Descent methods for optimization on homogeneous manifolds

In this article we present a framework for line search methods for optimization on smooth homogeneous manifolds, with particular emphasis to the Lie group of real orthogonal matrices. We propose strategies of univariate descent (UVD), methods. The main advantage of this approach is that the optimization problem is broken down into one-dimensional optimization problems, so that each optimization step involves little computation effort. In order to assess its numerical performance, we apply the devised method to eigen-problems as well as to independent component analysis in signal processing.     

Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing

This paper is dedicated to the statistical analysis of the space of multivariate normal distributions with an application to the processing of Diffusion Tensor Images (DTI). It relies on the differential geometrical properties of the underlying parameters space, endowed with a Riemannian metric, as well as on recent works that led to the generalization of the normal law on Riemannian manifolds. We review the geometrical properties of the space of multivariate normal distributions with zero mean vector and focus on an original characterization of the mean, covariance matrix and generalized normal law on that manifold. We extensively address the derivation of accurate and efficient numerical schemes to estimate these statistical parameters. A major application of the present work is related to the analysis and processing of DTI datasets and we show promising results on synthetic and real examples.     

Manifold models for signals and images

This article proposes a new class of models for natural signals and images. These models constrain the set of patches extracted from the data to analyze to be close to a low-dimensional manifold. This manifold structure is detailed for various ensembles suitable for natural signals, images and textures modeling. These manifolds provide a low-dimensional parameterization of the local geometry of these datasets. These manifold models can be used to regularize inverse problems in signal and image processing. The restored signal is represented as a smooth curve or surface traced on the manifold that matches the forward measurements. A manifold pursuit algorithm computes iteratively a solution of the manifold regularization problem. Numerical simulations on inpainting and compressive sensing inversion show that manifolds models bring an improvement for the recovery of data with geometrical features.     

Odd-length armlets with flipping property and its application in image compression

Armlet introduced by Lian is a novel property of multiwavelet designed for signal processing. In this paper, we first recall some concepts of armlet. Then, we construct a new class of analysis ready multiwavelets (armlets) with odd-length filters based on Lian’s method. To test the performance of armlet in application, we study the statistical analysis of their discrete multiwavelet transforms (DMWTs) and use them to compress images. In comparison with other wavelets, armlets show potential advantages.     

Codes for MIMO transmission based on orthogonal sequences

We consider MIMO communication systems with Rayleigh fading. We propose a new coded modulation based on orthogonal sequences and state a new decodability condition. We introduce concepts and constructions of permutation free (PF) and permutation and repetition free (PRF) codes. We also propose a construction of PRF codes with sign manipulation, whose code rate can exceed 1. For better analysis and construction of these codes we introduce a one-to-one mapping that transforms signal matrices to vectors over a finite field. We propose construction algorithms for PF and PRF codes. We build PF and PRF codes with large cardinality, which in several case achieve the maximum cardinality. Simulation of the constructed codes and estimation of their performance was done in Simulink environment. Results show high error-correcting capability, which often reaches that of STBC codes with full transmit diversity.     

Principal manifold learning by sparse grids

In this paper, we deal with the construction of lower-dimensional manifolds from high-dimensional data which is an important task in data mining, machine learning and statistics. Here, we consider principal manifolds as the minimum of a regularized, non-linear empirical quantization error functional. For the discretization we use a sparse grid method in latent parameter space. This approach avoids, to some extent, the curse of dimension of conventional grids like in the GTM approach. The arising non-linear problem is solved by a descent method which resembles the expectation maximization algorithm. We present our sparse grid principal manifold approach, discuss its properties and report on the results of numerical experiments for one-, two- and three-dimensional model problems.      

Linear and synchrosqueezed time–frequency representations revisited: Overview,standards of use,resolution, reconstruction,concentration, and algorithms

Time–frequency representations (TFRs) of signals, such as the windowed Fourier transform (WFT), wavelet transform (WT) and their synchrosqueezed versions (SWFT, SWT), provide powerful analysis tools. Here we present a thorough review of these TFRs, summarizing all practically relevant aspects of their use, reconsidering some conventions and introducing new concepts and procedures to advance their applicability and value. Furthermore, a detailed numerical and theoretical study of three specific questions is provided, relevant to the application of these methods, namely: the effects of the window/wavelet parameters on the resultant TFR; the relative performance of different approaches for estimating parameters of the components present in the signal from its TFR; and the advantages/drawbacks of synchrosqueezing. In particular, we show that the higher concentration of the synchrosqueezed transforms does not seem to imply better resolution properties, so that the SWFT and SWT do not appear to provide any significant advantages over the original WFT and WT apart from a more visually appealing pictures. The algorithms and Matlab codes used in this work, e.g. those for calculating (S)WFT and (S)WT, are freely available for download.     

