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\title{%
    Accelerated Filtering on Graphs using Lanczos method
    \\ \large Scientific Computing project report}
\author{Alberto Defendi}

\date{}

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\begin{document}
\maketitle

\begin{abstract}
    \noindent The Lanczos algorithm \ldots 
\end{abstract}

{\setlength{\parskip}{0em}
\tableofcontents}

\section{Introduction}

We introduce basic graph theory concepts and briefly overview the results used in the project experiments.

\subsection{Signal processing on graphs}

We consider a weighted undirected graph $ \G = (\V, \E, \mathcal{W}) $,
where $ \V \subset \R^N $ is the set of vertices, $ \E \subset \R^M $ is the set of edges, and $ \mathcal{W}
: \V \times \mathcal{V} \to \R$ is a weight function. The weight function can be
represented with a $ N \times N $ matrix $ W $ such that 
$W_{i,j} = 
\begin{cases} 
    W(v_i, v_j), & \text{if } (v_i, v_j) \in \E \\ 
    0, & \text{otherwise} 
\end{cases} 
\quad \text{for all } i,j = 1, \dots, |\V|
$,
and for our needs, we assume $ W $ to be symmetric, that is $ W_{i,j} =  W_{j,i} $.

In the study of signal processing on graph, we model the idea of sending a signal to a node as
assigning a value to a vertex with a function $ s : \V \to \R$, whose values can be represented as a vector
$ s = s(\V) = [s_1, \dots, s_N] \in \R^N $ where each entry $ s_i \defeq s(v_i)$ represents
the signal sent over a node $ v_i \in \V$. We also keep track of the weight of each vertex
with a function $ d(i) \defeq \sum_{j=1}^N W_{i,j} $, that we represent with the matrix $D =
\text{diag}(d(1), \dots, d(N))$.

In this setting, we introduce the graph Laplacian $\L$ defined as 
$ \L = D - W$, where $D $ is the diagonal degree matrix with entries  $D_{ii} = d(i)$. By
construction, $\L^T = \mathcal{L}$, thus the Lanczos method can be safely applied to our 
problem.

\begin{remark}
  The Laplacian $\L$ is symmetric and semi-definite (it is diagonally dominant), hence by the
  spectral theorem it admits the decomposition
  \begin{equation*}
    \L = U \Lambda U^{*},
  \end{equation*}
  where $ U = \left[ u_0,\dots,u_{N-1} \right] \in O(N)$ is called Fourier basis, and $ \Lambda =
  \text{diag}\left(\lambda_0, \dots, \lambda_{N-1}\right)$ is the matrix of eigenvalues of $ \L $,
  without loss of generality we can assume $ 0 =\lambda_0 \leq \lambda_1  \dots \leq \lambda_{N-1} $
  and the vectors in $ U $ to be in the same order that the eigenvalues.
  \footnote{this assumption is aligned with the PyGSP $ \text{compute\_fourier\_basis()} $ function
  implementation, used in this work to compute the matrix $ \Lambda $. Clarifying this assumption is
outside our scope.}.
\end{remark}

\begin{definition}[Graph signal]
  A graph signal is a continuous function $ g : \R^+ \to \R $.
\end{definition}
Diagonalising the Laplacian would give an easy way to compute the function $ g(\L) $ by
evaluating it on the eigenvalues of $ \L $, which means computing $ g(\L) = U g(\Lambda) U^{*} $.
In matrix notation, applying a filter to a graph $ \G $ corresponds to the operation
\begin{equation*}
  s^\prime \defeq g(\L)s = U g(\L) U^{*}s.
\end{equation*}
\begin{remark}[Computational cost]
Computing the Fourier basis for $ \L $ is computationally expensive for large graphs, thus the
choice of using the Lanczos method for $ g(\L) $, with its computational cost of $ O(M \dot |\E|) $,
it offers an efficient alternative in computing $ g(\L) $. However, storing the basis $ V_M $ costs MN
additional memory, that could be avoided using a two-step implementation, that we leave for future
work.
\end{remark}

\subsection{The Lanczos method}

\begin{definition}
 Given a matrix $ A \in \R^{N \times N} $ and a vector $ b \in R^N $, the Krylov subspace of order $j$ is defined as the set $ \mathcal{K}_j (A,b) = \{ b, Ab, A^2b, \dots,
A^{j-1}b\}  $. We represent the basis of this subspace in a matrix $ V_M = \left[ v_1,\dots,v_M
\right] \in \R^{N \times M}$.
\end{definition}
We now consider the Arnoldi relation
\begin{equation*}
  AV_j = V_jH_j + h_{j+1,j}v_{j+1}e_j^{*}, \\ \hspace{20pt}
  H_j = V_j^{*}AV_j =
  \begin{bmatrix}
    h_{11} & \dots & \dots & h_{1,j} \\
    h_{21} & h_{22} & & \vdots \\
           & \ddots & \ddots & \vdots \\
           & & h_{j,j-1} & h_{j,j}.
  \end{bmatrix}
\end{equation*}
Letting
\begin{equation*}
\alpha_j \defeq h_{j,j},\hspace{20pt}\beta_j \defeq h_{j-1,j},
\end{equation*}
if $ A $ is symmetric and positive defined, then so is $ H_j $. Because  $ H_j $ is both symmetric
and upper and lower Hessenberg matrix, then it is tridiagonal, and we refer to it as $ T_j $, hence
the Arnoldi relation takes the form:

\begin{equation*}
  AV_j = V_j \alpha_j  + \beta_j v_{j+1}e_j^{T}, \\ \hspace{20pt}
  T_j = V_j^{\top} A V_j = \begin{bmatrix}
    \alpha_1 & \beta_1 \\
    \beta_1 & \ddots & \ddots \\
      & \ddots & \ddots & \beta_{j-1} \\
      & & \beta_{j-1} & \alpha_j
  \end{bmatrix}.
\end{equation*}.

HERE PUT ALGORITHM






\section{Experiments}

\subsection{}

Consider the filter $ g : [0, \lambda_{\text{max}}] \to \R$ and a signal vector $ s \in \R^N $, by a
a result of Gallopolus and Saad (see ?) it holds\footnote{This results holds outside the graph
  signal processing context, that is for any function $f :\Omega \to \C$, where $ \Omega \subset \Lambda(A) $.} that
\begin{equation}
 g(\L)s \approx \norm{s}_2 V_M g(T_M) e_1 \eqdef g_M \label{eq:1}
\end{equation}

\begin{definition}(Errors)
  We define the Lanczos iteration error as $\norm{g_{M+j} - g_M}_2 $, where $ j $ is small, and the true
error as $ \norm{e_M} = \norm{g(\L)s - g_M} $.
\end{definition}
The scope of this experiment is verifying numerically equation \eqref{eq:1}

Studiamo i grafi di Erdos-Reiny e di tipo Sensors. Dal plot possiamo 
Figura (dida: Grafi di ER e sensor colorati in base al segnale (non filtrato, sopra) e filtrato
attraverso la valutazione $g(\L)s$.
test

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