The Mathematical Basis: Constraining Responses In Search & AI/LLM Spaces

“Finding a basis”.

“Looking for a basis.”

“What is the basis for this argument?”

“Is this cash basis or accrual?”

“Basis” is a term that’s ubiquitous in almost every profession.

From accounting to law to medicine to chemistry – and everything in between – its use may vary between fields, but fundamentally the word “basis” shares its root, much like most things in our world, in mathematics.

Looking at the general definition of “basis”, you’ll find words like “foundation” or “principal component” — seeking a basis means getting to the bottom of things, more or less.

Turning the lens a bit from the mathematics world, you’ll see this term show up in almost every branch of study – from topology, to linear algebra, algebra and analysis. As mathematics creates the foundation and guides the way for most professions, it serves – in a sense – as a meta-basis. A basis made up of many bases, one could say.

As search & AI/LLM spaces can be considered a sort of convergence of many fields of mathematics (calculus, linear algebra and probability theory mostly), it should be quite easy to not only find different bases within these spaces, but also leverage them to guide the SEO and marketing efforts, on the whole.

For those unfamiliar with the basis (or bases) and its use in mathematics, I’ll walk through some semi-formal concepts from different areas of the mathematics world, some common threads between them when stepping back and finally help connect the dots and walk them through search/AI/LLM spaces.

One could say this post is your basis for an understanding of bases – basically.

A Semi-Formal Look At Bases In Set Theory & Topology

I’ve written about topology in earlier posts and how it describes the relationship between objects in a space – and how they are preserved under transformation (bending, stretching, et. al. ). “Rubber sheet geometry” is commonly used to describe topology.

Topology is a very dense subject that extends beyond the scope of most of this site, so you can do some self-exploration if you want to look at more advanced topics, but you’ll find the concept of bases all the same.

Since topology can get abstract, in simple terms a basis in a topological space can be thought of as a subset of sets in space that form a covering for the entire topological space.

That is, if you join all of the subsets (otherwise known as a union), you can define the entire topological space.

Those subsets would be defined as the basis set of the topological space. Those subsets aren’t always unique – there can be many basis sets – infinite even.

There are also many different kinds of bases, but the basic (pun intended) idea behind bases in topology is that finding them is a way to understand and describe the entire topological space under study.

A Semi-Formal Look At Bases In Vector Spaces

Moving down into more structured spaces – like the vector spaces that we’ve come to know well in the AI/LLM spaces – we’ll find more familiar use of bases.

A basis of a vector space is a set of vectors that create the framework for the entire space. In other words, every vector in a space can be written as a combination of the basis vectors.

A simple example is the x-y coordinate system — every 2 dimensional vector can be written as a combination of an “x” coordinate and a “y” coordinate (5,9), (64, 208), et. al. In this case, (0, 1) and (1,0) are basis vectors. (5,9) can be written as (0,9) along the (0,1) basis vector and (5,0) along the (1,0) basis vector.

If you have a basis vector, no vector in the space can be described using any other vectors – it removes redundancy and everything in the space can be built using it.

Basis vectors extend beyond our 3-D world, as well, but the dimension of the basis vector is limited to the dimensionality of the space itself. An 11-dimensional space has a basis of 11 vectors, essentially.

Regardless of the shape or dimension of a vector space, each vector can be described – uniquely – by the basis vectors.

The Common Thread: The Basis For Bases

If you read closely above, finding bases means understanding the entire space – whether it’s a topological space, a vector space, an inner product space and even more interesting spaces I’ll touch on in future posts.

There can be several bases and changes of bases the more complex the space gets, but ultimately they play the same role: defining the entire space using a smaller subset of elements.

Bases In SEO

Keyword or keyword phrase research is something that is fundamental in SEO. Understanding relevant terminology used by a particular market or subset of users is an essential part of crafting a website to align with how folks might be searching for those particular terms or phrases.

I’ve found that keyword research boils down to looking for basis elements that can describe a particular subset of the overall search space.

If you can find those few basis terms, then you’ve done most of the work – every other term is some combination or iteration of those few terms or phrases, essentially. In topological terms, you’re finding some “covering” term set for a specialized search/candidate space (or subspace, as it were).

As things have progressed online and spaces have become more fluid, the rigid “keyword research” has moved toward more “topical” research as associative spaces can infer deeper meaning behind each search, rather than straight dictionary or lexical matching (addressable search spaces).

The idea is the same, however — finding those few, essential topics (or subtopics) that ultimately form the basis/covering for the resulting search space in that category.

When folks say they’re “covering a topic”, I always wonder if they know the connection to topology and bases… but I digress.

Bases In Search & AI/LLM Spaces And Measurement

I mentioned this briefly in a previous post, but measurement in search & AI/LLM spaces has to be defined by some basis – a subset of important elements in the input (prompt + context) that ultimately shape the results or responses.

Each prompt (and inferred attention/meaning) and contextual element can be thought of as a different dimension in a basis set — each constraining the response space in its own unique way, narrowing the starting point in the learned internal candidate space and the decoder as it navigates the candidate selection space to produce a “walk” as it chooses the next set of words or passages.

Ambiguous inputs (prompts/queries) or inputs that have less context attached naturally have fewer basis dimensions to form a response (simply in cardinality alone) – the “walk” is less deterministic because it has fewer constraints to guide its journey through the candidate space.

In a fresh search or prompt session, adding the input “bronco” without context may result in a response that includes “Ford Bronco” and its formal definition “an unbroken, wild horse”.

If previous searches or prompt/conversations include topics like “horse training” or “horse taming”, the space has that history as an additional dimension in its basis set, constraining (or, collapsing) the response walk into the more formal definition “wild horse”, et. al.

You can imagine, then, as a query/prompt session gets deeper and more detailed, these bases will change, shrink, rotate, expand and otherwise transform as needed to ultimately form the responses.

These bases formed by history, new context and related prompts create their own topological space, in a sense – with responses inheriting their changing bases throughout a session.

Understanding and predicting responses means understanding the topology and bases of the queries/prompts – and associated user groups – creating those contextualized input spaces, and how they constrain the “walks” through the internals of the AI/LLM search space that ultimately form results/responses.

Find the basis, understand the space.

Many more notes in the weeks to come.