#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"), 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.

== Optimizing a 1-bit sign-based quantization

Before we take a look at the higher order quantization cases, we will start with a very basic method of quantization: a quantizer, that only returns a symbol with a width of $1$ bit and uses the sign of the input value to determine the resulting bit symbol.

#figure(
  include("./../graphics/quantizers/bach/sign-based-overlay.typ"),
  caption: [1-bit quantizer with the PDF of a normal distribution]
)<fig:1-bit_normal>

If we overlay the PDF of a zero-mean Gaussian distributed variable $X$ with a sign-based quantizer function as shown in @fig:1-bit_normal, we can see that the expected value of the Gaussian distribution overlaps with the decision threshold of the sign-based quantizer.
Considering that the margin of error of the value $x$ is comparable with the one shown in @fig:tmhd_example_enroll, we can conclude that values of $X$ that reside near $0$ are to be considered more unreliable than values that are further away from the x-value 0.
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 $z$ as a linear combination of two realizations of $X$, $x_1$ and $x_2$ with a set of weights $h_1$ and $h_2$: 
$
z = h_1 dot x_1 + h_2 dot x_2 .
$<eq:lin_combs>

=== 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:
$
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).
$<eq:z_distribution>
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)) .
$<eq:half_normal>
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 #<eq:z_integral>
$
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.
$<eq:z_result>
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>

@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 $bold(z)$, 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.