7,658 research outputs found
Four lectures on probabilistic methods for data science
Methods of high-dimensional probability play a central role in applications
for statistics, signal processing theoretical computer science and related
fields. These lectures present a sample of particularly useful tools of
high-dimensional probability, focusing on the classical and matrix Bernstein's
inequality and the uniform matrix deviation inequality. We illustrate these
tools with applications for dimension reduction, network analysis, covariance
estimation, matrix completion and sparse signal recovery. The lectures are
geared towards beginning graduate students who have taken a rigorous course in
probability but may not have any experience in data science applications.Comment: Lectures given at 2016 PCMI Graduate Summer School in Mathematics of
Data. Some typos, inaccuracies fixe
A representer theorem for deep kernel learning
In this paper we provide a finite-sample and an infinite-sample representer
theorem for the concatenation of (linear combinations of) kernel functions of
reproducing kernel Hilbert spaces. These results serve as mathematical
foundation for the analysis of machine learning algorithms based on
compositions of functions. As a direct consequence in the finite-sample case,
the corresponding infinite-dimensional minimization problems can be recast into
(nonlinear) finite-dimensional minimization problems, which can be tackled with
nonlinear optimization algorithms. Moreover, we show how concatenated machine
learning problems can be reformulated as neural networks and how our
representer theorem applies to a broad class of state-of-the-art deep learning
methods
Estimation in high dimensions: a geometric perspective
This tutorial provides an exposition of a flexible geometric framework for
high dimensional estimation problems with constraints. The tutorial develops
geometric intuition about high dimensional sets, justifies it with some results
of asymptotic convex geometry, and demonstrates connections between geometric
results and estimation problems. The theory is illustrated with applications to
sparse recovery, matrix completion, quantization, linear and logistic
regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change
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