24,368 research outputs found
Curriculum Guidelines for Undergraduate Programs in Data Science
The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program
met for the purpose of composing guidelines for undergraduate programs in Data
Science. The group consisted of 25 undergraduate faculty from a variety of
institutions in the U.S., primarily from the disciplines of mathematics,
statistics and computer science. These guidelines are meant to provide some
structure for institutions planning for or revising a major in Data Science
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
On the Spectral Properties of Matrices Associated with Trend Filters
This paper is concerned with the spectral properties of matrices associated
with linear filters for the estimation of the underlying trend of a time
series. The interest lies in the fact that the eigenvectors can be interpreted
as the latent components of any time series that the filter smooths through the
corresponding eigenvalues. A difficulty arises because matrices associated with
trend filters are finite approximations of Toeplitz operators and therefore
very little is known about their eigenstructure, which also depends on the
boundary conditions or, equivalently, on the filters for trend estimation at
the end of the sample. Assuming reflecting boundary conditions, we derive a
time series decomposition in terms of periodic latent components and
corresponding smoothing eigenvalues. This decomposition depends on the local
polynomial regression estimator chosen for the interior. Otherwise, the
eigenvalue distribution is derived with an approximation measured by the size
of the perturbation that different boundary conditions apport to the
eigenvalues of matrices belonging to algebras with known spectral properties,
such as the Circulant or the Cosine. The analytical form of the eigenvectors is
then derived with an approximation that involves the extremes only. A further
topic investigated in the paper concerns a strategy for a filter design in the
time domain. Based on cut-off eigenvalues, new estimators are derived, that are
less variable and almost equally biased as the original estimator, based on all
the eigenvalues. Empirical examples illustrate the effectiveness of the method
Learning Model-Based Sparsity via Projected Gradient Descent
Several convex formulation methods have been proposed previously for
statistical estimation with structured sparsity as the prior. These methods
often require a carefully tuned regularization parameter, often a cumbersome or
heuristic exercise. Furthermore, the estimate that these methods produce might
not belong to the desired sparsity model, albeit accurately approximating the
true parameter. Therefore, greedy-type algorithms could often be more desirable
in estimating structured-sparse parameters. So far, these greedy methods have
mostly focused on linear statistical models. In this paper we study the
projected gradient descent with non-convex structured-sparse parameter model as
the constraint set. Should the cost function have a Stable Model-Restricted
Hessian the algorithm produces an approximation for the desired minimizer. As
an example we elaborate on application of the main results to estimation in
Generalized Linear Model
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