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We introduce the prime coset sum method for constructing tight wavelet frames, which allows one to construct nonseparable multivariate tight wavelet frames with prime dilation, using a univariate lowpass mask with this same prime dilation as input. This method relies on the idea of finding a sum of hermitian squares representation for a nonnegative trigonometric polynomial related to the sub-QMF condition for the lowpass mask. We prove the existence of these representations under some conditions on the input lowpass mask, utilizing the special structure of the recently introduced prime coset sum method, which is used to generate the lowpass masks in our construction. We also prove guarantees on the vanishing moments of the wavelets arising from this method, some of which hold more generally.

Constructing tight wavelet filter banks with prescribed directions is challenging. This paper presents a systematic method for designing a tight wavelet filter bank, given any prescribed directions. There are two types of wavelet filters in our tight wavelet filter bank. One type is entirely determined by the prescribed information about the directionality and makes the wavelet filter bank directional. The other type helps the wavelet filter bank to be tight. In addition to the flexibility in choosing the directions, our construction method has other useful properties. It works for any multi-dimension, and it allows the user to have any prescribed number of vanishing moments along the chosen directions. Furthermore, our tight wavelet filter banks have fast algorithms for analysis and synthesis. Concrete examples are given to illustrate our construction method and properties of resulting tight wavelet filter banks., Comment: 19 pages

In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer coefficients. Our result is general enough to show the invertibility of circulant matrices with any size and arrangement of entries. For example, using these conditions, we show the invertibility of the family of circulant matrices with particular forms of integers generated by a primitive element in $\mathbb{Z}_p$. Also, using a combinatorial structure of these sufficient conditions, we show invertibility for circulant $0, 1$-matrices., Comment: 18 pages

We present two methods for constructing new nonseparable multidimensional tight wavelet frames by combining the ideas of sum of squares representations of nonnegative trigonometric polynomials with the coset sum method of generating nonseparable multidimensional lowpass filters from univariate lowpass filters. In effect, these methods allow one to select a univariate lowpass filter and generate nonseparable multidimensional tight wavelet frames from it in any dimension n ≥ 2 , under certain conditions on the input filter which are given explicitly. We construct sum of hermitian squares representations for a particular class of trigonometric polynomials f in several variables, each related to a coset sum generated lowpass mask τ in that nonnegativity of f implies the sub-QMF condition for τ, in two ways: for interpolatory inputs to the coset sum method satisfying the univariate sub-QMF condition, we find this representation using the Fejer–Riesz Lemma; and in the general case, by writing f = x ⁎ P x , where x is a vector of complex exponential functions, and P is a constant positive semidefinite matrix that is constructed to reduce the number of generators in this representation. The generators of this representation of f may then be used to generate the filters in a tight wavelet frame with lowpass mask τ. Several examples of these representations and the corresponding frames are given throughout.

Tight wavelet frames are computationally and theoretically attractive, but most existing multivariate constructions have various drawbacks, including low vanishing moments for the wavelets, or a large number of wavelet masks. We further develop existing work combining sums of squares representations with tight wavelet frame construction, and present a new and general method for constructing such frames. Focusing on the case of box splines, we also demonstrate how the flexibility of our approach can lead to tight wavelet frames with high numbers of vanishing moments for all of the wavelet masks, while still having few highpass masks: in fact, we match the best known upper bound on the number of highpass masks for general box spline tight wavelet frame constructions, while typically achieving much better vanishing moments for all of the wavelet masks, proving a nontrivial lower bound on this quantity., Comment: 19 pages, 3 tables

In this paper, we review the scattering transform in the univariate setting. After reviewing its properties including translation invariance, stability under small diffeomorphism, and ability to carry high-frequency information, we investigate how these properties can be used in understanding the effect of the scattering transform when various types of signal deformation are considered. We find that, together with the Fourier transform modulus, the scattering transform can be used in classifying some of these deformations.

A wavelet-based distribution-free tabular CUSUM chart based on adaptive thresholding, is designed for rapidly detecting shifts in the mean of a high-dimensional profile whose noise components have a continuous nonsingular multivariate distribution. First computing a discrete wavelet transform of the noise vectors for randomly sampled Phase I (in-control) profiles, uses a matrix-regularization method to estimate the covariance matrix of the wavelet-transformed noise vectors; then, those vectors are aggregated (batched) so that the non-overlapping batch means of the wavelet-transformed noise vectors have manageable covariances. Lower and upper in-control thresholds are computed for the resulting batch means of the wavelet-transformed noise vectors using the associated marginal Cornish–Fisher expansions that have been suitably adjusted for between-component correlations. From the thresholded batch means of the wavelet-transformed noise vectors, Hotelling’s -type statistics are computed to set the parameters ...

