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<li><a href="../slides/transformer_1.remark.html">Transformer</a></li>
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network</a></li>
<li><a href="../slides/spectral.remark.html">Spectral methods</a></li>
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<h2 id="preface">Preface</h2>
<p>Graph laplacian is familiar to computation science researcher, with
which we can perform spectral analysis such as diffusion map, eigenmap
or spectral clustering. Here, we discuss how to generalize the graph
laplacian to it's high-order form, i.e., Hodge laplacian.</p>
<h2 id="graph-laplacian">Graph laplacian</h2>
<p>In the previous post we know that the graph laplacian can be obtianed
by degree matrix <span class="math inline">\(D\)</span> and adjacency
matrix <span class="math inline">\(A\)</span> such that: <span
class="math display">\[\begin{equation}
L_0 = D - A
\end{equation}\]</span></p>
<p>We can find the adjacency matrix and degree matrix of the graph blow
with nine vertices: <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/hodge_graph_9_nodes.svg" /></p>
<p>such that:</p>
<p><span class="math display">\[\begin{equation}
A = \begin{array}[r]{c |c c c c c c c c c }
& 1 & 2 & 3 & 4 & 5 & 6 & 7 &
8 & 9\\
\hline
1& 0 & 1 & 0 & 0 & 0 & 0 & 0
& 0 & 0\\
2& 1 & 0 & 1 & 0 & 0 & 0 & 0
& 0 & 0\\
3& 0 & 1 & 0 & 0 & 0 & 0 & 0
& 0 & 0\\
4& 0 & 0 & 0 & 0 & 1 & 1 & 0
& 0 & 0\\
5& 0 & 0 & 0 & 1 & 0 & 1 & 0
& 0 & 0\\
6& 0 & 0 & 0 & 1 & 1 & 0 & 1
& 0 & 0\\
7& 0 & 0 & 0 & 0 & 0 & 1 & 0
& 0 & 0\\
8& 0 & 0 & 0 & 0 & 0 & 0 & 0
& 0 & 1\\
9& 0 & 0 & 0 & 0 & 0 & 0 & 0
& 1 & 0\\
\end{array}
\end{equation}\]</span></p>
<p>and</p>
<p><span class="math display">\[\begin{equation}
D = \begin{array}[r]{c |c c c c c c c c c }
& 1 & 2 & 3 & 4 & 5 & 6 & 7 &
8 & 9\\
\hline
1& 1 & 0 & 0 & 0 & 0 & 0 & 0
& 0 & 0\\
2& 0 & 2 & 0 & 0 & 0 & 0 & 0
& 0 & 0\\
3& 0 & 0 & 1 & 0 & 0 & 0 & 0
& 0 & 0\\
4& 0 & 0 & 0 & 2 & 0 & 0 & 0
& 0 & 0\\
5& 0 & 0 & 0 & 0 & 2 & 0 & 0
& 0 & 0\\
6& 0 & 0 & 0 & 0 & 0 & 3 & 0
& 0 & 0\\
7& 0 & 0 & 0 & 0 & 0 & 0 & 1
& 0 & 0\\
8& 0 & 0 & 0 & 0 & 0 & 0 & 0
& 1 & 0\\
9& 0 & 0 & 0 & 0 & 0 & 0 & 0
& 0 & 1\\
\end{array}
\end{equation}\]</span></p>
<p>And the graph laplacian thus can be calculated as:</p>
<p><span class="math display">\[\begin{equation}
L_0 = D - A = \begin{array}[r]{c |c c c c c c c c c }
& 1 & 2 & 3 & 4 & 5 & 6 & 7 &
8 & 9\\
\hline
1& 1 & -1 & 0 & 0 & 0 & 0 & 0
& 0 & 0\\
2& -1 & 2 & -1 & 0 & 0 & 0 & 0
& 0 & 0\\
3& 0 & -1 & 1 & 0 & 0 & 0 & 0
& 0 & 0\\
4& 0 & 0 & 0 & 2 & -1 & -1 & 0
& 0 & 0\\
5& 0 & 0 & 0 & -1 & 2 & -1 & 0
& 0 & 0\\
6& 0 & 0 & 0 & -1 & -1 & 3 & -1
& 0 & 0\\
7& 0 & 0 & 0 & 0 & 0 & -1 & 1
& 0 & 0\\
8& 0 & 0 & 0 & 0 & 0 & 0 & 0
& 1 & -1\\
9& 0 & 0 & 0 & 0 & 0 & 0 & 0
& -1 & 1\\
\end{array}
\end{equation}\]</span></p>
<p>However, there's anoth way to represent the graph laplacian. Before
that, we first introduce the incidence matrix of a graph. First, we can
convert an undirected graph to be directed just assign the direction of
edges by book keeping order. Next, let's construct the incidence matrix,
each row represent a vertex, and each colum is an edge. We set the entry
to be 1 if the the edge enter the vertex -1 when leave the vertex. Set 0
if there's no connection between a vertex and an edge. Thus the
incidence matrix of a graph <span class="math inline">\(G\)</span> is
<span class="math inline">\(B\)</span> such that: <span
class="math display">\[\begin{equation}
B_1 =
\begin{array}[r]{c | c c c c c c c}
& \cdots & [4,5] & [4,6]& [5,6]&
[6,7]& \cdots\\
\hline
\vdots & \cdots & & \cdots & &
\cdots & \\
4 & \cdots & -1 & -1 & 0 & 0
& \cdots\\
5 & \cdots & 1 & 0 & -1 & 0 &
\cdots\\
6 & \cdots & 0 & 1 & 1 & -1 &
\cdots\\
7 & \cdots & 0 & 0 & 0 & 1 &
\cdots\\
\vdots & \cdots & & \cdots & &
\cdots & \\
\end{array}
\end{equation}\]</span> Next, let construct graph laplacian matrix using
<span class="math inline">\(B_1\)</span>. Actually, the graph laplacian
is： <span class="math display">\[\begin{equation}
L_0=B_1 B_1^\top = \begin{array}[r]{c |c c c c c c c c c }
& 1 & 2 & 3 & 4 & 5 & 6 & 7 &
8 & 9\\
\hline
1& 1 & -1 & 0 & 0 & 0 & 0 & 0
& 0 & 0\\
2& -1 & 2 & -1 & 0 & 0 & 0 & 0
& 0 & 0\\
3& 0 & -1 & 1 & 0 & 0 & 0 & 0
& 0 & 0\\
4& 0 & 0 & 0 & 2 & -1 & -1 & 0
& 0 & 0\\
5& 0 & 0 & 0 & -1 & 2 & -1 & 0
& 0 & 0\\
6& 0 & 0 & 0 & -1 & -1 & 3 & -1
& 0 & 0\\
7& 0 & 0 & 0 & 0 & 0 & -1 & 1
& 0 & 0\\
8& 0 & 0 & 0 & 0 & 0 & 0 & 0
& 1 & -1\\
9& 0 & 0 & 0 & 0 & 0 & 0 & 0
& -1 & 1\\
\end{array}
\end{equation}\]</span> As we can see that, <span
class="math inline">\(L_0\)</span> is the same as the that obtained from
degree matrix and adjacency matrix.</p>
<h2 id="zero-eigenvaules-of-graph-laplacian">Zero eigenvaules of graph
laplacian</h2>
<p>We know that for the application of diffusion map or spectral
analysis, we have to drop the first eigenvector due to its same values
and the eigenvaule is 0. The second eigenvector is also called fidler
vector. However, these applications usually created a connected graph.
Since the graph we show here is an unconnected graph. We can stop here
