Screen Content Image Segmentation Using Robust Regression and Sparse Decomposition

This paper considers how to separate text and/or graphics from smooth background in screen content and mixed document images and proposes two approaches to perform this segmentation task. The proposed methods make use of the fact that the background in each block is usually smoothly varying and can be modeled well by a linear combination of a few smoothly varying basis functions, while the foreground text and graphics create sharp discontinuity. The algorithms separate the background and foreground pixels by trying to fit background pixel values in the block into a smooth function using two different schemes. One is based on robust regression, where the inlier pixels will be considered as background, while remaining outlier pixels will be considered foreground. The second approach uses a sparse decomposition framework where the background and foreground layers are modeled with a smooth and sparse components respectively. These algorithms have been tested on images extracted from HEVC standard test sequences for screen content coding, and are shown to have superior performance over previous approaches. The proposed methods can be used in different applications such as text extraction, separate coding of background and foreground for compression of screen content, and medical image segmentation

Primary beam effects of radio astronomy antennas: I. Modelling the Karl G. Jansky Very Large Array (VLA) L-band beam using holography

Modern interferometric imaging relies on advanced calibration that incorporates direction-dependent effects. Their increasing number of antennas (e.g. in LOFAR, VLA, MeerKAT/SKA) and sensitivity are often tempered with the accuracy of their calibration. Beam accuracy drives particularly the capability for high dynamic range imaging (HDR - contrast > 1:$10^6$). The Radio Interferometric Measurement Equation (RIME) proposes a refined calibration framework for wide field of views (i.e. beyond the primary lobe and first null) using beam models. We have used holography data taken on 12 antennas of the Very Large Array (VLA) with two different approaches: a `data-driven' representation derived from Principal Component Analysis (PCA) and a projection on the Zernike polynomials. We determined sparse representations of the beam to encode its spatial and spectral variations. For each approach, we compressed the spatial and spectral distribution of coefficients using low-rank approximations. The spectral behaviour was encoded with a Discrete Cosine Transform (DCT). We compared our modelling to that of the Cassbeam software which provides a parametric model of the antenna and its radiated field. We present comparisons of the beam reconstruction fidelity vs. `compressibility'. We found that the PCA method provides the most accurate model. In the case of VLA antennas, we discuss the frequency ripple over L-band which is associated with a standing wave between antenna reflectors. The results are a series of coefficients that can easily be used `on-the-fly' in calibration pipelines to generate accurate beams at low computing costs.Comment: 16 pages, 15 figures, 2 online videos, Accepted in MNRAS. For the online videos, see http://www.youtube.com/watch?v=oc9lLxJy3Ow and http://www.youtube.com/watch?v=3RYrBxCyeb

Memory footprint reduction for the FFT-based volume integral equation method via tensor decompositions

We present a method of memory footprint reduction for FFT-based, electromagnetic (EM) volume integral equation (VIE) formulations. The arising Green's function tensors have low multilinear rank, which allows Tucker decomposition to be employed for their compression, thereby greatly reducing the required memory storage for numerical simulations. Consequently, the compressed components are able to fit inside a graphical processing unit (GPU) on which highly parallelized computations can vastly accelerate the iterative solution of the arising linear system. In addition, the element-wise products throughout the iterative solver's process require additional flops, thus, we provide a variety of novel and efficient methods that maintain the linear complexity of the classic element-wise product with an additional multiplicative small constant. We demonstrate the utility of our approach via its application to VIE simulations for the Magnetic Resonance Imaging (MRI) of a human head. For these simulations we report an order of magnitude acceleration over standard techniques.Comment: 11 pages, 10 figures, 5 tables, 2 algorithms, journa

Quantum compression relative to a set of measurements

In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and B\'ezout's theorem. The latter approach allows lifting the results from the single copy level to the case of multiple copies and from completely positive to merely positive maps.Comment: 40 pages. Minor clarifications in Section 9.2 and Section 7.

Active Subspace of Neural Networks: Structural Analysis and Universal Attacks

Active subspace is a model reduction method widely used in the uncertainty quantification community. In this paper, we propose analyzing the internal structure and vulnerability and deep neural networks using active subspace. Firstly, we employ the active subspace to measure the number of "active neurons" at each intermediate layer and reduce the number of neurons from several thousands to several dozens. This motivates us to change the network structure and to develop a new and more compact network, referred to as {ASNet}, that has significantly fewer model parameters. Secondly, we propose analyzing the vulnerability of a neural network using active subspace and finding an additive universal adversarial attack vector that can misclassify a dataset with a high probability. Our experiments on CIFAR-10 show that ASNet can achieve 23.98$\times$ parameter and 7.30$\times$ flops reduction. The universal active subspace attack vector can achieve around 20% higher attack ratio compared with the existing approach in all of our numerical experiments. The PyTorch codes for this paper are available online

