## About

Random Matrix Theory is an area of mathematics that deals with matrix-valued random variables. There are many interesting questions that can asked for these mathematical objects, and this theory has found numerous applications in science over the years, such as biology, quantum physics, computer science... The study of eigenvalues of random matrices has many roots, including early work of Wishart in statistics, Wigner in nuclear physics, as well as Goldstein and von Neumann in numerical analysis.

This 1-day Symposium on *Random Matrix Theory and its Applications*, organized by Andrew Blumberg, Mathieu Carrière, Ivan Corwin and Raul Rabadan, is part of the events supported by the **Columbia University Center for Topology of Cancer Evolution and Heterogeneity** (Director Dr Rabadan), which is part of the **National Cancer Institute's Physical Sciences in Oncology Network**, and the **"Probability and Society" initiative for Columbia**. The purpose of the symposium is to present foundations and applications of Random Matrix Theory in biology and computer science.

For any questions or concerns please contact Mathieu Carrière at mc4660@cumc.columbia.edu or (917)-941-5182

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## Speakers

**Andrew Blumberg, PhD** (University of Texas at Austin)

**Luis Aparicio, PhD** (Columbia University)

**Ivan Corwin, PhD** (Columbia University)

**Mor Nitzan, PhD** (Harvard University)

**Ben Landon, PhD** (MIT)

**Alex Bloemendal, PhD** (Broad Institute)

**Jeff Pennington, PhD** (Google Brain)

**Jonathan Bloom, PhD** (Broad Institute)

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## Symposium Program

__Friday November 1st:__

**9.30am - 10.00am** --- Coffee/Tea & Registration

**10.00am - 10.30am: Andrew Blumberg**, PhD, University of Texas at Austin.

*TBA*
TBA

**10.30am - 11.00am: Luis Aparicio**, PhD, Columbia University.

*TBA*
TBA

**11.00am - 11.30am ** --- Coffee break

**11.30am - 12.00pm: Ivan Corwin**, PhD, Columbia University.

*Products of thin random matrices and random walks in random media.*
I will consider the asymptotic behavior of products of independent thin (e.g. tridiagonal) random matrices. Matrix entries in these products can be interpreted as partition functions for random polymer models, or as transition probabilities for random walks in random media. Through special "exactly solvable" models, we will explain a "Kardar-Parisi-Zhang universality" conjecture for the asymptotic behavior of these products. We will also explain how these systems can be thought of as toy models for certain biological phenomena.

**12.00pm - 12.30pm: Mor Nitzan**, PhD, Harvard University.

*Revealing lineage-related signals in single-cell gene expression using random matrix theory*
The expression profiles of a population of single cells, generated for example by single cell RNA sequencing, contains rich information about the biological state of each cell, including cell type, location along the cell cycle, gene regulatory patterns and relative location in the tissue from which the cell was dissociated. A major challenge is to disentangle information about these different biological states from each other, including distinguishing them from cell lineage, since the correlation of expression patterns between different cells is necessarily contaminated by ancestry. Based on a recent advance in random matrix theory (Qin and Colwell 2018), we have shown that ancestral relationships between cells are expected to create a power law signature in the eigenvalue distribution of the covariance matrix of single-cell data. We demonstrate the existence of such signatures in single cell RNA sequencing data, and show that the genes driving them are indeed related to differentiation and developmental pathways. We illustrate these principles in diverse tissues and organisms, including the mammalian epidermis and lung, Drosophila whole-embryo, whole adult Hydra, yeast colonies, and cells undergoing iPSCs reprogramming. Based on these results, we can disentangle ancestry-related signals from other biological processes in single cell data. We illustrate the potential for such disentanglement for yeast single cell data, where we substantially improve the quality of reconstruction of gene regulatory networks by removing lineage-related effects.

**12.30pm - 2.00pm** --- Lunch break

**2.00pm - 2.30pm: Ben Landon**, PhD, MIT.

*Extremal eigenvalues of sparse random matrices*
Extremal eigenvalues of random matrices are of interest in statistical applications, and the random matrices that arise in such applications are may be sparse. We discuss the effect of sparsity on the asymptotic behavior of the extremal eigenvalues in some simple random matrix ensembles. Two such effects are higher order corrections to the spectral edges of Wigner's semicircle law, and a transition from Tracy-Widom to Gaussian fluctuations of the extremal eigenvalues.

**2.30pm - 3.00pm: Alex Bloemendal**, PhD, Broad Institute.

*TBA*
TBA

**3.00pm - 3.30pm ** --- Coffee break

**3.30pm - 4.00pm: Jeffrey Pennington**, PhD, Google Brain.

*Operator-valued free probability meets deep learning: training and generalization dynamics in high dimensions*
One of the distinguishing characteristics of modern deep learning systems is that they typically employ neural network architectures that utilize enormous numbers of parameters, often in the millions and sometimes even in the billions. While this paradigm has recently inspired a broad research effort on the properties of large networks, relatively little work has been devoted to the fact that these networks are often used to model large complex datasets, which may themselves contain millions or even billions of constraints. In this talk, I will present a formalism based on operator-valued free probability that enables exact predictions of training and generalization performance in the high-dimensional regime in which both the dataset size and the number of features tend to infinity. The analysis provides one of the first analytically tractable models that captures the effects of early stopping, over/under-parameterization, explicit regularization, and which exhibits the characteristic double-descent curve.

**4.00pm - 4.30pm: Jonathan Bloom**, PhD, Broad Institute.

*Loss landscapes, Morse theory, and linear autoencoders*
Random matrix theory has guided intuition on loss landscapes of deep neural networks, though rigorous connections invoke unrealistic assumptions. I’ll describe a simple model, the linear autoencoder, where the critical values are sums of eigenvalues of the data and the dynamics of learning may be understood through the topological lens of Morse theory. We recently extended this analysis to L2-regularized linear autoencoders, proving all critical points are symmetric, with implications for PCA algorithms and the biological plausibility of backprop. I'll speculate that Morse theory may give intuition complementary to RMT in deep models as well. Based on “Loss Landscapes of Regularized Linear Autoencoders” (ICML 2019) with Daniel Kunin, Aleksandrina Goeva, and Cotton Seed at the Broad Institute.

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## Location

**Columbia University**

**Lerner Hall 477**

2920 Broadway,

New York, NY 10027

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