to take a guess: Is the zero eigenvaule still there, or if so, how many
zero eigenvaules?</p>
<p>Now, let's perform the eigenvaule decomposition that: <span
class="math display">\[\begin{equation}
L_0 = Q\Lambda Q^\top
\end{equation}\]</span></p>
<p>Interestingly, the top 3 eigenvalues are all zero and we also have 3
connected components in the graph. Is this a coincidence or a property
of graph laplacian decomposition? <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/hodge_graph_9_nodes_eigenvalues.svg" /></p>
<p>Furthermore, the top 3 eigenvectors corresponding to the zero
eigenvaules are also interesting. Notice that the first eigenvector can
differentiate the components <span
class="math inline">\(\{4,5,6,7\}\)</span> from the rest vectices.
Likewise, the second eigenvector select the connected component <span
class="math inline">\(\{8,9\}\)</span> and the third choose the
component <span class="math inline">\(\{1,2,3\}\)</span>. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/hodge_graph_9_nodes_eigenvectors.svg" /></p>
<p>In fact, in the field of Topological Data Analysis (TDA), the <span
class="math inline">\(L_0\)</span> is the special case of hodge
laplacian (<span class="math inline">\(L_k\)</span>). The number of
connected components is call 0-dimensional cycles. And the graph
laplaican can capture these cycles. The 1-dimensional cycles are
correspond to holes, we will go into detail in the next section.</p>
<h2 id="hodge-laplacian">Hodge laplacian</h2>
<p>We can see the graph laplacian zero-order of hodge laplacian, and the
formalu can be represented as: <span
class="math display">\[\begin{equation}
L_0 = \mathbf{0}^\top \mathbf{0} + B_1 B_1^\top
\end{equation}\]</span></p>
<p>Similarly, we can obtian <span
class="math inline">\(L_1\)</span>:</p>
<p><span class="math display">\[\begin{equation}
L_1 = B_1^\top B_1 + B_2 B_2^\top
\end{equation}\]</span></p>
<p>You must ask where is <span class="math inline">\(B_2\)</span> coming
from? We know that <span class="math inline">\(B_1\)</span> captures the
relationship between vertices and edges. Thus, <span
class="math inline">\(B_2\)</span> captures the relationship between
edges and triangles. We can also define <span
class="math inline">\(B_3\)</span> to capture relationship between
triangles and tetrahedron and so on and so forth. So what is a triangle
or a tetrahedron in a graph? We would not go into the detail of the
thory of Simplex and Simplicial complex. Here we just need to know that
three connected vertices forms a triangle. Similarly, four connected
vertices forms a tetrahedron which is a high-order of triangles. To
define <span class="math inline">\(B_2\)</span>, of a graph <span
class="math inline">\(G\)</span>, we would check the direction of an
edge <span class="math inline">\(e_j\)</span> to the triangle <span
class="math inline">\(\bigtriangleup_q\)</span> it beblongs, if it has
the same direction as the triangle, the entry would be <span
class="math inline">\(1\)</span>, if the direction is opposite, the
entry would be -1, otherwise the entry would be zero. Specifically:</p>
<p><span class="math display">\[\begin{equation}
{B}_2[j, q] =
\begin{cases}
1 & \text{if } e_j \in \bigtriangleup_q \quad \text{with}\quad
\text{same}\quad \text{direction} \\
-1 & \text{if } e_j \in \bigtriangleup_q \quad \text{with}\quad
\text{opposite}\quad \text{direction} \\
0 & \text{otherwise}
\end{cases}
\end{equation}\]</span></p>
<p>With the definition, we can obtian <span
class="math inline">\(B_2\)</span> of the graph aforementioned: <span
class="math display">\[\begin{equation}
B_2 =
\begin{array}[r]{c | c }
& [4,5,6]\\
\hline
\vdots & \cdots \\
[4,5] & 1\\
[4,6] & -1\\
[5,6]& 1\\
[6,7] & 0\\
\vdots & \cdots \\
\end{array}
\end{equation}\]</span></p>
<p>We next introduce the normalized form and the decomposition of hodge
1-laplacian. The normalized form of hodge 1-laplacian is given:</p>
<p><span class="math display">\[\begin{equation}
\mathcal{L}_1 = {D}_2 {B}_1^\top {D}_1^{-1} {B}_1 + {B}_2 {D}_3
{B}_2^\top {D}_2^{-1}
\end{equation}\]</span> where <span
class="math inline">\(\mathbf{D}_1\)</span> is the vertices degree
matrix, <span class="math inline">\({D}_2\)</span> is <span
class="math inline">\(\max{(\text{diag}(|{B}_2| \mathbf{1}),
\mathbf{I})}\)</span> and <span class="math inline">\({D}_3\)</span> is
<span class="math inline">\(\frac{1}{3}\mathbf{I}\)</span>. Since the
normalized <span class="math inline">\(L_1\)</span> is not neccessarily
symmetric, we next need to define the symmetric normalized Hodge
1-Laplacian such that: <span class="math display">\[\begin{equation}
\begin{aligned}
\mathcal{L}_1^s
& = {D}_2^{-1/2} \mathcal{L}_1 {D}_2^{1/2}\\
& = {D}_2^{1/2} {B}_1^\top {D}_1^{-1} {B}_1 {D}_2^{1/2} +
{D}^{-1/2} {B}_2 {D}_3 {B}_2^\top {D}_2^{-1/2}
\end{aligned}
\end{equation}\]</span></p>
<p>We use the graph with three holes to present hodge 1-laplacian: <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/hodge_graph_3holes.svg" /></p>
<p>We next can perform eigenvalues decomposition on <span
