CMC Faculty Publications and ResearchCopyright (c) 2016 Claremont Colleges All rights reserved.
http://scholarship.claremont.edu/cmc_fac_pub
Recent documents in CMC Faculty Publications and Researchen-usFri, 16 Dec 2016 17:29:38 PST3600On Arithmetic Lattices in the Plane
http://scholarship.claremont.edu/cmc_fac_pub/450
http://scholarship.claremont.edu/cmc_fac_pub/450Fri, 09 Dec 2016 14:07:46 PST
We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the j-invariant corresponding to similarity classes of planar lattices.
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Lenny Fukshansky et al.Lattice Theory and Toeplitz Determinants
http://scholarship.claremont.edu/cmc_fac_pub/449
http://scholarship.claremont.edu/cmc_fac_pub/449Fri, 09 Dec 2016 14:07:37 PST
This is a survey of our recent joint investigations of lattices that are generated by finite Abelian groups. In the case of cyclic groups, the volume of a fundamental domain of such a lattice is a perturbed Toeplitz determinant with a simple Fisher-Hartwig symbol. For general groups, the situation is more complicated, but it can still be tackled by pure matrix theory. Our main result on the lattices under consideration states that they always have a basis of minimal vectors, while our results in the other direction concern exact and asymptotic formulas for perturbed Toeplitz determinants. The survey is a slightly modified version of the talk given by the first author at the Humboldt Kolleg and the IWOTA in Tbilisi in 2015. It is mainly for operator theorists and therefore also contains an introduction to the basics of lattice theory.
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Albrecht Böttcher et al.Spherical 2-Designs and Lattices from Abelian Groups
http://scholarship.claremont.edu/cmc_fac_pub/448
http://scholarship.claremont.edu/cmc_fac_pub/448Fri, 09 Dec 2016 14:07:30 PST
We consider lattices generated by finite Abelian groups. The main result says that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame.
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Albrecht Böttcher et al.On an Effective Variation of Kronecker's Approximation Theorem
http://scholarship.claremont.edu/cmc_fac_pub/447
http://scholarship.claremont.edu/cmc_fac_pub/447Fri, 09 Dec 2016 14:07:22 PST
Let Λ ⊂ Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z ⊂ Rn be the zero locus of a finite collection of polynomials such that Λ |⊂ Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1, . . . , Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε > 0 and a ∈ Rn, we prove the existence of a vector x ∈ Λ \ Z of explicitly bounded sup-norm such that ||Li(x) − ai|| < ε for each 1 ≤ i ≤ t, where || || stands for the distance to the nearest integer. The bound on sup-norm of x depends on Λ, K, Z, heights of linear forms, and ε. This presents a generalization of Kronecker’s approximation theorem, establishing an effective result on density of the image of Λ\Z under the linear forms L1, . . . , Lt in the t-torus Rt/Zt .
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Lenny FukshanskyStability of Ideal Lattices from Quadratic Number Fields
http://scholarship.claremont.edu/cmc_fac_pub/446
http://scholarship.claremont.edu/cmc_fac_pub/446Thu, 08 Dec 2016 12:55:15 PST
We study semi-stable ideal lattices coming from real quadratic number fields. Specifically, we demonstrate infinite families of semi-stable and unstable ideal lattices of trace type, establishing explicit conditions on the canonical basis of an ideal that ensure stability; in particular, our result implies that an ideal lattice of trace type coming from a real quadratic field is semi-stable with positive probability. We also briefly discuss the connection between stability and well-roundedness of Euclidean lattices.
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Lenny FukshanskyToeplitz Determinants with Perturbations in the Corners
http://scholarship.claremont.edu/cmc_fac_pub/445
http://scholarship.claremont.edu/cmc_fac_pub/445Thu, 08 Dec 2016 12:55:07 PST
The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher–Hartwig symbols. In that case we establish formulas for pure single Fisher–Hartwig singularities and for Hermitian matrices induced by general Fisher–Hartwig symbols.