Unsupervised manifold alignment using soft-assign technique

In this paper, we propose a robust unsupervised algorithm for automatic alignment of two manifolds in different datasets with possibly different dimensionalities. The significant contribution is that the proposed alignment algorithm is performed automatically without any assumptions on the correspondences between the two manifolds. For such purpose, we first automatically extract local feature histograms at each point of the manifolds and establish an initial similarity between the two datasets by matching their histogram-based features. Based on such similarity, an embedding space is estimated where the distance between the two manifolds is minimized while maximally retaining the original structure of the manifolds. The elegance of this idea is that such complicated problem is formulated as a generalized eigenvalue problem, which can be easily solved. The alignment process is achieved by iteratively increasing the sparsity of correspondence matrix until the two manifolds are correctly aligned and consequently one can reveal their joint structure. We demonstrate the effectiveness of our algorithm on different datasets by aligning protein structures, 3D face models and facial images of different subjects under pose and lighting variations. Finally, we also compare with a state-of-the-art algorithm and the results show the superiority of the proposed manifold alignment in terms of vision effect and numerical accuracy.     

滤波器组框架理论及其在图信号处理中的应用

滤波器组框架理论是应用数学、信号处理、图像处理和数字通信等领域的重要问题之一,对滤波器组框架的分析和设计问题进行研究有着重要的科学意义和应用前景.近年来,随着高维非规则化数据信息大量涌现,很多学者开始研究图信号处理的滤波器组方法.因此对滤波器组框架理论及其在图信号处理中的应用进行了综述研究.首先对传统滤波器组框架理论的基础知识进行概述,总结滤波器组框架分析与设计方法;然后重点介绍两类图信号处理架构以及图滤波器组的最新研究成果;最后对未来的研究进行展望.     

子波变换理论及其在信号处理中的应用

子波分析的形成是傅里叶分析发展史上里程碑式的进展，子波分析优于傅里叶交换的地方在于它在时域和频域同时具有良好的局部化性质，从而可以把分析的重点聚焦到任意的细节，被人们誉为数学显微镜，成为近年来在工具和方法上的重大突破。本文将子波理论中的主要定理、结论、变换特性和一些重要概念加以综述，以促进子波理论的应用。本文的重点在于介绍多分辨率分析和子波分析及其实现、子波变换及其算法、子波和滤波器组等的重要内容，并介绍其在信号处理中的应用及研究动态。     

新的连续自编码网络流形学习研究

发现高维观测数据空间的低维流形结构,是流形学习的主要目标。在前人利用神经网络进行非线性降维的基础上,提出一种新的连续自编码（Continuous Autoencoder,C-Autoencoder）网络,该方法特别采用CRBM（Continuous Restricted Boltzmann Machine）的网络结构,通过训练具有多个中间层的双向深层神经网络可将高维连续数据转换成低维嵌套并继而重构高维连续数据。特别地,这种连续自编码网络可以提供高维连续数据空间和低维嵌套结构的双向映射,不仅有效解决了大多数非线性降维方法所不具备的逆向映射问题,而且特别适用于高维连续数据的降维和重构。将C-Autoencoder用于人工连续数据的实验表明,C-Autoencoder不仅能发现嵌入在高维连续数据中的非线性流形结构,也能有效地从低维嵌套中恢复原始高维连续数据。     

A new approach to discover interlacing data structures in high-dimensional space

The discovery of structures hidden in high-dimensional data space is of great significance for understanding and further processing of the data. Real world datasets are often composed of multiple low dimensional patterns, the interlacement of which may impede our ability to understand the distribution rule of the data. Few of the existing methods focus on the detection and extraction of the manifolds representing distinct patterns. Inspired by the nonlinear dimensionality reduction method ISOmap, in this paper we present a novel approach called Multi-Manifold Partition to identify the interlacing low dimensional patterns. The algorithm has three steps: first a neighborhood graph is built to capture the intrinsic topological structure of the input data, then the dimensional uniformity of neighboring nodes is analyzed to discover the segments of patterns, finally the segments which are possibly from the same low-dimensional structure are combined to obtain a global representation of distribution rules. Experiments on synthetic data as well as real problems are reported. The results show that this new approach to exploratory data analysis is effective and may enhance our understanding of the data distribution.     

A new approach for rotation-invariant and noise-resistant texture analysis and classification

The analysis and classification of images, such as texture images, is one of the substantial and important fields in image processing. Due to destructive effects of image rotation and noise, the stability and efficiency of texture analysis and classification methods are an important research area. In this paper, a new method for texture analysis and classification has been proposed which is based on a particular combination of wavelet, ridgelet and Fourier transforms as well as support vector machine. The proposed method has been evaluated for 13 texture datasets produced by three original datasets containing 25 and 111 original textures from Brodatz database and 24 original textures from OUTEX database. These datasets comprise 415584 and 93600 rotated noise-free and noisy texture images for Brodatz database and also 49920 noisy and 4320 noise-free texture images for OUTEX database, respectively. Simulation results demonstrate the capability, efficiency and also stability of the proposed method especially for real-time rotation-invariant and noise-resistant texture analysis and classification.     

