Some progress on BACH, added pseudocode for better understanding

Session.vim

@@ -0,0 +1,176 @@
+let SessionLoad = 1
+let s:so_save = &g:so | let s:siso_save = &g:siso | setg so=0 siso=0 | setl so=-1 siso=-1
+let v:this_session=expand("<sfile>:p")
+silent only
+silent tabonly
+cd ~/thesis
+if expand('%') == '' && !&modified && line('$') <= 1 && getline(1) == ''
+  let s:wipebuf = bufnr('%')
+endif
+let s:shortmess_save = &shortmess
+if &shortmess =~ 'A'
+  set shortmess=aoOA
+else
+  set shortmess=aoO
+endif
+badd +128 content/BACH.typ
+badd +1 glossary.typ
+badd +1 bibliography.bib
+badd +0 \$
+badd +6 pseudocode/bach_1.typ
+badd +265 content/SMHD.typ
+badd +7 pseudocode/bach_find_best_appr.typ
+argglobal
+%argdel
+set stal=2
+tabnew +setlocal\ bufhidden=wipe
+tabnew +setlocal\ bufhidden=wipe
+tabnew +setlocal\ bufhidden=wipe
+tabrewind
+edit content/BACH.typ
+argglobal
+balt content/SMHD.typ
+setlocal fdm=manual
+setlocal fde=0
+setlocal fmr={{{,}}}
+setlocal fdi=#
+setlocal fdl=0
+setlocal fml=1
+setlocal fdn=20
+setlocal fen
+silent! normal! zE
+let &fdl = &fdl
+let s:l = 164 - ((38 * winheight(0) + 25) / 50)
+if s:l < 1 | let s:l = 1 | endif
+keepjumps exe s:l
+normal! zt
+keepjumps 164
+normal! 0
+tabnext
+edit glossary.typ
+argglobal
+balt content/BACH.typ
+setlocal fdm=manual
+setlocal fde=0
+setlocal fmr={{{,}}}
+setlocal fdi=#
+setlocal fdl=0
+setlocal fml=1
+setlocal fdn=20
+setlocal fen
+silent! normal! zE
+let &fdl = &fdl
+let s:l = 1 - ((0 * winheight(0) + 25) / 50)
+if s:l < 1 | let s:l = 1 | endif
+keepjumps exe s:l
+normal! zt
+keepjumps 1
+normal! 0
+tabnext
+edit bibliography.bib
+argglobal
+balt glossary.typ
+setlocal fdm=manual
+setlocal fde=0
+setlocal fmr={{{,}}}
+setlocal fdi=#
+setlocal fdl=0
+setlocal fml=1
+setlocal fdn=20
+setlocal fen
+silent! normal! zE
+let &fdl = &fdl
+let s:l = 1 - ((0 * winheight(0) + 25) / 50)
+if s:l < 1 | let s:l = 1 | endif
+keepjumps exe s:l
+normal! zt
+keepjumps 1
+normal! 015|
+tabnext
+edit pseudocode/bach_1.typ
+let s:save_splitbelow = &splitbelow
+let s:save_splitright = &splitright
+set splitbelow splitright
+wincmd _ | wincmd |
+split
+1wincmd k
+wincmd w
+let &splitbelow = s:save_splitbelow
+let &splitright = s:save_splitright
+wincmd t
+let s:save_winminheight = &winminheight
+let s:save_winminwidth = &winminwidth
+set winminheight=0
+set winheight=1
+set winminwidth=0
+set winwidth=1
+exe '1resize ' . ((&lines * 25 + 26) / 53)
+exe 'vert 1resize ' . ((&columns * 129 + 105) / 211)
+exe '2resize ' . ((&lines * 24 + 26) / 53)
+exe 'vert 2resize ' . ((&columns * 129 + 105) / 211)
+argglobal
+balt bibliography.bib
+setlocal fdm=manual
+setlocal fde=0
+setlocal fmr={{{,}}}
+setlocal fdi=#
+setlocal fdl=0
+setlocal fml=1
+setlocal fdn=20
+setlocal fen
+silent! normal! zE
+let &fdl = &fdl
+let s:l = 16 - ((15 * winheight(0) + 12) / 25)
+if s:l < 1 | let s:l = 1 | endif
+keepjumps exe s:l
+normal! zt
+keepjumps 16
+normal! 0
+wincmd w
+argglobal
+if bufexists(fnamemodify("pseudocode/bach_find_best_appr.typ", ":p")) | buffer pseudocode/bach_find_best_appr.typ | else | edit pseudocode/bach_find_best_appr.typ | endif
+if &buftype ==# 'terminal'
+  silent file pseudocode/bach_find_best_appr.typ
+endif
+balt pseudocode/bach_1.typ
+setlocal fdm=manual
+setlocal fde=0
+setlocal fmr={{{,}}}
+setlocal fdi=#
+setlocal fdl=0
+setlocal fml=1
+setlocal fdn=20
+setlocal fen
+silent! normal! zE
+let &fdl = &fdl
+let s:l = 7 - ((6 * winheight(0) + 12) / 24)
+if s:l < 1 | let s:l = 1 | endif
+keepjumps exe s:l
+normal! zt
+keepjumps 7
+normal! 031|
+wincmd w
+exe '1resize ' . ((&lines * 25 + 26) / 53)
+exe 'vert 1resize ' . ((&columns * 129 + 105) / 211)
+exe '2resize ' . ((&lines * 24 + 26) / 53)
+exe 'vert 2resize ' . ((&columns * 129 + 105) / 211)
+tabnext 1
+set stal=1
+if exists('s:wipebuf') && len(win_findbuf(s:wipebuf)) == 0 && getbufvar(s:wipebuf, '&buftype') isnot# 'terminal'
+  silent exe 'bwipe ' . s:wipebuf
+endif
+unlet! s:wipebuf
+set winheight=1 winwidth=20
+let &shortmess = s:shortmess_save
+let &winminheight = s:save_winminheight
+let &winminwidth = s:save_winminwidth
+let s:sx = expand("<sfile>:p:r")."x.vim"
+if filereadable(s:sx)
+  exe "source " . fnameescape(s:sx)
+endif
+let &g:so = s:so_save | let &g:siso = s:siso_save
+set hlsearch
+nohlsearch
+doautoall SessionLoadPost
+unlet SessionLoad
+" vim: set ft=vim :

