Into The Looking Glass: On Projections In AI/LLM Search Spaces
Published by Daniel A. Dahlen on
“Shadows in ghosts.”
A few weeks ago here I wrote about how AI/LLM search responses can resemble our traditional search foundations – “Markovian ghosts“, so to speak.
Markovian in the sense that responses conceptually modeled as “walks” through a contextual state space – each token selection moment governed by a transition matrix along a dynamically generated graph – shaped by the accumulated, compressed context available in that moment.
This is (generally) the same idea as web navigation and the rules governing the movement of users between web pages ( web pages as vertices and links as edges borrowed from graph theory ). An adjacency matrix tells you what pages are connected and a transition matrix tells you the probability of a user moving from one page to the next.
For me, it’s easy to see these “Markovian ghost” patterns in AI/LLM search spaces and surfaces – now carrying similar transition tendencies as walk trajectories unfold to reveal responses.
Same (or similar) paths, different presentation layer essentially – conversational responses vs a subset of links or other search features.
Those paths can be shaped and influenced by any number of contextual elements – including elements of the individual users performing the search.
Mentally, you can imagine a very bright light behind the user, casting a shadow – or projection – against these spaces that blends into the accumulated context (compressed memory) as response candidates are chosen during decoding. While AI/LLM search internals will have their own transition tendencies between tokens (tending to pass through dominant eigenstates, in a sense), the projection of user-dependent context can alter those transitions, effectively.
“Shadows in ghosts”.
Conceptual Projections
Stepping back a bit to build some intuition on projections (outside of the shadow analogy above), likely the place you’ve come across projections the most are with maps (and topology).
Whenever you take out a map, you’re looking at a 2-dimensional projection of our 3-dimensional globe.
Distortion occurs (see: Mercator projection), but these projections keep us from having to carry around a globe everywhere – it’s a compressed view that maintains the relationship between points (even if stretched). The terms chart, map and atlas all have formal definitions in the field of topology if you want to do a deep dive on that subject.
Another place you see projections are with mirrors (or “looking glass”). Instead of projecting something onto a surface, it reflects an virtual image back to the observer.
This is still a 2-dimensional projection of our 3-dimensional world, however it preserves much more detail than our shadow and map analogies. Largely the “z” dimension is hidden — we can still perceive depth relative to other elements in the image, but ultimately it’s no longer on the surface or can be measured in that “virtual subspace” of our reality.
Subspace Projections In Hilbert Spaces
On the topic of subspaces, last week I wrote a very brief summary of Hilbert spaces ( complete inner product spaces, with the norm defined by the inner product ).
Left out of that piece was the concept of subspaces – a smaller subset of the space that maintains all of the important properties ( completion, norm, inner product, et. al. ) of the larger Hilbert space.
Important concepts of closure and other properties can be explored, but for this post we’ll take a closer look at projections onto subspaces.
Going back to our map analogy above, our higher dimensional “real world” depiction of the globe is projected down into a lower dimensional 2-d subspace. Stepping back again, our 3-dimensional subspace we live in can be thought of a projection down from a higher dimensional Hilbert space.
Much like our mirror analogy, those extra dimensions are “hidden” in our world – they may still be present, but the projection reduces the effective dimensions we can perceive in our reality. (This is getting a bit philosophical here but the analogy holds, so bear with me.)
Projections In Graph Theory
Projections in graph theory work similarly as the subspace projections in Hilbert spaces — projection matrices applied to a graph reduces the graph to a smaller subset of the graph, leaving out unimportant or redundant items, leaving behind more stationary, independent areas of the graph.
An adjacency matrix describes the connections between nodes in a graph (links), a transition matrix describes the probability of moving between nodes in a graph (propagation) and a projection matrix simplifies the graph into its most important elements or regions (over time).
Thinking in terms of walks, the adjacency matrix tells you what walks are possible, the transition matrix tells you the probability of those walks occurring and applying a projection matrix tells you which walks are most prominent over time (revealing transition tendencies, essentially).
One can imagine a “projective walks” as walks that gravitate toward these more prominent nodes or vertices, over time (the roads *most* taken in defiance of Robert Frost, if you will).
Projections In SEO & AI/LLM Search Surfaces
Going back to my mental model, let’s imagine AI/LLM search responses as dynamic “walks” through Hilbert spaces – with each token (subword/word/concept/entity) being nodes on the walks and each token selection moment defined by a transition matrix (assigning probabilities to the “next nodes” at each step) with accumulated context compressed into each time step, resembling a Markov chain (tying in our web analogy) – and apply what we now know about projection matrices.
A fun property of projection matrices is that if you apply it multiple times, it will yield the same result. Relating this to the prompt-response pattern, you can think of projection matrices (also called operators – more on that in a future post) as questions/queries that have stable/consensus responses.
Things like “What is the capital of the United States?”. While the exact walk may vary slightly from response to response (think different arrangements of words), questions like these will almost always yield and project into the same stable space and respond: “Washington D. C.”.
A lot of what I’ve seen as “prompt engineering” is basically assembling projection-type operators (forming an input space on its own – more on that later) – “aiming at” responses with predictable transition tendencies (pushing into more consensus/dominant subspaces of each model).
In search, this can be thought of as mapping and classifying similar queries that have the same/consensus set of search results.
I looked at “search operators” differently over the years than most folks for this reason – but I digress.
User-Dependence (Observer Effect)
Where things get challenging is when user related elements are added into the context of the responses.
This is where the “shadow” of the user projection can subtly influence those responses — over time that user projection can embed itself into the vanilla projection baked into the model, shaping answers tailored to that user under different contextual situations.
Knowing a user is in the market for trucks may project into different subspaces and responses about “Colorado” (a type of Chevy Truck) as that has been cast into the space by that user, than a plain/vanilla conversation about the state of Colorado.
Responses reveal not only transition tendencies of the models, but also response tendencies by individual users at the helm of input side.
Understanding the starting point – user + contextual state – then, is just as important as the operator itself during any kind of measurement of these walk trajectories.
Mapping [ user + contextual states + operators ] into an aggregate projection operator space should allow us to reveal how response “walks” settle into dominant subspaces in the model for those operator sets. At the very least they can be used as “north stars” – a basis for different response measurements.
A stronger projection operator space (well structured, bounded) also can explain how some users project better, more reliable responses (or walks) — weaker projection operator spaces (ones that aren’t as structured or bounded) project into more unreliable subspaces and responses.
Strong operators vs weak operators (perhaps in the literal sense).
(I have more conceptual notes on conversational topology as well related to this matter – how that projection operator space can be transformed over time).
Final Takeaway
As mentioned early in this series of posts, it should now be apparent that knowing your users (AKA digital marketing) is – and always was – an important piece of understanding these spaces.
Hopefully after reading this you can have a mental model of *how* those users are affecting those results/responses.
Being included in “consensus walks” for users is great — being included in “personalized consensus walks” should be the aim for every SEO & marketer, in my opinion.
“Shadows in ghosts.”
The past few weeks have given you a lot to think about and absorb here, so I’ll be back in a few weeks with more updates and notes.
