From dd83a902f8e1df7dae48fbdfa327651935feacb5 Mon Sep 17 00:00:00 2001 From: Marius Drechsler Date: Sat, 17 Aug 2024 20:25:41 +0200 Subject: [PATCH] Some progress on BACH, added pseudocode for better understanding --- Session.vim | 176 ++++++++++++++++++++ bibliography.bib | 8 + content/BACH.typ | 43 ++++- graphics/quantizers/two-bit-enroll-real.typ | 43 +++++ main.pdf | Bin 1237187 -> 1249400 bytes pseudocode/bach_1.typ | 16 ++ pseudocode/bach_find_best_appr.typ | 12 ++ 7 files changed, 296 insertions(+), 2 deletions(-) create mode 100644 Session.vim create mode 100644 graphics/quantizers/two-bit-enroll-real.typ create mode 100644 pseudocode/bach_1.typ create mode 100644 pseudocode/bach_find_best_appr.typ diff --git a/Session.vim b/Session.vim new file mode 100644 index 0000000..c437a43 --- /dev/null +++ b/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(":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(":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 : diff --git a/bibliography.bib b/bibliography.bib index 82d7c66..6834725 100644 --- a/bibliography.bib +++ b/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} +} + diff --git a/content/BACH.typ b/content/BACH.typ index 4978461..41601cf 100644 --- a/content/BACH.typ +++ b/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. -$ +$ 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") +) -=== 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] +)] + +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") +) diff --git a/graphics/quantizers/two-bit-enroll-real.typ b/graphics/quantizers/two-bit-enroll-real.typ new file mode 100644 index 0000000..cea4ccc --- /dev/null +++ b/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", 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0000000..254d79b --- /dev/null +++ b/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* $bold(h)_"opt"$ +] diff --git a/pseudocode/bach_find_best_appr.typ b/pseudocode/bach_find_best_appr.typ new file mode 100644 index 0000000..bf04bb7 --- /dev/null +++ b/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 $z_"opt"$ +]

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