bibliography.bib

@@ -38,3 +38,11 @@
   year={2018},
   organization={IEEE}
 }
+
+@book{schmutz,
+  title={Introduction to mathematical statistics},
+  author={Hogg, Robert V and McKean, Joseph W and Craig, Allen T and others},
+  year={2013},
+  publisher={Pearson Education India}
+}
+

content/BACH.typ

@@ -110,7 +110,7 @@ Later we will be able to show that a higher number of summands for $z$ can provi
 We will define $z$ from now on as:
 $
 z = x_1 dot h_1 plus x_2 dot h_2 plus x_3 dot h_3.
-$
+$<eq:z_eq>
 
 We can now find the optimal linear combination $z_"opt"$ by finding the minimum of all distances to all optimal points defined as $bold(cal(o))$.
 The matrix that contains the distances of all linear combinations $bold(z)$ to all optimal points $bold(cal(o))$ is defined as: $bold(cal(A))$ with its entries $a_"ij" = abs(z_"i" - o_"j")$.\
@@ -119,7 +119,46 @@ $
 z_"opt" = op("argmin")(bold(cal(A)))
 = op("argmin")(mat(delim: "[", a_("00"), ..., a_("i0"); dots.v, dots.down, " "; a_"0j", " ", a_"ij" )).
 $
+#figure(
+  kind: "algorithm",
+  supplement: [Algorithm],
+  include("../pseudocode/bach_find_best_appr.typ")
+)<alg:best_appr>
 
-=== Algorithm definition
+@alg:best_appr shows a programmatic approach to find the set of weights for the best approximation. The algorithm returns a tuple consisting of the weight combination $bold(h)$ and the resulting value of the linear combination $z_"opt"$
+=== Realization of center point approximation
 
+As described earlier, we can define the ideal possible positions for the linear combination $z_"opt"$ to be the centers of the quantizer steps.
+Because the superposition of different linear combinations of normal distributions corresponds to a Gaussian Mixture Model, wherein finding the ideal set of points $bold(cal(o))$ analytically is impossible.
 
+Instead, we will first estimate $bold(cal(o))$ based on the normal distribution parameters after performing multiple convolutions with the input distribution $X$.
+The parameters of a multiple convoluted normal distribution is defined as: 
+$
+sum_(i=1)^(n) cal(N)(mu_i, sigma_i^2) tilde cal(N)(sum_(i=1)^n mu_i, sum_(i=1)^n sigma_i^2),
+$
+while $n$ defines the number of convolutions performed @schmutz.
+
+With this definition, we can define the parameters of the probability distribution $Z$ of the linear combinations $z$ based on the parameters of $X$, $mu_X$ and $sigma_X$: 
+
+$
+Z(mu_Z, sigma_Z^2) = Z(sum_(i=1^n) mu_X, sum_(i=1)^n sigma_X^2)
+$
+
+The parameters $mu_Z$ and $sigma_Z$ allow us to apply an inverse CDF on a multi-bit quantizer $cal(Q)(2, tilde(x))$ defined in the tilde-domain. 
+Our initial values for $bold(cal(o))_"first"$ can now be defined as the centers of the steps of the transformed quantizer function $cal(Q)(2, x)$.
+These points can be found easily but for the outermost center points whose quantizer steps have a bound $plus.minus infinity$.\
+However, we can still find these two remaining center points by artificially defining the outermost bounds of the quantizer as $frac(1, 2^M dot 4)$ and $frac((2^M dot 4)-1, 2^M dot 4)$ in the tilde-domain and also apply the inverse CDF to them. 
+
+#scale(x: 90%, y: 90%)[
+#figure(
+  include("../graphics/quantizers/two-bit-enroll-real.typ"),
+  caption: [Quantizer for the distribution resulting a triple convolution with distribution parameters $mu_X=0$ and $sigma_X=1$ with marked center points of the quantizer steps]
+)<fig:two-bit-enroll-find-centers>]
+
+We can now use an iterative algorithm that alternates between optimizing the quantizing bounds of $cal(Q)$ and our vector of optimal points $bold(cal(o))_"first"$.
+
+#figure(
+    kind: "algorithm",
+    supplement: [Algorithm],
+    include("../pseudocode/bach_1.typ")
+)

