More md -> mdx conversions.

Author: wiseodd

src/content/post/brouwers-fixed-point.md

@@ -0,0 +1,55 @@
+---
+title: "Brouwer's Fixed Point Theorem: A Proof with Reduced Homology"
+description: "A proof of special case (ball) of Brouwer's Fixed Point Theorem with Reduced Homology."
+publishDate: 2018-07-18 10:00
+tags: [math]
+---
+import BlogImage from '@/components/BlogImage.astro'
+
+This post is about the proof I found very interesting during the Topology course I took this semester. It highlights the application of Reduced Homology, which is a modification of Homology theory in Algebraic Topology. We will use two results from Reduced Homology as black-boxes for the proof. Everywhere, we will assume $ \mathbb{Q} $ is used as the coefficient of the Homology space.
+
+**Lemma 1 (Reduced Homology of spheres)**
+Given a $ d $-sphere $ \mathbb{S}^d $, then its reduced $ p $-th Homology space is:
+
+$$
+\tilde{H}_p(\mathbb{S}^d) = \begin{cases} \mathbb{Q}, & \text{if } p = d \\ 0, & \text{otherwise} \enspace . \end{cases}
+$$
+
+$$
+\qed
+$$
+
+**Lemma 2 (Reduced Homology of balls)**
+Given a $ d $-ball $ \mathbb{B}^d $, then its reduced $ p $-th Homology space is trivial, i.e. $\tilde{H}_p(\mathbb{B}^d) = 0 $, for any $ d $ and $ p $.
+
+$$
+\qed
+$$
+
+Equipped with these lemmas, we are ready to prove the special case of Brouwer's Fixed Point Theorem, where we consider map from a ball to itself.
+
+**Brouwer's Fixed Point Theorem**
+Given $ f: \mathbb{B}^{d+1} \to \mathbb{B}^{d+1} $ continuous, then there exists $ x
+\in \mathbb{B}^{d+1} $ such that $ f(x) = x $.
+
+_Proof._    For contradiction, assume $ \forall x \in \mathbb{B}^{d+1}: f(x) \neq x $. We construct a map $ r: \mathbb{B}^{d+1} \to \mathbb{S}^d $, casting ray from the ball to its shell by extending the line segment between $ x $ and $ f(x) $.
+
+<BlogImage imagePath='/img/brouwers-fixed-point/map_r.svg' altText="Casting a ray to the ball's shell." />
+
+Observe that $ r(x) $ is continuous because $ f(x) $ is. Also, $ x \in \mathbb{S}^d \implies r(x) = x $. Therefore we have the following commutative diagram.
+
+<BlogImage imagePath='/img/brouwers-fixed-point/comm_diag.svg' altText="Commutative diagram." />
+
+Above, $ i $ is inclusion map, and $ id $ is identity map. We then look of the Reduced Homology of the above, and this gives us the following commutative diagram.
+
+<BlogImage imagePath='/img/brouwers-fixed-point/comm_diag_hom.svg' altText="Reduced homology." />
+
+As the diagram commute, then $ \tilde{H}_d(\mathbb{S}^d) \xrightarrow{i^*} \tilde{H}_d(\mathbb{B}^{d+1}) \xrightarrow{r^*} \tilde{H}_d(\mathbb{S}^d) $ should be identity map on $ \tilde{H}_d(\mathbb{S}^d) $. By Lemma 2, $ \tilde{H}_d(\mathbb{B}^{d+1}) = 0 $. This implies $ \tilde{H}_d(\mathbb{S}^d) = 0 $. But this is a contradiction, as By Lemma 1, $ \tilde{H}_d(\mathbb{S}^d) = \mathbb{Q} $. Therefore there must be a fixed point.
+
+$$
+\qed
+$$
+
+## References
+
+1. Hatcher, Allen. "Algebraic topology." (2001).

src/content/post/brouwers-fixed-point.mdx

@@ -5,7 +5,7 @@ publishDate: 2018-03-11 07:00
 tags: [machine learning]
 ---
 
-Suppose we have a model parameterized by parameter vector \\( \theta \\) that models a distribution \\( p(x \vert \theta) \\). In frequentist statistics, the way we learn \\( \theta \\) is to maximize the likelihood \\( p(x \vert \theta) \\) wrt. parameter \\( \theta \\). To assess the goodness of our estimate of \\( \theta \\) we define a score function:
+Suppose we have a model parameterized by parameter vector $ \theta $ that models a distribution $ p(x \vert \theta) $. In frequentist statistics, the way we learn $ \theta $ is to maximize the likelihood $ p(x \vert \theta) $ wrt. parameter $ \theta $. To assess the goodness of our estimate of $ \theta $ we define a score function:
 
 $$
 s(\theta) = \nabla_{\theta} \log p(x \vert \theta) \, ,
@@ -16,7 +16,7 @@ that is, score function is the gradient of log likelihood function. The result a
 **Claim:**
 The expected value of score wrt. our model is zero.
 
