended-core task of NIST SRE'10 show that the proposed rapid SOP approach achieves considerable speed-up over the standard i-vector with comparable equal error rates.

One-Match-Ahead Forecasting in Two-Team Sports with Stacked Bayesian Regressions

Abstract There is a growing interest in applying machine learning algorithms to real-world examples by explicitly deriving models based on probabilistic reasoning. Sports analytics, being favoured mostly by the statistics community and less discussed in the machine learning community, becomes our focus in this paper. Specifically, we model two-team sports for the sake of one-match-ahead forecasting. We present a pioneering modeling approach based on stacked Bayesian regressions, in a way that winning probability can be calculated analytically. Benefiting from regression flexibility and high standard of performance, Sparse Spectrum Gaussian Process Regression (SSGPR) – an improved algorithm for the standard Gaussian Process Regression (GPR), was used to solve Bayesian regression tasks, resulting in a novel predictive model called TLGProb. For evaluation, TLGProb was applied to a popular sports event – National Basketball Association (NBA). Finally, 85.28% of the matches in NBA 2014/2015 regular season were correctly predicted by TLGProb, surpassing the existing predictive models for NBA.

Median Statistics Estimate of the Distance to the Galactic Center

We show that error distributions of a compilation of 28 recent independent measurements of the distance from the Sun to the Galactic center, R0, are wider than a standard Gaussian and best fit by an n = 4 Student’s t probability density function. Given this non-Gaussianity, the results of our median statistics analysis, summarized as (2σ error), probably provides the most reliable estimate of R0.

Electropolishing effect on roughness metrics of ground stainless steel: a length scale study

Electropolishing is a widely-used electrochemical surface finishing process for metals. The electropolishing of stainless steel has vast commercial application, such as improving corrosion resistance, improving cleanness, and brightening. The surface topography characterization is performed using several techniques with different lateral resolutions and length scales, from atomic force microscopy in the nano-scale (<0.1 µm) to stylus and optical profilometry in the micro- and mesoscales (0.1 µm–1 mm). This paper presents an experimental length scale study of the surface texture of ground stainless steel followed by an electropolishing process in the micro and meso lateral scales. Both stylus and optical profilometers are used, and multiple cut-off lengths of the standard Gaussian filter are adopted. While the commonly used roughness amplitude parameters (Ra, Rq and Rz) fail to characterize electropolished textures, the root mean square slope (RΔq) is found to better describe the electropolished surfaces and to be insensitive to scale.

Communication in the Presence of a State-Aware Adversary

We study communication systems over the state-dependent channels in the presence of a malicious state-aware jamming adversary. The channel has a memoryless state with an underlying distribution. The adversary introduces a jamming signal into the channel. The message and the entire state sequence are known non-causally to both the encoder and the adversary. This state-aware adversary may choose an arbitrary jamming vector depending on the message and the state vector. Taking an arbitrarily varying channel (AVC) approach, we consider two setups, namely, the discrete memoryless Gel’fand–Pinsker AVC and the additive white Gaussian dirty paper (DP) AVC. We determine the randomized coding capacity of both the AVCs under a maximum probability of error criterion. Similar to other randomized coding setups, we show that the capacity is the same even under the average probability of error criterion. Though the adversary can choose an arbitrary vector jamming strategy, we prove that the adversary cannot affect the rate any worse than when it employs a memoryless strategy, which depends only on the instantaneous state. Thus, the AVC capacity characterization is given in terms of the capacity of the worst memoryless channels with state, induced by the adversary employing such memoryless jamming strategies. For the DP-AVC, it is further shown that among memoryless jamming strategies, none impact the communication more than a memoryless Gaussian jamming strategy which completely disregards the knowledge of the state. Thus, the capacity of the DP-AVC equals that of a standard additive white Gaussian noise (AWGN) channel with two independent sources of AWGN, i.e., the channel noise and the jamming noise.

Dispersion managed dissipative solitons with higher order nonlinearity and frequency-selective feedback

We present the generation and interaction dynamics of dispersion managed soliton in a dissipative cubic-quintic nonlinear optical fiber coupled with frequency-selective feedback. Analytically, perturbative variational method is used to obtain evolution equations for different soliton parameters that are subsequently solved to study the propagation characteristics of the dispersion managed dissipative solitons (DMDS). The DMDSs show breather like characteristics and are found to be robust against certain level of initial noise. Both the standard Gaussian and more generic super-Gaussian pulse profile are shown to yield stable DMDS. Interaction of two DMDSs of same phase may lead to a bound state. By varying the temporal separation and phase difference between the DMDSs the bound state can be switched between low to high speed regime.

Triangular random matrices and biorthogonal ensembles

We study the singular values of certain triangular random matrices. When their elements are i.i.d. standard complex Gaussian random variables, the squares of the singular values form a biorthogonal ensemble, and with an appropriate change in the distribution of the diagonal elements, they give the biorthogonal Laguerre ensemble.