class="math inline">\(\mathcal{L}_1\)</span>: <span
class="math display">\[\begin{equation}
\begin{aligned}
\mathcal{L}_1
& = \mathbf{D}_2^{1/2} \mathcal{L}_1^s
\mathbf{D}_2^{-1/2} \\
& = \mathbf{D}_2^{1/2} Q \Lambda Q^\top
\mathbf{D}_2^{-1/2} \\
& = \mathbf{U} \Lambda \mathbf{U}^{-1}
\end{aligned}
\end{equation}\]</span> Interestingly, the top 3 eigenvector also all
zero which corresponding to the 3 holes, namely, the three 1-dimensional
cycles. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/hodge_graph_3holes_eigenvalues.svg" /></p>
<p>When it comes to the eigenvectors, we can also notice that the top
three eigenvectors are around the three holes. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/hodge_graph_3holes_eigenvectors.svg" /></p>
<p>In algebraic geometry, these the eigenvectors with zero eigenvaules
are called harmonic function or harmonic. These harmonic function around
holes is useful for some analysis like clustering to find the
1-dimensional cycles etc.</p>
<h2 id="conclusion">Conclusion</h2>
<p>Today, we review the graph laplacian, and we can find the zeor
eigenvalues and theirs corresponding eigenvectors can be used to find
connected components. The high order graph laplacian hodge laplacian
have the similar properties, we presented the hodge 1-laplaican and its
eigenvalues decomposition. We can find that the zero eigenvalues
indicates the number of holes of a graph. Furthermore, the corresponding
eigenvectors with zero eigenvalues are around holes.</p>
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<h2 id="diffussion-map">Diffussion map</h2>
<p>Eigenvalue decomposition of weighted graph laplacian is used to
obtain diffusion map. The asymmetric matrix or transition matrix <span
class="math inline">\(P=D^{-1}W\)</span>. The decomposition is: <span
class="math display">\[\begin{equation}
P = D^{-1/2}S\Lambda S^\top D^{1/2}
\end{equation}\]</span> In fact, the random walk on the graph would give
rise to the transition graph such that: <span
class="math display">\[\begin{equation}
p^{t+1}= p^t D^{-1}W = p^t P
\end{equation}\]</span> where <span class="math inline">\(p\)</span> is
a initial state of the graph (vertices weight). That is, after a certain
step of random walk on the graph, we would reach a steady state when any
more random walk would not change the weight. Let <span
class="math inline">\(Q=D^{-1/2}S\)</span>, then <span
class="math inline">\(Q^{-1}=S^{\top} D^{1/2}\)</span>, we can derive
<span class="math inline">\(P = Q\Lambda Q^{-1}\)</span>. The random
walk above mentioned can then be represent: <span
class="math display">\[\begin{equation}
\begin{aligned}
P^t &= Q\Lambda Q^{-1}Q\Lambda Q^{-1}\cdots Q\Lambda Q^{-1}
&= Q\Lambda^t Q
\end{aligned}
\end{equation}\]</span> Since <span
class="math inline">\(\Lambda\)</span> is diagnoal, the random walk on a
graph can be easily cacluted by using the eigenvalue decomposition
outcome. The column of <span class="math inline">\(Q\)</span> is the
diffusion map dimensions.</p>
<h2 id="diffusion-map-example">Diffusion map example</h2>
<p>To understand diffusion map, we here introduce an example data and
run diffusion map on it. The data is a single cell fibroblasts to neuron
data with 392 cells. We first download the data from NCBI and loaded the
data to memory:</p>
<figure class="highlight r"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line">import requests</span><br><span class="line">url <span class="operator">=</span> <span class="string">'https://www.ncbi.nlm.nih.gov/geo/download/?acc=GSE67310&format=file&file=GSE67310%5FiN%5Fdata%5Flog2FPKM%5Fannotated.txt.gz'</span></span><br><span class="line">r <span class="operator">=</span> requests.get<span class="punctuation">(</span>url<span class="punctuation">,</span> allow_redirects<span class="operator">=</span>True<span class="punctuation">)</span></span><br><span class="line">open<span class="punctuation">(</span><span class="string">'GSE67310.txt.gz'</span><span class="punctuation">,</span> <span class="string">'wb'</span><span class="punctuation">)</span>.write<span class="punctuation">(</span>r.content<span class="punctuation">)</span></span><br><span class="line">data <span class="operator">=</span> pd.read_csv<span class="punctuation">(</span><span class="string">'GSE67310.txt.gz'</span><span class="punctuation">,</span> sep<span class="operator">=</span><span class="string">'\t'</span><span class="punctuation">,</span> index_col<span class="operator">=</span><span class="number">0</span><span class="punctuation">)</span></span><br><span class="line">data <span class="operator">=</span> data.loc<span class="punctuation">[</span>data<span class="punctuation">[</span><span class="string">'assignment'</span><span class="punctuation">]</span> <span class="operator">!=</span><span class="string">'Fibroblast'</span><span class="punctuation">]</span></span><br><span class="line">group <span class="operator">=</span> data<span class="punctuation">[</span><span class="string">'assignment'</span><span class="punctuation">]</span></span><br></pre></td></tr></table></figure>
