7 research outputs found
Geodesic gaussian processes for the parametric reconstruction of a free-form surface
No full textReconstructing a free-form surface from 3-dimensional (3D) noisy measurements is a central problem in inspection, statistical quality control, and reverse engineering. We present a new method for the statistical reconstruction of a free-form surface patch based on 3D point cloud data. The surface is represented parametrically, with each of the three Cartesian coordinates (x, y, z) a function of surface coordinates (u, v), a model form compatible with computer-aided-design (CAD) models. This model form also avoids having to choose one Euclidean coordinate (say, z) as a “response” function of the other two coordinate “locations” (say, x and y), as commonly used in previous Euclidean kriging models of manufacturing data. The (u, v) surface coordinates are computed using parameterization algorithms from the manifold learning and computer graphics literature. These are then used as locations in a spatial Gaussian process model that considers correlations between two points on the surface a function of their geodesic distance on the surface, rather than a function of their Euclidean distances over the xy plane. We show how the proposed geodesic Gaussian process (GGP) approach better reconstructs the true surface, filtering the measurement noise, than when using a standard Euclidean kriging model of the “heights”, that is, z(x, y). The methodology is applied to simulated surface data and to a real dataset obtained with a noncontact laser scanner. Supplementary materials are available online
Fitting ARMA Time Series Models without Identification: A Proximal Approach
No full textFitting autoregressive moving average (ARMA) time series models requires
model identification before parameter estimation. Model identification involves
determining the order of the autoregressive and moving average components which
is generally performed by inspection of the autocorrelation and partial
autocorrelation functions or other offline methods. In this work, we regularize
the parameter estimation optimization problem with a non-smooth hierarchical
sparsity-inducing penalty based on two path graphs that allow performing model
identification and parameter estimation simultaneously. A proximal block
coordinate descent algorithm is then proposed to solve the underlying
optimization problem efficiently. The resulting model satisfies the required
stationarity and invertibility conditions for ARMA models. Numerical results
supporting the proposed method are also presented
Riemannian Stochastic Variance-Reduced Cubic Regularized Newton Method for Submanifold Optimization
No full textWe propose a stochastic variance-reduced cubic regularized Newton algorithm
to optimize the finite-sum problem over a Riemannian submanifold of the
Euclidean space. The proposed algorithm requires a full gradient and Hessian
update at the beginning of each epoch while it performs stochastic
variance-reduced updates in the iterations within each epoch. The iteration
complexity of to obtain an
-second-order stationary point, i.e., a point with
the Riemannian gradient norm upper bounded by and minimum eigenvalue
of Riemannian Hessian lower bounded by , is established when
the manifold is embedded in the Euclidean space. Furthermore, the paper
proposes a computationally more appealing modification of the algorithm which
only requires an inexact solution of the cubic regularized Newton subproblem
with the same iteration complexity. The proposed algorithm is evaluated and
compared with three other Riemannian second-order methods over two numerical
studies on estimating the inverse scale matrix of the multivariate
t-distribution on the manifold of symmetric positive definite matrices and
estimating the parameter of a linear classifier on the Sphere manifold
Riemannian Stochastic Gradient Method for Nested Composition Optimization
No full textThis work considers optimization of composition of functions in a nested form
over Riemannian manifolds where each function contains an expectation. This
type of problems is gaining popularity in applications such as policy
evaluation in reinforcement learning or model customization in meta-learning.
The standard Riemannian stochastic gradient methods for non-compositional
optimization cannot be directly applied as stochastic approximation of inner
functions create bias in the gradients of the outer functions. For two-level
composition optimization, we present a Riemannian Stochastic Composition
Gradient Descent (R-SCGD) method that finds an approximate stationary point,
with expected squared Riemannian gradient smaller than , in
calls to the stochastic gradient oracle of the outer
function and stochastic function and gradient oracles of the inner function.
Furthermore, we generalize the R-SCGD algorithms for problems with multi-level
nested compositional structures, with the same complexity of
for the first-order stochastic oracle. Finally, the performance of the R-SCGD
method is numerically evaluated over a policy evaluation problem in
reinforcement learning
Stochastic Optimization Algorithms for Problems with Controllable Biased Oracles
No full textMotivated by multiple emerging applications in machine learning, we consider
an optimization problem in a general form where the gradient of the objective
function is available through a biased stochastic oracle. We assume a
bias-control parameter can reduce the bias magnitude, however, a lower bias
requires more computation/samples. For instance, for two applications on
stochastic composition optimization and policy optimization for
infinite-horizon Markov decision processes, we show that the bias follows a
power law and exponential decay, respectively, as functions of their
corresponding bias control parameters. For problems with such gradient oracles,
the paper proposes stochastic algorithms that adjust the bias-control parameter
throughout the iterations. We analyze the nonasymptotic performance of the
proposed algorithms in the nonconvex regime and establish their sample or
bias-control computation complexities to obtain a stationary point. Finally, we
numerically evaluate the performance of the proposed algorithms over three
applications
Stochastic Composition Optimization of Functions without Lipschitz Continuous Gradient
No full textIn this paper, we study stochastic optimization of two-level composition of
functions without Lipschitz continuous gradient. The smoothness property is
generalized by the notion of relative smoothness which provokes the Bregman
gradient method. We propose three Stochastic Composition Bregman Gradient
algorithms for the three possible relatively smooth compositional scenarios and
provide their sample complexities to achieve an -approximate
stationary point. For the smooth of relatively smooth composition, the first
algorithm requires calls to the stochastic oracles of the
inner function value and gradient as well as the outer function gradient. When
both functions are relatively smooth, the second algorithm requires
calls to the inner function value stochastic oracle and
calls to the inner and outer functions gradients stochastic
oracles. We further improve the second algorithm by variance reduction for the
setting where just the inner function is smooth. The resulting algorithm
requires calls to the inner function value stochastic
oracle, calls to the inner function gradient and
calls to the outer function gradient stochastic oracles.
Finally, we numerically evaluate the performance of these three algorithms over
two different examples
