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Aaron Smith

Publications and source records attributed to Aaron Smith.

160 records · Page 9Linked to original sources
arXiv2016

On the Mixing Time of Kac's Walk and Other High-Dimensional Gibbs Samplers with Constraints

Determining the total variation mixing time of Kac's random walk on the special orthogonal group $\mathrm{SO}(n)$ has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not-amenable to existing techniques in random matrix theory. To circumvent this difficulty, we extend some recent bounds on the smallest singular values of matrices with independent entries to our setting. These bounds imply that the mixing time of Kac's walk on the group $\mathrm{SO}(n)$ is between $C_{1} n^{2}$ and $C_{2} n^{4} \log(n)$ for some explicit constants $0 < C_{1}, C_{2} < \infty$, substantially improving on the bound of $O(n^{5} \log(n)^{2})$ by Jiang. Our methods may also be applied to other high dimensional Gibbs samplers with constraints and thus are of independent interest. In addition to giving analytical bounds on the mixing time, our approach allows us to compute rigorous estimates of the mixing time by simulating the eigenvalues of a random matrix.

math.PR↗
arXiv2016

Evidence for a direct collapse black hole in the Lyman-alpha source CR7

Throughout the epoch of reionization the most luminous Lyα emitters are capable of ionizing their own local bubbles. The CR7 galaxy at $z \approx 6.6$ stands out for its combination of exceptionally bright Lyα and HeII 1640 Angstrom line emission but absence of metal lines. As a result CR7 may be the first viable candidate host of a young primordial starburst or direct collapse black hole. High-resolution spectroscopy reveals a +160 km s$^{-1}$ velocity offset between the Lyα and HeII line peaks while the spatial extent of the Lyα emitting region is $\sim 16$ kpc. The observables are indicative of an outflow signature produced by a strong central source. We present one-dimensional radiation-hydrodynamics simulations incorporating accurate Lyα feedback and ionizing radiation to investigate the nature of the CR7 source. We find that a Population III star cluster with $10^5$ K blackbody emission ionizes its environment too efficiently to generate a significant velocity offset. However, a massive black hole with a nonthermal Compton-thick spectrum is able to reproduce the Lyα signatures as a result of higher photon trapping and longer potential lifetime. For both sources, Lyα radiation pressure turns out to be dynamically important.

astro-ph.GA↗
arXiv2016

Kac's Walk on $n$-sphere mixes in $n\log n$ steps

Determining the mixing time of Kac's random walk on the sphere $\mathrm{S}^{n-1}$ is a long-standing open problem. We show that the total variation mixing time of Kac's walk on $\mathrm{S}^{n-1}$ is between $\frac{1}{2} \, n \log(n)$ and $200 \,n \log(n)$. Our bound is thus optimal up to a constant factor, improving on the best-known upper bound of $O(n^{5} \log(n)^{2})$ due to Jiang. Our main tool is a `non-Markovian' coupling recently introduced by the second author for obtaining the convergence rates of certain high dimensional Gibbs samplers in continuous state spaces.

math.PR↗
arXiv2016

The Use of a Single Pseudo-Sample in Approximate Bayesian Computation

We analyze the computational efficiency of approximate Bayesian computation (ABC), which approximates a likelihood function by drawing pseudo-samples from the associated model. For the rejection sampling version of ABC, it is known that multiple pseudo-samples cannot substantially increase (and can substantially decrease) the efficiency of the algorithm as compared to employing a high-variance estimate based on a single pseudo-sample. We show that this conclusion also holds for a Markov chain Monte Carlo version of ABC, implying that it is unnecessary to tune the number of pseudo-samples used in ABC-MCMC. This conclusion is in contrast to particle MCMC methods, for which increasing the number of particles can provide large gains in computational efficiency.

stat.CO↗
arXiv2016

Elementary Bounds On Mixing Times for Decomposable Markov Chains

Many finite-state reversible Markov chains can be naturally decomposed into "projection" and "restriction" chains. In this paper we provide bounds on the total variation mixing times of the original chain in terms of the mixing properties of these related chains. This paper is in the tradition of existing bounds on Poincare and log-Sobolev constants of Markov chains in terms of similar decompositions [JSTV04, MR02, MR06, MY09]. Our proofs are simple, relying largely on recent results relating hitting and mixing times of reversible Markov chains [PS13, Oli12]. We describe situations in which our results give substantially better bounds than those obtained by applying existing decomposition results and provide examples for illustration.

