###### tags: `one-offs` `linear algebra`
# A Mnemonic for Schur Complements
**Overview**: In this note, I describe a quick strategy for recalling the form of the Schur complement decomposition of a positive semidefinite matrix.
## The Technique
For some time, I had difficulty remembering the exact form of the Schur complement formula. I thus began working with the following mnemonic: given a positive semidefinite matrix $M$, write it in block form as
\begin{align}
M=\left[\begin{array}{cc}
A & B\\
B^{\top} & C
\end{array}\right],
\end{align}
where $A$ and $C$ are positive semidefinite. Writing the state variable as $z=\left(x,y\right)$, define
\begin{align}
f\left(z\right) &=\frac{1}{2}z^{\top}Mz \\
&=\frac{1}{2}\left(x^{\top}Ax+x^{\top}By+y^{\top}B^{\top}x+y^{\top}Cy\right).
\end{align}
Now, minimise $f\left(x,y\right)$ with respect to $y$, with $x$ held fixed. The first-order stationarity condition imposes that $B^{\top}x+Cy=0$, and so we set $y=-C^{-1}B^{\top}x$ to see that
\begin{align}
f\left(x,y\right) &\geqslant \frac{1}{2}\left(x^{\top}Ax-x^{\top}BC^{-1}B^{\top}x-\left(C^{-1}B^{\top}x\right)^{\top}B^{\top}x+\left(C^{-1}B^{\top}x\right)^{\top}C\left(C^{-1}B^{\top}x\right)\right) \\
&= \frac{1}{2}x^{\top}\left(A-BC^{-1}B^{\top}\right)x \\
&=: \frac{1}{2}x^{\top}Sx.
\end{align}
By construction, S must itself be positive semidefinite. Piecing things back together, one can then write that
\begin{align}
f\left(x,y\right) =\frac{1}{2}x^{\top}Sx+\frac{1}{2}\left(y+C^{-1}B^{\top}x\right)^{\top}C\left(y+C^{-1}B^{\top}x\right),
\end{align}
which somehow gives the full story: $M$ is PSD iff both of $S$ and $C$ are, and $S$ has this specific form.
I like this approach as a mnemonic because one simply has to remember “minimise $y$ given $x$”, and routine linear algebra does the rest of the work.

Overview: In this note, I log some basic observations about diffusion-based generative models.

8/14/2023Overview: In this note, I describe some aspects of hierarchical structure in MCMC algorithms, and how they can be of theoretical and practical relevance.

8/9/2023Overview: In this note, I discuss a recurrent question which can be used to generate research questions about methods of all sorts. I then discuss a specific instance of how this question has proved fruitful in the theory of optimisation algorithms. Methods and Approximations A nice story is that when Brad Efron derived the bootstrap, it was done in service of the question “What is the jackknife an approximation to?”. I can't help but agree that there's something quite exciting about research questions which have this same character of ''What is (this existing thing) an approximation to?''. One bonus tilt on this which I appreciate is that there can be multiple levels of approximation, and hence many answers to the same question. One well-known example is gradient descent, which can be viewed as an approximation to the proximal point method, which can then itself be viewed as an approximation to a gradient flow. There are probably even more stops along the way here. In this case, there is even the perspective that from the perspective of mathematical theory, there may be at least as much to be gained by stopping off at the proximal point interpretation, as there is from the gradient flow perspective. My experience is that generalist applied mathematicians get to grips with the gradient flow quickly, but optimisation theorists can squeeze more out of the PPM formulation. There is thus some hint that using this 'intermediate' approximation can be particularly insightful in its own right. It would be interesting to collect more examples with this character.

5/22/2023Overview: In this note, I prove Hoeffding's inequality from the perspectives of martingales and convex ordering. The Basic Construction Let $-\infty<a<x<b<\infty$, and define a random variable $M$ with law $M\left(x;a,b\right)$ by \begin{align} M=\begin{cases} a & \text{w.p. }\frac{b-x}{b-a}\ b & \text{w.p. }\frac{x-a}{b-a}. \end{cases}

5/22/2023
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