<p>From the code as we can see that, we extract the cell types from the
data and stores it into the variable <code>group</code>. The rest part
of the data is the normalized gene expression count matrix.</p>
<p>To do the pre-processing, we first get the log-normalized count
matrix <span class="math inline">\(X\)</span> and revert it to the
counts and apply log geomatric scaling to the count matrix as the
pre-processing then store it to matrix <span
class="math inline">\(Y\)</span>. <figure class="highlight r"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">X <span class="operator">=</span> np.array<span class="punctuation">(</span>data.iloc<span class="punctuation">[</span><span class="operator">:</span><span class="punctuation">,</span> <span class="number">5</span><span class="operator">:</span><span class="punctuation">]</span><span class="punctuation">)</span>.T</span><br><span class="line">X <span class="operator">=</span> np.power<span class="punctuation">(</span><span class="number">2</span><span class="punctuation">,</span> X<span class="punctuation">[</span>np.apply_along_axis<span class="punctuation">(</span>np.var<span class="punctuation">,</span> <span class="number">1</span><span class="punctuation">,</span> X<span class="punctuation">)</span><span class="operator">></span><span class="number">0</span><span class="punctuation">,</span> <span class="operator">:</span><span class="punctuation">]</span><span class="punctuation">)</span> <span class="operator">-</span> <span class="number">1</span></span><br><span class="line">Y <span class="operator">=</span> np.log<span class="punctuation">(</span>gscale<span class="punctuation">(</span>X<span class="operator">+</span><span class="number">0.5</span><span class="punctuation">)</span><span class="punctuation">)</span>.T</span><br></pre></td></tr></table></figure> The geomatric scaling
implementation is: <figure class="highlight r"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line">def gscale<span class="punctuation">(</span>X<span class="operator">:</span>np.ndarray<span class="punctuation">)</span> <span class="operator">-></span> np.ndarray<span class="operator">:</span></span><br><span class="line"> assert<span class="punctuation">(</span>X.all<span class="punctuation">(</span><span class="punctuation">)</span><span class="operator">>=</span><span class="number">0</span><span class="punctuation">)</span></span><br><span class="line"> div_ <span class="operator">=</span> np.divide<span class="punctuation">(</span>X.T<span class="punctuation">,</span> np.apply_along_axis<span class="punctuation">(</span>lambda x<span class="operator">:</span>np.exp<span class="punctuation">(</span>np.mean<span class="punctuation">(</span>np.log<span class="punctuation">(</span>x<span class="punctuation">)</span><span class="punctuation">)</span><span class="punctuation">)</span><span class="punctuation">,</span> <span class="number">1</span><span class="punctuation">,</span></span><br><span class="line"> X<span class="punctuation">)</span><span class="punctuation">)</span>.T</span><br><span class="line"> scale_ <span class="operator">=</span> np.apply_along_axis<span class="punctuation">(</span>np.median<span class="punctuation">,</span><span class="number">0</span><span class="punctuation">,</span> div_<span class="punctuation">)</span></span><br><span class="line"> sc <span class="operator">=</span> StandardScaler<span class="punctuation">(</span>with_mean<span class="operator">=</span>False<span class="punctuation">)</span></span><br><span class="line"> sc.fit<span class="punctuation">(</span>X<span class="punctuation">)</span></span><br><span class="line"> sc.scale_ <span class="operator">=</span> scale_</span><br><span class="line"> <span class="built_in">return</span> sc.transform<span class="punctuation">(</span>X<span class="punctuation">)</span></span><br></pre></td></tr></table></figure> After the pre-processing, we we next
run PCA to perform dimensionality reduction and use which to calculate
euclidean distances between cells, and then run diffusion map.</p>
<figure class="highlight r"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">pc <span class="operator">=</span> run_pca<span class="punctuation">(</span>Y<span class="punctuation">,</span> <span class="number">100</span><span class="punctuation">)</span></span><br><span class="line">R <span class="operator">=</span> distance_matrix<span class="punctuation">(</span>pc<span class="punctuation">,</span> pc<span class="punctuation">)</span></span><br><span class="line">d <span class="operator">=</span> diffusionMaps<span class="punctuation">(</span>R<span class="punctuation">,</span><span class="number">7</span><span class="punctuation">)</span></span><br></pre></td></tr></table></figure>
<p>We first take a look at the PCA figure which shows the first 2 PCs of
the dataset. PCA is a linear methods which cannot reflect cell fate
differentation process. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/fib2neuron_pca.png" />
Since the diffusion map applied the Gaussian kernal during the distances