math.PR↗
arXiv2015

Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations

We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis-Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler's exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the \emph{exact} posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this paper: when the likelihood has some \emph{local} regularity, the number of model evaluations per MCMC step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ODE and PDE inference problems, with both synthetic and real data.

stat.ME↗
arXiv2015

Ergodicity of Approximate MCMC Chains with Applications to Large Data Sets

In many modern applications, difficulty in evaluating the posterior density makes performing even a single MCMC step slow. This difficulty can be caused by intractable likelihood functions, but also appears for routine problems with large data sets. Many researchers have responded by running approximate versions of MCMC algorithms. In this note, we develop quantitative bounds for showing the ergodicity of these approximate samplers. We then use these bounds to study the bias-variance trade-off of approximate MCMC algorithms. We apply our results to simple versions of recently proposed algorithms, including a variant of the "austerity" framework of Korratikara et al.

math.ST↗
arXiv2015

The Lyman-alpha signature of the first galaxies

We present the Cosmic Lyman-$α$ Transfer code (COLT), a massively parallel Monte-Carlo radiative transfer code, to simulate Lyman-$α$ (Ly$α$) resonant scattering through neutral hydrogen as a probe of the first galaxies. We explore the interaction of centrally produced Ly$α$ radiation with the host galactic environment. Ly$α$ photons emitted from the luminous starburst region escape with characteristic features in the line profile depending on the density distribution, ionization structure, and bulk velocity fields. For example, anisotropic ionization exhibits a tall peak close to line centre with a skewed tail that drops off gradually. Idealized models of first galaxies explore the effect of mass, anisotropic H II regions, and radiation pressure driven winds on Ly$α$ observables. We employ mesh refinement to resolve critical structures. We also post-process an ab initio cosmological simulation and examine images captured at various escape distances within the 1 Mpc$^3$ comoving volume. Finally, we discuss the emergent spectra and surface brightness profiles of these objects in the context of high-$z$ observations. The first galaxies will likely be observed through the red damping wing of the Ly$α$ line. Observations will be biased toward galaxies with an intrinsic red peak located far from line centre that reside in extensive H II super bubbles, which allows Hubble flow to sufficiently redshift photons away from line centre and facilitate transmission through the intergalactic medium (IGM). Even with gravitational lensing to boost the luminosity this preliminary work indicates that Ly$α$ emission from stellar clusters within haloes of $M_{\rm vir}<10^9~{\rm M}_\odot$ is generally too faint to be detected by the James Webb Space Telescope (JWST).

astro-ph.CO↗
arXiv2015

Mixing times for a constrained Ising process on the torus at low density

We study a kinetically constrained Ising process (KCIP) associated with a graph G and density parameter p; this process is an interacting particle system with state space $\{0,1\}^{G}$. The stationary distribution of the KCIP Markov chain is the Binomial($|G|, p$) distribution on the number of particles, conditioned on having at least one particle. The `constraint' in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state `1'. The KCIP has been proposed by statistical physicists as a model for the glass transition, and more recently as a simple algorithm for data storage in computer networks. In this note, we study the mixing time of this process on the torus $G = \mathbb{Z}_{L}^{d}$, $d \geq 3$, in the low-density regime $p = \frac{c}{n}$ for arbitrary $0 < c < 1$; this regime is the subject of a conjecture of Aldous and is natural in the context of computer networks. Our results provide a counterexample to Aldous' conjecture, suggest a natural modifcation of the conjecture, and show that this modifcation is correct up to logarithmic factors. The methods developed in this paper also provide a strategy for tackling Aldous' conjecture for other graphs.

math.PR↗
arXiv2014

Finite Sample Properties of Adaptive Markov Chains via Curvature

Adaptive Markov chains are an important class of Monte Carlo methods for sampling from probability distributions. The time evolution of adaptive algorithms depends on past samples, and thus these algorithms are non-Markovian. Although there has been previous work establishing conditions for their ergodicity, not much is known theoretically about their finite sample properties. In this paper, using a notion of discrete Ricci curvature for Markov kernels introduced by Ollivier, we establish concentration inequalities and finite sample bounds for a class of adaptive Markov chains. After establishing some general results, we give quantitative bounds for `multi-level' adaptive algorithms such as the equi-energy sampler. We also provide the first rigorous proofs that the finite sample properties of an equi-energy sampler are superior to those of related parallel tempering and Metropolis-Hastings samplers after a learning period comparable to their mixing times.

