Math Requirements - bsiever/WUSTL-CSE-Curriculum GitHub Wiki

Outcomes:

Calc Series: fine
Prob/Stats: ESE 326 seems fine (uses applications/context)
- Math 3200: less applications; but introduces statistical software (R/SAS)
- peers: Stanford and CMU
Matrix Algebra:
- ESE 318 is listed as equivalent to Math 309
  - does not seem to be a good fit for CS students: TODO remove from requirements/webpage!
- room for improvement in terms of content covered
- content varies form instructor to instructor
- applications/context in CS is totally missing, however deemed important for CS students
TODO: identify content/learning outcomes that are missing, but required by upper-level courses
- input from faculty teaching upper-level courses is listed below.
- Google form

Probability and Statistics (Todd)

Matrix/Linear Algebra (Marion)

Calc Series (Jon)

What math content do our courses need?

List course(s) below and indicate if or to what extend the above courses cover the needs.

CSE347

CSE 247

Kunal:

The math requirements for 347 are not really covered well in the above listed content. In particular, we generally need students to be comfortable with discrete probability, combinatorics, how to write proofs and some number theory.

Discrete probability topics used: Indicator random variables, linearity of expectation, union bound, markov's inequality, Chebyshev and Chernoff bounds.
Combinatorics: Relatively simple counting arguments and upper and lower bounds on binomials.
How to write proofs: This is the most important aspect of 347 and the one the students are least prepared in. Mostly they need to know structural induction, contradiction, how to set up arguments, some reductions from one problem to another.
Number theory: Relatively simple modular algebra, some idea on how to manipulate mods and also how to do simple number theory proofs.

Some of this stuff is meant to be covered by 240, but the students are not really prepared for it when they come to 347. In particular, they find it hard to set up random variables and manipulate probabilities and expectations. Writing and understanding proofs is the biggest challenge.

Jeremy:

calc: ability to take a derivative to analytically minimize a function. prob/stat: knowing what expectation is, how to compute it, and its linearity; Bernoulli random variables; ability to compute simple conditional and marginal probabilities and understand simple random processes. linear algebra: know how to multiply two matrices or a vector and a matrix

Nice-to-have: calc: ability to evaluate simple integrals for, e.g., upper and lower bounds prob/stat: tail inequalities (Markov, Chebyshev), Bonferroni inequalities, maybe geometric random variables?

Brendan:

I am honestly more concerned about the material (counting, proof by induction, basic discrete probability) that should be acquired in CSE 240, but that students swear they've never seen before when they arrive in CSE 347.

CSE417T

Calc III (MATH 233), Probability and Statistics (e.g. ESE326, MATH 310), Matrix Algebra (MATH 309)

Sanmay:

All the stuff in the above classes.

Henry Chai:

Multivariable calculus

ability to compute gradients/partial derivatives
geometric understanding of inner/outer products

Probability theory

understanding of and ability to manipulate basic concepts e.g. expected value, variance
conditional probabilities and Bayes' theorem
familiarity with basic identities and distributions

Linear algebra

eigenvalue decomposition
semidefiniteness and its implications

the ability to reason probabilistically
the ability to prove theorems using the relevant mathematical concepts
strong intuitive grasp of the important ideas

CSE 452A (Tao)

Matrix Algebra (MATH 309)

Solving linear equations; basic matrix operations.
Being able to apply matrix algebra to problems in computer graphics, such as transformations and solving radiosity equations.

CSE 468T (Ron)

Calc III (MATH 233), Probability and Statistics (e.g. ESE326, MATH 310)

CSE511A (Will)

Calc III (MATH 233), Probability and Statistics (e.g. ESE326, MATH 310)

Calc 3: Enough calculus to understand gradient descent. Essentially, they just need to perform differentiations. Probability and Stats: Basic probability theory; stationary distributions

CSE517A (Marion)

Probability and Statistics (e.g. ESE326, MATH 310), Matrix Algebra (MATH 309)

Prob/Stats: joint, marginal, and conditional probability, independence, common distributions for random variables and vectors/multivariate distributions (normal as "one" example), expectation and variance

Learning outcomes: be aware of the different distributions for continuous, discrete, binary, and ordinal random variables; being able to derive Bayes Rule from the definition of conditional probabilities; understanding what independence means; being aware of the fact that marginal distributions are sums or integrals

Matrix Algebra: stating, manipulating, and optimizing over matrix equations; inner vs outer product; matrix rank; positive definiteness and how to show/check that a matrix is p(s)d; vector spaces and bases; matrix inverse and transpose; transpose of a product

Learning outcomes: be comfortable with handling matrices, including manipulating matrix equations and taking derivates wrt vectors; being able to compute expected values and variances of common distributions using the definitions; awareness that matrix dimensions need to be carefully defined and checked when dealing with matrix equations

Nice-to-have: Matrix Algebra: PCA/SVD, QR decomposition

CSE515T (Roman)

Calc III (MATH 233), Probability and Statistics (e.g. ESE326, MATH 310), Matrix Algebra (MATH 309), Only ESE 326 is listed, but MATH 233, 309 are also required implicitly.

