43,292 research outputs found
Precise Phase Transition of Total Variation Minimization
Characterizing the phase transitions of convex optimizations in recovering
structured signals or data is of central importance in compressed sensing,
machine learning and statistics. The phase transitions of many convex
optimization signal recovery methods such as minimization and nuclear
norm minimization are well understood through recent years' research. However,
rigorously characterizing the phase transition of total variation (TV)
minimization in recovering sparse-gradient signal is still open. In this paper,
we fully characterize the phase transition curve of the TV minimization. Our
proof builds on Donoho, Johnstone and Montanari's conjectured phase transition
curve for the TV approximate message passing algorithm (AMP), together with the
linkage between the minmax Mean Square Error of a denoising problem and the
high-dimensional convex geometry for TV minimization.Comment: 6 page
-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?
This paper investigates the problem of signal estimation from undersampled
noisy sub-Gaussian measurements under the assumption of a cosparse model. Based
on generalized notions of sparsity, we derive novel recovery guarantees for the
-analysis basis pursuit, enabling highly accurate predictions of its
sample complexity. The corresponding bounds on the number of required
measurements do explicitly depend on the Gram matrix of the analysis operator
and therefore particularly account for its mutual coherence structure. Our
findings defy conventional wisdom which promotes the sparsity of analysis
coefficients as the crucial quantity to study. In fact, this common paradigm
breaks down completely in many situations of practical interest, for instance,
when applying a redundant (multilevel) frame as analysis prior. By extensive
numerical experiments, we demonstrate that, in contrast, our theoretical
sampling-rate bounds reliably capture the recovery capability of various
examples, such as redundant Haar wavelets systems, total variation, or random
frames. The proofs of our main results build upon recent achievements in the
convex geometry of data mining problems. More precisely, we establish a
sophisticated upper bound on the conic Gaussian mean width that is associated
with the underlying -analysis polytope. Due to a novel localization
argument, it turns out that the presented framework naturally extends to stable
recovery, allowing us to incorporate compressible coefficient sequences as
well
Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
We use the conformal bootstrap to perform a precision study of the operator
spectrum of the critical 3d Ising model. We conjecture that the 3d Ising
spectrum minimizes the central charge c in the space of unitary solutions to
crossing symmetry. Because extremal solutions to crossing symmetry are uniquely
determined, we are able to precisely reconstruct the first several Z2-even
operator dimensions and their OPE coefficients. We observe that a sharp
transition in the operator spectrum occurs at the 3d Ising dimension
Delta_sigma=0.518154(15), and find strong numerical evidence that operators
decouple from the spectrum as one approaches the 3d Ising point. We compare
this behavior to the analogous situation in 2d, where the disappearance of
operators can be understood in terms of degenerate Virasoro representations.Comment: 55 pages, many figures; v2 - refs and comments added, to appear in a
special issue of J.Stat.Phys. in memory of Kenneth Wilso
Disordered jammed packings of frictionless spheres
At low volume fraction, disordered arrangements of frictionless spheres are
found in un--jammed states unable to support applied stresses, while at high
volume fraction they are found in jammed states with mechanical strength. Here
we show, focusing on the hard sphere zero pressure limit, that the transition
between un-jammed and jammed states does not occur at a single value of the
volume fraction, but in a whole volume fraction range. This result is obtained
via the direct numerical construction of disordered jammed states with a volume
fraction varying between two limits, and . We identify these
limits with the random loose packing volume fraction \rl and the random close
packing volume fraction \rc of frictionless spheres, respectively
On the Error in Phase Transition Computations for Compressed Sensing
Evaluating the statistical dimension is a common tool to determine the
asymptotic phase transition in compressed sensing problems with Gaussian
ensemble. Unfortunately, the exact evaluation of the statistical dimension is
very difficult and it has become standard to replace it with an upper-bound. To
ensure that this technique is suitable, [1] has introduced an upper-bound on
the gap between the statistical dimension and its approximation. In this work,
we first show that the error bound in [1] in some low-dimensional models such
as total variation and analysis minimization becomes poorly large.
Next, we develop a new error bound which significantly improves the estimation
gap compared to [1]. In particular, unlike the bound in [1] that is not
applicable to settings with overcomplete dictionaries, our bound exhibits a
decaying behavior in such cases
Sharp Time--Data Tradeoffs for Linear Inverse Problems
In this paper we characterize sharp time-data tradeoffs for optimization
problems used for solving linear inverse problems. We focus on the minimization
of a least-squares objective subject to a constraint defined as the sub-level
set of a penalty function. We present a unified convergence analysis of the
gradient projection algorithm applied to such problems. We sharply characterize
the convergence rate associated with a wide variety of random measurement
ensembles in terms of the number of measurements and structural complexity of
the signal with respect to the chosen penalty function. The results apply to
both convex and nonconvex constraints, demonstrating that a linear convergence
rate is attainable even though the least squares objective is not strongly
convex in these settings. When specialized to Gaussian measurements our results
show that such linear convergence occurs when the number of measurements is
merely 4 times the minimal number required to recover the desired signal at all
(a.k.a. the phase transition). We also achieve a slower but geometric rate of
convergence precisely above the phase transition point. Extensive numerical
results suggest that the derived rates exactly match the empirical performance
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