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Marius Drechsler 2024-08-19 21:02:18 +02:00
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content/BACH.typ
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@ -104,12 +104,12 @@ Its cardinality is $2^M$, while $M$ defines the number of bits we want to extrac
It has to be noted, that $bold(cal(o))$ consists of optimal values that we may not be able to exactly approximate using a linear combination based on weights and our given input values.
In comparison to the 1-bit sign-based quantization, we will not be able to find a linear combination of only two input values that approximates the optimal points we defined earlier.
Therefore, we will use -- without any loss of generality -- three summands for the linear combination as this give us more flexible control over the result of the linear combination with the helper data.
Therefore, we will use three or more summands for the linear combination as this give us more flexible control over the result of the linear combination with the helper data.
Later we will be able to show that a higher number of summands for $z$ can provide better approximations for the ideal values of $z$ at the expense of the number of available input values for the quantizer.
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.
z = sum_(i=3)^n x_i dot h_i
$<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 $bold(cal(o))$.
@ -161,4 +161,23 @@ We can now use an iterative algorithm that alternates between optimizing the qua
kind: "algorithm",
supplement: [Algorithm],
include("../pseudocode/bach_1.typ")
)
)<alg:bach_1>
We can see both of these alternating parts in @alg:bach_1_2[Lines] and @alg:bach_1_3[] of @alg:bach_1.
To optimize the quantizing bounds of $cal(Q)$, we will sort the values of all the resulting linear combinations $bold(z)_"opt"$ in ascending order.
Using the inverse @ecdf defined in @eq:ecdf_inverse, we can find new quantizer bounds based on $bold(z)_"opt"$ from the first iteration.
These bounds will then be used to define a new set of optimal points $bold(cal(o))$ used for the next iteration.
During every iteration of @alg:bach_1, we will store all weights $bold(h)$ used to generate the vector for optimal linear combinations $bold(z)_"opt"$.
The output of @alg:bach_1 is the vector of optimal weights $bold(h)_"opt"$.
$bold(h)_"opt"$ can now be used to complete the enrollment phase and quantize the values $bold(z)_"opt"$.
=== Maximum quantizing bound distance approximation
Instead of defining the optimal positions for $z$ with fixed values, we can also provide a more loose definition of $bold(cal(o))$.
Let's consider the following example:
== Experiments
== Results & Discussion

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content/introduction.typ
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@ -1,3 +1,4 @@
#import "@preview/glossarium:0.4.1": *
= Introduction
These are the introducing words
@ -39,6 +40,23 @@ This transformation can be performed using the function $xi = tilde(x)$. The key
Considering a normal distribution, the CDF is defined as
$ xi(frac(x - mu, sigma)) = frac(1, 2)[1 + \e\rf(frac(x - mu, sigma sqrt(2)))] $
==== ECDF
==== #gls("ecdf", display: "Empirical cumulative distribution function (eCDF)")
The @ecdf is constructed through sorting the empirical measurements of a distribution @dekking2005modern. Although less accurate, this method allows a more simple and less computationally complex way to transform real valued measurements into the Tilde-Domain. We will mainly use the eCDF in @chap:smhd because of the difficulty of finding an analytical description for the CDF of a Gaussian-Mixture.\
To apply it, we will sort the vector of realizations $bold(z)$ of a random distributed variable $Z$ in ascending order.
The function for an @ecdf can be defined as
$
xi_#gls("ecdf") (x) = frac("number of elements in " bold(z)", that" <= x, n) in [0, 1],
$<eq:ecdf_def>
where $n$ defines the number of elements in the vector $bold(z)$.
If the vector $bold(z)$ were to contain the elements $[1, 3, 4, 5, 7, 9, 10]$ and $x = 5$, @eq:ecdf_def would result to $xi_#gls("ecdf") (5) = frac(4, 7)$.\
The application of @eq:ecdf_def on $X$ will transform its values into the empirical tilde-domain.
We can also define an inverse @ecdf:
$
xi_#gls("ecdf")^(-1) (tilde(x)) = tilde(x) dot n
$<eq:ecdf_inverse>
The result of @eq:ecdf_inverse is the index $i$ of the element $z_i$ from the vector of realizations $bold(z)$.
The eCDF is constructed through sorting the empirical measurements of a distribution @dekking2005modern. Although less accurate, this method allows a more simple and less computationally complex way to transform real valued measurements into the Tilde-Domain. We will mainly use the eCDF in @chap:smhd because of the difficulty of finding an analytical description for the CDF of a Gaussian-Mixture.