16,921 research outputs found
Parameter-Dependent Lyapunov Functions for Linear Systems With Constant Uncertainties
Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties
A Geometric View on Constrained M-Estimators
We study the estimation error of constrained M-estimators, and derive
explicit upper bounds on the expected estimation error determined by the
Gaussian width of the constraint set. Both of the cases where the true
parameter is on the boundary of the constraint set (matched constraint), and
where the true parameter is strictly in the constraint set (mismatched
constraint) are considered. For both cases, we derive novel universal
estimation error bounds for regression in a generalized linear model with the
canonical link function. Our error bound for the mismatched constraint case is
minimax optimal in terms of its dependence on the sample size, for Gaussian
linear regression by the Lasso
Nonparametric estimation of the fragmentation kernel based on a PDE stationary distribution approximation
We consider a stochastic individual-based model in continuous time to
describe a size-structured population for cell divisions. This model is
motivated by the detection of cellular aging in biology. We address here the
problem of nonparametric estimation of the kernel ruling the divisions based on
the eigenvalue problem related to the asymptotic behavior in large population.
This inverse problem involves a multiplicative deconvolution operator. Using
Fourier technics we derive a nonparametric estimator whose consistency is
studied. The main difficulty comes from the non-standard equations connecting
the Fourier transforms of the kernel and the parameters of the model. A
numerical study is carried out and we pay special attention to the derivation
of bandwidths by using resampling
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
- …