calculation, it usually a better way to capture the cell differentation
events. Here we show diffusion dimension 1,2 and 1,3. It's clear that
1,2 captures the cell differentation from fibroblasts to neuron, and 1,3
captures the differentiation from firoblasts to myocytes. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/fib2neuron_diffusions.png" /></p>
<p>We can change the bandwidth from 7 to 100, which use the 100th
neareast neighbor as bandwidth instead of 7th. The following shows the
diffusion map that bandwidth=100. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/fib2neuron_bandwidth100.png" /></p>
<h2 id="pseudo-time-by-random-walk">pseudo time by random walk</h2>
<p>The eigenvalue decomposition of graph laplacian can also be used to
infer the pseudo time simulating the pseudo time of cell differentation
events. We here set top <span class="math inline">\(m\)</span> cells
<span class="math inline">\(1/m\)</span> of the progenitor cells (MEF)
and set other cell 0 as the initial state. Next, we perform random walk
to get new pesudotime <span class="math inline">\(u\)</span> such that:
<span class="math display">\[\begin{equation}
u = [\frac{1}{m}, \frac{1}{m}, \cdots, 0, \cdots 0](D^{-1}W)^t
\end{equation}\]</span> By testing different number of random walk steps
we can check the new pseudo time <span class="math inline">\(u\)</span>.
Here we show time step equals to 1, 10, 100 and 200. From the figure we
can notice that after certain step, the pseudo time will not change
anymore. That means the random walk reaches the steady state. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/fib2neuron_pseudotime.png" /></p>
<p>To test when can we reach the steady state, I use the graph we
mentioned in my last post <a
href="https://chengmingbo.github.io/2020/06/07/spectral/">Spectral
analysis (1)</a>:
<img src="https://cmb.oss-cn-qingdao.aliyuncs.com/a_graph.png" height="250px"></p>
<p>Here we random generate two initial states (<span
class="math inline">\(v_1\)</span>, <span
class="math inline">\(v_2\)</span>) to do the random walk such that:
<span class="math display">\[\begin{equation}
\begin{aligned}
v_1 &= [0.15,0.41,0.54,0.9,0.62,0.93,0.1,0.46,0.01,0.88]\\
v_2 &= [0.89,0.93,0.07,0.41,0.52,0.88,0.43,0.09,0.1,0.2]
\end{aligned}
\end{equation}\]</span></p>
<p>From the the figure below we can see that <span
class="math inline">\(v_1\)</span> reaches the steady state in 30 steps
whereas <span class="math inline">\(v_2\)</span> reaches steady state in
100 steps. In total, all these two initial state will reach the steady
state that would not change any longer. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/fib2neuron_steady.png" /></p>
<h2 id="spectral-clustering">spectral clustering</h2>
<p>The eigenvectors of graph Laplacian can also be used to do the
clustering. From the figure we can find clear cut using the 1st, 2nd and
the 3rd diffusion dimensions. In practice, we can use kmeans, DBSCAN,
leiden, louvain algorithm to perform clustering using the diffusion
dimensions. <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/fib2neuron_clustering.png" /></p>
<h2 id="appendix">Appendix</h2>
<h5 id="pca-function">PCA function</h5>
<figure class="highlight r"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line">def run_pca<span class="punctuation">(</span>mtx<span class="punctuation">,</span> n_components<span class="operator">=</span><span class="number">2</span><span class="punctuation">,</span> random_state<span class="operator">=</span><span class="number">2022</span><span class="punctuation">)</span><span class="operator">:</span></span><br><span class="line"> dm <span class="operator">=</span> None</span><br><span class="line"> <span class="keyword">if</span> scipy.sparse.issparse<span class="punctuation">(</span>mtx<span class="punctuation">)</span><span class="operator">:</span></span><br><span class="line"> clf <span class="operator">=</span> TruncatedSVD<span class="punctuation">(</span>n_components<span class="punctuation">,</span> random_state<span class="operator">=</span>random_state<span class="punctuation">)</span></span><br><span class="line"> dm <span class="operator">=</span> clf.fit_transform<span class="punctuation">(</span>mtx<span class="punctuation">)</span></span><br><span class="line"> <span class="keyword">else</span><span class="operator">:</span></span><br><span class="line"> pca <span class="operator">=</span> PCA<span class="punctuation">(</span>n_components<span class="operator">=</span>n_components<span class="punctuation">,</span> random_state<span class="operator">=</span>random_state<span class="punctuation">)</span></span><br><span class="line"> dm <span class="operator">=</span> pca.fit_transform<span class="punctuation">(</span>mtx<span class="punctuation">)</span></span><br><span class="line"> <span class="built_in">return</span> dm</span><br></pre></td></tr></table></figure>
<h5 id="affinity-matrix-function">Affinity matrix function</h5>
