Higher-order Fourier dimension and frequency decompositions

This paper continues work begun in \cite{M1}, in which we introduced a theory of Gowers uniformity norms for singular measures on $\mathbb{R}^d$. There, given a $d$-dimensional measure $\mu$, we introduced a $(k+1)d$-dimensional measure $\triangle^k\mu$, and developed a Uniformity norm $\|\mu\|_{U^k}$ whose $2^k$-th power is equivalent to $\triangle^k\mu([0,1]^{d(k+1)}$. In the present work, we introduce a fractal dimension associated to measures $\mu$ which we refer to as the $k$th-order Fourier dimension of $\mu$. This $k$-th order Fourier dimension is a normalization of the asymptotic decay rate of the Fourier transform of the measure $\int \triangle^k\mu(x;\cdot)\,dx$, and coincides with the classic Fourier dimension in the case that $k=1$. It provides quantitative control on the size of the $U^k$ norm. The main result of the present paper is that this higher-order Fourier dimension controls the rate at which $\|\mu-\mu_n\|_{U^k}\rightarrow 0$, where $\mu_n$ is an approximation to the measure $\mu$. This allows us to extract delicate information from the Fourier transform of a measure $\mu$ and the interactions of its frequency components, which is not available from the $L^p$ norms- or the decay- of the Fourier transform. In future work \cite{M4}, we apply this to obtain a differentiation theorem for singular measures

Towards a Fisher-information description of complexity in de Sitter universe

Recent developments on holography and quantum information physics suggest that quantum information theory come to play a fundamental role in understanding quantum gravity. Cosmology, on the other hand, plays a significant role in testing quantum gravity effects. How to apply this idea to a realistic universe is still missing. Here we show some concepts in quantum information theory have their cosmological descriptions. Particularly, we show complexity of a tensor network can be regarded as a Fisher information measure(FIM) of a dS universe, followed by several observations: (i) the holographic entanglement entropy has a tensor-network description and admits a information-theoretical interpretation, (ii) on-shell action of dS spacetime has a same description of FIM, (iii) complexity/action(CA) duality holds for dS spacetime. Our result is also valid for $f(R)$ gravity, whose FIM exhibits the same features of a recent proposed $L^n$ norm complexity.Comment: 18 pages, 3 figures. v2: improvements to presentation, fixes typos and matches published versio

R-Norm Information Measure with Applications in Multi Criteria Decision Making Technique under Intuitionistic Fuzzy Set Environment

The main aim of this research article is to define a new information measure for quantifying fuzziness in the intuitionistic fuzzy set environment. For this purpose, we present R-norm intuitionistic fuzzy measure that quantifies the amount of vagueness or fuzziness of a particular fuzzy set. We prove that this measure is a valid measure of intuitionistic fuzzy entropy by making it satisfy essential properties. Also, some mathematical properties are used to check the validation of the measure. In the end, a practical example of decision-making is illustrated in terms of Multi Criteria Decision Making problem that presents the application of the proposed measure

Long progressions in sets of fractional dimension

We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of Additive Combinatorics and uniformity norms) extension of the Fourier dimension of Geometric Measure Theory, and can be understood as asking that the uniformity norm of a measure, restricted to a given scale, decay as the scale increases. We further obtain quantitative information about the size and $L^p$ regularity of the set of common distances of the artihmetic progressions contained in the subsets of $\mathbb{R}$ under consideration

