Courses

Computational Statistics (Stat GR6104)

Spring 2024: Schedule

This is a Ph.D.-level course in computational statistics. A link to the most recent previous iteration of this course is here.

Note: instructor permission is required to take this class for students outside of the Statistics Ph.D. program.

Time: Tu 2:10
Place: Jerome L Greene Science Center, room L5-084
Professor: Liam Paninski; Email: liam at stat dot columbia dot edu. Hours by appointment.

Course goals: (partially adapted from the preface of Givens' and Hoeting's book): Computation plays a central role in modern statistics and machine learning. This course aims to cover topics needed to develop a broad working knowledge of modern computational statistics. We seek to develop a practical understanding of how and why existing methods work, enabling effective use of modern statistical methods. Achieving these goals requires familiarity with diverse topics in statistical computing, computational statistics, computer science, and numerical analysis. Our choice of topics reflects our view of what is central to this evolving field, and what will be interesting and useful. A key theme is scalability to problems of high dimensionality, which are of most interest to many recent applications.
Some important topics will be omitted because high-quality solutions are already available in most software. For example, the generation of pseudo-random numbers is a classic topic, but existing methods built in to standard software packages will suffice for our needs. On the other hand, we will spend a bit of time on some classical numerical linear algebra ideas, because choosing the right method for solving a linear equation (for example) can have a huge impact on the time it takes to solve a problem in practice, particularly if there is some special structure that we can exploit.

Audience: The course will be aimed at first- and second-year students in the Statistics Ph.D. program. Students from other departments or programs are welcome, space permitting; instructor permission required.

Background: The level of mathematics expected does not extend much beyond standard calculus and linear algebra. Breadth of mathematical training is more helpful than depth; we prefer to focus on the big picture of how algorithms work and to sweep under the rug some of the nitty-gritty numerical details. The expected level of statistics is equivalent to that obtained by a graduate student in his or her first year of study of the theory of statistics and probability. An understanding of maximum likelihood methods, Bayesian methods, elementary asymptotic theory, Markov chains, and linear models is most important.

Programming: With respect to computer programming, good students can learn as they go. We'll forgo much language-specific examples, algorithms, or coding; I won't be teaching much programming per se, but rather will focus on the overarching ideas and techniques. For the projects, I recommend you choose a high-level, interactive package that permits the flexible design of graphical displays and includes supporting statistics and probability functions, e.g., R, Python, or MATLAB.

Evaluation: Final grades will be based on class participation and a student project.


Topics:
Deterministic optimization
- Newton-Raphson, conjugate gradients, preconditioning, quasi-Newton methods, Fisher scoring, EM and its various derivatives
- Numerical recipes for linear algebra: matrix inverse, LU, Cholesky decompositions, low-rank updates, SVD, banded matrices, Toeplitz matrices and the FFT, Kronecker products (separable matrices), sparse matrix solvers
- Convex analysis: convex functions, duality, KKT conditions, interior point methods, projected gradients, augmented Lagrangian methods, convex relaxations
- Applications: support vector machines, splines, Gaussian processes, isotonic regression, LASSO and LARS regression

Graphical models: dynamic programming, hidden Markov models, forward-backward algorithm, Kalman filter, Markov random fields

Stochastic optimization: Robbins-Monro and Kiefer-Wolfowitz algorithms, simulated annealing, stochastic gradient methods

Deterministic integration: Gaussian quadrature, quasi-Monte Carlo. Application: expectation propagation

Monte Carlo methods
- Rejection sampling, importance sampling, variance reduction methods (Rao-Blackwellization, stratified sampling)
- MCMC methods: Gibbs sampling, Metropolis-Hastings, Langevin methods, Hamiltonian Monte Carlo, slice sampling. Implementation issues: burnin, monitoring convergence
- Sequential Monte Carlo (particle filtering)
- Variational and stochastic variational inference

References:
Givens and Hoeting (2005) Computational statistics
Robert and Casella (2004) Monte Carlo Statistical Methods
Boyd and Vandenberghe (2004), Convex Optimization.
Press et al, Numerical Recipes
Sun and Yuan (2006), Optimization theory and methods
Fletcher (2000) Practical methods of optimization
Searle (2006) Matrix Algebra Useful for Statistics
Spall (2003), Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control
Shewchuk (1994), An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
Boyd et al (2011), Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers

 Advanced Topics in Theoretical Neuroscience (Spring 2024)

Led by: Rainer Engelken ([email protected]) & Manuel Beiran ( [email protected])

 You may register via Courseworks, or contact us if you do not have a Columbia ID.

The classes will take place in person every Wednesday, 10:00-11:30 AM, location: L5-084 at Jerome L. Greene building. The course will begin next Wednesday January 24th and will continue on a weekly basis until May 1st, except for the winter break and the weeks overlapping with the Cosyne conference. The full schedule will be shared shortly. 

The course is taught by postdocs, and is intended to be interactive and oriented towards recent research advances. The topics will include: deep learning models for neuroscience, recurrent neural networks -dynamics and computation-, synaptic plasticity rules, and information theory.

