11,214 research outputs found
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions
We develop a robust uncertainty principle for finite signals in C^N which
states that for almost all subsets T,W of {0,...,N-1} such that |T|+|W| ~ (log
N)^(-1/2) N, there is no sigal f supported on T whose discrete Fourier
transform is supported on W. In fact, we can make the above uncertainty
principle quantitative in the sense that if f is supported on T, then only a
small percentage of the energy (less than half, say) of its Fourier transform
is concentrated on W.
As an application of this robust uncertainty principle (QRUP), we consider
the problem of decomposing a signal into a sparse superposition of spikes and
complex sinusoids. We show that if a generic signal f has a decomposition using
spike and frequency locations in T and W respectively, and obeying |T| + |W| <=
C (\log N)^{-1/2} N, then this is the unique sparsest possible decomposition
(all other decompositions have more non-zero terms). In addition, if |T| + |W|
<= C (\log N)^{-1} N, then this sparsest decomposition can be found by solving
a convex optimization problem.Comment: 25 pages, 9 figure
Stellar Kinematics of the Double Nucleus of M31
We report observations of the double nucleus of M31 with the f/48 long-slit
spectrograph of the HST Faint Object Camera. We obtain a total exposure of
19,000 sec. over 7 orbits, with the 0.063-arcsec-wide slit along the line
between the two brightness peaks (PA 42). A spectrum of Jupiter is used as a
spectral template. The rotation curve is resolved, and reaches a maximum
amplitude of ~250 km/s roughly 0.3 arcsec either side of a rotation center
lying between P1 and P2, 0.16 +/- 0.05 arcsec from the optically fainter P2. We
find the velocity dispersion to be < 250 km/s everywhere except for a narrow
``dispersion spike'', centered 0.06 +/- 0.03 arcsec on the anti-P1 side of P2,
in which sigma peaks at 440 +/- 70 km/s. At much lower confidence, we see local
disturbances to the rotation curve at P1 and P2, and an elevation in sigma at
P1. At very low significance we detect a weak asymmetry in the line-of-sight
velocity distribution opposite to the sense usually encountered. Convolving our
V and sigma profiles to CFHT resolution, we find good agreement with the
results of Kormendy & Bender (1998, preprint), though there is a 20%
discrepancy in the dispersion that cannot be attributed to the dispersion
spike. Our results are not consistent with the location of the maximum
dispersion as found by Bacon et al. We find that the sinking star cluster model
of Emsellem & Combes (1997) does not reproduce either the rotation curve or the
dispersion profile. The eccentric disk model of Tremaine (1995) fares better,
and can be improved somewhat by adjusting the original parameters. However,
detailed modeling will require dynamical models of significantly greater
realism.Comment: 29 pages, Latex, AASTeX v4.0, with 7 eps figures. To appear in The
Astronomical Journal, February 199
When do correlations increase with firing rates in recurrent networks?
A central question in neuroscience is to understand how noisy firing patterns are used to transmit information. Because neural spiking is noisy, spiking patterns are often quantified via pairwise correlations, or the probability that two cells will spike coincidentally, above and beyond their baseline firing rate. One observation frequently made in experiments, is that correlations can increase systematically with firing rate. Theoretical studies have determined that stimulus-dependent correlations that increase with firing rate can have beneficial effects on information coding; however, we still have an incomplete understanding of what circuit mechanisms do, or do not, produce this correlation-firing rate relationship. Here, we studied the relationship between pairwise correlations and firing rates in recurrently coupled excitatory-inhibitory spiking networks with conductance-based synapses. We found that with stronger excitatory coupling, a positive relationship emerged between pairwise correlations and firing rates. To explain these findings, we used linear response theory to predict the full correlation matrix and to decompose correlations in terms of graph motifs. We then used this decomposition to explain why covariation of correlations with firing rate—a relationship previously explained in feedforward networks driven by correlated input—emerges in some recurrent networks but not in others. Furthermore, when correlations covary with firing rate, this relationship is reflected in low-rank structure in the correlation matrix
Synchrosqueezed Wave Packet Transforms and Diffeomorphism Based Spectral Analysis for 1D General Mode Decompositions
This paper develops new theory and algorithms for 1D general mode
decompositions. First, we introduce the 1D synchrosqueezed wave packet
