Assumption, Challenge, and Resolution: Online Relationship Discovery

Assumption

We start from a single assumption: if an algorithm can understand the relationships in data, then it can do everything we normally train machine‑learning models to do.

Understanding relationships means capturing how variables constrain, influence, depend on, compete with, or reinforce each other across different forms: causal, temporal, probabilistic, structural, interactional, and compositional. Once this relational structure exists:

• Prediction is simulating the system forward
• Anomaly detection is detecting violations of learned constraints or distributions
• Classification is assigning a state to a region of relational space
• Clustering is grouping states or entities with similar relational structure

In this framing, these tasks are not separate learning problems. They are different queries over the same underlying structure. The only hard problem is discovering that structure.

Relationship structures considered

The system does not assume a single type of relationship. Instead, it maintains a broad hypothesis space of relationship structures that may coexist, overlap, or transform into one another as evidence accumulates. The primary structures considered include:

1. Causal – one variable directly produces change in another (intervention on X changes Y).
2. Correlational / Associative – variables move together without implied direction or mechanism.
3. Structural / Hierarchical – variables are nested or grouped (levels, containment, aggregation).
4. Temporal – relationships depend on ordering, lags, seasonality, or cycles over time.
5. Functional / Deterministic – one variable is an explicit function of others (exact or near‑exact constraints).
6. Probabilistic – variables shift the probability distribution of others rather than determining outcomes.
7. Compositional / Additive – variables form parts of a whole under conservation or accounting constraints.
8. Competitive / Antagonistic – variables compete for limited resources; gains in one reduce another.
9. Synergistic / Interactional – combined variables produce effects larger or qualitatively different than individual effects.
10. Mediating – the effect of one variable on another passes through an intermediate variable.
11. Moderating – the strength or direction of a relationship depends on a third variable.
12. Graph / Network – entities are related through explicit connections; structure itself carries information.
13. Similarity / Clustering – observations relate through resemblance rather than direct interaction.
14. Equilibrium / Attractor – system dynamics converge toward stable states or recurring regimes.
15. Logical / Boolean – rule‑based, conditional, or threshold relationships (if‑then structure).

These structures are not mutually exclusive. The same variables may participate in multiple relationship types at different scales or times. The goal of the system is not to label relationships once, but to maintain and revise a distribution over plausible structures.

Challenge

The challenge is combinatorial explosion.

With 50 variables, the number of possible subsets alone is 2^50. That is before considering:

• Pairwise and higher‑order interactions
• Temporal ordering and lag structure
• Mediation and moderation
• Non‑linear and conditional dependencies

Exhaustively searching this space is impossible. The problem becomes even harder under two constraints:

• Online learning: data arrives sequentially and decisions must be made early
• Cold start: there are no pretrained representations or labels to guide search

Trying to understand all variables and all interactions at full resolution from the start is computationally infeasible and statistically unstable. At the same time, ignoring interactions defeats the purpose of understanding relationships in the first place.

Rows as system states and constraints

The system adopts an explicit assumption about the structure of tabular data: columns represent variables, while rows represent observed system states. Rows are not treated as passive samples used only for fitting models, but as active constraints that test whether proposed relationships are consistent with reality.

Each hypothesized relationship is treated as a constraint that can be evaluated against a row. When a new row arrives, the system asks whether the current relational structure can explain that state. Relationships that consistently hold across many rows gain confidence, while relationships that repeatedly fail are weakened or discarded.

In this framing, rows serve three roles:

• Validation: a row tests whether existing relationships still hold
• Pruning: relationships that fail across many rows are eliminated early
• Signal generation: structured residuals across rows indicate where relationships are missing

Residuals are not treated as noise by default. Instead, the system tracks which variables repeatedly appear in unexplained behavior across rows. Consistent residual patterns suggest candidate variable combinations or group refinements to explore next.

By using rows as constraints, the system reduces blind combinatorial search. Relationships are proposed, tested, and either survive or die based on their consistency across observed states. This turns relationship discovery into a constraint‑satisfaction and hypothesis‑survival process rather than an exhaustive enumeration problem.

When not to group

Grouping is not a default behavior of the system. It is a response to combinatorial pressure, not a principle of learning. When the hypothesis space is small enough to be explored directly, grouping is unnecessary and can be actively harmful.

The system therefore avoids grouping under the following conditions:

• Small number of variables: when the number of variables is low (e.g. fewer than 8–10), explicit exploration of combinations and interactions is computationally feasible.
• Low interaction order: when meaningful structure is likely to be pairwise or low‑order, direct testing is cheaper and more reliable than abstraction.
• Strong, clean signals: when relationships are stable and consistently supported across rows, early compression risks hiding interpretable structure.
• Sufficient data per state: when rows provide enough coverage to strongly constrain hypotheses, explicit relational testing is preferred.

In these regimes, the system prioritizes direct combinatorial discovery:

• Relationships are tested explicitly against rows
• Constraints are learned at full resolution
• Grouping is deferred entirely

Only when the hypothesis space grows beyond feasible limits—due to variable count, interaction order, noise, or online constraints—does the system introduce grouping as a scaling mechanism.

This makes grouping conditional, adaptive, and reversible, rather than universal.

Resolution: grouping as a search strategy

To make relationship discovery tractable, the system uses grouping as an explicit search and compression strategy.

Instead of reasoning over all variables simultaneously, variables are first organized into a small number of coarse, provisional groups. For example, 50 variables may initially be grouped into 5 groups. These groups are formed using weak, low‑cost signals such as shared variance, temporal alignment, or distributional similarity.

Relationship discovery is then performed between groups, not between individual variables. This reduces the effective search space by orders of magnitude.

Coarse‑to‑fine exploration

Grouping is not the goal; it is a staging mechanism.

Once a group shows evidence of meaningful structure—such as strong interaction with other groups, internal inconsistency, or persistent unexplained behavior—the system zooms in. Within that group:

• Variables are rearranged, split, or re‑grouped
• Local interactions are explored at higher resolution
• Relationship types are hypothesized and tested

This process is recursive. The system moves from coarse understanding to fine detail only when the data justifies it.

Critical risk: early grouping error

Grouping compresses information before full understanding. This introduces a serious risk: important interactions may be averaged away or hidden inside incorrect abstractions.

In an online system, this is especially dangerous. Early grouping errors can bias all future learning if left unchecked.

How the system survives this risk

Grouping is treated as a hypothesis, not a commitment.

To avoid permanent blind spots, the system enforces three rules:

1. Reversibility

2. Groups can split, merge, or dissolve

3. No abstraction is final

4. Residual pressure

5. The system tracks what its current structure fails to explain

6. Persistent unexplained variance forces regrouping or refinement

7. Forced exploration

8. A small exploration budget is reserved for testing unlikely or cross‑group interactions

9. This prevents the system from locking into a wrong abstraction

What this approach is

This is not a single model and not a loss‑minimization pipeline.

It is an online hypothesis‑refinement process that continuously balances:

• Compression vs exploration
• Stability vs adaptability
• Coarse structure vs fine detail

When this balance is maintained, many traditional machine‑learning tasks stop being primary objectives and become secondary byproducts of understanding structure.

Summary

• Assumption: understanding relationships subsumes most ML tasks
• Challenge: combinatorial explosion makes naïve discovery impossible
• Resolution: hierarchical grouping with reversible, coarse‑to‑fine exploration

The system does not guarantee truth. It maintains the best relational structure it can, given the data it has seen so far, and continuously revises it as new evidence arrives.