arXiv subjects

Alan Edelman

Publications and source records attributed to Alan Edelman.

At least 73 records · Page 4Linked to original sources

arXiv2016

Accelerated Convolutions for Efficient Multi-Scale Time to Contact Computation in Julia

Convolutions have long been regarded as fundamental to applied mathematics, physics and engineering. Their mathematical elegance allows for common tasks such as numerical differentiation to be computed efficiently on large data sets. Efficient computation of convolutions is critical to artificial intelligence in real-time applications, like machine vision, where convolutions must be continuously and efficiently computed on tens to hundreds of kilobytes per second. In this paper, we explore how convolutions are used in fundamental machine vision applications. We present an accelerated n-dimensional convolution package in the high performance computing language, Julia, and demonstrate its efficacy in solving the time to contact problem for machine vision. Results are measured against synthetically generated videos and quantitatively assessed according to their mean squared error from the ground truth. We achieve over an order of magnitude decrease in compute time and allocated memory for comparable machine vision applications. All code is packaged and integrated into the official Julia Package Manager to be used in various other scenarios.

cs.CV↗

arXiv2016

Julia Implementation of the Dynamic Distributed Dimensional Data Model

Julia is a new language for writing data analysis programs that are easy to implement and run at high performance. Similarly, the Dynamic Distributed Dimensional Data Model (D4M) aims to clarify data analysis operations while retaining strong performance. D4M accomplishes these goals through a composable, unified data model on associative arrays. In this work, we present an implementation of D4M in Julia and describe how it enables and facilitates data analysis. Several experiments showcase scalable performance in our new Julia version as compared to the original Matlab implementation.

cs.MS↗

arXiv2016

Beyond universality in random matrix theory

In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the $1/N$ expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives valuable information about the smallest singular value not seen in universality laws. In particular, we show the dependence on the fourth moment (or the kurtosis) of the entries. This work makes use of the so-called complex Gaussian divisible ensembles for both Wigner and sample covariance matrices.

math.PR↗

arXiv2015

Julia: A Fresh Approach to Numerical Computing

Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is designed to be easy and fast. Julia questions notions generally held as "laws of nature" by practitioners of numerical computing: 1. High-level dynamic programs have to be slow. 2. One must prototype in one language and then rewrite in another language for speed or deployment, and 3. There are parts of a system for the programmer, and other parts best left untouched as they are built by the experts. We introduce the Julia programming language and its design --- a dance between specialization and abstraction. Specialization allows for custom treatment. Multiple dispatch, a technique from computer science, picks the right algorithm for the right circumstance. Abstraction, what good computation is really about, recognizes what remains the same after differences are stripped away. Abstractions in mathematics are captured as code through another technique from computer science, generic programming. Julia shows that one can have machine performance without sacrificing human convenience.

cs.MS↗

arXiv2015

Integral geometry for Markov chain Monte Carlo: overcoming the curse of search-subspace dimensionality

We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold $\mathcal{M}$. Our method exploits the symmetries of the algorithms' isotropic random search-direction subspaces to analytically average out the variance in the intersection volume caused by the orientation of the search-subspace with respect to the manifold $\mathcal{M}$ it intersects. This variance can grow exponentially with the dimension of the search-subspace, greatly slowing down the algorithm. Eliminating this variance allows us to use search-subspaces of dimensions many times greater than would otherwise be possible, allowing us to sample very rare events that a lower-dimensional search-subspace would be unlikely to intersect. To extend this method to events that are rare for reasons other than their support $\mathcal{M}$ having a lower dimension, we formulate and prove a new theorem in integral geometry that makes use of the curvature form of the Chern-Gauss-Bonnet theorem to reweight sampling probabilities. On the side, we also apply our theorem to obtain new theoretical bounds for the volumes of real algebraic manifolds. Finally, we demonstrate the computational effectiveness and speedup of our method by numerically applying it to the conditional stochastic Airy operator sampling problem in random matrix theory.

math.PR↗

arXiv2015

The Singular Values of the GUE (Less is More)

Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large $n$ limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter $\pm$1/2. Similarly, we write the absolute value of the determinant of the $n\times{}n$ GUE as a product n independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.

math.PR↗

arXiv2015

Infinite Random Matrix Theory, Tridiagonal Bordered Toeplitz Matrices, and the Moment Problem

The four major asymptotic level density laws of random matrix theory may all be showcased though their Jacobi parameter representation as having a bordered Toeplitz form. We compare and contrast these laws, completing and exploring their representations in one place. Inspired by the bordered Toeplitz form, we propose an algorithm for the finite moment problem by proposing a solution whose density has a bordered Toeplitz form.

math.PR↗

arXiv2015

Random Triangle Theory with Geometry and Applications

What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to $(0,0)$ or reformulation as a 2x2 matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage other to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: Triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the "square root ellipticity statistic" of random matrix theory. Another proof connects the Hopf map to the SVD of 2 by 2 matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight to both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed and applications are proposed.

math.HO↗

arXiv2014

Parallel Prefix Polymorphism Permits Parallelization, Presentation & Proof

Polymorphism in programming languages enables code reuse. Here, we show that polymorphism has broad applicability far beyond computations for technical computing: parallelism in distributed computing, presentation of visualizations of runtime data flow, and proofs for formal verification of correctness. The ability to reuse a single codebase for all these purposes provides new ways to understand and verify parallel programs.

cs.PL↗

arXiv2014

Array operators using multiple dispatch: a design methodology for array implementations in dynamic languages

