18,575 research outputs found
Statistics for the Luria-Delbr\"uck distribution
The Luria-Delbr\"uck distribution is a classical model of mutations in cell
kinetics. It is obtained as a limit when the probability of mutation tends to
zero and the number of divisions to infinity. It can be interpreted as a
compound Poisson distribution (for the number of mutations) of exponential
mixtures (for the developing time of mutant clones) of geometric distributions
(for the number of cells produced by a mutant clone in a given time). The
probabilistic interpretation, and a rigourous proof of convergence in the
general case, are deduced from classical results on Bellman-Harris branching
processes. The two parameters of the Luria-Delbr\"uck distribution are the
expected number of mutations, which is the parameter of interest, and the
relative fitness of normal cells compared to mutants, which is the heavy tail
exponent. Both can be simultaneously estimated by the maximum likehood method.
However, the computation becomes numerically unstable as soon as the maximal
value of the sample is large, which occurs frequently due to the heavy tail
property. Based on the empirical generating function, robust estimators are
proposed and their asymptotic variance is given. They are comparable in
precision to maximum likelihood estimators, with a much broader range of
calculability, a better numerical stability, and a negligible computing time
Tail bounds for all eigenvalues of a sum of random matrices
This work introduces the minimax Laplace transform method, a modification of
the cumulant-based matrix Laplace transform method developed in "User-friendly
tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both
upper and lower bounds on each eigenvalue of a sum of random self-adjoint
matrices. This machinery is used to derive eigenvalue analogues of the
classical Chernoff, Bennett, and Bernstein bounds.
Two examples demonstrate the efficacy of the minimax Laplace transform. The
first concerns the effects of column sparsification on the spectrum of a matrix
with orthonormal rows. Here, the behavior of the singular values can be
described in terms of coherence-like quantities. The second example addresses
the question of relative accuracy in the estimation of eigenvalues of the
covariance matrix of a random process. Standard results on the convergence of
sample covariance matrices provide bounds on the number of samples needed to
obtain relative accuracy in the spectral norm, but these results only guarantee
relative accuracy in the estimate of the maximum eigenvalue. The minimax
Laplace transform argument establishes that if the lowest eigenvalues decay
sufficiently fast, on the order of (K^2*r*log(p))/eps^2 samples, where K is the
condition number of an optimal rank-r approximation to C, are sufficient to
ensure that the dominant r eigenvalues of the covariance matrix of a N(0, C)
random vector are estimated to within a factor of 1+-eps with high probability.Comment: 20 pages, 1 figure, see also arXiv:1004.4389v
Statistical estimation of a growth-fragmentation model observed on a genealogical tree
We model the growth of a cell population by a piecewise deterministic Markov
branching tree. Each cell splits into two offsprings at a division rate
that depends on its size . The size of each cell grows exponentially in
time, at a rate that varies for each individual. We show that the mean
empirical measure of the model satisfies a growth-fragmentation type equation
if structured in both size and growth rate as state variables. We construct a
nonparametric estimator of the division rate based on the observation of
the population over different sampling schemes of size on the genealogical
tree. Our estimator nearly achieves the rate in squared-loss
error asymptotically. When the growth rate is assumed to be identical for every
cell, we retrieve the classical growth-fragmentation model and our estimator
improves on the rate obtained in \cite{DHRR, DPZ} through
indirect observation schemes. Our method is consistently tested numerically and
implemented on {\it Escherichia coli} data.Comment: 46 pages, 4 figure
Asynchronous Distributed Optimization over Lossy Networks via Relaxed ADMM: Stability and Linear Convergence
In this work we focus on the problem of minimizing the sum of convex cost
functions in a distributed fashion over a peer-to-peer network. In particular,
we are interested in the case in which communications between nodes are prone
to failures and the agents are not synchronized among themselves. We address
the problem proposing a modified version of the relaxed ADMM, which corresponds
to the Peaceman-Rachford splitting method applied to the dual. By exploiting
results from operator theory, we are able to prove the almost sure convergence
of the proposed algorithm under general assumptions on the distribution of
communication loss and node activation events. By further assuming the cost
functions to be strongly convex, we prove the linear convergence of the
algorithm in mean to a neighborhood of the optimal solution, and provide an
upper bound to the convergence rate. Finally, we present numerical results
testing the proposed method in different scenarios.Comment: To appear in IEEE Transactions on Automatic Contro
Fast Non-Parametric Learning to Accelerate Mixed-Integer Programming for Online Hybrid Model Predictive Control
Today's fast linear algebra and numerical optimization tools have pushed the
frontier of model predictive control (MPC) forward, to the efficient control of
highly nonlinear and hybrid systems. The field of hybrid MPC has demonstrated
that exact optimal control law can be computed, e.g., by mixed-integer
programming (MIP) under piecewise-affine (PWA) system models. Despite the
elegant theory, online solving hybrid MPC is still out of reach for many
applications. We aim to speed up MIP by combining geometric insights from
hybrid MPC, a simple-yet-effective learning algorithm, and MIP warm start
techniques. Following a line of work in approximate explicit MPC, the proposed
learning-control algorithm, LNMS, gains computational advantage over MIP at
little cost and is straightforward for practitioners to implement
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