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- Well-rounded lattices (3)
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Articles 1 - 30 of 35
Full-Text Articles in Physical Sciences and Mathematics
Combinatorial Proofs Of Fibonomial Identities, Arthur Benjamin, Elizabeth Reiland
Combinatorial Proofs Of Fibonomial Identities, Arthur Benjamin, Elizabeth Reiland
All HMC Faculty Publications and Research
We provide a list of simple looking identities that are still in need of combinatorial proof.
Review: Crystal Bases Of Q-Deformed Kac Modules Over The Quantum Superalgebras Uq(Gl(Mln)), Gizem Karaali
Review: Crystal Bases Of Q-Deformed Kac Modules Over The Quantum Superalgebras Uq(Gl(Mln)), Gizem Karaali
Pomona Faculty Publications and Research
No abstract provided.
Op-Ed: Solve This Math Problem: The Gender Gap, Francis Su
Op-Ed: Solve This Math Problem: The Gender Gap, Francis Su
All HMC Faculty Publications and Research
Women may not face such blatant impediments to doing math and science today. But Mirzakhani's achievement aside, we are still a long way from adequately recognizing the outstanding work of women.
Math Talk: Preparing Your Conference Presentation, Gizem Karaali
Math Talk: Preparing Your Conference Presentation, Gizem Karaali
Pomona Faculty Publications and Research
If you are a typical reader of this blog, then you recently wrapped up your finals week and then dutifully made a summer plan. And then came the summer. Your plan may have involved working on a manuscript, preparing for a qualifying exam or a new course coming up in the fall, drafting a grant proposal, learning a new language (human or machine), eating kale in four different forms, and perhaps some fun times under the sun. Some, like me, also made plans to travel to conferences and give talks. Gearing up to get ready for my first conference of …
Four Quotient Set Gems, Stephan Ramon Garcia, Michael Someck '14, Bob Lutz '13, Bryan Brown '15, Michael Dairyko '13
Four Quotient Set Gems, Stephan Ramon Garcia, Michael Someck '14, Bob Lutz '13, Bryan Brown '15, Michael Dairyko '13
Pomona Faculty Publications and Research
Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the MONTHLY, despite its intense coverage of quotient sets over the years.
Exponential Decay Of Reconstruction Error From Binary Measurements Of Sparse Signals, Richard Baraniuk, Simon Foucart, Deanna Needell, Yaniv Plan, Mary Wootters
Exponential Decay Of Reconstruction Error From Binary Measurements Of Sparse Signals, Richard Baraniuk, Simon Foucart, Deanna Needell, Yaniv Plan, Mary Wootters
CMC Faculty Publications and Research
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 …
Near Oracle Performance And Block Analysis Of Signal Space Greedy Methods, Raja Giryes, Deanna Needell
Near Oracle Performance And Block Analysis Of Signal Space Greedy Methods, Raja Giryes, Deanna Needell
CMC Faculty Publications and Research
Compressive sampling (CoSa) is a new methodology which demonstrates that sparse signals can be recovered from a small number of linear measurements. Greedy algorithms like CoSaMP have been designed for this recovery, and variants of these methods have been adapted to the case where sparsity is with respect to some arbitrary dictionary rather than an orthonormal basis. In this work we present an analysis of the so-called Signal Space CoSaMP method when the measurements are corrupted with mean-zero white Gaussian noise. We establish near-oracle performance for recovery of signals sparse in some arbitrary dictionary. In addition, we analyze the block …
Linear Convergence Of Stochastic Iterative Greedy Algorithms With Sparse Constraints, Nam Nguyen, Deanna Needell, Tina Woolf
Linear Convergence Of Stochastic Iterative Greedy Algorithms With Sparse Constraints, Nam Nguyen, Deanna Needell, Tina Woolf
CMC Faculty Publications and Research
Motivated by recent work on stochastic gradient descent methods, we develop two stochastic variants of greedy algorithms for possibly non-convex optimization problems with sparsity constraints. We prove linear convergence in expectation to the solution within a specified tolerance. This generalized framework applies to problems such as sparse signal recovery in compressed sensing, low-rank matrix recovery, and co-variance matrix estimation, giving methods with provable convergence guarantees that often outperform their deterministic counterparts. We also analyze the settings where gradients and projections can only be computed approximately, and prove the methods are robust to these approximations. We include many numerical experiments which …
Lattices From Elliptic Curves Over Finite Fields, Lenny Fukshansky, Hiren Maharaj
Lattices From Elliptic Curves Over Finite Fields, Lenny Fukshansky, Hiren Maharaj
CMC Faculty Publications and Research
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.
