Beauty is Compressible Complexity

A new Definition of Beauty informed by Machine Learning. And why Caravaggio is better than Basquiat

What makes this…

Judith Beheading Holofernes, Caravaggio

…better than this?

Untitled, Jean-Michel Basquiat

You may find Caravaggio’s Judith Beheading Holofernes an obviously more beautiful and better work of art than Basquiat’s garish Untitled. But says who? Is beauty not subjective? Is all art not a matter of taste? Does beauty not reside squarely in the eye of the beholder? On the other hand, is Basquiat not blatantly stretching the bounds of our credulity? To answer the question of objective ranking of the quality of art, one would need a definition of beauty and some measure of good art.

I was thinking about these questions while also thinking about the progress in machine learning. This lead to a cross-fertilization event. Therefore in this essay, I will make the case that beauty and art are based on compression of complexity. I will show how this view explains previously unexplained phenomena. Finally, the concept will allow me to argue that there is a degree of objectivity in beauty and art, and a basis for objectively judging art’s quality.

Compression and decompression: Learning and Imagining

Learning means compressing complexity. To learn is to take data, find a function that can generate features of the data, and encode the function in a generalized form. When a basic neural network learns to distinguish cats from dogs it compresses the recognizable features of these animals. A method to recognize whiskers and pointy ears is imprinted into the neural network model.

\(Examples \ast Neural\ Network \rightarrow Object\ Names\)

The advances in AI of the last decade, from AlexNet to GPT-4, boil down to improving the algorithm for estimating the weights (parameters) of the model. The weights encode the features of the world learned by the model. In other words, the advances in AI are advances in recognizing and compressing features of the world.

The process can also be inverted:

\(Object\ Names \ast Neural\ Network \rightarrow Examples\)

The labels (“cat”, “dog”) can be fed back to the model that will then generate images from the encoded features. This can be thought of as decompression - or imagination. Imagination is generation of objects by decompression of the features of the world that are encoded in the AI model (or human brain).

Beauty is compressible complexity

I would like to postulate that beautiful is that whose features allow the human brain to easily compress its great complexity. Let’s go right into a few examples of beauty and ugliness, to illustrate the notion, before we formalize the idea.

Example 1: Architecture - Compressible Complexity vs Incoherent Plainness

Hungarian Parliament Building

The Hungarian National Assembly in Budapest has a highly complex shape. But the intricate symmetry, the triangular shape, the repeated columns and arches and spires, all join together to make it easy to recognize, to encode in the brain, and to imagine in the future. A person who sees it will never mistake it for a different building, and will be able to imagine it in some detail.

Scottish Parliament Building

On the other hand The Scottish Parliament Building in Edinburgh is a chaotic combination of random shapes at random angles. The shapes are disconnected from each another while indistinct in aggregate. It is impossible to be certain where the building starts and where it ends. At the same time it holds no great complexity within it. The shapes of each constituent building are plain sections of cubes and cylinders, unlike the ornate gothic windows and buttresses of Budapest.

Example 2: Nature - Compressible Complexity vs Super-Compressible Plainness

The natural beauty can also be ranked by its compressible complexity.

Yosemite National Park, California

The Tunnel View over the Yosemite Valley immediately speaks to the mind with its complex near-symmetry and point-projection symmetry. A longer examination reveals new features that fit into the landscape and can easily be incorporated into the mental image - the sunshine on El Cap, the Bridalveil waterfall, the Half Dome summit in the distance. Further inspection shows yet more - the autumn leaves in the valley, the winter snow at the peaks. The picture is highly complex, but easy to compress and remember without much loss of detail.

Los Carneros, California

The field in Los Carneros, used from a different angle for the default desktop in Windows XP, is pretty and extremely compressible. But it is not beautiful, at least not as much as Yosemite. That’s because it doesn’t hold complexity. Looking at it does not reveal to the brain any new features that can be stored and used. A simple field for simple minds.

Example 3: Architecture - Compressible Complexity vs Incoherent Complexity

Let’s apply the concept to literature.

The Lord of the Rings has a complex storyline over many hundreds of pages, unfamiliar names, and imaginary places, a rich mythology, and nuanced references to the real world. But is woven together in a way that makes it impossible to forget. It is universally loved by people of different ages and intelligences. It spawned an entire genre of literature.

Here is what Wikipedia has to say about it James Joyce’s Finnegans Wake: “[…] It has also been regarded as an attempt by Joyce to combine many of his prior aesthetic ideas, with references to other works and outside ideas woven into the text; Joyce declared that "every syllable can be justified". Due to its linguistic experiments, stream of consciousness writing style, literary allusions, free dream associations, and abandonment of narrative conventions, Finnegans Wake has been agreed to be a work largely unread by the general public”

We call beautiful - a building, a landscape, a book - when it contains a great complexity that yields itself to be efficiently compressed and stored in memory or subconscious.

Compressible complexity defined

My thinking here is partly informed by Jurgen Schmidhuber’s theory of beauty which is definitely worth looking into.

Complexity can be intuitively understood as the amount and variety of features within the dataset. Compressibility is the ability of the observer to efficiently encode the features of the object.

Compressibility of an object depends on both the object and the observer. The object may have features that can be recognized and efficiently encoded. But at the same time the observer may not have the right compression algorithm to recognize the features. This is why people often need art education and repeated exposure to develop an appreciation for specific art genres.

Why have minds evolved to take pleasure from observing beauty? It is important for a mind to continuously improve its compression algorithm to recognize more features of the world. After all the ability to recognize new features may come useful in the future. Therefore the mind should seek out opportunities for training. Schmidhuber defines interestingness as the first-derivative of beauty: “[…] as the learning agent improves its compression algorithm, formerly apparently random data parts become subjectively more regular and beautiful”.

Here are some examples of what I identify as compressible complexity across domains. See if you find them beautiful:

• Landscape: Mountain valleys, tree-lined waterfronts, large waterfalls, city skylines

• Faces: Symmetry, regular proportions, clear and smooth skin

• Poetry: Short forms able to bring about a great emotion or a profound truth

• Architecture: Intricate symmetry, golden ratio, repetition of sub-elements, product-landscape fit: e.g. Alhambra, Golden Temple of Amritsar, Monticello

• Painting: Representational art filtered through the eyes of the artist: e.g. Impressionism, Renaissance, Neoclassicism

• Mathematics: Profound realizations in simple terms: e.g. Euler equation

• Science: Simple explanations with great explanatory power: e.g. e=mc2

This is the end of Part 1. In Part 2 I will go deeper into this paradigm. I will define art by expanding the idea of compressible complexity. We will see if we can stretch the concept yet further - to ethics. And finally I will examine if we can objectively rank Caravaggio over Basquiat