Laplacian pyramid based Laurent polynomial (LP$^2$) matrices are generated by Laurent polynomial column vectors and have long been studied in connection with Laplacian pyramidal algorithms in Signal Processing. In this paper, we investigate when such matrices are scalable, that is when right multiplication by Laurent polynomial diagonal matrices results in paraunitary matrices. The notion of scalability has recently been introduced in the context of finite frame theory and can be considered as a preconditioning method for frames. This paper significantly extends the current research on scalable frames to the setting of polyphase representations of filter banks. Furthermore, as applications of our main results we propose new construction methods for tight wavelet filter banks and tight wavelet frames., Version accepted for publication in SIAM Journal on Matrix Analysis and Applications

In this chapter we give an overview of a method recently developed for designing wavelet filter banks via the Quillen-Suslin Theorem for Laurent polynomials. In this method, the Quillen-Suslin Theorem is used to transform vectors with Laurent polynomial entries to other vectors with Laurent polynomial entries so that the matrix analysis tools that were not readily available for the vectors before the transformation can now be employed. As a result, a powerful and general method for designing non-redundant wavelet filter banks is obtained. In particular, the vanishing moments of the resulting wavelet filter banks can be controlled in a very simple way, which is especially advantageous compared to other existing methods for the multi-dimensional cases.

In this paper we present a new approach for constructing the wavelet filter bank. Our approach enables constructing nonseparable multidimensional non-redundant wavelet filter banks with FIR filters using the Quillen-Suslin Theorem for Laurent polynomials. Our construction method presents some advantages over the traditional methods of multidimensional wavelet filter bank design. First, it works for any spatial dimension and for any sampling matrix. Second, it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition. Third, it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter.

DNA copy number alterations (CNAs) are the main genomic events that occur during the initiation and development of cancer. Distinguishing driver aberrant regions from passenger regions, which might contain candidate target genes for cancer therapies, is an important issue. Several methods for identifying cancer-driver genes from multiple cancer patients have been developed for single nucleotide polymorphism (SNP) arrays. However, for NGS data, methods for the SNP array cannot be directly applied because of different characteristics of NGS such as higher resolutions of data without predefined probes and incorrectly mapped reads to reference genomes. In this study, we developed a wavelet-based method for identification of focal genomic alterations for sequencing data (WIFA-Seq). We applied WIFA-Seq to whole genome sequencing data from glioblastoma multiforme, ovarian serous cystadenocarcinoma and lung adenocarcinoma, and identified focal genomic alterations, which contain candidate cancer-related genes as well as previously known cancer-driver genes.

DNA copy number alterations (CNAs) are the main genomic events that occur during the initiation and development of cancer. Distinguishing driver aberrant regions from passenger regions, which might contain candidate target genes for cancer therapies, is an important issue. Several methods for identifying cancer-driver genes from multiple cancer patients have been developed for single nucleotide polymorphism (SNP) arrays. However, for NGS data, methods for the SNP array cannot be directly applied because of different characteristics of NGS such as higher resolutions of data without predefined probes and incorrectly mapped reads to reference genomes. In this study, we developed a wavelet-based method for identification of focal genomic alterations for sequencing data (WIFA-Seq). We applied WIFA-Seq to whole genome sequencing data from glioblastoma multiforme, ovarian serous cystadenocarcinoma and lung adenocarcinoma, and identified focal genomic alterations, which contain candidate cancer-related genes as well as previously known cancer-driver genes.

WDFTC is a wavelet-based distribution-free CUSUM chart for detecting shifts in the mean of a profile with noisy components. Exploiting a discrete wavelet transform (DWT) of the mean in-control profile, WDFTC selects a reduced-dimension vector of the associated DWT components from which the mean in-control profile can be approximated with minimal weighted relative reconstruction error. Based on randomly sampled Phase I (in-control) profiles, the covariance matrix of the corresponding reduced-dimension DWT vectors is estimated using a matrix-regularisation method; then the DWT vectors are aggregated (batched) so that the non-overlapping batch means of the reduced-dimension DWT vectors have manageable covariances. To monitor shifts in the mean profile during Phase II operation, WDFTC computes a Hotelling's T 2-type statistic from successive non-overlapping batch means and applies a CUSUM procedure to those statistics, where the associated control limits are evaluated analytically from the Phase I data. Exper...