graphics/quantizers/two-bit-enroll-real.typ

@@ -0,0 +1,43 @@
+#import "@preview/cetz:0.2.2": canvas, plot, decorations, draw
+
+#let line_style = (stroke: (paint: black, thickness: 2pt))
+#let dashed = (stroke: (dash: "dashed"))
+#canvas({
+  plot.plot(size: (8,6),
+    legend: "legend.south",
+    name: "plot",
+    x-tick-step: 1,
+   // x-ticks: ((-1.168, [-1.16]), (1.168, [1.16])),
+    y-label: $cal(Q)(2, x)$,
+    x-label: $x$,
+    y-tick-step: none,
+    y-ticks: ((0.25, [00]), (0.5, [01]), (0.75, [10]), (1, [11])),
+    axis-style: "left",
+    x-min: -3,
+    x-max: 3,
+    y-min: 0,
+    y-max: 1,{
+    plot.add(((-3,0.25), (-1.168,0.25), (-1.168,0.5), (0, 0.5), (0, 0.75), (1.168,0.75), (1.168, 1), (3, 1)), style: line_style)
+    plot.add(((-2.657, 0), (-2.657, 1)), style: (stroke: (paint: red)), label: [Artificial quantizer bounds])
+    plot.add(((2.657, 0), (2.657, 1)), style: (stroke: (paint: red)))
+    plot.add-hline(0.25, 0.5, 0.75, 1, style: dashed)
+    plot.add-vline(-1.168, 1.168, style: dashed)
+    plot.add-anchor("uc01", (-0.584, 0.5))
+    plot.add-anchor("lc01", (-0.584, 0))
+    plot.add-anchor("uc10", (0.584, 0.75))
+    plot.add-anchor("lc10", (0.584, 0))
+
+    plot.add-anchor("uc00", (-1.9125, 0.25))
+    plot.add-anchor("lc00", (-1.9125, 0))
+    plot.add-anchor("uc11", (1.9125, 1))
+    plot.add-anchor("lc11", (1.9125, 0))
+
+  })
+
+  draw.line("plot.uc01", ((), "|-", "plot.lc01"), mark: (end: ">"))
+  draw.line("plot.uc10", ((), "|-", "plot.lc10"), mark: (end: ">"))
+  draw.line("plot.uc00", ((), "|-", "plot.lc00"), mark: (end: ">"))
+  draw.line("plot.uc11", ((), "|-", "plot.lc11"), mark: (end: ">"))
+
+
+})

pseudocode/bach_1.typ

@@ -0,0 +1,16 @@
+#import "@preview/lovelace:0.3.0": *
+
+
+#pseudocode-list(booktabs: true, numbered-title: [Center Point Approximation])[
+  + *input*: $bold(cal(o))_"first", bold(x), t, M$
+  + *lists*: optimal weights $bold(h)_"opt"$
+  + $bold(cal(o)) arrow.l bold(cal(o))_"first"$
+  + *repeat* t times: 
+    + *perform* @alg:best_appr for all input values with $bold(cal(o))$:
+      + *update* $bold(h)_"opt"$ with returned weights
+      + $bold(z)_"opt" arrow.l$ all returned linear combinations 
+    + *sort* $bold(z)_"opt"$ in ascending order
+    + *define* new quantizer $cal(Q)^*$ using the @ecdf based on $bold(z)_"opt"$
+    + *update* $bold(cal(o))$ with newly found quantizer step centers
+  + *return* $bold(h)_"opt"$ 
+]

pseudocode/bach_find_best_appr.typ

@@ -0,0 +1,12 @@
+#import "@preview/lovelace:0.3.0": *
+
+#pseudocode-list(booktabs: true, numbered-title: [Find best approximation])[
+  + *inputs*:
+    + $bold(y)$ input values for linear combinations
+    + $bold(cal(o))$ list of optimal points
+  + *output*: $(bold(h), z_"opt")$
+    //+ n number of summands in linear combination
+  + *calculate* all possible linear combinations $bold(z)$ with @eq:z_eq
+  + *calculate* matrix $bold(cal(A))$ with $a_"ij" = abs(z_i - cal(o)_j)$
+  + *return* weights $bold(h)$ for $z_"opt" = op("argmin")(bold(cal(A)))$ and $z_"opt"$
+]