Equation from Teacher’s Choice

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So I have been wanting to understand Category Theory (see [1], [2], [3]) mainly because I thought it would help me understand advanced functional programming in Haskell and because of the book “Physics, topology, logic and computation: a Rosetta Stone” by Baez and Stay (2011). A number of the constructs in Haskell were derived from Category Theory so that was enough motivation for me to learn it. But also, monoidal categories are useful in several fields: physics, computer science, pure mathematics, and statistics. The biggest application for me is the idea of generalizing probabilistic graphical models through either Ologs or monoidal categories / tensor networks (see Brendan Fong’s thesis “Causal Theories: A Categorical Perspective on Bayesian Networks“).

To start my investigation of Category Theory, I began with the $20 thin book “Basic Category Theory for Computer Scientists” by Benjamin Pierce (see also the free online version “A taste of category theory for computer scientists” (1988)).

I really enjoyed the book. Dr. Pierce’s style is a little informal compared to pure math books like Mac Lane’s “Categories for the Working Mathematician” (pdf / Amazon), but I enjoy that more relaxed style of writing when I am first learning a field.

The biggest obstacle for learning category theory is the fact that category theory generalizes a lot of areas of pure mathematics like topology, abstract algebra, and geometry. It’s hard to generalize before you have examples to generalize, but the examples being generalized in category theory are mostly from higher level mathematics found in senior level undergraduate and graduate level courses. Pierce ameliorates this problem by introducing some of the most basic categories first: sets, ordered sets, partially ordered sets, groups, monoids, vector spaces, measure spaces, topological spaces, proofs, and a simple functional computer language. He takes the time to explicitly define most of these ideas, so, in theory, you could read this book without a background in theoretical mathematics, but it would be hard.

After defining categories and introducing the most basic categories, Pierce describes and defines the most basic ideas in category theory: subcategories, commutative diagrams, monomorphisms, epimorphisms, isomorphisms, initial/terminal objects, products, coproducts, universal constructions, equalizers, pullbacks, pushouts, limits, cones, colimits, cocones, exponentiation, and closed Cartesian categories. These ideas are spelled out over the thirty pages of chapter one including illuminating homework exercises. The homework exercises varied significantly in difficulty. Many of the exercises were trivial and there are two or three that I am still working on despite investing several hours of thought. Generally, I found the exercises to be a bit harder than those in Mac Lane’s book, but Pierce’s book required less of a background in mathematics. A couple of the exercises were incorrectly stated or impossible.

Chapter two introduced functors, natural transformations, adjoints, and F-algebras. After reading this chapter, I was finally able to understand the definition of monads which are an important part of the computer language Haskell! Pierce provides many examples of each of these ideas and enjoyable homework exercises to increase understanding. Pierce’s definition of adjoints is much easier to understand than the standard definitions using counit adjunction or Hom Sets.

The last major chapter concerns applications of category theory to computer science–specifically lambda-calculus and programming language design.

The first two chapters of the book give a reasonable condensed introduction to category theory for students that have taken a course in abstract algebra. A course in topology or linear algebra would be another useful prerequisite. I carried around the light 100 page book for a few months so that I could learn something whenever I had some extra time. I had hoped that when I had proven that several functors where monads, I would then really understand monads, but a full understanding still eludes me. Similarly, I have proven that several functor pairs are adjoint, but I still don’t feel as though I understand adjoint functors. I guess more work and learning are needed.

Have a great year. – Cheers, Hein

Orthogonal functions are very useful and rather easy to use. The most famous ones are sin(n x) and cos(n x) on the interval [$-\pi$, $\pi$]. It is extremely easy to do linear regression with orthogonal functions.

Linear regression involves approximating a known function $f(x)$ as the weighted sum of other functions. This shows up all the time in various disguises in machine learning:

- When using AdaBoosting we perform a weighted sum of weak classifiers.
- Support vector machines find a linear combination of the features that separates the data into two classes.
- Each node in a neural net typically computes a weighted sum of the nodes in the prior layer.

For orthogonal functions, computing the linear regression is simple. Suppose that the features are $f_1, f_2, \ldots, f_n$ and the functions $f_i$ are orthonormal, then the best approximation of the function $g(x)$ is the same as the solution to the linear regression

$$ g(x) \approx \sum_{i=1}^n \alpha_i f_i(x)$$

where $\alpha_i$ is the “inner product” of $f_i$ and $g$ denoted

$\alpha_i= \langle\;g, f_i\rangle = \int g(x) f(x) w(x) dx.$ (Two function are orthogonal if their inner product (or dot product) is zero. The set of orthogonal functions $\{f_i\}$ is called orthonormal if $<f_i, f_i>=1$ for all $i$.) The function $w(x)$ is a positive weight function which is typically just $w(x)=1$.