Block Kaczmarz Method With Inequalities, Jonathan Briskman, Deanna Needell
Block Kaczmarz Method With Inequalities, Jonathan Briskman, Deanna Needell
CMC Faculty Publications and Research
The randomized Kaczmarz method is an iterative algorithm that solves overdetermined systems of linear equations. Recently, the method was extended to systems of equalities and inequalities by Leventhal and Lewis. Even more recently, Needell and Tropp provided an analysis of a block version of the method for systems of linear equations. This paper considers the use of a block type method for systems of mixed equalities and inequalities, bridging these two bodies of work. We show that utilizing a matrix paving over the equalities of the system can lead to significantly improved convergence, and prove a linear convergence rate as …
Energy Driven Pattern Formation In Planar Dipole-Dipole Systems In The Presence Of Weak Noise, Jaron P. Kent-Dobias '14, Andrew Bernoff
Energy Driven Pattern Formation In Planar Dipole-Dipole Systems In The Presence Of Weak Noise, Jaron P. Kent-Dobias '14, Andrew Bernoff
All HMC Faculty Publications and Research
We study pattern formation in planar fluid systems driven by intermolecular cohesion (which manifests as a line tension) and dipole-dipole repulsion which are observed in physical systems including ferrofluids in Hele-Shaw cells and Langmuir layers. When the dipolar repulsion is sufficiently strong, domains undergo forked branching reminiscent of viscous fingering. A known difficulty with these models is that the energy associated with dipole-dipole interactions is singular at small distances. Following previous work, we demonstrate how to ameliorate this singularity and show that in the macroscopic limit, only the relative scale of the microscopic details of a system are relevant, and …
Two-Part Reconstruction With Noisy-Sudocodes, Yanting Ma, Dror Baron, Deanna Needell
Two-Part Reconstruction With Noisy-Sudocodes, Yanting Ma, Dror Baron, Deanna Needell
CMC Faculty Publications and Research
We develop a two-part reconstruction framework for signal recovery in compressed sensing (CS), where a fast algorithm is applied to provide partial recovery in Part 1, and a CS algorithm is applied to complete the residual problem in Part 2. Partitioning the reconstruction process into two complementary parts provides a natural trade-off between runtime and reconstruction quality. To exploit the advantages of the two-part framework, we propose a Noisy-Sudocodes algorithm that performs two-part reconstruction of sparse signals in the presence of measurement noise. Specifically, we design a fast algorithm for Part 1 of Noisy-Sudocodes that identifies the zero coefficients of …
Why You Need A Summer Plan, Gizem Karaali
Why You Need A Summer Plan, Gizem Karaali
Pomona Faculty Publications and Research
In the last decade, for many times, I have tasted first-hand the end-of-summer blues I described above. I have spent many early spring months dreaming of all that I would be doing when the summer arrives, only to realize that it was already September and I had not much to show for for the months in between. I have also observed many of my peers going through similar things, and I just assumed for years that this was how it had to be. But then some time in the middle of the tenure track, I decided to try approaching my …
Improving Image Clustering Using Sparse Text And The Wisdom Of The Crowds, Anna Ma, Arjuna Flenner, Deanna Needell, Allon G. Percus
Improving Image Clustering Using Sparse Text And The Wisdom Of The Crowds, Anna Ma, Arjuna Flenner, Deanna Needell, Allon G. Percus
CMC Faculty Publications and Research
We propose a method to improve image clustering using sparse text and the wisdom of the crowds. In particular, we present a method to fuse two different kinds of document features, image and text features, and use a common dictionary or “wisdom of the crowds” as the connection between the two different kinds of documents. With the proposed fusion matrix, we use topic modeling via non-negative matrix factorization to cluster documents.
