The Artist’s Fingerprint: Kernel Intuition For AI & LLM Search Measurement
Published by Daniel A. Dahlen on
Picasso. Rembrandt. Van Gogh. da Vinci.
Famous artists with distinctive names and incredibly valuable artwork to match. So valuable, in fact, that copies or replicas are often created in an attempt to capture that value – many, likely with less-than-noble intentions rather than simply being inspired by their work.
While provenance data and related documentation can create a paper trail for collectors to verify originality, deeper still exists the artist’s unique fingerprints: hidden brushstroke characteristics, pressure variations, complex ridges and valleys and other patterns embedded into their work.
These analog fingerprints are incredibly difficult to reproduce, meaning it can add even more validation of authorship for a collector of those pieces. One can’t see or observe these fingerprints when looking at the artwork from afar (in all likelihood) — one has to scope down to microscopic levels to map those hidden ridges and valleys in the artists’ brushstrokes.
With that microscopic map assembled, one can compare the pieces’ brushstroke tendencies with some baseline tendency figures and compute a similarity score. I wrote about similarity scores in an earlier post (scroll back if you want a refresher there), but using the generalized dot product – called the inner product – one can take two pieces of information (after mapping data into a vector space) to produce this similarity score.
As you can imagine, the data compressed into those data points can be incredibly complex, representing not just points in space but often functions and other objects — which is why we need the inner product to compute the similarity.
Turning the similarity calculation across all of the data points into a function itself, we get the mathematical kernel (loosely speaking).
The process of mapping complex data (like our brushstroke data) into a high dimensional space and computing the inner product to find hidden patterns is something called the “kernel trick”.
If you’ve followed along on this blog, you know I often avoid gimmicks or “tricks” in the SEO/search world – but this trick is safe, I assure you.
Building Kernel Intuition
If mapping data into a high dimensional space to find hidden patterns and connections sounds familiar, it’s because you’ve likely come across it in any study of large language models and AI.
The process of tokenizing inputs (words and word pieces) into a high dimensional vector space, coupled with the attention mechanism (that finds and connects the hidden patterns based on similarity scores in those spaces) is – loosely – the same conceptual process as the kernel trick.
(Strictly speaking, with the nuances of the engineering of the attention scores, this doesn’t directly map (the softmax, normalization, et al.) but abstracting the process, it’s essentially the same idea.)
If you take two words and perform a raw similarity score, no pattern may be evident — take those same two words (along with their surrounding context), cast them into a high dimensional space and perform a similarity measurement (the kernel trick) you may reveal hidden patterns that would otherwise not be discovered.
That’s the fun and intuition behind kernel methods.
The Reproducing Kernel Hilbert Space (RKHS)
As the kernel defines an inner product over a vector space (where vectors can be functions themselves), we naturally find ourselves in Hilbert space territory ( go back and scan my Hilbert primer if you’ve missed that ).
This Hilbert Space defined by the kernel – which, again, acts as a set of “similarity rules” about the hidden patterns or tendencies in the source data (like our “artist’s fingerprint” data) – can be used to evaluate new data (our test piece of art).
The complexity of the source data means it can look at several (perhaps thousands) of dimensions in evaluating new data, but with the kernel trick, it boils down to a simple inner product calculation, which can be used to “reproduce” a score for that data based on its available features/dimensions ( and potentially mark it as authentic or inauthentic – “van Gogh” or “van No”, so to speak ).
As this is meant as a primer on RKHS, the intricacies can be left to you to dig into, but for purpose of this post it can be boiled down to: cast complex data into a high dimensional space, apply a similarity function to the points in those space and use that as a basis to evaluate, score and classify new data.
Kernel Methods Over Operator Spaces
In that post I mentioned how abstracting individual prompts into operators allows us to leverage the precision of operator theory (machinery like spectral theory) to stabilize our measurement methodologies.
Too much focus has been on the outputs – without a firm handle on the input side of things, it’s a bit of a fool’s errand to accurately measure and predict the output side of things, in my opinion; modeling prompts as operators gives us that handle.
The complexity and vastness of the prompt space is not unlike our artist brushstroke data — but by modeling prompts as operators (looking at what they do, rather than what they are — like grouping brushstroke data by pressure sensitivity and other characteristics) and leverage kernel methods, we can find the hidden patterns that exist on the input side of things.
If You’ve Worked In SEO, This Should Feel Familiar
The kernel trick and RKHS may feel familiar to those reading from the search world.
If you look closely this is simply an abstraction of query classification and topic classification more or less.
Queries and topics were more… simple objects. A few keywords or relevant details about a topic given some context were grouped together into clusters — there were nuances and interpretations of those clusters, of course, but it was still very much WYSIWYG.
As the input (prompt) side becomes more capable of handling complex, often longer prompts – that can be compositions of different classifications – the need to abstract those classifications becomes vital.
Without that abstraction into spaces like the RKHS (and similar), we have no way of understanding the behavior of new prompts — and no way of capturing any stable measurement methods.
Up Next: Operator Space Construction And Some New Visibility Metrics
Over the next few weeks I’ll be adding some more rigor to this model (and connecting previous ideas on eigenvectors, projections and Markovian ghosts/walks) – first by offering some thoughts on the operator space abstraction and construction.
Then I’ll be introducing an important note on a better visibility metric (current visibility metrics are largely outdated and need an important updated approach).
Every artist leaves their own hidden signature in their work — often it requires some advanced machinery for it to become more “visible” from the outside looking in.
What model signatures will become visible in light of this new context?
