Fast and Robust High-Dimensional Sparse Representation Recovery Using Generalized SL0

Sparse representation can be described in high dimensions and used in many applications, including MRI imaging and radar imaging. In some cases, methods have been proposed to solve the high-dimensional sparse representation problem, but main solution is converting high-dimensional problem into one-dimension. Solving the equivalent problem had very high computational complexity. In this paper, the problem of high-dimensional sparse representation is formulated generally based on the theory of tensors, and a method for solving it based on SL0 (Smoothed Least zero-nor) is presented. Also, the uniqueness conditions for solution of the problem are considered in the high-dimensions. At the end of the paper, some numerical experiments are performed to evaluate the efficiency of the proposed algorithm and the results are presented

Group lifting structures for multirate filter banks I: Uniqueness of lifting factorizations

Group lifting structures are introduced to provide an algebraic framework for studying lifting factorizations of two-channel perfect reconstruction finite-impulse-response (FIR) filter banks. The lifting factorizations generated by a group lifting structure are characterized by Abelian groups of lower and upper triangular lifting matrices, an Abelian group of unimodular gain scaling matrices, and a set of base filter banks. Examples of group lifting structures are given for linear phase lifting factorizations of the two nontrivial classes of two-channel linear phase FIR filter banks, the whole- and half-sample symmetric classes, including both the reversible and irreversible cases. This covers the lifting specifications for whole-sample symmetric filter banks in Parts 1 and 2 of the ISO/IEC JPEG 2000 still image coding standard. The theory is used to address the uniqueness of lifting factorizations. With no constraints on the lifting process, it is shown that lifting factorizations are highly nonunique. When certain hypotheses developed in the paper are satisfied, however, lifting factorizations generated by a group lifting structure are shown to be unique. A companion paper applies the uniqueness results proven in this paper to the linear phase group lifting structures for whole- and half-sample symmetric filter banks.Comment: 19 pages, 3 figure

On the group-theoretic structure of lifted filter banks

The polyphase-with-advance matrix representations of whole-sample symmetric (WS) unimodular filter banks form a multiplicative matrix Laurent polynomial group. Elements of this group can always be factored into lifting matrices with half-sample symmetric (HS) off-diagonal lifting filters; such linear phase lifting factorizations are specified in the ISO/IEC JPEG 2000 image coding standard. Half-sample symmetric unimodular filter banks do not form a group, but such filter banks can be partially factored into a cascade of whole-sample antisymmetric (WA) lifting matrices starting from a concentric, equal-length HS base filter bank. An algebraic framework called a group lifting structure has been introduced to formalize the group-theoretic aspects of matrix lifting factorizations. Despite their pronounced differences, it has been shown that the group lifting structures for both the WS and HS classes satisfy a polyphase order-increasing property that implies uniqueness ("modulo rescaling") of irreducible group lifting factorizations in both group lifting structures. These unique factorization results can in turn be used to characterize the group-theoretic structure of the groups generated by the WS and HS group lifting structures.Comment: Book chapter: 22 pages, 6 figures. Expository overview of recent research, including arXiv:1309.7665. Version 2: added BibTeX citation (BibTeX_citation.txt) and conference presentation slides (Brislawn_AMS_ABQ_2014_slides.pdf, 328 KB) as ancillary file

Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields

This paper concerns the reconstruction of a complex-valued anisotropic tensor \gamma=\sigma+\i\omega\varepsilon from knowledge of several internal magnetic fields $H$, where $H$ satisfies the anisotropic Maxwell system on a bounded domain with prescribed boundary conditions. We show that $\gamma$ can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition of $H$. A minimum number of 6 such functionals is sufficient to obtain a local reconstruction of $\gamma$. In the special case where $\gamma$ is close to a scalar tensor, boundary conditions are chosen by means of complex geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a Runge approximation property is used to obtain partial results. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.Comment: 24 pages, submitted to Inverse Problems and Imagin