We introduce new methodologies for the construction of high-performance very local Riesz wavelet bases of $L_2({\mathbb{R}^n})$ in arbitrarily high spatial dimension $n$. The localness $L$ of the representation is measured as the sum of the volumes of the supports of the underlying mother wavelets; small localness number is one of the sought-for properties in wavelet constructions. Our constructs are very simple and they are based on our recent framelet construction methods: the CAMP scheme and the L-CAMP scheme. Within our general methodology, the subclass of piecewise-constant constructions is the most local one. It includes Riesz wavelet bases with any performance grade and in any spatial dimension. In this subclass, the Riesz wavelet basis with Jackson-type performance $k$ (namely, with $k$ vanishing moments) has localness score $L

ErbB3 is noted as one of major causes of acquired cetuximab resistance in colorectal and head and neck cancers. ErbB3 causes activation of alternative signaling pathways that bypass the original target and sustained PI3K/AKT activation, and these are associated with cetuximab resistance. We confirmed the induced ErbB3 activation by the cetuximab treatment in FaDu head and neck squamous-cell carcinoma (HNSCC) xenograft model. Immunoblot analysis was shown that twice weekly 10 mg/kg of cetuximab treatment upregulated ErbB3 expression and phosphorylation even though tumor growth was well controlled. To investigate whether ErbB3 activation by ligand, heregulin (HRG,) might induce cetuximab resistance, two cetuximab-sensitive colorectal cancer cell lines, DiFi and LIM1215, were treated 0-25 ng/mL of HRG. HRG induced dose-dependent cetuximab resistance in cell proliferation assay, and this was reversed via ISU104 treatment. To evaluate if ISU104 could overcome resistance to cetuximab in vivo, acquired cetuximab-resistant FaDu xenograft model was established. Tumors that had acquired resistance to 5 mg/kg of cetuximab treatment were significantly regressed by replaced treatment of ISU104 alone (10 mg/kg) or combination treatment of ISU104 and cetuximab, while mice continued on cetuximab only showed uncontrolled tumor growth like vehicle-treated group. Our results suggest that ISU104 effectively overcomes cetuximab resistance and may provide clinical benefit to cetuximab-resistant patients. Citation Format: Miyoung Kim, Youngmi Hur, Mirim Hong, Youngsoo Sohn, Kum-Joo Shin, Kyungyong Kim, Seung-Beom Hong, Kum-Joo Shin, Donggoo Bae. ISU104, a fully human anti-ErbB3 antibody, overcomes acquired cetuximab resistance [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2018; 2018 Apr 14-18; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2018;78(13 Suppl):Abstract nr 836.

Members of the epidermal growth factor receptor family (ErbB family) are known as potent mediators in the development and progression of cancer. Activated ErbBs recruit various adaptors and signaling molecules through the phosphorylated cytoplasmic domain, which further leads to activation of downstream oncogenic signaling pathways. There are approved therapeutics for ErbB1 (EGFR) and ErbB2 (HER2) in the treatment of human cancers, while monoclonal antibodies targeting ErbB3 are just undergoing clinical trials. ISU104 is a fully human anti-ErbB3 antibody isolated from phage display antibody library. We performed the hydrogen/deuterium exchange mass spectrometry (HDX-MS) analysis with ErbB3 extracellular domain and ISU104, and that indicated that ISU104 mainly binds with domain 3 and weakly interacts with domain 1 of ErbB3. The binding property of ISU104 induced dose-dependent inhibition of ligand (heregulin, HRG) binding, blocking of dimerization of ErbB3 with other ErbBs and subsequently inactivated the downstream signaling of ErbB3. Also, ISU104 occasioned internalization of ErbB3 from plasma membrane, and downregulated the expression level of ErbB3. We demonstrated the biologic effect of ISU104 in several ErbB3-expressing cancer cell lines, including head and neck squamous cell carcinoma (HNSCC) and breast cancers. ISU104 completely suppressed the HRG-induced ErbB3/AKT phosphorylation, reduced cell proliferation and survival. Next, we evaluated efficacy of ISU104 in multiple xenograft models. Mice were treated with 10 mg/kg of ISU104 twice weekly. ISU104 regressed tumor growth in FaDu HNSCC model, and showed more than 70% tumor growth inhibition (TGI) in CAL27 (HNSCC), BxPC3 (pancreatic cancer), MDA-MB-468 (breast cancer), A549 (skin cancer), and BT474 (breast cancer) models. Our results suggest that ISU104 effectively blocks activation of ErbB3 and the downstream pathway by ErbB3, and may provide clinical benefit to ErbB3-activated patients. Citation Format: Miyoung Kim, Youngmi Hur, Sohyeon Seo, Heynjeong Lim, Kyungyong Kim, Youngsoo Sohn, Seung-Beom Hong, Donggoo Bae. ISU104, a fully human antibody targeting a specific epitope on the ErbB3, displays potent inhibition of tumor growth in multiple xenograft tumor models [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2018; 2018 Apr 14-18; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2018;78(13 Suppl):Abstract nr 830.