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Albrecht Böttcher et al.On Lattices Generated by Finite Abelian Groups
http://scholarship.claremont.edu/cmc_fac_pub/444
http://scholarship.claremont.edu/cmc_fac_pub/444Thu, 08 Dec 2016 12:54:58 PST
This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In the case where the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by M. Sha for the covering radius in the case of the Barnes lattices. Also discussed are properties of the automorphism groups of these lattices.
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Albrecht Böttcher et al.Permutation Invariant Lattices
http://scholarship.claremont.edu/cmc_fac_pub/443
http://scholarship.claremont.edu/cmc_fac_pub/443Thu, 08 Dec 2016 12:54:50 PST
We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ ∈ Sn, we study properties of the set of all lattices closed under the action of τ: we call such lattices τ-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio in [8, 9], which we previously studied in [1]. Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all τ-invariant lattices in Rn has positive co-dimension (and hence comprises zero proportion) for all τ different from an n-cycle.
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Lenny Fukshansky et al.Height Bounds on Zeros of Quadratic Forms Over Q-bar
http://scholarship.claremont.edu/cmc_fac_pub/442
http://scholarship.claremont.edu/cmc_fac_pub/442Thu, 08 Dec 2016 12:54:42 PST
In this paper we establish three results on small-height zeros of quadratic polynomials over Q. For a single quadratic form in N ≥ 2 variables on a subspace of Q N , we prove an upper bound on the height of a smallest nontrivial zero outside of an algebraic set under the assumption that such a zero exists. For a system of k quadratic forms on an L-dimensional subspace of Q N , N ≥ L ≥ k(k+1) 2 + 1, we prove existence of a nontrivial simultaneous small-height zero. For a system of one or two inhomogeneous quadratic and m linear polynomials in N ≥ m + 4 variables, we obtain upper bounds on the height of a smallest simultaneous zero, if such a zero exists. Our investigation extends previous results on small zeros of quadratic forms, including Cassels’ theorem and its various generalizations and contributes to the literature of so-called “absolute” Diophantine results with respect to height. All bounds on height are explicit.
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Lenny FukshanskyTotally Isotropic Subspaces of Small Height in Quadratic Spaces
http://scholarship.claremont.edu/cmc_fac_pub/441
http://scholarship.claremont.edu/cmc_fac_pub/441Thu, 08 Dec 2016 12:54:34 PST
Let K be a global field or Q, F a nonzero quadratic form on KN , N ≥ 2, and V a subspace of KN . We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of (V, F) such that each such family spans V as a K-vector space. This result generalizes and extends a well known theorem of J. Vaaler [16] and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels’ theorem on small zeros of quadratic forms. All bounds on height are explicit.
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Wai Kiu Chan et al.Lattices from Hermitian Function Fields
http://scholarship.claremont.edu/cmc_fac_pub/440
http://scholarship.claremont.edu/cmc_fac_pub/440Thu, 08 Dec 2016 12:54:27 PST
We consider the well-known Rosenbloom–Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of Gerhard Hiss on the factorization of functions with particular divisor support into lines and their inverses.
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Albrecht Böttcher et al.Small Zeros of Quadratic Forms Outside a Union of Varieties
http://scholarship.claremont.edu/cmc_fac_pub/439
http://scholarship.claremont.edu/cmc_fac_pub/439Wed, 23 Nov 2016 12:58:33 PST
Let be a quadratic form in variables defined on a vector space over a global field , and be a finite union of varieties defined by families of homogeneous polynomials over . We show that if contains a nontrivial zero of , then there exists a linearly independent collection of small-height zeros of in , where the height bound does not depend on the height of , only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace of the quadratic space such that is not contained in . Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit.
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Wai Kiu Chan et al.Lattices from Elliptic Curves over Finite Fields
http://scholarship.claremont.edu/cmc_fac_pub/438
http://scholarship.claremont.edu/cmc_fac_pub/438Wed, 23 Nov 2016 12:58:25 PST
In their well known book Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities.