<figure class="highlight r"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br></pre></td><td class="code"><pre><span class="line">def affinity<span class="punctuation">(</span>R<span class="punctuation">,</span> k<span class="operator">=</span><span class="number">7</span><span class="punctuation">,</span> sigma<span class="operator">=</span>None<span class="punctuation">,</span> <span class="built_in">log</span><span class="operator">=</span>False<span class="punctuation">)</span><span class="operator">:</span></span><br><span class="line"> <span class="string">""</span><span class="string">"</span></span><br><span class="line"><span class="string"> Gaussian affinity matrix constructor</span></span><br><span class="line"><span class="string"> W = exp(-r_{ij}^2/sigma)</span></span><br><span class="line"><span class="string"> "</span><span class="string">""</span></span><br><span class="line"> def top_k<span class="punctuation">(</span>lst<span class="punctuation">,</span> k<span class="operator">=</span><span class="number">1</span><span class="punctuation">)</span><span class="operator">:</span></span><br><span class="line"> assert<span class="punctuation">(</span>len<span class="punctuation">(</span>lst<span class="punctuation">)</span> <span class="operator">></span>k<span class="punctuation">)</span></span><br><span class="line"> <span class="built_in">return</span> np.partition<span class="punctuation">(</span>lst<span class="punctuation">,</span> k<span class="punctuation">)</span><span class="punctuation">[</span>k<span class="punctuation">]</span></span><br><span class="line"></span><br><span class="line"> R <span class="operator">=</span> np.array<span class="punctuation">(</span>R<span class="punctuation">)</span></span><br><span class="line"> <span class="keyword">if</span> not sigma<span class="operator">:</span></span><br><span class="line"> s <span class="operator">=</span> <span class="punctuation">[</span>top_k<span class="punctuation">(</span>R<span class="punctuation">[</span><span class="operator">:</span><span class="punctuation">,</span> i<span class="punctuation">]</span><span class="punctuation">,</span> k<span class="operator">=</span>k<span class="punctuation">)</span> <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span><span class="punctuation">(</span>R.shape<span class="punctuation">[</span><span class="number">1</span><span class="punctuation">]</span><span class="punctuation">)</span><span class="punctuation">]</span></span><br><span class="line"> S <span class="operator">=</span> np.sqrt<span class="punctuation">(</span>np.outer<span class="punctuation">(</span>s<span class="punctuation">,</span> s<span class="punctuation">)</span><span class="punctuation">)</span></span><br><span class="line"> <span class="keyword">else</span><span class="operator">:</span></span><br><span class="line"> S <span class="operator">=</span> sigma</span><br><span class="line"> logW <span class="operator">=</span> <span class="operator">-</span>np.power<span class="punctuation">(</span>np.divide<span class="punctuation">(</span>R<span class="punctuation">,</span> S<span class="punctuation">)</span><span class="punctuation">,</span> <span class="number">2</span><span class="punctuation">)</span></span><br><span class="line"> <span class="keyword">if</span> <span class="built_in">log</span><span class="operator">:</span></span><br><span class="line"> <span class="built_in">return</span> logW</span><br><span class="line"> <span class="built_in">return</span> np.exp<span class="punctuation">(</span>logW<span class="punctuation">)</span></span><br></pre></td></tr></table></figure>
<h5 id="diffusion-map-function">Diffusion map function</h5>
<figure class="highlight r"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br></pre></td><td class="code"><pre><span class="line">def diffusionMaps<span class="punctuation">(</span>R<span class="punctuation">,</span>k<span class="operator">=</span><span class="number">7</span><span class="punctuation">,</span>sigma<span class="operator">=</span>None<span class="punctuation">)</span><span class="operator">:</span></span><br><span class="line"> <span class="string">""</span><span class="string">"</span></span><br><span class="line"><span class="string"> Diffusion map(Coifman, 2005)</span></span><br><span class="line"><span class="string"> https://en.wikipedia.org/wiki/Diffusion_map</span></span><br><span class="line"><span class="string"> ----------</span></span><br><span class="line"><span class="string"> dic:</span></span><br><span class="line"><span class="string"> psi: right eigvector of P = D^{-1/2} * evec</span></span><br><span class="line"><span class="string"> phi: left eigvector of P = D^{1/2} * evec</span></span><br><span class="line"><span class="string"> eig: eigenvalues</span></span><br><span class="line"><span class="string"> "</span><span class="string">""</span></span><br><span class="line"> k<span class="operator">=</span>k<span class="operator">-</span><span class="number">1</span> <span class="comment">## k is R version minus 1 for the index</span></span><br><span class="line"> logW <span class="operator">=</span> affinity<span