117,640 research outputs found
A Unified Analysis of Multi-task Functional Linear Regression Models with Manifold Constraint and Composite Quadratic Penalty
This work studies the multi-task functional linear regression models where
both the covariates and the unknown regression coefficients (called slope
functions) are curves. For slope function estimation, we employ penalized
splines to balance bias, variance, and computational complexity. The power of
multi-task learning is brought in by imposing additional structures over the
slope functions. We propose a general model with double regularization over the
spline coefficient matrix: i) a matrix manifold constraint, and ii) a composite
penalty as a summation of quadratic terms. Many multi-task learning approaches
can be treated as special cases of this proposed model, such as a reduced-rank
model and a graph Laplacian regularized model. We show the composite penalty
induces a specific norm, which helps to quantify the manifold curvature and
determine the corresponding proper subset in the manifold tangent space. The
complexity of tangent space subset is then bridged to the complexity of
geodesic neighbor via generic chaining. A unified convergence upper bound is
obtained and specifically applied to the reduced-rank model and the graph
Laplacian regularized model. The phase transition behaviors for the estimators
are examined as we vary the configurations of model parameters
Neural Graphical Models
Probabilistic Graphical Models are often used to understand dynamics of a
system. They can model relationships between features (nodes) and the
underlying distribution. Theoretically these models can represent very complex
dependency functions, but in practice often simplifying assumptions are made
due to computational limitations associated with graph operations. In this work
we introduce Neural Graphical Models (NGMs) which attempt to represent complex
feature dependencies with reasonable computational costs. Given a graph of
feature relationships and corresponding samples, we capture the dependency
structure between the features along with their complex function
representations by using a neural network as a multi-task learning framework.
We provide efficient learning, inference and sampling algorithms. NGMs can fit
generic graph structures including directed, undirected and mixed-edge graphs
as well as support mixed input data types. We present empirical studies that
show NGMs' capability to represent Gaussian graphical models, perform inference
analysis of a lung cancer data and extract insights from a real world infant
mortality data provided by Centers for Disease Control and Prevention
Probing the arrangement of hyperplanes
AbstractIn this paper we investigate the combinatorial complexity of an algorithm to determine the geometry and the topology related to an arrangement of hyperplanes in multi-dimensional Euclidean space from the “probing” on the arrangement. The “probing” by a flat means the operation from which we can obtain the intersection of the flat and the arrangement. For a finite set H of hyperplanes in Ed, we obtain the worst-case number of fixed direction line probes and that of flat probes to determine a generic line of H and H itself. We also mention the bound for the computational complexity of these algorithms based on the efficient line probing algorithm which uses the dual transform to compute a generic line of H.We also consider the problem to approximate arrangements by extending the point probing model, which have connections with computational learning theory such as learning a network of threshold functions, and introduce the vertical probing model and the level probing model. It is shown that the former is closely related to the finger probing for a polyhedron and that the latter depends on the dual graph of the arrangement.The probing for an arrangement can be used to obtain the solution for a given system of algebraic equations by decomposing the μ-resultant into linear factors. It also has interesting applications in robotics such as a motion planning using an ultrasonic device that can detect the distances to obstacles along a specified direction
Software Engineering and Complexity in Effective Algebraic Geometry
We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
arXiv:1201.434
Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part I: template-based generic programming
An approach for incorporating embedded simulation and analysis capabilities
in complex simulation codes through template-based generic programming is
presented. This approach relies on templating and operator overloading within
the C++ language to transform a given calculation into one that can compute a
variety of additional quantities that are necessary for many state-of-the-art
simulation and analysis algorithms. An approach for incorporating these ideas
into complex simulation codes through general graph-based assembly is also
presented. These ideas have been implemented within a set of packages in the
Trilinos framework and are demonstrated on a simple problem from chemical
engineering
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