Symmetric Neighborhood Research Articles

INVERSE THEOREMS FOR SETS AND MEASURES OF POLYNOMIAL GROWTH

We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$ , showing that such sets are controlled by (a bounded number of translates of) a coset nilprogression in a certain precise sense. This description recovers some previous results of Breuillard–Green–Tao and Breuillard–Tointon concerning sets of polynomial growth; we are also able to describe the subsequent growth of $|A^m|$ fairly explicitly for $m \geq n$ , at least when $A$ is a symmetric neighbourhood of the identity. We also obtain an analogous description of symmetric probability measures $\mu $ whose $n$ -fold convolutions $\mu ^{*n}$ obey the condition $\| \mu ^{*n} \|_{\ell ^2}^{-2} \leq n^d \|\mu \|_{\ell ^2}^{-2}$ . In the abelian case, this description recovers the inverse Littlewood–Offord theorem of Nguyen–Vu, and gives a ‘symmetrized’ variant of a recent non-abelian inverse Littlewood–Offord theorem of Tiep–Vu. Our main tool to establish these results is the inverse theorem of Breuillard, Green, and the author that describes the structure of approximate groups.

Dynamic texture recognition based on completed volume local binary pattern

Dynamic textures are the sequences of images of moving scenes having some stationary properties in time; these include sea-waves, smoke, foliage, whirlwind etc. In recent years, dynamic texture description and recognition has attracted growing attention. This paper presents a more effective completed modeling of volume local binary pattern (VLBP) to recognize dynamic textures. Due to the dependency on local binary pattern (LBP), traditional VLBP also suffers with noise sensitivity and may give the same LBP code to different structural patterns; thus limiting the discriminating power of a texture descriptor. To represent temporal textures we have used VLBP. Local region of a volume is represented using its center frame and local sign magnitude difference with the circularly symmetric neighborhood. We have proposed a new contrast operator to complement the sign information of the temporal texture. To add additional discriminative information, volume center pixel information is also fused with the sign magnitude difference of texture. By combining these features into hybrid distributions we get higher classification accuracy for rotation invariant texture classification. Experimental results on UCLA and Dyntex databases show that the proposed approach provides better performance in comparison to the existing approaches.

Completeness on locally convex cones

We investigate complete and compact subsets for the lower, upper and symmetric topologies of a locally convex cone and prove that weakly closed sets will be weakly compact, whenever they are weakly precompact. This leads to the weak* compactness of the polars of neighborhoods and weak compactness of the lower, upper and symmetric neighborhoods.

On symmetric neighborhood assignments

In this paper, properties of symmetric neighborhood assignments are discussed. We show that in some results of Balogh and Gruenhage, the family of spherical neighborhoods in a metric space can be generalized to a symmetric open neighborhood assignments in any topology space. By a simple example, we answer a question raised by Hung negatively. We also discuss two questions raised by Nagata about metrization and symmetric neighborhood assignments. By giving new characterizations of strongly paracompact metrizable spaces and orthocompact Moore spaces respectively, we show that answer of the first question is negative, while the second question is undecidable under ZFC.

Outlier Identification using Symmetric Neighborhoods

In recent research, outlier mining has been widely used in areas such as telecommunication, health care, finance and network intrusion detection. By applying outlier mining in network anomaly detection, rarely occurring attacks can be identified. In this paper we propose an outlier mining method based on symmetric neighborhood relationships. We evaluate the method with UCI ML Repository datasets, benchmark dataset KDD Cup 1999 and real time intrusion datasets. The experimental results are compared with existing approaches and performance is excellent.

ON A NONABELIAN BALOG–SZEMERÉDI-TYPE LEMMA

AbstractWe show that if G is a group and A⊂G is a finite set with ∣A2∣≤K∣A∣, then there is a symmetric neighbourhood of the identity S such that Sk⊂A2A−2 and ∣S∣≥exp (−KO(k))∣A∣.