Multivariate calculus: integration, differentiation, gradients, optimization. Matrix algebra: solving linear systems, matrix factorization, eigendecompositions.

Probability: discrete and continuous random variables, expectation, variance, probability density functions, common distributions (uniform, normal, etc.).

Machine learning: supervised machine learning, regression, classification, least squares, linear methods.

Numerically stable computation.

Ultimately: mathematical maturity.

CSE559A (Ayan)

Probability and Statistics (e.g. ESE326, MATH 310), Matrix Algebra (MATH 309)

Prob/Stats: Distributions of discrete and continuous random variables, multi-variate distributions for random vectors, computing distributions of random variables that are functions of other random variables. Matrix Algebra: Matrix vector operations (transpose, multiplication, inner product, inverse, etc.), methods for solving linear systems, eigen and singular-value decompositions, rank of a matrix and singular matrices, linear independence, linear spaces and subspaces, orthogonal and orthonormal basis vectors, orthonormal/unitary transforms.

CSE 538T (Roch)

Probability and Statistics (e.g. ESE326, MATH 310)

Sample space & events; conditional probabilities; various probability laws, e.g., bayes law; discrete and continuous random variables; probabilities and densities; expectation & variance; joint probabilities; generating functions; random sums of random variables; sample paths, convergence and averages

The ability to compute metrics of interest and understand which notion of probability apply in different settings.

Nice-to-have: measure theory; various notions of convergence, e.g., in distribution, in probability, almost surely; Laplace transforms, confidence intervals and various notions of statistical significance

Commets: My experience based on the sample of students I have interacted with is that most are totally ill-prepared when it comes to mastering basic probability concepts and tools

CSE 585T (Bek also for large-scale optimization (new/future course))

Probability and Statistics (e.g. ESE326, MATH 310), Matrix Algebra (MATH 309)

Eigenvalues/eigenvectors; subspaces; matrix norms; various norm inequalities; SVD; various multivariable probability distributions (Normal and Laplace); mean, variance, and covariance; conditional mean, variance, and covariance; statistical independence

Be comfortable to think in finite (n-) dimensional inner-product spaces and basic manipulations of multivariable probability distributions

CSE 544T - Special Topics in Computer Science Theory (Brendan)

Probability and Statistics (e.g. ESE326, MATH 310), Matrix Algebra (MATH 309)

matrix/linear algebra: spectral decomposition (eigenvectors/values); prob/stats: expected value/moments, standard estimators (e.g., sample mean), concentration inequalities (e.g., Chebyshev)

matrix/linear algebra: ability to recognize when a linear transformation can be represented by a spectral decomposition and to write a formula for the decomposition. prob/stats: comfort with calculations involving expected value, ability to apply concentration inequalities to derive high-probability bounds/confidence intervals.

Nice-to-have: matrix/linear algebra: Cramer's rule; prob/stats: proof of Central Limit Theorem

CSE 543T

CSE 514A (Weixiong)

Calc III (MATH 233), Probability and Statistics (e.g. ESE326, MATH 310), cse247, cse131

knowledge of data structures/algorithms, probability and statistics, one programming language.
knowledge of matrix algebra is a plus
basic data analytic skills

CSE 516A (Sanmay)

Calc III (MATH 233), Probability and Statistics (e.g. ESE326, MATH 310)

Basically, I expect some mathematical maturity, and ability to reason algorithmically. If students are not comfortable with calculus and probability, or with thinking about algorithms, they will have a hard time in this class.

Some prior exposure to artificial intelligence, machine learning, game theory, and microeconomics may be helpful, but is not required.

CSE 518A (CJ)

Calc III (MATH 233), Probability and Statistics (e.g. ESE326, MATH 310), Matrix Algebra (MATH 309), CSE247

mostly require basic understanding of the concepts (e.g., when I write a conditional probability, they need to know what it means)

Identified missing math content in CSE requirements

Integration and Approximation/Numerical Methods