For example, if we try to approximate the function $g(x)=x^2$ on the interval [-1,1] by combining the features $f_1(x) = 1/\sqrt(2)$ and $f_2(x) = x \sqrt{3 \over 2} $, then

$$ g(x) \approx \sum_{i=1}^n \alpha_i f_i(x)$$

where

$$\alpha_1 = \langle g, f_1\rangle = \int_{-1}^1 g(x) f_1(x) dx = \int_{-1}^1 x^2 1/\sqrt2 dx = \sqrt{2}/3,$$

$$\alpha_2 = \langle g, f_2\rangle = \int_{-1}^1 g(x) f_2(x) dx= \int_{-1}^1 x^2 \sqrt{3/2}\; x dx =0,$$

and the best approximation is then $\sum_{i=1}^n \alpha_i f_i(x) = \alpha_1 f_1(x) = 1/3$.

The other day I needed to do a linear regression and I wanted to use orthogonal functions, so I started by using the trigonometric functions to approximate my function $g(x)$. Using trig functions to approximate another function on a closed interval like [-1,1] is the same as computing the Fourier Series. The resulting approximations were not good enough at x=-1 and x=1, so I wanted to change the weight function $w(x)$. Now assume that $u_1(x), u_2(x), \ldots, u_n(x)$ are orthonormal with the weight function w(x) on [-1,1]. Then

$$ \int_{-1}^1 u_i(x) u_j(x) w(x) dx = \delta_{i,j}$$

where $\delta_{i,j}$ is one if $i=j$ and zero otherwise. If we make a change of variable $x=y(t)$, then

$$ \int_{y^{-1}(-1)}^{y^{-1}(1)} u_i(y(t)) u_j(y(t)) w(y(t)) y'(t) dx = \delta_{i,j}.$$

It would be nice if

$$(1)\quad w(y(t)) y'(t)=1,$$

because then I could just set $u_i(y(t))$ equal to a trig function and the resulting $u_i(x)$ would be orthonormal with respect to the weight function $w(x)$. Condition (1) on $y$ is really an ordinary differential equation

$$y’ = {1\over{w(y)}}.$$

In order to get a better fit at $x=\pm 1$, I tried two weight functions $w(x) = {1\over{(1-x)(1+x)}}$ and $w(x) = {1\over{\sqrt{(1-x)(1+x)}}}$. For the former, the differential equation has solutions that look similar to the hyperbolic tangent function. The latter $w(x)$ yields the nice solution

$$y(t) = \sin( c + t).$$

So now if I choose $c=\pi/2$, then $y(t) = \cos(t)$, $y^{-1}(-1) = \pi$, $y^{-1}(1)=0$, so I need functions that are orthonormal on the interval $[-\pi, 0]$. The functions $\cos(n x)$ are orthogonal on that interval, but I need to divide by $\sqrt{\pi/2}$ to make them orthornormal. So, I set $u_n(y(t)) = \sqrt{2/\pi}\cos(n\;t)$ and now

$$ \int_{-1}^1 {{u_i(x) u_j(x)}\over {\sqrt{(1-x)(1+x)}}} dx = \delta_{i,j}.$$

The $u_n(x)$ functions are orthonormal with respect to the weight $w(x)$ which is exactly what I wanted.

Solving for $u(x)$ yields

$$u_n(x) = \sqrt{2/\pi} \cos( n \;\arccos(x)),$$

$$u_0(x) = \sqrt{2/\pi}, u_1(x) = \sqrt{2/\pi} x,$$

$$u_2(x) = \sqrt{2/\pi}(2x^2-1), u_3(x) = \sqrt{2/\pi}(4x^3-3x),\ldots.$$

If you happen to be an approximation theorist, you would probably notice that these are the Chebyshev polynomials multiplied by $\sqrt{2/\pi}$. I was rather surprised when I stumbled across them, but when I showed this to my friend Ludmil, he assured me that this accidental derivation was obvious. (But still fun )

You can find anything on the web. Earlier today, I noticed that the Sterling approximation and the Gaussian distribution both had a $\sqrt{2 \pi}$ in them, so I started thinking that maybe you could apply Sterling’s approximation to the binomial distribution with $p=1/2$ and large $n$ and the central limit theorem to derive the Gaussian distribution. After making a few calculations it seemed to be working and I thought, “Hey, maybe someone else has done this.” Sure enough, Jake Hoffman did it and posted it here. The internet is pretty cool.