Robust color texture features based on ranklets and discrete Fourier transform

We present a set of multiscale, multidirectional, rotation-invariant features for color texture characterization. The proposed model is based on the ranklet transform, a technique relying on the calculation of the relative rank of the intensity level of neighboring pixels. Color and texture are merged into a compact descriptor by computing the ranklet transform of each color channel separately and of couples of color channels jointly. Robustness against rotation is based on the use of circularly symmetric neighborhoods together with the discrete Fourier transform. Experimental results demonstrate that the approach shows good robustness and accuracy.

Ordering process of self-organizing maps improved by asymmetric neighborhood function

The Self-organizing map (SOM) is an unsupervised learning method based on the neural computation, which has found wide applications. However, the learning process sometime takes multi-stable states, within which the map is trapped to an undesirable disordered state including topological defects on the map. These topological defects critically aggravate the performance of the SOM. In order to overcome this problem, we propose to introduce an asymmetric neighborhood function for the SOM algorithm. Compared with the conventional symmetric one, the asymmetric neighborhood function accelerates the ordering process even in the presence of the defect. However, this asymmetry tends to generate a distorted map. This can be suppressed by an improved method of the asymmetric neighborhood function. In the case of one-dimensional SOM, it is found that the required steps for perfect ordering is numerically shown to be reduced from O(N (3)) to O(N (2)). We also discuss the ordering process of a twisted state in two-dimensional SOM, which can not be rectified by the ordinary symmetric neighborhood function.

Further development of multiple centrality correctors for interior point methods

This paper addresses the role of centrality in the implementation of interior point methods. We provide theoretical arguments to justify the use of a symmetric neighbourhood, and translate them into computational practice leading to a new insight into the role of re-centering in the implementation of interior point methods. Second-order correctors, such as Mehrotra's predictor---corrector, can occasionally fail: we derive a remedy to such difficulties from a new interpretation of multiple centrality correctors. Through extensive numerical experience we show that the proposed centrality correcting scheme leads to noteworthy savings over second-order predictor---corrector technique and previous implementations of multiple centrality correctors.

Self-Organizing Maps with Asymmetric Neighborhood Function

The self-organizing map (SOM) is an unsupervised learning method as well as a type of nonlinear principal component analysis that forms a topologically ordered mapping from the high-dimensional data space to a low-dimensional representation space. It has recently found wide applications in such areas as visualization, classification, and mining of various data. However, when the data sets to be processed are very large, a copious amount of time is often required to train the map, which seems to restrict the range of putative applications. One of the major culprits for this slow ordering time is that a kind of topological defect (e.g., a kink in one dimension or a twist in two dimensions) gets created in the map during training. Once such a defect appears in the map during training, the ordered map cannot be obtained until the defect is eliminated, for which the number of iterations required is typically several times larger than in the absence of the defect. In order to overcome this weakness, we propose that an asymmetric neighborhood function be used for the SOM algorithm. Compared with the commonly used symmetric neighborhood function, we found that an asymmetric neighborhood function accelerates the ordering process of the SOM algorithm, though this asymmetry tends to distort the generated ordered map. We demonstrate that the distortion of the map can be suppressed by improving the asymmetric neighborhood function SOM algorithm. The number of learning steps required for perfect ordering in the case of the one-dimensional SOM is numerically shown to be reduced from O(N(3)) to O(N(2)) with an asymmetric neighborhood function, even when the improved algorithm is used to get the final map without distortion.

Intensity-driven adaptive-neighborhood technique for polarimetric and interferometric SAR parameters estimation

In this paper, a new method to filter coherency matrices of polarimetric or interferometric data is presented. For each pixel, an adaptive neighborhood (AN) is determined by a region growing technique driven exclusively by the intensity image information. All the available intensity images of the polarimetric and interferometric terms are fused in the region growing process to ensure the validity of the stationarity assumption. Afterward, all the pixels within the obtained AN are used to yield the filtered values of the polarimetric and interferometric coherency matrices, which can be derived either by direct complex multilooking or from the locally linear minimum mean-squared error (LLMMSE) estimator. The entropy/alpha/anisotropy decomposition is then applied to the estimated polarimetric coherency matrices, and coherence optimization is performed on the estimated polarimetric and interferometric coherency matrices. Using this decomposition, unsupervised classification for land applications by an iterative algorithm based on a complex Wishart density function is also applied. The method has been tested on airborne high-resolution polarimetric interferometric synthetic aperture radar (POL-InSAR) images (Oberpfaffenhofen area-German Space Agency). For comparison purposes, the two estimation techniques (complex multilooking and LLMMSE) were tested using three different spatial supports: a fix-sized symmetric neighborhood (boxcar filter), directional nonsymmetric windows, and the proposed AN. Subjective and objective performance analysis, including coherence edge detection, receiver operating characteristics plots, and bias reduction tables, recommends the proposed algorithm as an effective POL-InSAR postprocessing technique.