Schedule: 

Neural chaos

Jan 24th - JLG L5-084

Rainer Engelken: Lyapunov exponents in spiking networks

Learning in neural networks

Jan 31st - JLG L5-084

Kaushik Lakshminarasimhan: Reinforcement learning in recurrent neural networks

Feb 7th - JLG L5-084

Owen Marschall: Algorithms for training recurrent neural networks

Neural processing in feed-forward networks

Feb 14th - JLG L5-084: 

W Jeffrey Johnston: Information theory for sensory processing

Feb 21st - JLG L5-084:

Samuel Muscinelli: Random expansion in biological neural networks

Feb 28st - JLG L5-084: no class (CoSyNe)

Mar 6th - JLG L5-084: no class (CoSyNe)

Mar 13th - JLG L5-084: no class (spring break)

Control theory

Mar 20th - JLG L5-084: 

Bin Wang: Predictive Coding and its Network Models 

Computations in recurrent networks

Mar 27th - JLG L5-084:

Ji Xia: Non-normal dynamics

Apr 3rd - JLG L5-084: 

Manuel Beiran: Ring attractor models

Neural representations

Apr 10th - JLG L5-084:

Tahereh Toosi: Deep neural networks as models in neuroscience

Apr 17rd - JLG L5-084:

Valeria Fascianelli & Lorenzo Posani: Geometry of neural representations

Student presentations

Apr 24th - JLG L5-084

Advanced Theory, Spring 2023

Please register via Courseworks, the course is also open to external guests

Time: Wednesdays, 10.00-11.30 am
Place: Jerome L. Greene Science Center, L4.078 (will eventually be L5.084, but check email announcements)

For the zoom link contact [email protected]

Schedule:

Information theory
Jan 25 Information theory (Jeff Johnston) Material
Feb 01 Fisher information (Jeff Johnston)
Feb 08 Gaussian Information bottleneck (Rainer Engelken)

Learning systems
Feb 15 Expectation Maximization (Ji Xia)
Feb 22 Reinforcement learning I (Kaushik Lakshminarasimhan)
Mar 01 Reinforcement learning II (Kaushik Lakshminarasimhan)
Mar 08 No session (Cosyne)
Mar 15 No session (Spring break)
Mar 22 Feedforward architectures I (Samuel Muscinelli)
29 Mar Feedforward architectures II (Samuel Muscinelli)

Neural dynamics and computations
Apr 05 Oscillations and Wilson-Cowan model (Patricia Cooney)
Apr 12 Mean-field theory and perturbative approaches (Agostina Palmigiano)
Apr 19 Low-rank neural networks I (Manuel Beiran)
Apr 26  Low-rank neural networks II (Manuel Beiran)
May 3 Single-neuron computations (Salomon Muller)
May 10 Student presentations

Mathematical Tools for Theoretical Neuroscience (NBHV GU4359)

Spring 2023:  Download schedule

Class Assistants:
Ines Aitsahalia ([email protected])
Elom Amematsro ([email protected])
Ching Fang ([email protected])
Christine Liu ([email protected])
Amol Pasarkar ([email protected]) 

Faculty contact: Prof. Ken Miller ([email protected])

Time: Tuesdays, Thursdays 12:10 - 1:25pm
Place: L5-084
Webpage: CourseWorks (announcements, assignments, readings)
Credits: 3 

Description: An introduction to mathematical concepts used in theoretical neuroscience aimed to give a minimal requisite background for NBHV G4360, Introduction to Theoretical Neuroscience. The target audience is students with limited mathematical background who are interested in rapidly acquiring the vocabulary and basic mathematical skills for studying theoretical neuroscience, or who wish to gain a deeper exposure to mathematical concepts than offered by NBHV G4360. Topics include single- and multivariable calculus, linear algebra, differential equations, signals and systems, and probability. Examples and applications are drawn primarily from theoretical and computational neuroscience. 

Prerequisites: Basic prior exposure to trigonometry, calculus, and vector/matrix operations at the high school level.

Registration: Undergraduate and graduate students must register on SSOL. *If you’re only interested in attending a subset of lectures, register Pass/Fail and contact the TAs.

Neuroscience and Philosophy (GU4500), Spring 2023

Time: Thursdays, 2:10-4:00
Place: Jerome L. Greene Science Center, L5-084
Professor: John Morrison, Email: [email protected], office hours 

Description: This course is about the philosophical foundations of cognitive neuroscience. Cognitive neuroscientists often describe the brain as representing and inferring. It is their way of describing the overall function of the brain’s activity, an abstraction away from detailed neural recordings. But, because there are no settled definitions of representation and inference, there are often no objective grounds for these descriptions. As a result, they are often treated as casual glosses rather than as rigorous analyses. Philosophers have proposed a number of (somewhat) rigorous definitions, but they rarely say anything about the brain. The goal of this course is to survey the philosophical literature and consider which definitions, if any, might be useful to cognitive neuroscientists. I will begin each class with a description of a different definition from the philosophical literature. We will then rely on our collective expertise to assess its potential applications to the brain. This course is for graduate students in Neurobiology and Behavior. No prior background in philosophy will be assumed.

