I am interested in understanding machine learning and optimization from geometric and algebraic perspectives. My research generally involves using geometry and high-dimensional statistics to understand and develop machine learning concepts and algorithms. I tend to care about things such as the generalization and robustness of machine learning methods, and their susceptibility to altered or missing data. Previously, I worked on problems in computational algebra and number theory.

**Preprints**

**Approximate Gradient Coding via Sparse Random Graphs
**Z. Charles, D. Papailiopoulos, J. Ellenberg

[arXiv:1711.06771]

**Stability and Generalization of Learning Algorithms that Converge to Global Optima
**Z. Charles, D. Papailiopoulos

[arXiv:1710.08402]

**Exploiting Algebraic Structure in Global Optimization and the Belgian Chocolate Problem**

Z. Charles, N. Boston

[arXiv:1708.08114]

**Subspace Clustering with Missing and Corrupted Data**

Z. Charles, A. Jalali, R. Willett

[arXiv:1707.02461]

**Publications**

**Efficiently Finding All Power Flow Solutions to Tree Networks
**Z. Charles, A. Zachariah

To appear in Allerton, 2017.

**Generating Random Factored Ideals in Number Fields**

*Z. Charles*

To appear in Mathematics of Computation, 2017. [arXiv:1612.06260]

**Nonpositive Eigenvalues of Hollow, Symmetric, Nonnegative Matrices
**

*Z. Charles, M. Farber, C. Johnson, L. Kennedy-Shaffer*

SIAM Journal on Matrix Analysis and Applications, 2013.

**Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues
**

*Z. Charles, M. Farber, C. Johnson, L. Kennedy-Shaffer*

Discrete Mathematics, 2013. [arXiv:1108.4810]

**The Relation Between the Diagonal Entries and the Eigenvalues of a Symmetric Matrix, Based upon the Sign Pattern of its Off-Diagonal Entries
**

*Z. Charles, M. Farber, C. Johnson, L. Kennedy-Shaffer*

Linear Algebra and its Applications, 2013.