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NOTICES

SECURITIES: This booklet is not an offer, or a solicitation for an offer, to enter into any transaction. It is solely for informational purposes, only to describe a set of algorithms that

implement machine learning and deep learning (the “algorithms”).

Black Tree

AutoML

Vectorized Deep Learning

Charles Davi

July 18, 2022

Abstract

In a series of lemmas and corollaries, I proved that under certain rea-

sonable assumptions, you can classify and cluster datasets with literally

perfect accuracy. Of course, real world datasets don’t perfectly conform to

the assumptions, but my work nonetheless shows, that worst-case polyno-

mial runtime algorithms can produce astonishingly high accuracies. This

results in run-times that are simply incomparable to any other approach

to A.I. of which I’m aware, with classiﬁcations at times taking seconds

over datasets comprised of tens of millions of vectors, even when run on

consumer devices. Below is a summary of the results of this model as

applied to benchmark datasets, including UCI and MNIST datasets, as

well as several novel datasets rooted in thermodynamics. All of the code

necessary to follow along is available on my ResearchGate Homepage, and

www.blacktreeautoml.com.

1

1 Introduction

In a series of lemmas and corollaries (see, “Analyzing Dataset Consistency”

[1]), I proved that given certain reasonable assumptions about a dataset, simple

algorithms can classify and cluster with literally perfect accuracy (see, speciﬁ-

cally, Lemmas 1.1 and 2.3 of [1]). Of course, real world datasets don’t always

conform to the assumptions, but my work nonetheless shows, that worst-case

polynomial runtime algorithms can produce astonishingly high accuracies, as a

general matter. This results in run-times that are simply incomparable to any

other approach to deep learning of which I’m aware, with classiﬁcations at times

taking seconds over datasets comprised of tens of millions of vectors, even when

run on consumer devices. Below is a summary of the results of this model as

applied to UCI and MNIST datasets, as well as several novel datasets rooted in

thermodynamics. All of the code necessary to follow along is available on my

ResearchGate Homepage, and www.blacktreeautoml.com.

For a mathematically rigorous, theoretical explanation, of why these algo-

rithms work, see [1]. For an in-depth, practical explanation of how these al-

gorithms work, including applications to other datasets, see “A New Model of

Artiﬁcial Intelligence” [2].

1.1 Results

As a general matter, my work seeks to make maximum use of data compression,

and parallel computing, taking worst-case polynomial runtime algorithms, pro-

ducing, at times, best-case constant runtime algorithms, that also, at times, run

on a small fraction of the input data. The net result is astonishingly accurate

and eﬃcient Deep Learning software, that is so simple and universal, it can run

in a point-and-click GUI.

Figure 1: Runtime with 10 Columns Figure 2: Runtime with 15 Columns

2

Even when running on consumer devices, Black Tree’s runtimes are simply

incomparable to typical Deep Learning techniques, such as Neural Networks,

and Figures 1 and 2 above show the runtimes (in seconds) of Black Tree’s fully

vectorized Delta Clustering algorithm, running on a MacBook Air 1.3 GHz Intel

Core i5, as a function of the number of rows, given datasets with 10 columns

(left) and 15 columns (right), respectively. In the worst case (i.e., with no

parallel computing), Black Tree’s algorithms are all polynomial in runtime as a

function of the number of rows and columns.

1.2 Spherical Clustering

Almost all of my classiﬁcation algorithms ﬁrst make use of clustering, and so

I’ll spend some time describing that process. My basic clustering method is in

essence simple:

1. Fix a radius of delta (the calculation is described below);

2. Then, for each row of the dataset (treated as a vector), ﬁnd all other rows

(again, treated as vectors), that are within delta of that row (i.e., contained in

the sphere of radius delta, with an origin of the vector in question).

This will generate a spherical cluster, for each row of the dataset, and there-

fore, a distribution of classes within each such cluster.

The iterative version of this method has a linear runtime as a function of

(M1) ⇥N,whereMis the number of rows and Nis the number of columns

(note, we simply take the norm of the di↵erence between a given vector, and

all other vectors in the dataset). The fully vectorized version of this algorithm

has a constant runtime, because all rows are independent of each other, and all

columns are independent of each other, and you can, therefore, take the norm

of the di↵erence between a given row and all other rows, simultaneously. As a

result, the parallel runtime is constant.

1.3 Calculating Delta

My simplest clustering methods use a supervised calculation of delta: simply

increase delta some ﬁxed number of times, beginning at delta equals zero, using

a ﬁxed increment, until you encounter your ﬁrst error, which is deﬁned by the

cluster in question containing a vector that is not of the same class as the origin

vector (see Section 2.2 below). This will of course produce clusters that contain

a single class of data, though it could be the case, that you have a cluster of

one for a given row (i.e., the cluster contains only the origin).

3