Format: Each lecture will begin with a summary of the books and articles listed below. But I don’t expect you to read any of them. Many of them are (unnecessarily!) dense and inaccessible. Plus, they total hundreds of pages. It’s my job to distill them into a few basic points. If you want to learn more, reach out and I will recommend particular chapters, papers, etc.

Evaluation: Graduate students in Neurobiology and Behavior should write a 15-page term paper applying one of the philosophical definitions to a particular experiment, preferably an experiment relevant to their own research. Other students should consult with the professor to identify an alternative form of evaluation. Undergraduates will be required to write at least two papers.

View Course Schedule

Advanced Theory, Spring 2022

Time: Tuesdays, 10.00 am
Place: Jerome L Greene Science Center, L7-119

For the zoom link contact [email protected]

Schedule:

Methods for decoding and interpreting neural data
Feb 01 Methods on calculating receptive fields (Ji Xia) Material
Feb 08 Information theory (Jeff Johnston)
Feb 15 cancelled

Neural representations
Feb 22 Geometry of abstractions (theory session) (Valeria Fascianelli) slides
Mar 01 Geometry of abstractions (hands-on session) (Lorenzo Posani)

Network dynamics
Mar 08 Solving very nonlinear problems with Homotopy Analysis Method (Serena Di Santo) notes notebook
Mar 29 Forgetting in attractor networks (Samuel Muscinelli)
Apr 05 Mean-field models of network dynamics (Alessandro Sanzeni + Mario Dipoppa)

Dynamics of learning
Apr 12 Learning dynamics in feedforward neural networks (Manuel Beiran + Rainer Engelken)
Apr 19 Learning dynamics recurrent neural networks (Manuel Beiran + Rainer Engelken)

Causality
Apr 26 Introduction to some causality issues in neuroscience (Laureline Logiaco)
May 3 Causality and latent variable models (Josh Glaser)
May 10 student presentations
May 17 Bonus session: Many methods, one problem: modern inference techniques as applied to linear regression (Juri Minxha)

Mathematical Tools for Theoretical Neuroscience (NBHV GU4359)

Spring 2022: Download schedule

Teaching Assistants:
Ching Fang [email protected]
Jack Lindsey [email protected]
Zhenrui Liao [email protected]
Amol Pasarkar [email protected]
Dan Tyulmankov [email protected]

Recitation instructor: David Clark [email protected] 

Faculty contact: Prof. Ken Miller* [email protected] 

(*Please contact Prof. Miller to sign add/drop forms and other items which require faculty permission)

Time:
Lectures: Tues/Thurs 10:10 - 11:25a
Office hours: Thurs 11:30a - 12:30p
Recitations: Alternating Fridays 10 - 11a

Location: Zoom

Webpage: CourseWorks

Credits: 3 

Description: An introduction to mathematical concepts used in theoretical neuroscience aimed to give a minimal requisite background for NBHV G4360, Introduction to Theoretical Neuroscience. The target audience is students with limited mathematical background who are interested in rapidly acquiring the vocabulary and basic mathematical skills for studying theoretical neuroscience, or who wish to gain a deeper exposure to mathematical concepts than offered by NBHV G4360. Topics include single- and multivariable calculus, linear algebra, differential equations, signals and systems, and probability. Examples and applications are drawn primarily from theoretical and computational neuroscience. 

Prerequisites: Basic prior exposure to trigonometry, calculus, and vector/matrix operations at the high school level.

Mathematical Tools for Theoretical Neuroscience (NBHV GU4359)

Faculty: Ken Miller ([email protected]) 

Instructor: Dan Tyulmankov ([email protected]) 

Teaching Assistant: Amin Nejatbakhsh ([email protected]) (Office Hours: Thursdays 1-2pm)

Time: Tuesdays, Thursdays 8:40a-10:00a

Place: JLGSC L5-084 Zoom (https://columbiauniversity.zoom.us/j/489220180)

Webpage: CourseWorks 

Credits: 3 credits, pass/fail only (Register on SSOL. Do not register for a grade.)

Description: An introduction to mathematical concepts used in theoretical neuroscience aimed to give a minimal requisite background for NBHV G4360, Introduction to Theoretical Neuroscience. The target audience is students with limited mathematical background who are interested in rapidly acquiring the vocabulary and basic mathematical skills for studying theoretical neuroscience, or who wish to gain a deeper exposure to mathematical concepts than offered by NBHV G4360. Topics include single- and multivariable calculus, linear algebra, differential equations, dynamical systems, and probability. Examples and applications are drawn primarily from theoretical and computational neuroscience. 

Prerequisites: Basic prior exposure to trigonometry, calculus, and vector/matrix operations at the high school level

Download schedule