From b0399c46759577faf86c62f1d5f085a9e1731b1f Mon Sep 17 00:00:00 2001 From: Marius Drechsler Date: Mon, 12 Aug 2024 20:28:04 +0200 Subject: [PATCH] Continued BACH --- content/BACH.typ | 72 +- .../calculate_function_values.py | 11 + data/z_distribution/flake.lock | 61 + data/z_distribution/flake.nix | 32 + data/z_distribution/z_distribution.csv | 1000 +++++++++++++++++ graphics/plots/z_distribution.typ | 26 + main.pdf | Bin 1201215 -> 1224567 bytes 7 files changed, 1193 insertions(+), 9 deletions(-) create mode 100644 data/z_distribution/calculate_function_values.py create mode 100644 data/z_distribution/flake.lock create mode 100644 data/z_distribution/flake.nix create mode 100644 data/z_distribution/z_distribution.csv create mode 100644 graphics/plots/z_distribution.typ diff --git a/content/BACH.typ b/content/BACH.typ index 7e2cd1f..e1b02aa 100644 --- a/content/BACH.typ +++ b/content/BACH.typ @@ -1,8 +1,8 @@ -#import "@preview/glossarium:0.4.1": * +#import "@preview/glossarium:0.4.1": * = Boundary Adaptive Clustering with Helper Data -Instead of generating helper-data to improve the quantization process itself, like in #gls("smhdt"), we can also try to find helper-data before performing enrollment that will optimize our input values before the quantization step to minimize the risk of bit and symbol errors during the reconstruction phase. +Instead of generating helper-data to improve the quantization process itself, like in #gls("smhdt"), or using some kind of error correcting code after the quantization process, we can also try to find helper-data before performing the quantization that will optimize our input values before quantizing them to minimize the risk of bit and symbol errors during the reconstruction phase. Since this #gls("hda") modifies the input values before the quantization takes place, we will consider the input values as zero-mean Gaussian distributed and not use a CDF to transform these values into the tilde-domain. @@ -20,17 +20,71 @@ Considering that the margin of error of the value $x$ is comparable with the one This means that the quantizer used here is very unreliable without generated helper-data. Now, to increase the reliability of this quantizer, we can try to move our input values further away from the value $x = 0$. -To do so, we can define a new input value $x^"lin"$ as a linear combination of two realizations of $X$, $x_1$ and $x_2$ with a set of weights $h_1$ and $h_2$: +To do so, we can define a new input value $z$ as a linear combination of two realizations of $X$, $x_1$ and $x_2$ with a set of weights $h_1$ and $h_2$: $ -x^"lin" = h_1 dot x_1 + h_2 dot x_2 . +z = h_1 dot x_1 + h_2 dot x_2 . $ -We can define the vector of all possible linear combinations $bold(x^"lin")$ as the vector-matrix multiplication of the two input values $x_i$ and the matrix of all weight combinations: + +=== Derivation of the resulting distribution + +To find a description for the random distribution $Z$ of $z$ we can interpret this process mathematically as a maximisation of a sum. +This can be realized by replacing the values of $x_i$ with their absolute values: $ -bold(x^"lin") &= vec(x_1, x_2) dot mat(delim: "[", h_1, -h_1, h_1, -h_1; h_2, h_2, -h_2, -h_2)\ +z = abs(x_1) + abs(x_2) +$ +Taking into account, that $x_i$ are realizations of a normal distribution -- that we can assume without loss of generality to have its expected value at $x=0$ and a standard deviation of $sigma = 1$ -- we can define the overall resulting