The theory of elementary landscapes

When joined to a stipulated neighborhood diagraph, an objective functin defined on the solution space of a real combinatorial optimization problem forms a landscape. Grover shows that landscapes satisfying a certain difference equation have properties favorable to local search. Studying only symmetric and regular neighborhood diagraphs, Stadler defines elementary landscapes as those which can be realized as an eigenvector of the Laplacian of the neighborhood diagraph, and shows that such landscapes satisfy Grover's difference equation. Recent developments in algebraic graph theory support a new definition of the graph Laplacian which we use to extend the notion of elementary landscapes to neighborhood diagraphs which may be neither regular nor symmetric. This paper uses the new definition to extend the notion of elementary landscapes so that they characterize landscapes satisfying Grover's wave equation. We extend some known results to these more general elementary landscapes and analyse the types which may occur.

Tabu search and finite convergence

We establish finite convergence for some tabu search algorithms based on recency memory or frequency memory, distinguishing between symmetric and asymmetric neighborhood structures. These are the first demonstrations of explicit bounds provided by such forms of memory, and their finiteness suggests an important distinction between these ideas and those underlying certain “probabilistic” procedures such as annealing. We also show how an associated reverse elimination memory can be used to create a new type of tree search. Finally, we give designs for more efficient forms of convergent tabu search.

THE MAXIMUM OF THE Γ-MAX-SUBADDITIVE FUNCTIONALS ON A CLASS OF TOPOLOGICAL GROUPS

It is shown that if a “max-subadditive functional” p(x) defined on some symmetric neighborhood U¯0 of zero vector θ in a “b.f.-toplological group” X is “upper semi-cotinuous” at a point x0∈U¯0, or “lower semi-continuous” in some neighborhood V(x0)⊂U0 and X is of second category; then p(x) can attain its supremum in U¯0. And there is a similar conclusion for the Γ-max-subadditive functional when its supremum is 0 and if U0 is “pseudo-bounded” set in X.

Image restoration and reconstruction with a Bayesian approach.

We have extended Johnson's Bayesian method for image restoration and reconstruction by introducing diagonal line sites, using symmetric neighborhood configurations, and employing an additional hyperparameter for estimation of line sites. A general formulation for arbitrary neighborhood configurations was derived. The major part of this paper deals with the conduct of computer simulations intended to examine the effect of the hyperparameters, the diagonal line sites, and the size of the neighborhood configuration on the performance of the proposed Bayesian method. We show that, for optimal performance, distinct hyperparameters should be used for the intensity sites and line sites. The results also suggest that a large neighborhood configuration should be used. By comparing the near-optimal restored images, we demonstrated that the use of diagonal line sites, along with the symmetric configurations thus made possible, can effectively remove the blocky edge artifacts and produce images of better quality. When the method was applied to positron emission tomography (PET) image reconstruction, our results showed that the quality of the reconstructed images was improved for both computer-simulated and real patient PET data.