Low-Rank Tucker Approximation of a Tensor From Streaming Data

This paper describes a new algorithm for computing a low-Tucker-rank approximation of a tensor. The method applies a randomized linear map to the tensor to obtain a sketch that captures the important directions within each mode, as well as the interactions among the modes. The sketch can be extracted from streaming or distributed data or with a single pass over the tensor, and it uses storage proportional to the degrees of freedom in the output Tucker approximation. The algorithm does not require a second pass over the tensor, although it can exploit another view to compute a superior approximation. The paper provides a rigorous theoretical guarantee on the approximation error. Extensive numerical experiments show that that the algorithm produces useful results that improve on the state of the art for streaming Tucker decomposition.Comment: 34 pages, 14 figure

Non-Orthogonal Tensor Diagonalization

Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It is generalization of approximate joint diagonalization (AJD) of a set of matrices. In particular, we derive (1) a new algorithm for symmetric AJD, which is called two-sided symmetric diagonalization of order-three tensor, (2) a similar algorithm for non-symmetric AJD, also called general two-sided diagonalization of an order-3 tensor, and (3) an algorithm for three-sided diagonalization of order-3 or order-4 tensors. The latter two algorithms may serve for canonical polyadic (CP) tensor decomposition, and they can outperform other CP tensor decomposition methods in terms of computational speed under the restriction that the tensor rank does not exceed the tensor multilinear rank. Finally, we propose (4) similar algorithms for tensor block diagonalization, which is related to the tensor block-term decomposition.Comment: The manuscript was revised deeply, but the main idea is the same. The algorithm has changed significantl

Fourth-order Tensors with Multidimensional Discrete Transforms

The big data era is swamping areas including data analysis, machine/deep learning, signal processing, statistics, scientific computing, and cloud computing. The multidimensional feature and huge volume of big data put urgent requirements to the development of multilinear modeling tools and efficient algorithms. In this paper, we build a novel multilinear tensor space that supports useful algorithms such as SVD and QR, while generalizing the matrix space to fourth-order tensors was believed to be challenging. Specifically, given any multidimensional discrete transform, we show that fourth-order tensors are bilinear operators on a space of matrices. First, we take a transform-based approach to construct a new tensor space by defining a new multiplication operation and tensor products, and accordingly the analogous concepts: identity, inverse, transpose, linear combinations, and orthogonality. Secondly, we define the $\mathcal{L}$-SVD for fourth-order tensors and present an efficient algorithm, where the tensor case requires a stronger condition for unique decomposition than the matrix case. Thirdly, we define the tensor $\mathcal{L}$-QR decomposition and propose a Householder QR algorithm to avoid the catastrophic cancellation problem associated with the conventional Gram-Schmidt process. Finally, we validate our schemes on video compression and one-shot face recognition. For video compression, compared with the existing tSVD, the proposed $\mathcal{L}$-SVD achieves $3\sim 10$dB gains in RSE, while the running time is reduced by about $50\%$ and $87.5\%$, respectively. For one-shot face recognition, the recognition rate is increased by about $10\% \sim 20\%$

Essential dimensions of algebraic groups and a resolution theorem for G-varieties

Let G be an algebraic group and let X be a generically free G-variety. We show that X can be transformed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety X' with the following property: the stabilizer of every point of X' is isomorphic to a semidirect product of a unipotent group U and a diagonalizable group A. As an application of this and related results, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.Comment: This revision contains new lower bounds for essential dimensions of algebraic groups of types A_n and E_7. AMS LaTeX 1.1, 42 pages. Paper by Zinovy Reichstein and Boris Youssi, includes an appendix by J\'anos Koll\'ar and Endre Szab\'o. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.htm

Computation of extremal eigenvalues of high-dimensional lattice-theoretic tensors via tensor-train decompositions

This paper lies in the intersection of several fields: number theory, lattice theory, multilinear algebra, and scientific computing. We adapt existing solution algorithms for tensor eigenvalue problems to the tensor-train framework. As an application, we consider eigenvalue problems associated with a class of lattice-theoretic meet and join tensors, which may be regarded as multidimensional extensions of the classically studied meet and join matrices such as GCD and LCM matrices, respectively. In order to effectively apply the solution algorithms, we show that meet tensors have an explicit low-rank tensor-train decomposition with sparse tensor-train cores with respect to the dimension. Moreover, this representation is independent of tensor order, which eliminates the so-called curse of dimensionality from the numerical analysis of these objects and makes the solution of tensor eigenvalue problems tractable with increasing dimensionality and order. For LCM tensors it is shown that a tensor-train decomposition with an a priori known TT rank exists under certain assumptions. We present a series of easily reproducible numerical examples covering tensor eigenvalue and generalized eigenvalue problems that serve as future benchmarks. The numerical results are used to assess the sharpness of existing theoretical estimates.Comment: 23 pages, 7 figure