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Understanding this hidden structure could help us visualize data, remove noise, compare examples, and build machine-learning systems that are faster, more reliable, and easier to understand.

In this project, we will try to answer: When can we discover the hidden shape of data accurately and efficiently?

This is a difficult problem. In the most general setting, learning the full shape may require a very large amount of data and computation. Real data are also noisy, so observations may not lie exactly on a clean surface. Even deciding how many underlying dimensions the data have can be challenging.

Tags: Python, Basic Programming, Linear Algebra, Calculus, Statistics, Machine Learning, Optimization, All Years

This project wants to answer: Why does sparse regression often work well in practice, even when the usual theoretical assumptions do not clearly apply?

We will study this question using ideas from geometry, statistics, and optimization. Here, geometry means thinking about variables as directions or points in space. For example, two variables that contain almost the same information can be viewed as pointing in nearly the same direction. This viewpoint may help us understand when sparse regression makes reliable predictions, when it selects meaningful variables, and when its answer is unstable.

Tags: Python, Basic Programming, Linear Algebra, Statistics, Calculus, Optimization, Machine Learning, All Years

In this project, we will explore how machine learning can help astronomers find and study interesting objects or events. For example, a model might be used to classify astronomical objects, identify unusual observations, detect rare events, study populations of galaxies or galaxy clusters, or uncover patterns in the shape and organization of these systems. It may also help researchers understand the different stages or components of events such as gamma-ray bursts. The exact scientific question will depend on the available datasets and discussions with collaborators in astronomy and cosmology. There are opportunities to collaborate with astrophysicists and cosmologists in institutes like Perimeter Institute and Vera Rubin Observatory in medium and/or longer term.

Tags: Python, Basic Programming, Data Structures, Algorithms, Statistics, Linear Algebra, Calculus, Machine Learning, Astronomy, All Years

For secure multiparty computation (MPC), our goal is for parties 1 to n to securely compute f(x1, …, xn) where xi is the private input of party i. Our security condition is for the messages each party sends and receives during the computation of f to reveal no more information than its input and output.  This allows the parties to collaboratively compute a function over their private inputs while maintaining privacy.

Traditionally, MPC algorithms have a fixed runtime that depends only on input size rather than the specific input since otherwise the runtime would leak information about the private input. However, for non-private algorithms, there are practical algorithms with a runtime that is both random and low in expectation. One example that has been successfully adapted to the MPC setting is quicksort, which is an algorithm whose random runtime is independent of the input list. Our goal in this project is to adapt another algorithm with random runtime that is independent of the specific input and benchmark it against private deterministic versions of the same algorithm. A successful implementation could enable adaptation of richer algorithm classes to the private setting.

Tags: Algorithms, Statistics, Security, All Years