Evidence of phase transitions in heart period dynamics

Complexity measures of non-linear dynamics are a useful tool for quantifying observed stretching, folding, scaling and mixing processes in the Takens-reconstructed state space of heart period dynamics. Although such measures are not suited to provide evidence of deterministic chaos or to estimate true fractal dimensions and Lyapunov spectra in heart period time series, they allow the classification of RR dynamics and the identification of changes in RR complexity (RRC). The aim of this study was to develop appropriate measures and examine their utility in identifying the physiological effect of changes between the sleeping and waking state. Twenty-four hour electrocardiography (EEG) recordings and diaries noting their waking/sleeping period were obtained from 78 healthy subjects, aged 20 to 55 years. The approximate information dimension (ApD1) and the approximate Kolmogorov entropy (ApEn), introduced by Pincus, Kaplan and others, were modified in order to allow the calculation of strictly local values. That is, the local or pointwise dimensions and entropies were calculated for each reference vector with respect to its symmetric neighbourhood in time. For each subject the values for the local measures were averaged for 10-min periods, resulting in 144 global values over 24 h. Similarly, low- and high-frequency spectral parameters were calculated. All measures were examined and compared for the waking and the sleeping periods. All complexity measures as well as to a lesser degree high-frequency power showed a linear dependency on mean RR interval with a large individual variation. For the RRC measures this linear correlation was separated into two different clusters corresponding to the sleeping and waking periods. In almost all cases the correlation was greater in the waking period. In particular, in many cases no correlation was observed in the sleeping period. However, the r values for LF were appreciably lower and indicated solely a weak relationship to the RR interval in the waking period. Analysis of variance combining mean RR interval with RRC or spectral parameters singly and in couples revealed that the best separation with respect to physiological state could be achieved with the complexity measures, in particular with ApEn. The results show evidence of at least two dynamical regimes (phases) of heart period dynamics and a close but different functional relationship within the phases between RR interval and RR complexity. The separation between these regimes and the relatively sudden shift from one regime to the other suggest the existence of a phase transition with respect to waking and sleeping periods in terms of synergetics.

Parallel algorithms for image enhancement and segmentation by region growing, with an experimental study

This paper presents efficient and portable implementations of a powerful image enhancement process, the Symmetric Neighborhood Filter (SNF), and an image segmentation technique that makes use of the SNF and a variant of the conventional connected components algorithm which we call δ-Connected Components. We use efficient techniques for distributing and coalescing data as well as efficient combinations of task and data parallelism. The image segmentation algorithm makes use of an efficient connected components algorithm based on a novel approach for parallel merging. The algorithms have been coded in Split-C and run on a variety of platforms, including the Thinking Machines CM-5, IBM SP-1 and SP-2, Cray Research T3D, Meiko Scientific CS-2, Intel Paragon, and workstation clusters. Our experimental results are consistent with the theoretical analysis (and provide the best known execution times for segmentation, even when compared with machine-specific implementations). Our test data include difficult images from the Landsat Thematic Mapper (TM) satellite data.

Local and global majorities revisited

In this article we recall the interesting problem about local and global proportionalities in ball rings posed by Fishburn et al. (1986). For the symmetric neighborhood case, we decrease the upper bounds (which were conjectured to be tight) by giving a uniform construction for the three subcases distinguished in the original paper. Furthermore, we describe our technique of obtaining upper bounds because it may be reused for the study of other instances of the original problem.

The finite part of singular integrals in several complex variables

A divergent integral can sometimes be handled by assigning to it as its value the finite part in the sense of Hadamard. This is done by expanding the integral over the complement of a symmetric neighborhood of a singularity in powers of the radius, and throwing away the negative powers. In this paper the finite part of a singular integral of Cauchy type is defined, and this is then used to describe the boundary behavior of derivatives of a Cauchy-type integral. The finite part of a singular integral of Bochner-Martinelli type is studied, and an extension of the Plemelj jump formulas is shown to hold.

Code vector density in topographic mappings: Scalar case

The author derives some new results that build on his earlier work (1989) of combining vector quantization (VQ) theory and topographic mapping (TM) theory. A VQ model (with a noisy transmission medium) is used to model the processes that occur in TMs, which leads to the standard TM training algorithm, albeit with a slight modification to the encoding process. To emphasize this difference, the model is called a topographic vector quantizer (TVQ). In the continuum limit of the one-dimensional (scalar) TVQ. It is found that the density of code vectors is proportional to P(x)(a) (alpha=1/3) assuming that the transmission medium introduces additive noise with a zero-mean, symmetric, monotically decreasing probability density. This result is dramatically different from the result that is predicted when the standard TM training algorithm is used with a uniform symmetric neighborhood [-n, +n], and it is noted that this difference arises entirely from using minimum distortion rather than nearest neighbor encoding.