Being Robust (in High Dimensions) Can Be Practical

Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.Comment: Appeared in ICML 201

Relating vesicle shapes in pyroclasts to eruption styles

Vesicles in pyroclasts provide a direct record of conduit conditions during explosive volcanic eruptions. Although their numbers and sizes are used routinely to infer aspects of eruption dynamics, vesicle shape remains an underutilized parameter. We have quantified vesicle shapes in pyroclasts from fall deposits of seven explosive eruptions of different styles, using the dimensionless shape factor , a measure of the degree of complexity of the bounding surface of an object. For each of the seven eruptions, we have also estimated the capillary number, Ca, from the magma expansion velocity through coupled diffusive bubble growth and conduit flow modeling. We find that Ω is smaller for eruptions with Ca 1 than for eruptions with Ca 1. Consistent with previous studies, we interpret these results as an expression of the relative importance of structural changes during magma decompression and bubble growth, such as coalescence and shape relaxation of bubbles by capillary stresses. Among the samples analyzed, Strombolian and Hawaiian fire-fountain eruptions have Ca 1, in contrast to Vulcanian, Plinian, and ultraplinian eruptions. Interestingly, the basaltic Plinian eruptions of Tarawera volcano, New Zealand in 1886 and Mt. Etna, Italy in 122 BC, for which the cause of intense explosive activity has been controversial, are also characterized by Ca 1 and larger values of Ω than Strombolian and Hawaiian style (fire fountain) eruptions. We interpret this to be the consequence of syn-eruptive magma crystallization, resulting in high magma viscosity and reduced rates of bubble growth. Our model results indicate that during these basaltic Plinian eruptions, buildup of bubble overpressure resulted in brittle magma fragmentation.National Science Foundation EAR-1019872National Science Foundation EAR-081033

Parallel software tools at Langley Research Center

This document gives a brief overview of parallel software tools available on the Intel iPSC/860 parallel computer at Langley Research Center. It is intended to provide a source of information that is somewhat more concise than vendor-supplied material on the purpose and use of various tools. Each of the chapters on tools is organized in a similar manner covering an overview of the functionality, access information, how to effectively use the tool, observations about the tool and how it compares to similar software, known problems or shortfalls with the software, and reference documentation. It is primarily intended for users of the iPSC/860 at Langley Research Center and is appropriate for both the experienced and novice user

Extended states in 1D lattices: application to quasiperiodic copper-mean chain

The question of the conditions under which 1D systems support extended electronic eigenstates is addressed in a very general context. Using real space renormalisation group arguments we discuss the precise criteria for determining the entire spertrum of extended eigenstates and the corresponding eigenfunctions in disordered as well as quasiperiodic systems. For purposes of illustration we calculate a few selected eigenvalues and the corresponding extended eigenfunctions for the quasiperiodic copper-mean chain. So far, for the infinite copper-mean chain, only a single energy has been numerically shown to support an extended eigenstate [ You et al. (1991)] : we show analytically that there is in fact an infinite number of extended eigenstates in this lattice which form fragmented minibands.Comment: 10 pages + 2 figures available on request; LaTeX version 2.0

INFOPHARE: Newsletter of the Phare Information Office. July 1995 Issue 8.

We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise | where an "-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error O(ϵ) in the total variation distance, which is optimal up to a universal constant that is independent of the dimension. In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of p 2 and the running time is polynomial in d and 1/ϵ. When both the mean and covariance are unknown, the running time is polynomial in d and quasipolynomial in 1/ϵ. Moreover all of our algorithms require only a polynomial number of samples. Our work shows that the same sorts of error guarantees that were established over fifty years ago in the one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings

Spawning rings of exceptional points out of Dirac cones

The Dirac cone underlies many unique electronic properties of graphene and topological insulators, and its band structure--two conical bands touching at a single point--has also been realized for photons in waveguide arrays, atoms in optical lattices, and through accidental degeneracy. Deformations of the Dirac cone often reveal intriguing properties; an example is the quantum Hall effect, where a constant magnetic field breaks the Dirac cone into isolated Landau levels. A seemingly unrelated phenomenon is the exceptional point, also known as the parity-time symmetry breaking point, where two resonances coincide in both their positions and widths. Exceptional points lead to counter-intuitive phenomena such as loss-induced transparency, unidirectional transmission or reflection, and lasers with reversed pump dependence or single-mode operation. These two fields of research are in fact connected: here we discover the ability of a Dirac cone to evolve into a ring of exceptional points, which we call an "exceptional ring." We experimentally demonstrate this concept in a photonic crystal slab. Angle-resolved reflection measurements of the photonic crystal slab reveal that the peaks of reflectivity follow the conical band structure of a Dirac cone from accidental degeneracy, whereas the complex eigenvalues of the system are deformed into a two-dimensional flat band enclosed by an exceptional ring. This deformation arises from the dissimilar radiation rates of dipole and quadrupole resonances, which play a role analogous to the loss and gain in parity-time symmetric systems. Our results indicate that the radiation that exists in any open system can fundamentally alter its physical properties in ways previously expected only in the presence of material loss and gain