Arrays are such a rich and fundamental data type that they tend to be built into a language, either in the compiler or in a large low-level library. Defining this functionality at the user level instead provides greater flexibility for application domains not envisioned by the language designer. Only a few languages, such as C++ and Haskell, provide the necessary power to define $n$-dimensional arrays, but these systems rely on compile-time abstraction, sacrificing some flexibility. In contrast, dynamic languages make it straightforward for the user to define any behavior they might want, but at the possible expense of performance. As part of the Julia language project, we have developed an approach that yields a novel trade-off between flexibility and compile-time analysis. The core abstraction we use is multiple dispatch. We have come to believe that while multiple dispatch has not been especially popular in most kinds of programming, technical computing is its killer application. By expressing key functions such as array indexing using multi-method signatures, a surprising range of behaviors can be obtained, in a way that is both relatively easy to write and amenable to compiler analysis. The compact factoring of concerns provided by these methods makes it easier for user-defined types to behave consistently with types in the standard library.

cs.PL↗

arXiv2013

The Beta-MANOVA Ensemble with General Covariance

We find the joint generalized singular value distribution and largest generalized singular value distributions of the $β$-MANOVA ensemble with positive diagonal covariance, which is general. This has been done for the continuous $β> 0$ case for identity covariance (in eigenvalue form), and by setting the covariance to $I$ in our model we get another version. For the diagonal covariance case, it has only been done for $β= 1,2,4$ cases (real, complex, and quaternion matrix entries). This is in a way the first second-order $β$-ensemble, since the sampler for the generalized singular values of the $β$-MANOVA with diagonal covariance calls the sampler for the eigenvalues of the $β$-Wishart with diagonal covariance of Forrester and Dubbs-Edelman-Koev-Venkataramana. We use a conjecture of MacDonald proven by Baker and Forrester concerning an integral of a hypergeometric function and a theorem of Kaneko concerning an integral of Jack polynomials to derive our generalized singular value distributions. In addition we use many identities from Forrester's {\it Log-Gases and Random Matrices}. We supply numerical evidence that our theorems are correct.

math.PR↗

arXiv2013

Partial freeness of random matrices

We investigate the implications of free probability for random matrices. From rules for calculating all possible joint moments of two free random matrices, we develop a notion of partial freeness which is quantified by the breakdown of these rules. We provide a combinatorial interpretation for partial freeness as the presence of closed paths in Hilbert space defined by particular joint moments. We also discuss how asymptotic moment expansions provide an error term on the density of states. We present MATLAB code for the calculation of moments and free cumulants of arbitrary random matrices.

math.PR↗

arXiv2013

The Beta-Wishart Ensemble

This paper proves a matrix model for the Wishart Ensemble with general covariance and general dimension parameter beta. In so doing, we introduce a new and elegant definition of Jack polynomials.

math.PR↗

arXiv2012

Julia: A Fast Dynamic Language for Technical Computing

Dynamic languages have become popular for scientific computing. They are generally considered highly productive, but lacking in performance. This paper presents Julia, a new dynamic language for technical computing, designed for performance from the beginning by adapting and extending modern programming language techniques. A design based on generic functions and a rich type system simultaneously enables an expressive programming model and successful type inference, leading to good performance for a wide range of programs. This makes it possible for much of the Julia library to be written in Julia itself, while also incorporating best-of-breed C and Fortran libraries.

cs.PL↗

arXiv2012

Condition Numbers of Indefinite Rank 2 Ghost Wishart Matrices

We define an indefinite Wishart matrix as a matrix of the form A=W^{T}WΣ, where Σis an indefinite diagonal matrix and W is a matrix of independent standard normals. We focus on the case where W is L by 2 which has engineering applications. We obtain the distribution of the ratio of the eigenvalues of A. This distribution can be "folded" to give the distribution of the condition number. We calculate formulas for W real (β=1), complex (β=2), quaternionic (β=4) or any ghost 0<β<\infty. We then corroborate our work by comparing them against numerical experiments.

math.ST↗

arXiv2012

Isotropic Entanglement

The method of "Isotropic Entanglement" (IE), inspired by Free Probability Theory and Random Matrix Theory, predicts the eigenvalue distribution of quantum many-body (spin) systems with generic interactions. At the heart is a "Slider", which interpolates between two extrema by matching fourth moments. The first extreme treats the non-commuting terms classically and the second treats them isotropically. Isotropic means that the eigenvectors are in generic positions. We prove Matching Three Moments and Slider Theorems and further prove that the interpolation is universal, i.e., independent of the choice of local terms. Our examples show that IE provides an accurate picture well beyond what one expects from the first four moments alone.

quant-ph↗

arXiv2012

Error analysis of free probability approximations to the density of states of disordered systems

Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approximate the ensemble- averaged density of states without exact diagonalization. We present an error analysis that quantifies the accuracy using a generalized moment expansion, allowing us to distinguish between different approximations. We identify an approximation that is accurate to the eighth moment across all noise strengths, and contrast this with the perturbation theory and isotropic entanglement theory.

cond-mat.dis-nn↗

arXiv2011

Density of States of Quantum Spin Systems from Isotropic Entanglement

We propose a method which we call "Isotropic Entanglement" (IE), that predicts the eigenvalue distribution of quantum many body (spin) systems (QMBS) with generic interactions. We interpolate between two known approximations by matching fourth moments. Though, such problems can be QMA-complete, our examples show that IE provides an accurate picture of the spectra well beyond what one expects from the first four moments alone. We further show that the interpolation is universal, i.e., independent of the choice of local terms.

cond-mat.str-el↗