Deterministic equivalents for certain functionals of large random matrices

Consider an $N\times n$ random matrix $Y_n=(Y^n_{ij})$ where the entries are given by $Y^n_{ij}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X^n_{ij}$, the $X^n_{ij}$ being independent and identically distributed, centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic $N\times n$ matrix A_n whose columns and rows are uniformly bounded in the Euclidean norm. Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a deterministic $N\times N$ matrix-valued function T_n(z) analytic in $\mathbb{C}-\mathbb{R}^+$ such that, almost surely, $$\lim_{n\to+\infty,N/n\to c}\biggl(\frac{1}{N}\operatorname {Trace}(\Sigma_n\Sigma_n^T-zI_N)^{-1}-\frac{1}{N}\operatorname {Trace}T_n(z)\biggr)=0.$$ Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of $\Sigma_n\Sigma_n^T$. For each n, the entries of matrix T_n(z) are defined as the unique solutions of a certain system of nonlinear functional equations. It is also proved that $\frac{1}{N}\operatorname {Trace} T_n(z)$ is the Stieltjes transform of a probability measure $\pi_n(d\lambda)$, and that for every bounded continuous function f, the following convergence holds almost surely $$\frac{1}{N}\sum_{k=1}^Nf(\lambda_k)-\int_0^{\infty}f(\lambda)\pi _n(d\lambda)\mathop {\longrightarrow}_{n\to\infty}0,$$ where the $(\lambda_k)_{1\le k\le N}$ are the eigenvalues of $\Sigma_n\Sigma_n^T$. This work is motivated by the context of performance evaluation of multiple inputs/multiple output (MIMO) wireless digital communication channels. As an application, we derive a deterministic equivalent to the mutual information: $$C_n(\sigma^2)=\frac{1}{N}\mathbb{E}\log \det\biggl(I_N+\frac{\Sigma_n\Sigma_n^T}{\sigma^2}\biggr),$$ where $\sigma^2$ is a known parameter.Comment: Published at http://dx.doi.org/10.1214/105051606000000925 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fusing R features and local features with context-aware kernels for action recognition

The performance of action recognition in video sequences depends significantly on the representation of actions and the similarity measurement between the representations. In this paper, we combine two kinds of features extracted from the spatio-temporal interest points with context-aware kernels for action recognition. For the action representation, local cuboid features extracted around interest points are very popular using a Bag of Visual Words (BOVW) model. Such representations, however, ignore potentially valuable information about the global spatio-temporal distribution of interest points. We propose a new global feature to capture the detailed geometrical distribution of interest points. It is calculated by using the 3D R transform which is defined as an extended 3D discrete Radon transform, followed by the application of a two-directional two-dimensional principal component analysis. For the similarity measurement, we model a video set as an optimized probabilistic hypergraph and propose a context-aware kernel to measure high order relationships among videos. The context-aware kernel is more robust to the noise and outliers in the data than the traditional context-free kernel which just considers the pairwise relationships between videos. The hyperedges of the hypergraph are constructed based on a learnt Mahalanobis distance metric. Any disturbing information from other classes is excluded from each hyperedge. Finally, a multiple kernel learning algorithm is designed by integrating the l2 norm regularization into a linear SVM classifier to fuse the R feature and the BOVW representation for action recognition. Experimental results on several datasets demonstrate the effectiveness of the proposed approach for action recognition

Random matrices, large deviations and reflected Brownian motion

In this thesis we present results in large deviations theory, free probability and the theory of reflected Brownian motion. We study the large deviations behaviour of the block structure of a non-crossing partition chosen uniformly at random. This allows us to apply the free momentcumulant formula of Speicher to express the spectral radius of a non-commutative random variable in terms of its free cumulants. Next the distributions of three quadratic functionals of the free Brownian bridge are studied: the square norm, the signature and the Lévy area of the free Brownian bridge. We introduce two representation of the free Brownian bridge as series involving free semicircular variables, analogous to classical results due to Lévy and Kac. The latter representation extends to all semicircular processes. For each of the three quadratic functionals we give the R-transform, from which we extract information about the distribution, including free infinite divisibility and smoothness of the density. We also apply our result about the spectral radius to compute the maximum of the support for Lévy area and square norm. In both cases we obtain implicit equations. The final chapter of the thesis is devoted to the study of a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. This is motivated by recent developments in the theory of directed polymer and percolation models, in which existence of an invariant measure in product form plays a role. Informally, RBM is defined by running a standard Brownian motion in the polyhedral domain and giving it a singular drift whenever it hits one of the boundaries, kicking the process back into the interior. Our process is obtained by replacing this singular drift by a continuous one, involving a continuous potential. RBM has an invariant measure in product form if and only if a certain skew-symmetry condition holds. We show that this result extends to our generalisation. Applications include examples motivated by queueing theory, Brownian motion with rank-dependent drift and a process with close connections to the δ-Bose gas

Quantum Correlations in Multipartite Quantum Systems

We review some concepts and properties of quantum correlations, in particular multipartite measures, geometric measures and monogamy relations. We also discuss the relation between classical and total correlationsComment: to be published as a chapter of the book "Lectures on general quantum correlations and their applications" edited by F. Fanchini, D. Soares-Pinto, and G. Adesso (Springer, 2017