The median of an exponential family and the normal law

Let P be a probability on the real line generating a natural exponential family (Pt)t∈R. We show that the property that t is a median of Pt for all t characterizes P as the standard Gaussian law N(0,1).

Electronic Band Structure of Helical Polyisocyanides.

Restricted Hartree-Fock computations are reported for a methyl isocyanide polymer (repeating unit -C═N-CH3), whose most stable conformation is expected to be a helical chain. The computations used a standard contracted Gaussian orbital set at the computational levels STO-3G, 3-21G, 6-31G, and 6-31G**, and studies were made for two line-group configurations motivated by earlier work and by studies of space-filling molecular models: (1) A structure of line-group symmetry L95, containing a 9-fold screw axis with atoms displaced in the axial direction by 5/9 times the lattice constant, and (2) a structure of symmetry L41 that had been proposed, containing a 4-fold screw axis with translation by 1/4 of the lattice constant. Full use of the line-group symmetry was employed to cause most of the computational complexity to depend only on the size of the asymmetric repeating unit. Data reported include computed bond properties, atomic charge distribution, longitudinal polarizability, band structure, and the convoluted density of states. Most features of the description were found to be insensitive to the level of computational approximation. The work also illustrates the importance of exploiting line-group symmetry to extend the range of polymer structural problems that can be treated computationally.

Local Comparability of Measures, Averaging and Maximal Averaging Operators

We explore the consequences for the boundedness properties of averaging and maximal averaging operators, of the following local comparabiliity condition for measures: Intersecting balls of the same radius have comparable sizes. Since in geometrically doubling spaces this property yields the same results as doubling, we study under which circumstances it is equivalent to the latter condition, and when it is more general. We also study the concrete case of the standard gaussian measure, where this property fails, but nevertheles averaging operators are uniformly bounded, with respect to the radius, in $L^1$. However, such bounds grow exponentially fast with the dimension, very much unlike the case of Lebesgue measure.

Quantum diffusion during inflation and primordial black holes

We calculate the full probability density function (PDF) of inflationary curvature perturbations, even in the presence of large quantum backreaction. Making use of the stochastic-δ N formalism, two complementary methods are developed, one based on solving an ordinary differential equation for the characteristic function of the PDF, and the other based on solving a heat equation for the PDF directly. In the classical limit where quantum diffusion is small, we develop an expansion scheme that not only recovers the standard Gaussian PDF at leading order, but also allows us to calculate the first non-Gaussian corrections to the usual result. In the opposite limit where quantum diffusion is large, we find that the PDF is given by an elliptic theta function, which is fully characterised by the ratio between the squared width and height (in Planck mass units) of the region where stochastic effects dominate. We then apply these results to the calculation of the mass fraction of primordial black holes from inflation, and show that no more than ∼ 1 e-fold can be spent in regions of the potential dominated by quantum diffusion. We explain how this requirement constrains inflationary potentials with two examples.

Variational Bayesian multinomial logistic Gaussian process classification

The multinomial logistic Gaussian process is a flexible non-parametric model for multi-class classification tasks. These tasks are often involved in solving a pattern recognition problem in real life. In such contexts, the multinomial logistic function (or softmax function) is usually assumed to be the likelihood function. But, exact inferences for this model have proved challenging problem because it requires high-dimensional integration. In this paper, we propose approximate variational Bayesian inference for the multinomial logistic Gaussian process model. First, we compute the second-order approximation for the logarithm of the logistic likelihood function using Taylor series expansion, and derive the posterior distributions of all hidden variables and model parameters using the variational Bayesian inference method. Second, we derive the predictive distribution of the latent classification variable corresponding to the relevant test data point using the characteristics of the Cauchy product for a standard Gaussian process using a learning model parameter. We conducted experiments to verify the effectiveness of the proposed model using a number of synthetic and real datasets. The results show that the proposed model has superior classification capability to existing methods.

Multifeature Anisotropic Orthogonal Gaussian Process for Automatic Age Estimation

Automatic age estimation is an important yet challenging problem. It has many promising applications in social media. Of the existing age estimation algorithms, the personalized approaches are among the most popular ones. However, most person-specific approaches rely heavily on the availability of training images across different ages for a single subject, which is usually difficult to satisfy in practical application of age estimation. To address this limitation, we first propose a new model called Orthogonal Gaussian Process (OGP), which is not restricted by the number of training samples per person. In addition, without sacrifice of discriminative power, OGP is much more computationally efficient than the standard Gaussian Process. Based on OGP, we then develop an effective age estimation approach, namely anisotropic OGP (A-OGP), to further reduce the estimation error. A-OGP is based on an anisotropic noise level learning scheme that contributes to better age estimation performance. To finally optimize the performance of age estimation, we propose a multifeature A-OGP fusion framework that uses multiple features combined with a random sampling method in the feature space. Extensive experiments on several public domain face aging datasets (FG-NET, MORPH Album1, and MORPH Album 2) are conducted to demonstrate the state-of-the-art estimation accuracy of our new algorithms.