I was reading MarkCC’s gripe about the misuse of the word dimension and it reminded me about the Hausdorff dimension. Generally speaking, the Hausdorff dimension matches our normal intuitive definition of dimension, i.e the H-dimension of a smooth curve or line is 1, the H-dimension of a smooth surface is 2, the H-dimension of the interior of a cube is 3. But for fractals, the H-dimension can be a non-integer. For the Cantor set, the H-dimension is 0.63. For more information, check out the Wikipedia article

http://en.wikipedia.org/wiki/Hausdorff_dimension

Check out Terence Tao‘s wonderful post “Matrix identities as derivatives of determinant identities“.

Thank you to Freakonometrics for pointing me toward the book “Proofs without words” by Rodger Nelson. Might be a nice Christmas present

Matroids are an abstraction of Vector Spaces. Vectors spaces have addition and scalar multiplication and the axioms of vectors spaces lead to the concepts of subspaces, rank, and independent sets. All Vector Spaces are Matroids, but Matroids only retain the properties of independence, rank, and subspaces. The idea of the Matroid closure of a set is the same as the span of the set in the Vector Space.

- counting
- zero
- integer decimal positional notation 100, 1000, …
- the four arithmetic operations + – * /
- fractions
- decimal notation 0.1, 0.01, …
- basic propositional logic (Modus ponens, contrapositive, If-then, and, or, nand, …)
- negative numbers
- equivalence classes
- equality & substitution
- basic algebra – idea of variables, equations, …
- the idea of probability
- commutative and associative properties
- distributive property
- powers (squared, cubed,…), – compound interest (miracle of)
- scientific notation 1.3e6 = 1,300,000
- polynomials
- first order predicate logic
- infinity
- irrational numbers
- Demorgan’s laws
- statistical independence
- the notion of a function
- square root (cube root, …)
- inequalities (list of inequalities)
- power laws (i.e. $a^b a^c = a^{b+c}$ )
- Cartesian coordinate plane
- basic set theory
- random variable
- probability distribution
- histogram
- the mean, expected value & strong law of large numbers
- the graph of a function
- standard deviation
- Pythagorean theorem
- vectors and vector spaces
- limits
- real numbers as limits of fractions, the least upper bound
- continuity
- $R^n$, Euclidean Space, and Hilbert spaces (inner or dot product)
- derivative
- correlation
- central limit theorem, Gaussian Distribution, Properties of Guassains.
- integrals
- chain rule
- modular arithmetic
- sine cosine tangent
- $\pi$, circumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,…
- linear regression
- Taylor’s theorem
- the number e and the exponential function
- Rolle’s theorem, Karush–Kuhn–Tucker conditions, derivative is zero at the maximum
- the notion of linearity
- Big O notation
- injective (one-to-one) / surjective (onto) functions
- imaginary numbers
- symmetry
- Euler’s Formula $e^{i \pi} + 1 = 0$
- Fourier transform, convolution in time domain is the product in the frequency domain (& vice versa), the FFT
- fundamental theorem of calculus
- logarithms
- matrices
- conic sections
- Boolean algebra
- Cauchy–Schwarz inequality
- binomial theorem – Pascal’s triangle
- the determinant
- ordinary differential equation (ODE)
- mode (maximum likelihood estimator)
- cosine law
- prime numbers
- linear independence
- Jacobian
- fundamental theorem of arithmetic
- duality – (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, …)
- intermediate value theorem
- eigenvalues
- median
- entropy
- KL distance
- binomial distribution
- Bayes’ theorem
- $2^{10} \approx 1000$
- compactness, Heine – Borel theorem
- metric space, Triangle Inequality
- Projections, Best Approximation
- $1/(1-X) = 1 + X + X^2 + \ldots$
- partial differential equations
- quadratic formula
- Reisz representation theorem
- Fubini’s theorem
- the ideas of groups, semigroups, monoids, rings, …
- Singular Value Decomposition
- numeric integration – trapezoidal rule, Simpson’s rule, …
- mutual information
- Plancherel’s theorem
- matrix condition number
- integration by parts
- Euler’s method for numerical integration of ODEs (and improved Euler & Runge–Kutta)
- pigeon hole principle

Gauss did a lot of math, but sometimes I am surprised when I find something new to me done by Gauss

$\tan(x) = { 1 \over{1/x \ – {1\over{3/x \ – {1\over{5/x\ – \ldots}}}}}}$