29,536 research outputs found
Sparse kernel density construction using orthogonal forward regression with leave-one-out test score and local regularization
The paper presents an efficient construction algorithm for obtaining sparse kernel density estimates based on a regression approach that directly optimizes model generalization capability. Computational efficiency of the density construction is ensured using an orthogonal forward regression, and the algorithm incrementally minimizes the leave-one-out test score. A local regularization method is incorporated naturally into the density construction process to further enforce sparsity. An additional advantage of the proposed algorithm is that it is fully automatic and the user is not required to specify any criterion to terminate the density construction procedure. This is in contrast to an existing state-of-art kernel density estimation method using the support vector machine (SVM), where the user is required to specify some critical algorithm parameter. Several examples are included to demonstrate the ability of the proposed algorithm to effectively construct a very sparse kernel density estimate with comparable accuracy to that of the full sample optimized Parzen window density estimate. Our experimental results also demonstrate that the proposed algorithm compares favourably with the SVM method, in terms of both test accuracy and sparsity, for constructing kernel density estimates
Batch Nonlinear Continuous-Time Trajectory Estimation as Exactly Sparse Gaussian Process Regression
In this paper, we revisit batch state estimation through the lens of Gaussian
process (GP) regression. We consider continuous-discrete estimation problems
wherein a trajectory is viewed as a one-dimensional GP, with time as the
independent variable. Our continuous-time prior can be defined by any
nonlinear, time-varying stochastic differential equation driven by white noise;
this allows the possibility of smoothing our trajectory estimates using a
variety of vehicle dynamics models (e.g., `constant-velocity'). We show that
this class of prior results in an inverse kernel matrix (i.e., covariance
matrix between all pairs of measurement times) that is exactly sparse
(block-tridiagonal) and that this can be exploited to carry out GP regression
(and interpolation) very efficiently. When the prior is based on a linear,
time-varying stochastic differential equation and the measurement model is also
linear, this GP approach is equivalent to classical, discrete-time smoothing
(at the measurement times); when a nonlinearity is present, we iterate over the
whole trajectory to maximize accuracy. We test the approach experimentally on a
simultaneous trajectory estimation and mapping problem using a mobile robot
dataset.Comment: Submitted to Autonomous Robots on 20 November 2014, manuscript #
AURO-D-14-00185, 16 pages, 7 figure
Marginal integration for nonparametric causal inference
We consider the problem of inferring the total causal effect of a single
variable intervention on a (response) variable of interest. We propose a
certain marginal integration regression technique for a very general class of
potentially nonlinear structural equation models (SEMs) with known structure,
or at least known superset of adjustment variables: we call the procedure
S-mint regression. We easily derive that it achieves the convergence rate as
for nonparametric regression: for example, single variable intervention effects
can be estimated with convergence rate assuming smoothness with
twice differentiable functions. Our result can also be seen as a major
robustness property with respect to model misspecification which goes much
beyond the notion of double robustness. Furthermore, when the structure of the
SEM is not known, we can estimate (the equivalence class of) the directed
acyclic graph corresponding to the SEM, and then proceed by using S-mint based
on these estimates. We empirically compare the S-mint regression method with
more classical approaches and argue that the former is indeed more robust, more
reliable and substantially simpler.Comment: 40 pages, 14 figure
Regularized Nonparametric Volterra Kernel Estimation
In this paper, the regularization approach introduced recently for
nonparametric estimation of linear systems is extended to the estimation of
nonlinear systems modelled as Volterra series. The kernels of order higher than
one, representing higher dimensional impulse responses in the series, are
considered to be realizations of multidimensional Gaussian processes. Based on
this, prior information about the structure of the Volterra kernel is
introduced via an appropriate penalization term in the least squares cost
function. It is shown that the proposed method is able to deliver accurate
estimates of the Volterra kernels even in the case of a small amount of data
points
Kernel methods in machine learning
We review machine learning methods employing positive definite kernels. These
methods formulate learning and estimation problems in a reproducing kernel
Hilbert space (RKHS) of functions defined on the data domain, expanded in terms
of a kernel. Working in linear spaces of function has the benefit of
facilitating the construction and analysis of learning algorithms while at the
same time allowing large classes of functions. The latter include nonlinear
functions as well as functions defined on nonvectorial data. We cover a wide
range of methods, ranging from binary classifiers to sophisticated methods for
estimation with structured data.Comment: Published in at http://dx.doi.org/10.1214/009053607000000677 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- âŠ