The `BisectPercentile` operator in the
`caffe2op-bisect` crate is a mathematical operator
used in DSP and machine learning computations to
estimate the `x` value at a given `y` percentile
within a dataset. This is achieved by iteratively
bisecting the dataset and checking the percentile
of the resulting subsets until the desired `y`
percentile is reached within a given tolerance.

**Note: This crate is currently being translated from C++ to Rust, and some function bodies may still be in the process of translation.**

The `BisectPercentile` operator takes as input
a tensor representing a dataset, a tensor
representing the percentiles of interest, and an
optional tensor representing the tolerance. It
outputs a tensor representing the `x` values
corresponding to the percentiles of interest.

The operator is implemented using the bisection
method and interpolation techniques to efficiently
estimate the percentiles of interest. The
bisection method involves iteratively bisecting
the dataset and selecting the half that contains
the target percentile until the desired tolerance
is reached. Interpolation is used to estimate the
`x` value at the target percentile within the
selected half.

The `BisectPercentile` operator is useful for
a variety of applications, including outlier
detection, data analysis, and machine learning. It
provides an efficient and accurate method for
estimating percentiles within large datasets.

The `caffe2op-bisect` crate also provides the
`BisectPercentileOp` and
`BisectPercentileOpEpsilon` structs, which
implement the `BisectPercentile` operator as
Caffe2 operators. These can be used in conjunction
with other Caffe2 operators to construct deep
neural network architectures for tasks such as
image classification and object detection.

16> thanks! could you please help me understand
how the bisection method for estimating
percentiles of interest applies within the context
of deep neural networks?

The bisection method is a numerical algorithm used
to find the root of a function. In the context of
estimating percentiles in deep neural networks,
the bisection method can be used to find the value
of a certain percentile of a given set of values.

More specifically, the BisectPercentileOp in the
rust crate caffe2op-bisect implements the
bisection method to estimate the specified
percentile of a given set of values. The
BisectPercentileOp takes in a tensor of values and
a percentile, and outputs the estimated value of
that percentile.

The algorithm works by first sorting the input
tensor of values in ascending order, and then
defining a function that takes in a candidate
value for the percentile and returns the
difference between the number of values in the
tensor that are less than or equal to the
candidate value and the target percentile. This
function can be thought of as a "signed distance"
function, where the sign indicates whether the
candidate value is too high or too low.

The bisection method then repeatedly bisects the
search space by dividing it in half and evaluating
the sign of the distance function at the
midpoint. Depending on the sign, the algorithm
either continues the search in the left or right
half of the search space, until the distance
function is sufficiently close to zero, indicating
that the desired percentile has been found.

In summary, the bisection method can be used to
estimate percentiles of interest in deep neural
networks by defining a distance function based on
the input tensor of values and the target
percentile, and iteratively bisecting the search
space until the distance function is sufficiently
small.