random distribution $Z$ to be: +$ +Z = abs(X) + abs(X). +$ +We will redefine $abs(X)$ as a half-normal distribution $Y$ whose PDF is +$ +f_Y(y, sigma) &= frac(sqrt(2), sigma sqrt(pi)) lr(exp(-frac(y^2, 2 sigma^2)) mid(|))_(sigma = 1), y >= 0 \ +&= sqrt(frac(2, pi)) exp(- frac(y^2, sigma^2)) . +$ +Now, $Z$ simplifies to +$ +Z = Y + Y. +$ +We can assume that the realizations of $Y$ are independent of each other. +The PDF of the addition of these two distributions can be described through the convolution of their respective PDFs: +$ +f_Z(z) &= integral_0^z f_Y (y) f_Y (z-y) \dy\ +&= integral_0^z [sqrt(2/pi) exp(-frac(y^2,2)) sqrt(2/pi) exp(-frac((z-x)^2, 2))] \dx\ +&= 2/pi integral_0^z exp(- frac(x^2 + (z-x)^2, 2)) \dx # +$ +Evaluating the integral of @eq:z_integral, we can now describe the resulting distribution of this maximisation process analytically: +$ +f_Z = 2/sqrt(pi) exp(-frac(2^2, 4)) "erf"(z/2) z >= 0. +$ +Our derivation of $f_Z$ currently only accounts for the addition of positive values of $x_i$, but two negative $x_i$ values would also return the maximal distance to the coordinate origin. +The derivation for the corresponding PDF is identical, except that the half-normal distribution @eq:half_normal is mirrored around the y-axis. +Because the resulting PDF $f_Z^"neg"$ is a mirrored variant of $f_Z$ and $f_Z$ is symmetrical arranged around the origin, we can define a new PDF $f_Z^*$ as +$ +f_Z^* (z) = abs(f_Z (z)), +$ +on the entire z-axis. +$f_Z^* (z)$ now describes the final random distribution after the application of our optimization of the input values $x_i$. +#figure( + include("../graphics/plots/z_distribution.typ"), + caption: [Optimized input values $z$ overlaid with sign-based quantizer $cal(Q)$] +) + +@fig:z_pdf shows two key properties of this optimization: +1. Adjusting the input values using the method described above does not require any adjustment of the decision threshold of the sign-based quantizer. +2. The resulting PDF + +=== Generating helper-data + +To find the optimal set of helper-data that will result in the distribution shown in @fig:z_pdf, we can define the vector of all possible linear combinations $bold(z)$ as the vector-matrix multiplication of the two input values $x_i$ and the matrix of all weight combinations: +$ +bold(z) &= vec(x_1, x_2) dot mat(delim: "[", h_1, -h_1, h_1, -h_1; h_2, h_2, -h_2, -h_2)\ &= vec(x_1, x_2) dot mat(delim: "[", +1, -1, +1, -1; +1, +1, -1, -1) $ -We will choose the optimal weights based on the highest absolute value of $bold(x^"lin")$, as that value will be the furthest away from $0$. -We may encounter two entries in $bold(x^"lin")$ that both have the same highest absolute value. +We will choose the optimal weights based on the highest absolute value of $bold(z)$, as that value will be the furthest away from $0$. +We may encounter two entries in $bold(z)$ that both have the same highest absolute value. +In that case, we will choose the combination of weights randomly out of our possible options. + +If we take a look at the dimensionality of the matrix of all weight combinations, we notice that we will need to store $log_2(2) = 1$ helper-data bit. +In fact, we will show later, that the amount of helper-data bits used by this HDA is directly linked to the number of input values used instead of the number of bits we want to extract during quantization. -Lets take a look at the resulting random distribution of this process: diff --git a/data/z_distribution/calculate_function_values.py