16910052
On best restricted range approximation in continuous complex-valued function spaces
To provide a Kolmogorov-type condition for characterizing a best approximation in a continuous complex-valued function space, it is usually assumed that the family of closed convex sets in the complex plane used to restrict the range satisfies a strong interior-point condition, and this excludes the interesting case when some @W"t is a line-segment or a singleton. The main aim of the present paper is to remove this restriction by virtue of a study of the notion of the strong CHIP for an infinite system of closed convex sets in a continuous complex-valued function space.
10976471
Derivatives of Generalized Distance Functions and Existence of Generalized Nearest Points
The relationship between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. Let G be any nonempty closed subset in a compact locally uniformly convex Banach space. It is proved that if the one-sided directional derivative of the generalized distance function associated to G at x equals to 1 or -1, then the generalized nearest points to x from G exist. We also give a partial answer (Theorem 3.5) to the open problem put forward by S. Fitzpatrick (1989, Bull. Austral. Math. Soc.39, 233-238).
2028079
Strong uniqueness of the restricted Chebyshev center with respect to an RS-set in a Banach space
Let G be a strict RS-set (resp. an RS-set) in X and let F be a bounded (resp. totally bounded) subset of X satisfying rG (F) < rx (F), where rG (F) is the restricted Chebyshev radius of F with respect to G. It is shown that the restricted Chebyshev center of F with respect to G is strongly unique in the case when X is a real Banach space, and that, under some additional convexity assumptions, the restricted Chebyshev center of F with respect to G is strongly unique of order α ≥ 2 in the case when X is a complex Banach space.
17566608
Porosity of mutually nearest and mutually furthest points in Banach spaces
Let X be a real strictly convex and Kadec Banach space and G a nonempty closed relatively boundedly weakly compact subset of X. Let B(X) (resp. K(X)) be the family of nonempty bounded closed (resp. compact) subsets of X endowed with the Hausdorff distance and let BG(X) denote the closure of the set {A ∈ B(X) : A ∩ G = φ} and KG(X) = BG(X)∩K(X). We introduce the admissible family A of B(X) and prove that EAo(G) (resp. EoA(G)), the set of all subsets F ∈ A ⊆ BG(X) (resp. F ∈ A ⊆ BG(X)) such that the minimization problem min(F, G) (resp. the maximization problem max(F, G)) is well-posed, is a dense Gδ-subset of A. Furthermore, when X is uniformly convex, we prove that A\EAo(G) and A\EoA(G) are σ-porous in A.
10820274
Simultaneous and exact interval estimates for the contrast of two groups based on an extremely high dimensional variable: application to mass spec data
MOTIVATION
Analysis of high-throughput proteomic/genomic data, in particular, surface-enhanced laser desorption/ionization time-of-flight mass spectrometry (SELDI-TOF MS) data and microarray data, has led to a multitude of techniques aimed at identifying potential biomarkers. Most of the statistical techniques for comparing two groups are based on qualitative measures such as P-value. A quantitative way such as interval estimation for the contrasts of two groups is more appealing.


RESULTS
We have devised a simultaneous confidence bands method capable of detecting potential biomarkers, while controlling for overall confidence coverage level, in high-dimensional datasets that discriminate two treatment groups using a permutation scheme. For example, for the SELDI-TOF MS data, we deal with the entire spectrum simultaneously and construct (1 - alpha) confidence bands for the mean differences between groups. Furthermore, peaks were identified based on the maximal differences between the groups as determined by the confidence bands. The analysis method herein described gives both qualitative (P-value) and quantitative data (magnitude of difference). The Clinical Proteomics Programs Databank's ovarian cancer dataset and data from in-house samples containing known spiked-in proteins were analyzed. We were able to identify potential biomarkers similar to those described in previous analysis of the ovarian cancer data, however, while these markers are highly significant between cancer and normal groups, our analysis indicated the absolute difference between the two groups was minimal. In addition, we found additional markers than those previously described with greater differences in average intensities. The proposed confidence bands method successfully detected the spiked-in peaks, as well as, secondary peaks generated by adducts and double-charged species. We also illustrate our method utilizing paired gene expression data from a prostate cancer microarray experiment by constructing confidence bands for the fold changes between cancer and normal samples.


