582 research outputs found
New Uniform Bounds for Almost Lossless Analog Compression
Wu and Verd\'u developed a theory of almost lossless analog compression,
where one imposes various regularity conditions on the compressor and the
decompressor with the input signal being modelled by a (typically
infinite-entropy) stationary stochastic process. In this work we consider all
stationary stochastic processes with trajectories in a prescribed set
of (bi)infinite sequences and find
uniform lower and upper bounds for certain compression rates in terms of metric
mean dimension and mean box dimension. An essential tool is the recent
Lindenstrauss-Tsukamoto variational principle expressing metric mean dimension
in terms of rate-distortion functions.Comment: This paper is going to be presented at 2019 IEEE International
Symposium on Information Theory. It is a short version of arXiv:1812.0045
Lossless Linear Analog Compression
We establish the fundamental limits of lossless linear analog compression by
considering the recovery of random vectors
from the noiseless linear
measurements
with
measurement matrix . Specifically,
for a random vector of arbitrary
distribution we show that can be recovered with
zero error probability from
linear measurements,
where denotes the lower
modified Minkowski dimension and the infimum is over all sets
with . This achievability statement holds for Lebesgue almost all measurement
matrices . We then show that -rectifiable random vectors---a
stochastic generalization of -sparse vectors---can be recovered with zero
error probability from linear measurements. From classical compressed
sensing theory we would expect to be necessary for successful
recovery of . Surprisingly, certain classes of
-rectifiable random vectors can be recovered from fewer than
measurements. Imposing an additional regularity condition on the distribution
of -rectifiable random vectors , we do get the
expected converse result of measurements being necessary. The resulting
class of random vectors appears to be new and will be referred to as
-analytic random vectors
Compression-Based Compressed Sensing
Modern compression algorithms exploit complex structures that are present in
signals to describe them very efficiently. On the other hand, the field of
compressed sensing is built upon the observation that "structured" signals can
be recovered from their under-determined set of linear projections. Currently,
there is a large gap between the complexity of the structures studied in the
area of compressed sensing and those employed by the state-of-the-art
compression codes. Recent results in the literature on deterministic signals
aim at bridging this gap through devising compressed sensing decoders that
employ compression codes. This paper focuses on structured stochastic processes
and studies the application of rate-distortion codes to compressed sensing of
such signals. The performance of the formerly-proposed compressible signal
pursuit (CSP) algorithm is studied in this stochastic setting. It is proved
that in the very low distortion regime, as the blocklength grows to infinity,
the CSP algorithm reliably and robustly recovers instances of a stationary
process from random linear projections as long as their count is slightly more
than times the rate-distortion dimension (RDD) of the source. It is also
shown that under some regularity conditions, the RDD of a stationary process is
equal to its information dimension (ID). This connection establishes the
optimality of the CSP algorithm at least for memoryless stationary sources, for
which the fundamental limits are known. Finally, it is shown that the CSP
algorithm combined by a family of universal variable-length fixed-distortion
compression codes yields a family of universal compressed sensing recovery
algorithms
Polarization of the Renyi Information Dimension with Applications to Compressed Sensing
In this paper, we show that the Hadamard matrix acts as an extractor over the
reals of the Renyi information dimension (RID), in an analogous way to how it
acts as an extractor of the discrete entropy over finite fields. More
precisely, we prove that the RID of an i.i.d. sequence of mixture random
variables polarizes to the extremal values of 0 and 1 (corresponding to
discrete and continuous distributions) when transformed by a Hadamard matrix.
Further, we prove that the polarization pattern of the RID admits a closed form
expression and follows exactly the Binary Erasure Channel (BEC) polarization
pattern in the discrete setting. We also extend the results from the single- to
the multi-terminal setting, obtaining a Slepian-Wolf counterpart of the RID
polarization. We discuss applications of the RID polarization to Compressed
Sensing of i.i.d. sources. In particular, we use the RID polarization to
construct a family of deterministic -valued sensing matrices for
Compressed Sensing. We run numerical simulations to compare the performance of
the resulting matrices with that of random Gaussian and random Hadamard
matrices. The results indicate that the proposed matrices afford competitive
performances while being explicitly constructed.Comment: 12 pages, 2 figure
Metric mean dimension and analog compression
Wu and Verd\'u developed a theory of almost lossless analog compression,
where one imposes various regularity conditions on the compressor and the
decompressor with the input signal being modelled by a (typically
infinite-entropy) stationary stochastic process. In this work we consider all
stationary stochastic processes with trajectories in a prescribed set of
(bi-)infinite sequences and find uniform lower and upper bounds for certain
compression rates in terms of metric mean dimension and mean box dimension. An
essential tool is the recent Lindenstrauss-Tsukamoto variational principle
expressing metric mean dimension in terms of rate-distortion functions. We
obtain also lower bounds on compression rates for a fixed stationary process in
terms of the rate-distortion dimension rates and study several examples.Comment: v3: Accepted for publication in IEEE Transactions on Information
Theory. Additional examples were added. Material have been reorganized (with
some parts removed). Minor mistakes were correcte
Lossless Analog Compression
We establish the fundamental limits of lossless analog compression by
considering the recovery of arbitrary m-dimensional real random vectors x from
the noiseless linear measurements y=Ax with n x m measurement matrix A. Our
theory is inspired by the groundbreaking work of Wu and Verdu (2010) on almost
lossless analog compression, but applies to the nonasymptotic, i.e., fixed-m
case, and considers zero error probability. Specifically, our achievability
result states that, for almost all A, the random vector x can be recovered with
zero error probability provided that n > K(x), where K(x) is given by the
infimum of the lower modified Minkowski dimension over all support sets U of x.