class="punctuation">(</span>R<span class="punctuation">,</span>k<span class="punctuation">,</span>sigma<span class="punctuation">,</span><span class="built_in">log</span><span class="operator">=</span>True<span class="punctuation">)</span></span><br><span class="line"> rs <span class="operator">=</span> np.exp<span class="punctuation">(</span><span class="punctuation">[</span>logsumexp<span class="punctuation">(</span>logW<span class="punctuation">[</span>i<span class="punctuation">,</span><span class="operator">:</span><span class="punctuation">]</span><span class="punctuation">)</span> <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span><span class="punctuation">(</span>logW.shape<span class="punctuation">[</span><span class="number">0</span><span class="punctuation">]</span><span class="punctuation">)</span><span class="punctuation">]</span><span class="punctuation">)</span> <span class="comment">## dii=\sum_j</span></span><br><span class="line"> w_<span class="punctuation">{</span>i<span class="punctuation">,</span>j<span class="punctuation">}</span></span><br><span class="line"> D <span class="operator">=</span> np.diag<span class="punctuation">(</span>np.sqrt<span class="punctuation">(</span>rs<span class="punctuation">)</span><span class="punctuation">)</span></span><br><span class="line"> <span class="comment">## D^{1/2}</span></span><br><span class="line"> Dinv <span class="operator">=</span> np.diag<span class="punctuation">(</span><span class="number">1</span><span class="operator">/</span>np.sqrt<span class="punctuation">(</span>rs<span class="punctuation">)</span><span class="punctuation">)</span> <span class="comment">##D^{-1/2}</span></span><br><span class="line"> Ms <span class="operator">=</span> Dinv <span class="operator">@</span> np.exp<span class="punctuation">(</span>logW<span class="punctuation">)</span> <span class="operator">@</span> Dinv <span class="comment">## D^{-1/2} W D^{-1/2}</span></span><br><span class="line"> e <span class="operator">=</span> np.linalg.eigh<span class="punctuation">(</span>Ms<span class="punctuation">)</span> <span class="comment">## eigen decomposition of P'</span></span><br><span class="line"> evalue<span class="operator">=</span> e<span class="punctuation">[</span><span class="number">0</span><span class="punctuation">]</span><span class="punctuation">[</span><span class="operator">::</span><span class="operator">-</span><span class="number">1</span><span class="punctuation">]</span></span><br><span class="line"> evec <span class="operator">=</span> np.flip<span class="punctuation">(</span>e<span class="punctuation">[</span><span class="number">1</span><span class="punctuation">]</span><span class="punctuation">,</span> axis<span class="operator">=</span><span class="number">1</span><span class="punctuation">)</span></span><br><span class="line"> s <span class="operator">=</span> np.sum<span class="punctuation">(</span>np.sqrt<span class="punctuation">(</span>rs<span class="punctuation">)</span> <span class="operator">*</span> evec<span class="punctuation">[</span><span class="operator">:</span><span class="punctuation">,</span><span class="number">0</span><span class="punctuation">]</span><span class="punctuation">)</span> <span class="comment"># scaling</span></span><br><span class="line"> <span class="comment"># Phi is orthonormal under the weighted inner product</span></span><br><span class="line"> <span class="comment">#0:Psi, 1:Phi, 2:eig</span></span><br><span class="line"> dic <span class="operator">=</span> <span class="punctuation">{</span><span class="string">'psi'</span><span class="operator">:</span>s <span class="operator">*</span> Dinv<span class="operator">@</span>evec<span class="punctuation">,</span> <span class="string">'phi'</span><span class="operator">:</span> <span class="punctuation">(</span><span class="number">1</span><span class="operator">/</span>s<span class="punctuation">)</span><span class="operator">*</span>D<span class="operator">@</span>evec<span class="punctuation">,</span> <span class="string">"eig"</span><span class="operator">:</span> evalue<span class="punctuation">}</span></span><br><span class="line"> <span class="built_in">return</span> dic</span><br></pre></td></tr></table></figure>
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<a href="/2020/06/07/spectral/" class="post-title-link" itemprop="url">Spectral analysis (1)</a>
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<h2 id="introduction">Introduction</h2>
<p>Newton used glass separted the white sunlight into red, orange,
yellow, green, blue, indigo and violet single colors. The light spectrum
analysis can help scientists to interpret the world. For example, we can
detect the elements of our solar system as well as far stars in the
universe. The spectrum analysis is also a field in mathmatjics. In the
graph thoery field, Laplacian matrix is used to represented a graph. We
can obtian features from the undirected graph below (<a
target="_blank" rel="noopener" href="https://en.wikipedia.org/wiki/Laplacian_matrix">wikipedia</a>).