b/data/z_distribution/calculate_function_values.py new file mode 100644 index 0000000..0ffdd0b --- /dev/null +++ b/data/z_distribution/calculate_function_values.py @@ -0,0 +1,11 @@ +import csv +import numpy as np +import math + +with open('z_distribution.csv', "w") as csvfile: + writer = csv.writer(csvfile) + for i in np.linspace(-10, 10, 1000): + calculation = 2/math.sqrt(math.pi) * math.exp(-(i**2)/4) * math.erf(i/2) + if i < 0: + calculation = calculation * -1 + writer.writerow([i, calculation]) diff --git a/data/z_distribution/flake.lock b/data/z_distribution/flake.lock new file mode 100644 index 0000000..ae74e2e --- /dev/null +++ b/data/z_distribution/flake.lock @@ -0,0 +1,61 @@ +{ + "nodes": { + "flake-utils": { + "inputs": { + "systems": "systems" + }, + "locked": { + "lastModified": 1710146030, + "narHash": "sha256-SZ5L6eA7HJ/nmkzGG7/ISclqe6oZdOZTNoesiInkXPQ=", + "owner": "numtide", + "repo": "flake-utils", + "rev": "b1d9ab70662946ef0850d488da1c9019f3a9752a", + "type": "github" + }, + "original": { + "owner": "numtide", + "repo": "flake-utils", + "type": "github" + } + }, + "nixpkgs": { + "locked": { + "lastModified": 1723175592, + "narHash": "sha256-M0xJ3FbDUc4fRZ84dPGx5VvgFsOzds77KiBMW/mMTnI=", + "owner": "nixos", + "repo": "nixpkgs", + "rev": "5e0ca22929f3342b19569b21b2f3462f053e497b", + "type": "github" + }, + "original": { + "owner": "nixos", + "ref": "nixos-unstable", + "repo": "nixpkgs", + "type": "github" + } + }, + "root": { + "inputs": { + "flake-utils": "flake-utils", + "nixpkgs": "nixpkgs" + } + }, + "systems": { + "locked": { + "lastModified": 1681028828, + "narHash": "sha256-Vy1rq5AaRuLzOxct8nz4T6wlgyUR7zLU309k9mBC768=", + "owner": "nix-systems", + "repo": "default", + "rev": "da67096a3b9bf56a91d16901293e51ba5b49a27e", + "type": "github" + }, + "original": { + "owner": "nix-systems", + "repo": "default", + "type": "github" + } + } + }, + "root": "root", + "version": 7 +} diff --git a/data/z_distribution/flake.nix b/data/z_distribution/flake.nix new file mode 100644 index 0000000..3eddf4b --- /dev/null +++ b/data/z_distribution/flake.nix @@ -0,0 +1,32 @@ +{ + inputs = { + nixpkgs.url = "github:nixos/nixpkgs/nixos-unstable"; + flake-utils.url = "github:numtide/flake-utils"; + }; + + outputs = { self, nixpkgs, flake-utils, ... }: flake-utils.lib.eachDefaultSystem (system: + let + pkgs = import nixpkgs { + inherit system; + }; + in { + devShell = pkgs.mkShell rec { + buildInputs = with pkgs; [ + ]; + nativeBuildInputs = with pkgs; [ + python312Packages.pandas + python312Packages.matplotlib + python312Packages.imageio + python312Packages.requests + python312Packages.csvw + python312 + ]; + }; + + packages.default = pkgs.mkDerivation {}; + + + + } + ); +} diff --git a/data/z_distribution/z_distribution.csv b/data/z_distribution/z_distribution.csv new file mode 100644 index 0000000..8f17016 --- /dev/null +++ b/data/z_distribution/z_distribution.csv @@ -0,0 +1,1000 @@ +-10.0,1.5670866530993244e-11 +-9.97997997997998,1.731898421572835e-11 +-9.95995995995996,1.913660039744377e-11 +-9.93993993993994,2.1140737179795716e-11 +-9.91991991991992,2.3350083243845715e-11 +-9.8998998998999,2.57851526231251e-11 +-9.87987987987988,2.8468458165120247e-11 +-9.85985985985986,3.142470099460374e-11 +-9.83983983983984,3.4680977407952604e-11 +-9.81981981981982,3.826700475075735e-11 +-9.7997997997998,4.2215367964389906e-11 +-9.77977977977978,4.656178863156737e-11 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