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Lenny Fukshansky et al.On the Geometry of Cyclic Lattices
http://scholarship.claremont.edu/cmc_fac_pub/437
http://scholarship.claremont.edu/cmc_fac_pub/437Wed, 23 Nov 2016 12:58:17 PST
Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices. Our main result is a counting estimate for the number of well-rounded cyclic lattices, indicating that well-rounded lattices are more common among cyclic lattices than generically. We also show that SVP is equivalent to SIVP on a positive proportion of Minkowskian well-rounded cyclic lattices in every dimension. As an example, we demonstrate an explicit construction of a family of such lattices on which this equivalence holds. To conclude, we introduce a class of sublattices of ZN closed under the action of subgroups of the permutation group SN, which are a natural generalization of cyclic lattices, and show that our results extend to all such lattices closed under the action of any N-cycle.
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Lenny Fukshansky et al.A Sampling Kaczmarz-Motzkin Algorithm for Linear Feasibility
http://scholarship.claremont.edu/cmc_fac_pub/436
http://scholarship.claremont.edu/cmc_fac_pub/436Fri, 11 Nov 2016 16:06:54 PST
We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. We obtain a family of algorithms that generalize and extend both projection-based techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods.
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Jesus A. De Loera et al.Weighted ℓ1-Minimization for Sparse Recovery under Arbitrary Prior Information
http://scholarship.claremont.edu/cmc_fac_pub/435
http://scholarship.claremont.edu/cmc_fac_pub/435Fri, 11 Nov 2016 16:06:46 PST
Weighted ℓ1-minimization has been studied as a technique for the reconstruction of a sparse signal from compressively sampled measurements when prior information about the signal, in the form of a support estimate, is available. In this work, we study the recovery conditions and the associated recovery guarantees of weighted ℓ1-minimization when arbitrarily many distinct weights are permitted. For example, such a setup might be used when one has multiple estimates for the support of a signal, and these estimates have varying degrees of accuracy. Our analysis yields an extension to existing works that assume only a single constant weight is used. We include numerical experiments, with both synthetic signals and real video data, that demonstrate the benefits of allowing non-uniform weights in the reconstruction procedure.
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Deanna Needell et al.A note on practical approximate projection schemes in signal space methods
http://scholarship.claremont.edu/cmc_fac_pub/434
http://scholarship.claremont.edu/cmc_fac_pub/434Fri, 11 Nov 2016 16:06:38 PST
Compressive sensing (CS) is a new technology which allows the acquisition of signals directly in compressed form, using far fewer measurements than traditional theory dictates. Recently, many socalled signal space methods have been developed to extend this body of work to signals sparse in arbitrary dictionaries rather than orthonormal bases. In doing so, CS can be utilized in a much broader array of practical settings. Often, such approaches often rely on the ability to optimally project a signal onto a small number of dictionary atoms. Such optimal, or even approximate, projections have been difficult to derive theoretically. Nonetheless, it has been observed experimentally that conventional CS approaches can be used for such projections, and still provide accurate signal recovery. In this letter, we summarize the empirical evidence and clearly demonstrate for what signal types certain CS methods may be used as approximate projections. In addition, we provide theoretical guarantees for such methods for certain sparse signal structures. Our theoretical results match those observed in experimental studies, and we thus establish both experimentally and theoretically that these CS methods can be used in this context.
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Xiaoyi Gu et al.Exponential decay of reconstruction error from binary measurements of sparse signals
http://scholarship.claremont.edu/cmc_fac_pub/433
http://scholarship.claremont.edu/cmc_fac_pub/433Fri, 11 Nov 2016 16:06:30 PST
Binary measurements arise naturally in a variety of statistical and engineering applications. They may be inherent to the problem—e.g., in determining the relationship between genetics and the presence or absence of a disease—or they may be a result of extreme quantization. A recent influx of literature has suggested that using prior signal information can greatly improve the ability to reconstruct a signal from binary measurements. This is exemplified by onebit compressed sensing, which takes the compressed sensing model but assumes that only the sign of each measurement is retained. It has recently been shown that the number of one-bit measurements required for signal estimation mirrors that of unquantized compressed sensing. Indeed, s-sparse signals in R n can be estimated (up to normalization) from Ω(s log(n/s)) one-bit measurements. Nevertheless, controlling the precise accuracy of the error estimate remains an open challenge. In this paper, we focus on optimizing the decay of the error as a function of the oversampling factor λ := m/(s log(n/s)), where m is the number of measurements. It is known that the error in reconstructing sparse signals from standard one-bit measurements is bounded below by Ω(λ −1 ). Without adjusting the measurement procedure, reducing this polynomial error decay rate is impossible. However, we show that an adaptive choice of the thresholds used for quantization may lower the error rate to e −Ω(λ) . This improves upon guarantees for other methods of adaptive thresholding as proposed in Sigma-Delta quantization. We develop a general recursive strategy to achieve this exponential decay and two specific polynomialtime algorithms which fall into this framework, one based on convex programming and one on hard thresholding. This work is inspired by the one-bit compressed sensing model, in which the engineer controls the measurement procedure. Nevertheless, the principle is extendable to signal reconstruction problems in a variety of binary statistical models as well as statistical estimation problems like logistic regression.