<img src="https://cmb.oss-cn-qingdao.aliyuncs.com/a_graph.png" height="300px"></p>
<p>For example, we can check the degree of each vertex, which forms our
Degree matrix <span class="math inline">\(D\)</span> such that: <span
class="math display">\[\begin{equation}
D = \begin{pmatrix}
2 & 0 & 0 & 0 & 0 & 0\\
0 & 3 & 0 & 0 & 0 & 0\\
0 & 0 & 2 & 0 & 0 & 0\\
0 & 0 & 0 & 3 & 0 &0\\
0 & 0 & 0 & 0 & 3 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{pmatrix}
\end{equation}\]</span></p>
<p>By checking the connection between all pairs nodes, we can create a
Adjacency matrix:</p>
<p><span class="math display">\[\begin{equation}
A = \begin{pmatrix}
0 & 1 & 0 & 0 & 1 & 0\\
1 & 0 & 1 & 0 & 1 & 0\\
0 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 1 &1\\
1 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0
\end{pmatrix}
\end{equation}\]</span></p>
<h2 id="graph-laplacian">graph laplacian</h2>
<p>The graph Laplacian <span class="math inline">\(L\)</span> extracts
all useful information from a graph which is: <span
class="math display">\[\begin{equation}
L=D-A =\begin{pmatrix}
2 & -1 & 0 & 0 & -1 & 0\\
-1 & 3 & -1 & 0 & -1 & 0\\
0 & -1 & 2 & -1 & 0 & 0\\
0 & 0 & -1 & 3 & -1 & -1\\
-1 & -1 & 0 & -1 & 3 & 0\\
0 & 0 & 0 & -1 & 0 & 1
\end{pmatrix}
\end{equation}\]</span></p>
<p>In fact, the graph Laplaican matrix is symmetric and also positive
semidefinite (PSD), which means if we perform eigenvalue decomposition,
the eigen values are all real and nonnegative. We can normlize the graph
Laplaican by left multiplying the <span
class="math inline">\(D^{-1}\)</span> which will give rise to all <span
class="math inline">\(1\)</span>s of the diagnal entries of the matrix,
namely: <span class="math display">\[\begin{equation}
\text{norm}({L}) = D^{-1}L = \begin{pmatrix}1 & -0.50 & 0
& 0 & -0.50 & 0\\
-0.33 & 1 & -0.33 & 0 & -0.33 & 0\\
0 & -0.50 & 1 & -0.50 & 0 & 0\\
0 & 0 & -0.33 & 1 & -0.33 &-0.33\\
-0.33 & -0.33 & 0 & -0.33 & 1 & 0\\
0 & 0 & 0 & -1 & 0 & 1
\end{pmatrix}
\end{equation}\]</span> This matrix is called transition matrix.
However, the matrix would not keep the symmetric and the PSD property
after the normalization. We will come back to discuss the spectral for
the normalized graph Laplacian.</p>
<h2 id="weighted-graph">Weighted graph</h2>
<p>In practice, graph are usually weighted. The weight between vertices
can be euclidean distance or other measures. The figure blow shows the
weights of the graph. Here we apply gussian kernel to the euclidean
distances between vertices such that: <img
src="https://cmb.oss-cn-qingdao.aliyuncs.com/a_graph_weighted.png"
alt="weighted graph" /></p>
<p><span class="math display">\[\begin{equation}
w_{ij} =\exp (-r_{ij}^2/\sigma) = \exp \big(\frac{-\|x_i -
x_j\|^2}{\sigma_i\sigma_j}\big)
\end{equation}\]</span> where <span
class="math inline">\(r_{ij}\)</span> is the euclidean distance between
vetex <span class="math inline">\(i\)</span> and <span
class="math inline">\(j\)</span>, and <span
class="math inline">\(sigma\)</span> controls the bandwidth. We call the
matrix <span class="math inline">\(W=(w_{ij})\)</span> gaussian affinity
matrix such that: <span class="math display">\[\begin{equation}
\begin{pmatrix}
0 & w_{12} & w_{13} & w_{14} & w_{15}& w_{16}\\
w_{21} & 0 & w_{23} & w_{24} & w_{25}& w_{26}\\
w_{31} & w_{32} & 0 & w_{34} & w_{35}& w_{36}\\
w_{41} & w_{42} & w_{43} & 0 & w_{45}& w_{46}\\
w_{51} & w_{52} & w_{53} & w_{54} & 0& w_{56}\\
w_{61} & w_{62} & w_{63} & w_{64} & w_{65}& 0
\end{pmatrix}
\end{equation}\]</span> The guassian kernel would enlarge the distance
between too far vertices. Similar to unweighted matrix, we can also
construct graph Laplaican matrix using the gaussian affinity matrix.
First, we need find the weighted degree based on <span
class="math inline">\(W\)</span> such that: <span
class="math display">\[\begin{equation}
d_{ii} = \sum_j{w_{ij}}
\end{equation}\]</span> With the diagnal degree matrix and affinity
matrix, we now can have the weighted laplacian that: <span
class="math display">\[\begin{equation}
L = D - W
\end{equation}\]</span> Likewise, we next give the normalized form of
Laplacian such that: <span class="math display">\[\begin{equation}