Review: Truncated Toeplitz Operators Of Finite Rank, Stephan Ramon Garcia
Review: Truncated Toeplitz Operators Of Finite Rank, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
The Fundamental Principle Of Productivity: What They Don't Teach You In Graduate School, Gizem Karaali
The Fundamental Principle Of Productivity: What They Don't Teach You In Graduate School, Gizem Karaali
Pomona Faculty Publications and Research
But through the years, I have read many books, ranging from self-help guides to creative writing manuals, attended workshops, frequented relevant websites, and reflected upon my own personal experiences. As a result, I have collected together a few ideas and tools on productivity and time management that have been working well for me. In this post I want to share with you the most basic of these principles, with the hope that it may assist you in your life, at whatever stage you are. Whether you are a graduate student, a postdoctoral researcher, a junior mathematician on the tenure track, …
The Scientist–Reporter Collaboration: A Guide To Working With The Press, Rachel Levy, Flora Lichtman, David L. Hu
The Scientist–Reporter Collaboration: A Guide To Working With The Press, Rachel Levy, Flora Lichtman, David L. Hu
All HMC Faculty Publications and Research
Science, technology, engineering, and mathematics (STEM) to the public can be challenging. Often, the language that researchers use among themselves is technical and difficult for non-experts to decipher. But as you probably know, communicating your research to non-experts is becoming mandatory. In a direct sense, funding agencies often require outreach for grant fulfillment. There are indirect benefits as well: Conveying the joy of discovery and the relevance of scientific results builds scientific literacy among the public---which of course includes both students who will eventually do research of their own and people who elect the policy makers who allocate funding. How …
Aftermath: Every Math Major Should Take A Public-Speaking Course, Rachel Levy
Aftermath: Every Math Major Should Take A Public-Speaking Course, Rachel Levy
All HMC Faculty Publications and Research
Rachel Levy argues that all mathematics majors should learn the art of public speaking.
Guaranteed Sparse Signal Recovery With Highly Coherent Sensing Matrices, Guangliang Chen, Atul Divekar, Deanna Needell
Guaranteed Sparse Signal Recovery With Highly Coherent Sensing Matrices, Guangliang Chen, Atul Divekar, Deanna Needell
CMC Faculty Publications and Research
Compressive sensing is a methodology for the reconstruction of sparse or compressible signals using far fewer samples than required by the Nyquist criterion. However, many of the results in compressive sensing concern random sampling matrices such as Gaussian and Bernoulli matrices. In common physically feasible signal acquisition and reconstruction scenarios such as super-resolution of images, the sensing matrix has a non-random structure with highly correlated columns. Here we present a compressive sensing type recovery algorithm, called Partial Inversion (PartInv), that overcomes the correlations among the columns. We provide theoretical justification as well as empirical comparisons.
