11,477 research outputs found
A Molecular Implementation of the Least Mean Squares Estimator
In order to function reliably, synthetic molecular circuits require
mechanisms that allow them to adapt to environmental disturbances. Least mean
squares (LMS) schemes, such as commonly encountered in signal processing and
control, provide a powerful means to accomplish that goal. In this paper we
show how the traditional LMS algorithm can be implemented at the molecular
level using only a few elementary biomolecular reactions. We demonstrate our
approach using several simulation studies and discuss its relevance to
synthetic biology.Comment: Molecular circuits, synthetic biology, least mean squares estimator,
adaptive system
Structural Variability from Noisy Tomographic Projections
In cryo-electron microscopy, the 3D electric potentials of an ensemble of
molecules are projected along arbitrary viewing directions to yield noisy 2D
images. The volume maps representing these potentials typically exhibit a great
deal of structural variability, which is described by their 3D covariance
matrix. Typically, this covariance matrix is approximately low-rank and can be
used to cluster the volumes or estimate the intrinsic geometry of the
conformation space. We formulate the estimation of this covariance matrix as a
linear inverse problem, yielding a consistent least-squares estimator. For
images of size -by- pixels, we propose an algorithm for calculating this
covariance estimator with computational complexity
, where the condition number
is empirically in the range --. Its efficiency relies on the
observation that the normal equations are equivalent to a deconvolution problem
in 6D. This is then solved by the conjugate gradient method with an appropriate
circulant preconditioner. The result is the first computationally efficient
algorithm for consistent estimation of 3D covariance from noisy projections. It
also compares favorably in runtime with respect to previously proposed
non-consistent estimators. Motivated by the recent success of eigenvalue
shrinkage procedures for high-dimensional covariance matrices, we introduce a
shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We
evaluate our methods on simulated datasets and achieve classification results
comparable to state-of-the-art methods in shorter running time. We also present
results on clustering volumes in an experimental dataset, illustrating the
power of the proposed algorithm for practical determination of structural
variability.Comment: 52 pages, 11 figure
Lecture notes on ridge regression
The linear regression model cannot be fitted to high-dimensional data, as the
high-dimensionality brings about empirical non-identifiability. Penalized
regression overcomes this non-identifiability by augmentation of the loss
function by a penalty (i.e. a function of regression coefficients). The ridge
penalty is the sum of squared regression coefficients, giving rise to ridge
regression. Here many aspect of ridge regression are reviewed e.g. moments,
mean squared error, its equivalence to constrained estimation, and its relation
to Bayesian regression. Finally, its behaviour and use are illustrated in
simulation and on omics data. Subsequently, ridge regression is generalized to
allow for a more general penalty. The ridge penalization framework is then
translated to logistic regression and its properties are shown to carry over.
To contrast ridge penalized estimation, the final chapter introduces its lasso
counterpart
Zero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to Forces
A simple and stable method for computing accurate expectation values of
observable with Variational Monte Carlo (VMC) or Diffusion Monte Carlo (DMC)
algorithms is presented. The basic idea consists in replacing the usual
``bare'' estimator associated with the observable by an improved or
``renormalized'' estimator. Using this estimator more accurate averages are
obtained: Not only the statistical fluctuations are reduced but also the
systematic error (bias) associated with the approximate VMC or (fixed-node) DMC
probability densities. It is shown that improved estimators obey a
Zero-Variance Zero-Bias (ZVZB) property similar to the usual Zero-Variance
Zero-Bias property of the energy with the local energy as improved estimator.
Using this property improved estimators can be optimized and the resulting
accuracy on expectation values may reach the remarkable accuracy obtained for
total energies. As an important example, we present the application of our
formalism to the computation of forces in molecular systems. Calculations of
the entire force curve of the H,LiH, and Li molecules are presented.
Spectroscopic constants (equilibrium distance) and (harmonic
frequency) are also computed. The equilibrium distances are obtained with a
relative error smaller than 1%, while the harmonic frequencies are computed
with an error of about 10%
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