We then particularize this achievability result to the class of s-rectifiable
random vectors as introduced in Koliander et al. (2016); these are random
vectors of absolutely continuous distribution---with respect to the
s-dimensional Hausdorff measure---supported on countable unions of
s-dimensional differentiable submanifolds of the m-dimensional real coordinate
space. Countable unions of differentiable submanifolds include essentially all
signal models used in the compressed sensing literature. Specifically, we prove
that, for almost all A, s-rectifiable random vectors x can be recovered with
zero error probability from n>s linear measurements. This threshold is,
however, found not to be tight as exemplified by the construction of an
s-rectifiable random vector that can be recovered with zero error probability
from n<s linear measurements. This leads us to the introduction of the new
class of s-analytic random vectors, which admit a strong converse in the sense
of n greater than or equal to s being necessary for recovery with probability
of error smaller than one. The central conceptual tools in the development of
our theory are geometric measure theory and the theory of real analytic
functions
Achieving the Fundamental Limit of Lossless Analog Compression via Polarization
In this paper, we study the lossless analog compression for i.i.d.
nonsingular signals via the polarization-based framework. We prove that for
nonsingular source, the error probability of maximum a posteriori (MAP)
estimation polarizes under the Hadamard transform, which extends the
polarization phenomenon to analog domain. Building on this insight, we propose
partial Hadamard compression and develop the corresponding analog successive
cancellation (SC) decoder. The proposed scheme consists of deterministic
measurement matrices and non-iterative reconstruction algorithm, providing
benefits in both space and computational complexity. Using the polarization of
error probability, we prove that our approach achieves the
information-theoretical limit for lossless analog compression developed by Wu
and Verdu.Comment: 48 pages, 5 figures. This work was presented in part at the 2023 IEEE
Global Communications Conferenc
Worst-Case Analysis of Electrical and Electronic Equipment via Affine Arithmetic
In the design and fabrication process of electronic equipment, there are many unkown parameters which significantly affect the product performance. Some uncertainties are due to manufacturing process fluctuations, while others due to the environment such as operating temperature, voltage, and various ambient aging stressors. It is desirable to consider these uncertainties to ensure product performance, improve yield, and reduce design cost. Since direct electromagnetic compatibility measurements impact on both cost and time-to-market, there has been a growing demand for the availability of tools enabling the simulation of electrical and electronic equipment with the inclusion of the effects of system uncertainties.
In this framework, the assessment of device response is no longer regarded as deterministic but as a random process. It is traditionally analyzed using the Monte Carlo or other sampling-based methods. The drawback of the above methods is large number of required samples to converge, which are time-consuming for practical applications. As an alternative, the inherent worst-case approaches such as interval analysis directly provide an estimation of the true bounds of the responses. However, such approaches might provide unnecessarily strict margins, which are very unlikely to occur. A recent technique, affine arithmetic, advances the interval based methods by means of handling correlated intervals. However, it still leads to over-conservatism due to the inability of considering probability information.
The objective of this thesis is to improve the accuracy of the affine arithmetic and broaden its application in frequency-domain analysis. We first extend the existing literature results to the efficient time-domain analysis of lumped circuits considering the uncertainties. Then we provide an extension of the basic affine arithmetic to the frequency-domain simulation of circuits. Classical tools for circuit analysis are used within a modified affine framework accounting for complex algebra and uncertainty interval partitioning for the accurate and efficient computation of the worst case bounds of the responses of both lumped and distributed circuits.
The performance of the proposed approach is investigated through extensive simulations in several case studies. The simulation results are compared with the Monte Carlo method in terms of both simulation time and accuracy
Supernova / Acceleration Probe: A Satellite Experiment to Study the Nature of the Dark Energy
The Supernova / Acceleration Probe (SNAP) is a proposed space-based
experiment designed to study the dark energy and alternative explanations of
the acceleration of the Universe's expansion by performing a series of
complementary systematics-controlled measurements. We describe a
self-consistent reference mission design for building a Type Ia supernova
Hubble diagram and for performing a wide-area weak gravitational lensing study.
A 2-m wide-field telescope feeds a focal plane consisting of a 0.7
square-degree imager tiled with equal areas of optical CCDs and near infrared
sensors, and a high-efficiency low-resolution integral field spectrograph. The
SNAP mission will obtain high-signal-to-noise calibrated light-curves and
spectra for several thousand supernovae at redshifts between z=0.1 and 1.7. A
wide-field survey covering one thousand square degrees resolves ~100 galaxies
per square arcminute. If we assume we live in a cosmological-constant-dominated
Universe, the matter density, dark energy density, and flatness of space can
all be measured with SNAP supernova and weak-lensing measurements to a
systematics-limited accuracy of 1%. For a flat universe, the
density-to-pressure ratio of dark energy can be similarly measured to 5% for
the present value w0 and ~0.1 for the time variation w'. The large survey area,
depth, spatial resolution, time-sampling, and nine-band optical to NIR
photometry will support additional independent and/or complementary dark-energy
measurement approaches as well as a broad range of auxiliary science programs.
(Abridged)Comment: 40 pages, 18 figures, submitted to PASP, http://snap.lbl.go
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