math.PR↗
arXiv2014

A Gibbs sampler on the $n$-simplex

We determine the mixing time of a simple Gibbs sampler on the unit simplex, confirming a conjecture of Aldous. The upper bound is based on a two-step coupling, where the first step is a simple contraction argument and the second step is a non-Markovian coupling. We also present a MCMC-based perfect sampling algorithm based on our proof which can be applied with Gibbs samplers that are harder to analyze.

math.PR↗
arXiv2013

Comparison Theory for Markov Chains on Different State Spaces and Application to Random Walk on Derangements

Let $X_{t}$ and $Y_{t}$ be two Markov chains, on state spaces $Ω\subset \hatΩ$. In this paper, we discuss how to prove bounds on the spectrum of $X_{t}$ based on bounds on the spectrum of $Y_{t}$. This generalizes work of Diaconis, Saloff-Coste, Yuen and others on comparison of chains in the case $Ω= \hatΩ$. The main tool is the extension of functions from the smaller space to the larger, which allows comparison of the entire spectrum of the two chains. The theory is used to give quick analyses of several chains without symmetry. The main application is to a `random transposition' walk on derangements.

math.PR↗
arXiv2012

The Cutoff Phenomenon for Random Birth and Death Chains

For any distribution $π$ with support equal to $[n] = \{1, 2,..., n \}$, we study the set $\mathcal{A}_π$ of tridiagonal stochastic matrices $K$ satisfying $π(i) K[i,j] = π(j) K[j,i]$ for all $i, j \in [n]$. These matrices correspond to birth and death chains with stationary distribution $π$. We study matrices $K$ drawn uniformly from $\mathcal{A}_π$, following the work of Diaconis and Wood on the case $π(i) = \frac{1}{n}$. We analyze a `block sampler' version of their algorithm for drawing from $\mathcal{A}_π$ at random, and use results from this analysis to draw conclusions about typical matrices. The main result is a soft argument comparing cutoff for sequences of random birth and death chains to cutoff for a special family of birth and death chains with the same stationary distributions.

math.PR↗
arXiv2012

The Impact of Assuming Flatness in the Determination of Neutrino Properties from Cosmological Data

Cosmological data have provided new constraints on the number of neutrino species and the neutrino mass. However these constraints depend on assumptions related to the underlying cosmology. Since a correlation is expected between the number of effective neutrinos N_{eff}, the neutrino mass \sum m_ν, and the curvature of the universe Ω_k, it is useful to investigate the current constraints in the framework of a non-flat universe. In this paper we update the constraints on neutrino parameters by making use of the latest cosmic microwave background (CMB) data from the ACT and SPT experiments and consider the possibility of a universe with non-zero curvature. We first place new constraints on N_{eff} and Ω_k, with N_{eff} = 4.03 +/- 0.45 and 10^3 Ω_k = -4.46 +/- 5.24. Thus, even when Ω_k is allowed to vary, N_{eff} = 3 is still disfavored with 95% confidence. We then investigate the correlation between neutrino mass and curvature that shifts the 95% upper limit of \sum m_ν< 0.45 eV to \sum m_ν< 0.95 eV. Thus, the impact of assuming flatness in neutrino cosmology is significant and an essential consideration with future experiments.

astro-ph.CO↗
arXiv2011

Three-Axis Measurement and Cancellation of Background Magnetic Fields to less than 50 uG in a Cold Atom Experiment

Many experiments involving cold and ultracold atomic gases require very precise control of magnetic fields that couple to and drive the atomic spins. Examples include quantum control of atomic spins, quantum control and quantum simulation in optical lattices, and studies of spinor Bose condensates. This makes accurate cancellation of the (generally time dependent) background magnetic field a critical factor in such experiments. We describe a technique that uses the atomic spins themselves to measure DC and AC components of the background field independently along three orthogonal axes, with a resolution of a few tens of uG in a bandwidth of ~1 kHz. Once measured, the background field can be cancelled with three pairs of compensating coils driven by arbitrary waveform generators. In our laboratory, the magnetic field environment is sufficiently stable for the procedure to reduce the field along each axis to less than ~50 uG rms, corresponding to a suppression of the AC part by about one order of magnitude. This suggests our approach can provide access to a new low-field regime in cold-atom experiments.

physics.atom-ph↗