2,817 research outputs found
Machine Learning, Quantum Mechanics, and Chemical Compound Space
We review recent studies dealing with the generation of machine learning
models of molecular and solid properties. The models are trained and validated
using standard quantum chemistry results obtained for organic molecules and
materials selected from chemical space at random
Calibrated Multivariate Regression with Application to Neural Semantic Basis Discovery
We propose a calibrated multivariate regression method named CMR for fitting
high dimensional multivariate regression models. Compared with existing
methods, CMR calibrates regularization for each regression task with respect to
its noise level so that it simultaneously attains improved finite-sample
performance and tuning insensitiveness. Theoretically, we provide sufficient
conditions under which CMR achieves the optimal rate of convergence in
parameter estimation. Computationally, we propose an efficient smoothed
proximal gradient algorithm with a worst-case numerical rate of convergence
\cO(1/\epsilon), where is a pre-specified accuracy of the
objective function value. We conduct thorough numerical simulations to
illustrate that CMR consistently outperforms other high dimensional
multivariate regression methods. We also apply CMR to solve a brain activity
prediction problem and find that it is as competitive as a handcrafted model
created by human experts. The R package \texttt{camel} implementing the
proposed method is available on the Comprehensive R Archive Network
\url{http://cran.r-project.org/web/packages/camel/}.Comment: Journal of Machine Learning Research, 201
A fast semi-direct least squares algorithm for hierarchically block separable matrices
We present a fast algorithm for linear least squares problems governed by
hierarchically block separable (HBS) matrices. Such matrices are generally
dense but data-sparse and can describe many important operators including those
derived from asymptotically smooth radial kernels that are not too oscillatory.
The algorithm is based on a recursive skeletonization procedure that exposes
this sparsity and solves the dense least squares problem as a larger,
equality-constrained, sparse one. It relies on a sparse QR factorization
coupled with iterative weighted least squares methods. In essence, our scheme
consists of a direct component, comprised of matrix compression and
factorization, followed by an iterative component to enforce certain equality
constraints. At most two iterations are typically required for problems that
are not too ill-conditioned. For an HBS matrix with
having bounded off-diagonal block rank, the algorithm has optimal complexity. If the rank increases with the spatial dimension as is
common for operators that are singular at the origin, then this becomes
in 1D, in 2D, and
in 3D. We illustrate the performance of the method on
both over- and underdetermined systems in a variety of settings, with an
emphasis on radial basis function approximation and efficient updating and
downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App
Elastic net prefiltering for two class classification
A two-stage linear-in-the-parameter model construction algorithm is proposed aimed at noisy two-class classification problems. The purpose of the first stage is to produce a prefiltered signal that is used as the desired output for the second stage which constructs a sparse linear-in-the-parameter classifier. The prefiltering stage is a two-level process aimed at maximizing a model’s generalization capability, in which a new elastic-net model identification algorithm using singular value decomposition is employed at the lower level, and then, two regularization parameters are optimized using a particle-swarm-optimization algorithm at the upper level by minimizing the leave-one-out (LOO) misclassification rate. It is shown that the LOO misclassification rate based on the resultant prefiltered signal can be analytically computed without splitting the data set, and the associated computational cost is minimal due to orthogonality. The second stage of sparse classifier construction is based on orthogonal forward regression with the D-optimality algorithm. Extensive simulations of this approach for noisy data sets illustrate the competitiveness of this approach to classification of noisy data problems
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