-_Proof._    Below, the gradient is wrt. \\( \theta \\).
+_Proof._    Below, the gradient is wrt. $ \theta $.
 
 $$
 \begin{align}
@@ -40,13 +40,13 @@ $$
     \mathop{\mathbb{E}}_{p(x \vert \theta)} \left[ (s(\theta) - 0) \, (s(\theta) - 0)^{\text{T}} \right] \, .
 $$
 
-We can then see it as an information. The covariance of score function above is the definition of Fisher Information. As we assume \\( \theta \\) is a vector, the Fisher Information is in a matrix form, called Fisher Information Matrix:
+We can then see it as an information. The covariance of score function above is the definition of Fisher Information. As we assume $ \theta $ is a vector, the Fisher Information is in a matrix form, called Fisher Information Matrix:
 
 $$
     \text{F} = \mathop{\mathbb{E}}_{p(x \vert \theta)} \left[ \nabla \log p(x \vert \theta) \, \nabla \log p(x \vert \theta)^{\text{T}} \right] \, .
 $$
 
-However, usually our likelihood function is complicated and computing the expectation is intractable. We can approximate the expectation in \\( \text{F} \\) using empirical distribution \\( \hat{q}(x) \\), which is given by our training data \\( X = \\{ x_1, x_2, \cdots, x_N \\} \\). In this form, \\( \text{F} \\) is called Empirical Fisher:
+However, usually our likelihood function is complicated and computing the expectation is intractable. We can approximate the expectation in $ \text{F} $ using empirical distribution $ \hat{q}(x) $, which is given by our training data $ X = \{ x_1, x_2, \cdots, x_N \} $. In this form, $ \text{F} $ is called Empirical Fisher:
 
 $$
 \begin{align}
@@ -56,10 +56,10 @@ $$
 
 ## Fisher and Hessian
 
-One property of \\( \text{F} \\) that is not obvious is that it has the interpretation of being the negative expected Hessian of our model's log likelihood.
+One property of $ \text{F} $ that is not obvious is that it has the interpretation of being the negative expected Hessian of our model's log likelihood.
 
 **Claim:**
-The negative expected Hessian of log likelihood is equal to the Fisher Information Matrix \\( \text{F} \\).
+The negative expected Hessian of log likelihood is equal to the Fisher Information Matrix $ \text{F} $.
 
 _Proof._    The Hessian of the log likelihood is given by the Jacobian of its gradient:
 
@@ -85,19 +85,19 @@ $$
 \end{align}
 $$
 
-Thus we have \\( \text{F} = -\mathop{\mathbb{E}}\_{p(x \vert \theta)} \left[ \text{H}\_{\log p(x \vert \theta)} \right] \\).
+Thus we have $ \text{F} = -\mathop{\mathbb{E}}\_{p(x \vert \theta)} \left[ \text{H}\_{\log p(x \vert \theta)} \right] $.
 
 $$
 \qed
 $$
 
-Indeed knowing this result, we can see the role of \\( \text{F} \\) as a measure of curvature of the log likelihood function.
+Indeed knowing this result, we can see the role of $ \text{F} $ as a measure of curvature of the log likelihood function.
 
 ## Conclusion
 
-Fisher Information Matrix is defined as the covariance of score function. It is a curvature matrix and has interpretation as the negative expected Hessian of log likelihood function. Thus the immediate application of \\( \text{F} \\) is as drop-in replacement of \\( \text{H} \\) in second order optimization methods.
+Fisher Information Matrix is defined as the covariance of score function. It is a curvature matrix and has interpretation as the negative expected Hessian of log likelihood function. Thus the immediate application of $ \text{F} $ is as drop-in replacement of $ \text{H} $ in second order optimization methods.
 
