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Frank Gao
Professor of Mathematics
Department of Mathematics
University of Idaho
Moscow, ID 83844-1103
(208) 885-5274

I received my Ph.D. in mathematics from the University of Connecticut and joined the faculty of the University of Idaho in 1999. I research on Theory of Probability, Approximation Theory, Non-Parametric Estimation, and Deep Neural Networks. In my free time, I enjoy organic gardening, and write poetry.


Universities exist because of the students. As a university professor I strive to be a good teacher. What makes a good teacher? I believe that a good mathematical science teacher must: 1) set up a good example of loving mathematical science; 2) motivate students' learning and further stimulate their interest in the field; and 3) effectively deliver course material. In my class, students can expect to see: 1) My excitement of doing mathematical science; 2) My constant encouragement of studying mathematical science; 3) My hand-written classnotes posted on Blackboard after each lecture. I highly value the importance of homework assignments and believes that the amount and level of difficulties of the homework problems directly affect the effectiveness of students' practice. Students who plan to take my class should expect to do weekly homework assignments with a reasonable amount of problems including some challenging ones.

In spring 2020, I will teach Analysis of Deep Learning (MF 3:30-4:45 PM. LOCATION: TLC 247).

Deep Learning is a machine learning technique that is penetrating almost all scientific areas that involve big data. In this course, through some basis deep neural network models, I will explain the roles of essential components of deep learning. With the knowledge learned from this course, the students will be able to implement the common neural networks, customize and design neural networks for their specific applications, and analyze the performance of the models.

This course will be taught in Python and Pytorch. Prior experience on Python is helpful, but not required. Linear algebra (Math 330) background is needed to understand the analysis in most lectures. Two of the lectures require Calculus III (Math 275). Two lectures require some background in probability and statistics (Stat 301). Students who do not have any machine learning or Python background are advised to register for the Pass/Fail session (Math 404-06). The topics are:

- Python and Pytorch basics for deep learning
- Machine learning basics on working with real data
- A minimal neural network with essential components, applied to MNIST dataset
- Distances; Cross Entropy Loss and its probabilistic interpretation
- Backpropagation and stochastic gradient descent method
- Choosing Activation functions; customizing loss function
- Neural network examples on time series data and voice data
- Convolutional Layers
- Some popular models on image classifications, applied to cifar 10 dataset
- Analyzing neural network performance and uncertainty quantification
- Batch normalization, whitening and techniques on preventing vanishing gradients
- Dimension reduction techniques and popular practices on avoiding overfitting
- Faster optimizers
- Generative Adversarial Networks (GANs)


I am currently working on problems in two different areas:

1. Metric entropy and its applications in non-parametric estimation
Consider the following problem: How many fire stations are needed in the New York city so that every house can be reached by a fire truck within 1 minute after receiving a fire alarm? The problem is complicated as the time a fire truck takes to travel from A to B depends on road condition, trafic, weather, events and many other things. If we redefine the distance between A and B as the expected traveling time, the problem can be then formulated as follows: Given a bounded (two-dimensional) set in a metric spaces, how many balls of radius r in this metric space are needed to cover the set? The latter is a typical metric entropy problem, except that in metric entropy, the set is usually high dimensional, or infinite dimensional (such as a set of functions), which the distance is clearly defined. Here is an open problem: How many probability distributions on the d-dimensional unit cube should be sampled, so that any unknown probability distributions on the d-dimensional unit cube is within L(2)-distance r from the ones you have sampled? (For the best known results, please see Metric Entropy of High-Dimensional Distributions.) This is only one of the many examples of the open problems I am interested in. These problems have one thing in common, that is, they have applications in non-parametric estimation in statistics. Indeed, metric entropy gauges the geometric complexity of these function spaces, while the geometric complexity determines the best rate of convergence of an statistical estimator can possibly achieve.
Motivated graduate students are welcome to this area of research.

2. Deep learning
Deep neural network is one of the current top choices for analyzing high-dimensional data. Given a dataset, what neural network architecture should we use to make best prediction? What depth? how many neurons? Currently, all these are mainly determined by modelers' experience. My group attempts to answer these questions using mathematical and statistical tools. We are working on three different but interrelated projects:
Approximation Theory of Neural Networks:
Under various assumptions on the shapes of the level sets of the multivariate functions under consideration, we study the upper bound for the network complexity.
Analytically architecting deep neural networks:
Given a high-dimensional dataset, We propose to use statistical analysis to quantify the structure complexity of the dataset, then use the quantative characterization of the data structure to design neural network architectures using approximation theory.
Representations of high-dimensional data:
Most current neural networks require the input data to be low-rank tensors. For example, a multi-channel image is input as a 3-D tensor. In real-world applications, effectively expressing the data in tensor forms could be challenging. For example, there does not seem to have a ready to use effective way to express a protein structure or a nonoporous Metal-Organic Framework (MOF) material in tensor form. Also, in practice, expressing a categorical variable that takes many possible values usign the commonly used one-hot method could unnecessarily increase the neural network complexity. In this area, Our group aims to find a unified approach for a tensor representation of the data that keeos the essential structural complexity of the original data. As a specific application of this approach, we are developing a tensor representation of MOF materials.

Currently, the group has four members: Frank Gao (PI), Boyu Zhang (postdoc), Daniel Furman (graduate) and Zachary Sugano (undergraduate), under an NSF grant support. We are looking for one more undergraduate research assistant.


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