A new wavelet-based methodology for representing data on regular grids is introduced and studied. The main attraction of this new "local compression-alignment-modified- prediction (L-CAMP)" methodology is in the way it scales with the spatial dimension, making it, thus, highly suitable for the representation of high dimensional data. The specific highlights of the L-CAMP methodology are three. First, it is computed and inverted by fast algorithms with linear complexity and very small constants; moreover, the constants in the complexity bound decay, rather than grow, with the spatial dimension. Second, the representation is accompanied by solid mathematical theory that reveals its performance in terms of the maximal level of smoothness that is accurately encoded by the representation. Third, the localness of the representation, measured as the sum of the volumes of the supports of the underlying mother wavelets, is extreme. An illustration of this last property is done by comparing the L-CAMP system that is marked in this paper as V with the widely used tensor-product biorthogonal 9/7. Both are essentially equivalent in terms of performance. However, the L-CAMP V has in 10D localness score 575 000 000 000.

A new conformational neutralizable epitope is created on heptocyte growth factor (HGF), when it interacts with its receptor, cMet. By immunizing rabbits with HGF-cMet complex, we successfully generated a monoclonal antibody (SFN68) that inhibits HGF-cMet interaction, and blocks the biological function mediated by HGF. To define the epitope, we screened out an epitope-mimicking peptide, KSLSRHDHIHHH, from a phage display of combinatorial peptide library. In molecular mimicry this peptide bound to cMet and inhibited HGF-cMet interaction. No humoral response was induced to this epitope-mimicking peptide when immunization was done with HGF alone.

A finite frame is said to be scalable if its vectors can be rescaled so that the resulting set of vectors is a tight frame. The theory of scalable frame has been extended to the setting of Laplacian pyramids which are based on (rectangular) paraunitary matrices whose column vectors are Laurent polynomial vectors. This is equivalent to scaling the polyphase matrices of the associated filter banks. Consequently, tight wavelet frames can be constructed by appropriately scaling the columns of these paraunitary matrices by diagonal matrices whose diagonal entries are square magnitude of Laurent polynomials. In this paper we present examples of tight wavelet frames constructed in this manner and discuss some of their properties in comparison to the (non tight) wavelet frames they arise from.

As constructing multi-D wavelets remains a challenging problem, we propose a new method called prime coset sum to construct multi-D wavelets. Our method provides a systematic way to construct multi-D non-separable wavelet filter banks from two 1-D low-pass filters, with one of which being interpolatory. Our method has many important features including the following: 1) it works for any spatial dimension, and any prime scalar dilation; 2) the vanishing moments of the multi-D wavelet filter banks are guaranteed by certain properties of the initial 1-D low-pass filters, and furthermore; 3) the resulting multi-D wavelet filter banks are associated with fast algorithms that are faster than the existing fast tensor product algorithms.

In this paper we design thin n-D wavelet filters. Our wavelet filters are thin in the sense that each filter is essentially a 1-D filter, which is supported on a straight line. We first use the coset sum, a recently developed alternative to the tensor product, in order to obtain thin n-D wavelet filters that can capture directional information in 2n − 1 different directions. Furthermore the choice of directions is quite flexible, and it can be made so that there is no strong directional bias along lines parallel to the coordinate direction. One limitation of thin n-D wavelet filters constructed by the coset sum method is that they can capture only 2n − 1 directions. In order to overcome this limitation we discuss how to generalize the coset sum method so that thin n-D Haar wavelet filters with more than 2n − 1 directions can be obtained.

Background Copy number aberrations (CNAs) are an important molecular signature in cancer initiation, development, and progression. However, these aberrations span a wide range of chromosomes, making it hard to distinguish cancer related genes from other genes that are not closely related to cancer but are located in broadly aberrant regions. With the current availability of high-resolution data sets such as single nucleotide polymorphism (SNP) microarrays, it has become an important issue to develop a computational method to detect driving genes related to cancer development located in the focal regions of CNAs. Results In this study, we introduce a novel method referred to as the wavelet-based identification of focal genomic aberrations (WIFA). The use of the wavelet analysis, because it is a multi-resolution approach, makes it possible to effectively identify focal genomic aberrations in broadly aberrant regions. The proposed method integrates multiple cancer samples so that it enables the detection of the consistent aberrations across multiple samples. We then apply this method to glioblastoma multiforme and lung cancer data sets from the SNP microarray platform. Through this process, we confirm the ability to detect previously known cancer related genes from both cancer types with high accuracy. Also, the application of this approach to a lung cancer data set identifies focal amplification regions that contain known oncogenes, though these regions are not reported using a recent CNAs detecting algorithm GISTIC: SMAD7 (chr18q21.1) and FGF10 (chr5p12). Conclusions Our results suggest that WIFA can be used to reveal cancer related genes in various cancer data sets.