Greedy Signal Space Methods For Incoherence And Beyond, Raja Giryes, Deanna Needell
Greedy Signal Space Methods For Incoherence And Beyond, Raja Giryes, Deanna Needell
CMC Faculty Publications and Research
Compressive sampling (CoSa) has provided many methods for signal recovery of signals compressible with respect to an orthonormal basis. However, modern applications have sparked the emergence of approaches for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary. Recently, several "signal-space" greedy methods have been proposed to address signal recovery in this setting. However, such methods inherently rely on the existence of fast and accurate projections which allow one to identify the most relevant atoms in a dictionary for any given signal, up to a very strict accuracy. When the dictionary is highly overcomplete, …
Review: The Relationships Among Multiplicities Of A J-Self-Adjoint Differential Operator's Eigenvalue, Stephan Ramon Garcia
Review: The Relationships Among Multiplicities Of A J-Self-Adjoint Differential Operator's Eigenvalue, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Stochastic Gradient Descent, Weighted Sampling, And The Randomized Kaczmarz Algorithm, Deanna Needell, Nathan Srebro, Rachel Ward
Stochastic Gradient Descent, Weighted Sampling, And The Randomized Kaczmarz Algorithm, Deanna Needell, Nathan Srebro, Rachel Ward
CMC Faculty Publications and Research
We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning (L/µ) 2 (where L is a bound on the smoothness and µ on the strong convexity) to a linear dependence on L/µ. Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other …
Randomized Block Kaczmarz Method With Projection For Solving Least Squares, Deanna Needell, Ran Zhao, Anastasios Zouzias
Randomized Block Kaczmarz Method With Projection For Solving Least Squares, Deanna Needell, Ran Zhao, Anastasios Zouzias
CMC Faculty Publications and Research
The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax = b. The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix A and converges exponentially in expectation to the solution of a consistent system. In this paper we analyze two block versions of the method each with a randomized projection, that converge in expectation to the least squares solution of inconsistent systems. Our approach utilizes a paving of the matrix A to guarantee exponential convergence, …
Paved With Good Intentions: Analysis Of A Randomized Block Kaczmarz Method, Deanna Needell, Joel A. Tropp
Paved With Good Intentions: Analysis Of A Randomized Block Kaczmarz Method, Deanna Needell, Joel A. Tropp
CMC Faculty Publications and Research
The block Kaczmarz method is an iterative scheme for solving overdetermined least-squares problems. At each step, the algorithm projects the current iterate onto the solution space of a subset of the constraints. This paper describes a block Kaczmarz algorithm that uses a randomized control scheme to choose the subset at each step. This algorithm is the first block Kaczmarz method with an (expected) linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and its submatrices. The analysis reveals that the algorithm is most effective when it is given a good row paving …
Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09
Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle '09, Blake Mellor, Matt Pittluck, Xan Vongsathorn '09
Pomona Faculty Publications and Research
We characterize which automorphisms of an arbitrary complete bipartite graph Kn,m can be induced by a homeomorphism of some embedding of the graph in S3.
An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin
An Extremal Problem For Characteristic Functions, Stephan Ramon Garcia, Isabelle Chalendar, Williams T. Ross, Dan Timotin
Pomona Faculty Publications and Research
Suppose E is a subset of the unit circle T and Hinfinity C Linfinity is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to znHinfinity. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.
Mathematical And Physical Aspects Of Complex Symmetric Operators, Stephan Ramon Garcia, Emil Prodan, Mihai Putinar
Mathematical And Physical Aspects Of Complex Symmetric Operators, Stephan Ramon Garcia, Emil Prodan, Mihai Putinar
Pomona Faculty Publications and Research
Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables.
Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño
Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño
All HMC Faculty Publications and Research
We prove the existence of a positive solution for a class of nonlin- ear elliptic systems with Neumann boundary conditions. The proof combines extensive use of a priori estimates for elliptic problems with Neumann boundary condition and Krasnoselskii's compression-expansion theorem
The Apple Doesn’T Fall Far From The (Metric) Tree: Equivalence Of Definitions, Asuman Güven Aksoy, Sixian Jin
The Apple Doesn’T Fall Far From The (Metric) Tree: Equivalence Of Definitions, Asuman Güven Aksoy, Sixian Jin
CMC Faculty Publications and Research
In this paper we prove the equivalence of definitions for metric trees and for δ-Hperbolic spaces. We point out how these equivalences can be used to understand the geometric and metric properties of δ-Hperbolic spaces and its relation to CAT(κ) spaces.
On Approximation Schemes And Compactness, Asuman Güven Aksoy, Jose M. Almira
On Approximation Schemes And Compactness, Asuman Güven Aksoy, Jose M. Almira
CMC Faculty Publications and Research
We present an overview of some results about characterization of compactness in which the concept of approximation scheme has had a role. In particular, we present several results that were proved by the second author, jointly with Luther, a decade ago, when these authors were working on a very general theory of approximation spaces. We then introduce and show the basic properties of a new concept of compactness, which was studied by the first author in the eighties, by using a generalized concept of approximation scheme and its associated Kolmogorov numbers, which generalizes the classical concept of compactness.