Compact Factorization of Matrices Using Generalized Round-Rank

Matrix factorization is a well-studied task in machine learning for compactly representing large, noisy data. In our approach, instead of using the traditional concept of matrix rank, we define a new notion of link-rank based on a non-linear link function used within factorization. In particular, by applying the round function on a factorization to obtain ordinal-valued matrices, we introduce generalized round-rank (GRR). We show that not only are there many full-rank matrices that are low GRR, but further, that these matrices cannot be approximated well by low-rank linear factorization. We provide uniqueness conditions of this formulation and provide gradient descent-based algorithms. Finally, we present experiments on real-world datasets to demonstrate that the GRR-based factorization is significantly more accurate than linear factorization, while converging faster and using lower rank representations

Phaseless Reconstruction from Space-Time Samples

Phaseless reconstruction from space-time samples is a nonlinear problem of recovering a function $x$ in a Hilbert space $\mathcal{H}$ from the modulus of linear measurements $\{\lvert \langle x, \phi_i\rangle \rvert$, $\ldots$, $\lvert \langle A^{L_i}x, \phi_i \rangle \rvert : i \in\mathscr I\}$, where $\{\phi_i; i \in\mathscr I\}\subset \mathcal{H}$ is a set of functionals on $\mathcal{H}$, and $A$ is a bounded operator on $\mathcal{H}$ that acts as an evolution operator. In this paper, we provide various sufficient or necessary conditions for solving this problem, which has connections to $X$-ray crystallography, the scattering transform, and deep learning.Comment: 23 pages, 4 figure

Reconstruction of a polynomial from its Radon projections

A polynomial of degree $n$ in two variables is shown to be uniquely determined by its Radon projections taken over $[n/2]+1$ parallel lines in each of the $(2[(n+1)/2]+1)$ equidistant directions along the unit circle.Comment: SIAM J. Math. Anal. (to appear

Compressed Factorization: Fast and Accurate Low-Rank Factorization of Compressively-Sensed Data

What learning algorithms can be run directly on compressively-sensed data? In this work, we consider the question of accurately and efficiently computing low-rank matrix or tensor factorizations given data compressed via random projections. We examine the approach of first performing factorization in the compressed domain, and then reconstructing the original high-dimensional factors from the recovered (compressed) factors. In both the matrix and tensor settings, we establish conditions under which this natural approach will provably recover the original factors. While it is well-known that random projections preserve a number of geometric properties of a dataset, our work can be viewed as showing that they can also preserve certain solutions of non-convex, NP-Hard problems like non-negative matrix factorization. We support these theoretical results with experiments on synthetic data and demonstrate the practical applicability of compressed factorization on real-world gene expression and EEG time series datasets.Comment: Updates for ICML'19 camera-read

Average Case Recovery Analysis of Tomographic Compressive Sensing

The reconstruction of three-dimensional sparse volume functions from few tomographic projections constitutes a challenging problem in image reconstruction and turns out to be a particular instance problem of compressive sensing. The tomographic measurement matrix encodes the incidence relation of the imaging process, and therefore is not subject to design up to small perturbations of non-zero entries. We present an average case analysis of the recovery properties and a corresponding tail bound to establish weak thresholds, in excellent agreement with numerical experiments. Our result improve the state-of-the-art of tomographic imaging in experimental fluid dynamics by a factor of three

Phase Transitions and Cosparse Tomographic Recovery of Compound Solid Bodies from Few Projections

We study unique recovery of cosparse signals from limited-angle tomographic measurements of two- and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality $\ell$ with respect to the discrete gradient operator. We relate $\ell$ both to the number of measurements and to a nullspace condition with respect to the measurement matrix, so as to achieve unique recovery by linear programming. These results are supported by comprehensive numerical experiments that show a high correlation of performance in practice and theoretical predictions. Despite poor properties of the measurement matrix from the viewpoint of compressed sensing, the class of uniquely recoverable signals basically seems large enough to cover practical applications, like contactless quality inspection of compound solid bodies composed of few materials