transform and prove that it is able to estimate the instantaneous information
of well-separated modes from their superposition accurately. The
synchrosqueezed wave packet transform has a better resolution than the
synchrosqueezed wavelet transform in the time-frequency domain for separating
high frequency modes. Second, we present a new approach based on
diffeomorphisms for the spectral analysis of general shape functions. These two
methods lead to a framework for general mode decompositions under a weak
well-separation condition and a well different condition. Numerical examples of
synthetic and real data are provided to demonstrate the fruitful applications
of these methods.Comment: 39 page
Bell-shaped nonstationary refinable ripplets
We study the approximation properties of the class of nonstationary refinable
ripplets introduced in \cite{GP08}. These functions are solution of an infinite
set of nonstationary refinable equations and are defined through sequences of
scaling masks that have an explicit expression. Moreover, they are
variation-diminishing and highly localized in the scale-time plane, properties
that make them particularly attractive in applications. Here, we prove that
they enjoy Strang-Fix conditions and convolution and differentiation rules and
that they are bell-shaped. Then, we construct the corresponding minimally
supported nonstationary prewavelets and give an iterative algorithm to evaluate
the prewavelet masks. Finally, we give a procedure to construct the associated
nonstationary biorthogonal bases and filters to be used in efficient
decomposition and reconstruction algorithms. As an example, we calculate the
prewavelet masks and the nonstationary biorthogonal filter pairs corresponding
to the nonstationary scaling functions in the class and construct the
corresponding prewavelets and biorthogonal bases. A simple test showing their
good performances in the analysis of a spike-like signal is also presented.
Keywords: total positivity, variation-dimishing, refinable ripplet, bell-shaped
function, nonstationary prewavelet, nonstationary biorthogonal basisComment: 30 pages, 10 figure
Data-driven multivariate and multiscale methods for brain computer interface
This thesis focuses on the development of data-driven multivariate and multiscale methods
for brain computer interface (BCI) systems. The electroencephalogram (EEG), the
most convenient means to measure neurophysiological activity due to its noninvasive nature,
is mainly considered. The nonlinearity and nonstationarity inherent in EEG and its
multichannel recording nature require a new set of data-driven multivariate techniques to
estimate more accurately features for enhanced BCI operation. Also, a long term goal
is to enable an alternative EEG recording strategy for achieving long-term and portable
monitoring.
Empirical mode decomposition (EMD) and local mean decomposition (LMD), fully
data-driven adaptive tools, are considered to decompose the nonlinear and nonstationary
EEG signal into a set of components which are highly localised in time and frequency. It
is shown that the complex and multivariate extensions of EMD, which can exploit common
oscillatory modes within multivariate (multichannel) data, can be used to accurately
estimate and compare the amplitude and phase information among multiple sources, a
key for the feature extraction of BCI system. A complex extension of local mean decomposition
is also introduced and its operation is illustrated on two channel neuronal
spike streams. Common spatial pattern (CSP), a standard feature extraction technique
for BCI application, is also extended to complex domain using the augmented complex
statistics. Depending on the circularity/noncircularity of a complex signal, one of the
complex CSP algorithms can be chosen to produce the best classification performance
between two different EEG classes.
Using these complex and multivariate algorithms, two cognitive brain studies are
investigated for more natural and intuitive design of advanced BCI systems. Firstly, a Yarbus-style auditory selective attention experiment is introduced to measure the user
attention to a sound source among a mixture of sound stimuli, which is aimed at improving
the usefulness of hearing instruments such as hearing aid. Secondly, emotion experiments
elicited by taste and taste recall are examined to determine the pleasure and displeasure
of a food for the implementation of affective computing. The separation between two
emotional responses is examined using real and complex-valued common spatial pattern
methods.
Finally, we introduce a novel approach to brain monitoring based on EEG recordings
from within the ear canal, embedded on a custom made hearing aid earplug. The new
platform promises the possibility of both short- and long-term continuous use for standard
brain monitoring and interfacing applications
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