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Richard Baraniuk et al.Rows vs. Columns: Randomized Kaczmarz or Gauss-Seidel for Ridge Regression
http://scholarship.claremont.edu/cmc_fac_pub/432
http://scholarship.claremont.edu/cmc_fac_pub/432Fri, 11 Nov 2016 16:06:22 PST
The Kaczmarz and Gauss-Seidel methods aim to solve a linear m × n system Xβ = y by iteratively refining the solution estimate; the former uses random rows of X to update β given the corresponding equations and the latter uses random columns of X to update corresponding coordinates in β. Interest in these methods was recently revitalized by a proof of Strohmer and Vershynin showing linear convergence in expectation for a randomized Kaczmarz method variant (RK), and a similar result for the randomized Gauss-Seidel algorithm (RGS) was later proved by Lewis and Leventhal. Recent work unified the analysis of these algorithms for the overcomplete and undercomplete systems, showing convergence to the ordinary least squares (OLS) solution and the minimum Euclidean norm solution respectively. This paper considers the natural follow-up to the OLS problem, ridge regression, which solves (X∗X +λI)β = X∗y. We present particular variants of RK and RGS for solving this system and derive their convergence rates. We compare these to a recent proposal by Ivanov and Zhdanov to solve this system, that can be interpreted as randomly sampling both rows and columns, which we argue is often suboptimal. Instead, we claim that one should always use RGS (columns) when m > n and RK (rows) when m < n. This difference in behavior is simply related to the minimum eigenvalue of two related positive semidefinite matrices, X∗X +λIn and XX∗ +λIm when m > n or m < n.
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Ahmed Hefny et al.A Practical Study of Longitudinal Reference Based Compressed Sensing for MRI
http://scholarship.claremont.edu/cmc_fac_pub/431
http://scholarship.claremont.edu/cmc_fac_pub/431Fri, 11 Nov 2016 16:06:14 PST
Compressed sensing (CS) is a new signal acquisition paradigm that enables the reconstruction of signals and images from a low number of samples. A particularly exciting application of CS is Magnetic Resonance Imaging (MRI), where CS significantly speeds up scan time by requiring far fewer measurements than standard MRI techniques. Such a reduction in sampling time leads to less power consumption, less need for patient sedation, and more accurate images. This accuracy increase is especially pronounced in pediatric MRI where patients have trouble being still for long scan periods. Although such gains are already significant, even further improvements can be made by utilizing past MRI scans of the same patient. Many patients require repeated scans over a period of time in order to track illnesses and the prior scans can be used as references for the current image. This allows samples to be taken adaptively, based on both the prior scan and the current measurements. Work by Weizman [20] has shown that so-called reference based adaptive-weighted temporal Compressed Sensing MRI (LACS-MRI) requires far fewer samples than standard Compressed Sensing (CS) to achieve the same reconstruction signal-to-noise ratio (RSNR). The method uses a mixture of reference-based and adaptive-sampling. In this work, we test this methodology by using various adaptive sensing schemes, reconstruction methods, and image types. We create a thorough catalog of reconstruction behavior and success rates that is interesting from a mathematical point of view and is useful for practitioners. We also solve a grayscale compensation toy problem that supports the insensitivity of LACS-MRI to changes in MRI acquisition parameters and thus showcases the reliability of LACS-MRI in possible clinical situations.
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Samuel Birns et al.