基于可变形多尺度变换的几何不变鲁棒图像水印算法

尽管过去十多年鲁棒数字水印取得了长足的进展,然而如何有效地抵抗几何变换仍然是鲁棒数字水印的关键问题之一. 本文通过设计具有平移不变性、可旋转和可缩放特性的可变形多尺度变换(Deformable multi-scale transform, DMST)来抵抗全局几何攻击. 基于此变换,本文从理论上推导几何同步机制,然后进一步利用它并辅助于模板来设计能有效估计几何攻击参数的算法. 此外,将常规(双)正交小波域隐马尔科夫模型进一步推广到可旋转小波域,以提高水印的检测性能. 实验结果表明本文算法对于常见信号处理攻击、全局几何攻击及其联合攻击具有很好的鲁棒性.     

基于聚类过采样和自动编码器的网络入侵检测方法

近年来,随着互联网技术的不断发展,入侵检测在维护网络空间安全方面发挥着越来越重要的作用。但是,由于网络入侵行为的数据稀疏性,已有的检测方法对于海量流量数据的检测效果较差,模型准确率、F-measure等指标数值较低,并且高维数据处理的成本过高。为了解决这些问题,本文提出了一种基于稀疏异常样本数据场景下的新型深度神经网络入侵检测方法,该方法能够有效地识别不平衡数据集中的异常行为。本文首先使用k均值综合少数过采样方法来处理不平衡的流量数据,解决网络流量数据类别分布不平衡问题,平衡网络流量数据分布。再采用自动编码器来处理海量高维数据并训练检测模型,来提升海量高维流量中异常行为的检测精度,并在两个真实典型的入侵检测数据集上进行了大量的实验。实验结果表明,本文所提出的方法在两个真实典型数据集上的检测准确率分别为99.06%和99.16%, F-measure分别为99.15%和98.22%。相比于常用的欠采样和过采样方法, k均值综合少数过采样技术能够有效地解决网络流量数据类别分布不平衡的问题,提升模型对低频攻击行为的检测效果。同时,与已有的网络入侵检测方法相比,本文所提出的方法在准确率、F-m...     

LoHDP: Adaptive local differential privacy for high-dimensional data publishing

The increasing availability of high-dimensional data collected from numerous users has led to the need for multi-dimensional data publishing methods that protect individual privacy. In this paper, we investigate the use of local differential privacy for such purposes. Existing solutions calculate pairwise attribute marginals to construct probabilistic graphical models for generating attribute clusters. These models are then used to derive low-dimensional marginals of these clusters, allowing for an approximation of the distribution of the original dataset and the generation of synthetic datasets. Existing solutions have limitations in computing the marginals of pairwise attributes and multi-dimensional distribution on attribute clusters, as well as constructing relational dependency graphs that contain large clusters. To address these problems, we propose LoHDP, a high-dimensional data publishing method composed of adaptive marginal computing and an effective attribute clustering method. The adaptive local marginal calculates any k-dimensional marginals required in the algorithm. In particular, methods such as sampling-based randomized response are used instead of privacy budget splits to perturb user data. The attribute clustering method measures the correlation between pairwise attributes using an effective method, reduces the search space during the construction of the dependency graph using high-pass filtering technology, and realizes dimensionality reduction by combining sufficient triangulation operation. We demonstrate through extensive experiments on real datasets that our LoHDP method outperforms existing methods in terms of synthetic dataset quality.     

Multipartite quantum systems and symplectic toric manifolds

In this paper we study the geometrical structures of multi-qubit states based on symplectic toric manifolds. After a short review of symplectic toric manifolds, we discuss the space of a single quantum state in terms of these manifolds. We also investigate entangled multipartite states based on moment map and Delzant’s construction of toric manifolds and algebraic toric varieties.     

Association rule mining algorithms on high-dimensional datasets

The science of bioinformatics has been accelerating at a fast pace, introducing more features and handling bigger volumes. However, these swift changes have, at the same time, posed challenges to data mining applications, in particular efficient association rule mining. Many data mining algorithms for high-dimensional datasets have been put forward, but the sheer numbers of these algorithms with varying features and application scenarios have complicated making suitable choices. Therefore, we present a general survey of multiple association rule mining algorithms applicable to high-dimensional datasets. The main characteristics and relative merits of these algorithms are explained, as well, pointing out areas for improvement and optimization strategies that might be better adapted to high-dimensional datasets, according to previous studies. Generally speaking, association rule mining algorithms that merge diverse optimization methods with advanced computer techniques can better balance scalability and interpretability.