-One of the most exciting results of \\( \text{F} \\) is that it has connection to KL-divergence. This gives rise to natural gradient method, which we shall discuss further in the next article.
+One of the most exciting results of $ \text{F} $ is that it has connection to KL-divergence. This gives rise to natural gradient method, which we shall discuss further in the next article.
 
 ## References
 

src/content/post/fisher-information.md src/content/post/fisher-information.mdxsrc/content/post/fisher-information.md renamed to src/content/post/fisher-information.mdx

@@ -0,0 +1,60 @@
+---
+title: 'Reduced Betti number of sphere: Mayer-Vietoris Theorem'
+description: 'A proof of reduced homology of sphere with Mayer-Vietoris sequence.'
+publishDate: 2018-07-23 10:00
+tags: [math]
+---
+
+import BlogImage from '@/components/BlogImage.astro'
+
+In the previous post about Brouwer's Fixed Point Theorem, we used two black boxes. In this post we will prove the slight variation of those black boxes. We will start with the simplest lemma first: the reduced homology of balls.
+
+**Lemma 2 (Reduced homology of balls)**  
+Given a $ d $-ball $ \mathbb{B}^d $, then its reduced $ p $-th homology space is trivial, i.e. $\tilde{H}_p(\mathbb{B}^d) = 0 $, for any $ d $ and $ p $.
+
+_Proof._    Observe that $ \mathbb{B}^d $ is contractible, i.e. homotopy equivalent to a point. Assuming we use coefficient $ \mathbb{Q} $, we know the zero-th homology space of point is $ H_0(\, \cdot \,, \mathbb{Q}) = \mathbb{Q} $, and trivial otherwise, i.e. $ H_p (\, \cdot \,, \mathbb{Q}) = 0 \enspace \forall p \geq 1 $.
+
+In the reduced homology, therefore $ \tilde{H}_0(\, \cdot \,, \mathbb{Q}) = 0 $. Thus the reduced homology of balls is trivial for all $ d, p $.
+
+$$
+\qed
+$$
+
+**Corollary 1 (Reduced Betti numbers of balls)**  
+The $ p $-th reduced Betti numbers of $ \mathbb{B}^d $ is zero for all $d, p$.
+
+$$
+\qed
+$$
+
+Now, we are ready to prove the main theme of this post.
+
+**Lemma 1 (Reduced Betti numbers of spheres)**  
+Given a $ d $-sphere $ \mathbb{S}^d $, then its $ p $-th reduced Betti number is:
+
+$$
+\tilde{\beta}_p(\mathbb{S}^d) = \begin{cases} 1, & \text{if } p = d \\ 0, & \text{otherwise} \enspace . \end{cases}
+$$
+
+_Proof._    We use "divide-and-conquer" approach to apply Mayer-Vietoris Theorem. We cut the sphere along the equator and note that the upper and lower portion of the sphere is just a disk, and the intersection between those two parts is a circle (sphere one dimension down), as shown in the figure below.
+
+<BlogImage
+ imagePath='/img/mayer-vietoris-sphere/sphere.svg'
+ altText='Mayer-Vietoris on a sphere.'
+/>
+
+By Mayer-Vietoris Theorem, we have a long exact sequence in the form of:
+
+$$
+\dots \longrightarrow \tilde{H}_p(\mathbb{S}^{d-1}) \longrightarrow \tilde{H}_p(\mathbb{B}^d) \oplus \tilde{H}_p(\mathbb{B}^d) \longrightarrow \tilde{H}_p(\mathbb{S}^d) \longrightarrow \tilde{H}_{p-1}(\mathbb{S}^{d-1}) \longrightarrow \dots \enspace .
+$$
+
+By Corollary 1, $ \tilde{H}_p(\mathbb{B}^d) \oplus \tilde{H}_p(\mathbb{B}^d) = \tilde{H}_{p-1}(\mathbb{B}^d) \oplus \tilde{H}_{p-1}(\mathbb{B}^d) = 0 $. As the sequence is exact, therefore $ \tilde{H}_p(\mathbb{S}^d) \longrightarrow \tilde{H}_{p-1}(\mathbb{S}^{d-1}) $ is a bijection, and thus an isomorphism. Then by induction with base case of $ \mathbb{S}^0 $, we conclude that the claim holds.
+
+$$
+\qed
+$$
+
+## References
+
+1. Hatcher, Allen. "Algebraic topology." (2001).