AVAILABILITY
R-package, 'seie.zip' (license: GNU GPL), is publiclly available at http://research2.dfci.harvard.edu/dfci/MS_spike-in_data/
40002746
Soft computing approach to feature extraction
Based on both wavelet theory and fuzzy theory, a soft computing system (SCS) is proposed for feature extraction of signals. The proposed SCS approach possesses the advantages of soft decision-making on wavelet coefficients for feature extraction, adaptive selectivity of mapping factors to coarse-to-fine resolution, and compact form of feature representation with the SCS feature-extractor. Fuzzy sets are used to provide a robust representation for signal information, and wavelet transform is used to decompose a signal into detail and approximation signals. At a given resolution, the detail and approximation signals are inputted to the proposed SCS to extract signal features at that resolution level. The sensitivity in feature extraction of the proposed approach can be adapted by tuning the fuzzy sets for the detail and approximation signals. At different resolutions, the signal can be examined and suitable features can be extracted. Examples of both one-dimensional signals and two-dimensional fingerprint images are used to illustrate the proposed soft computing approach for feature extraction and pattern recognition. The results show that the extracted features are sensitive enough to distinguish the similar signal from the different ones and are robust enough to tolerate noise corruption.
9155774
Image Registration Using Wavelet-Based Motion Model
An image registration algorithm is developed to estimate dense motion vectors between two images using the coarse-to-fine wavelet-based motion model. This motion model is described by a linear combination of hierarchical basis functions proposed by Cai and Wang (SIAM Numer. Anal., 33(3):937–970, 1996). The coarser-scale basis function has larger support while the finer-scale basis function has smaller support. With these variable supports in full resolution, the basis functions serve as large-to-small windows so that the global and local information can be incorporated concurrently for image matching, especially for recovering motion vectors containing large displacements. To evaluate the accuracy of the wavelet-based method, two sets of test images were experimented using both the wavelet-based method and a leading pyramid spline-based method by Szeliski et al. (International Journal of Computer Vision, 22(3):199–218, 1996). One set of test images, taken from Barron et al. (International Journal of Computer Vision, 12:43–77, 1994), contains small displacements. The other set exhibits low texture or spatial aliasing after image blurring and contains large displacements. The experimental results showed that our wavelet-based method produced better motion estimates with error distributions having a smaller mean and smaller standard deviation.
8510271
Supporting Efficient Record Linkage for Large Data Sets Using Mapping Techniques
This paper describes an efficient approach to record linkage. Given two lists of records, the record-linkage problem consists of determining all pairs that are similar to each other, where the overall similarity between two records is defined based on domain-specific similarities over individual attributes. The record-linkage problem arises naturally in the context of data cleansing that usually precedes data analysis and mining. Since the scalability issue of record linkage was addressed in [21], the repertoire of database techniques dealing with multidimensional data sets has significantly increased. Specifically, many effective and efficient approaches for distance-preserving transforms and similarity joins have been developed. Based on these advances, we explore a novel approach to record linkage. For each attribute of records, we first map values to a multidimensional Euclidean space that preserves domain-specific similarity. Many mapping algorithms can be applied, and we use the Fastmap approach [16] as an example. Given the merging rule that defines when two records are similar based on their attribute-level similarities, a set of attributes are chosen along which the merge will proceed. A multidimensional similarity join over the chosen attributes is used to find similar pairs of records. Our extensive experiments using real data sets show that our solution has very good efficiency and recall.
15246731
Constructions of Self-complementary Circulants with No Multiplicative Isomorphisms
The main topic of the paper is the question of the existence of self-complementary Cayley graphs Cay(G, S) with the property S??=G#\S for all ??Aut(G). We answer this question in the positive by constructing an infinite family of self-complementary circulants with this property. Moreover, we obtain a complete classification of primes p for which there exist self-complementary circulants of order p2with this property.