Knowledge Graphs That Know What They Don't Know.

Click to Toggle: Pins vs. Clouds

Four vehicle types are known precisely. A vague report just says "a vehicle."

Traditional embeddings place every entity at a single fixed point — ambiguity has nowhere to go.

A natural concept forms a convex set: any point between two members also belongs to the category.

Plain English: if "orange" is a valid colour and "yellow" is too, then any blend of the two is still a valid colour. Concepts aren't points — they're regions with soft edges, exactly what a Gaussian distribution is.

Entities as Gaussians

Each entity is a distribution — the mean is the best-guess position, the variance encodes uncertainty.

Real-world example: "Challenger 2 Tank" → a small, tight cloud (we're certain). "Armoured Vehicle" → a large, diffuse cloud (could be almost anything). Small \(\sigma^2\) = confident. Large \(\sigma^2\) = uncertain.

Speed Through Geometry

Bayesian methods are powerful but usually too slow — they rely on sampling millions of simulations. PICASO skips simulation entirely: because clouds are Gaussian, interactions have a direct geometric formula.

Plain English: to predict a missing fact, PICASO rotates, scales and translates the head entity's cloud. The uncertainty from both entity and relation combine in one formula — no simulation, no sampling. The result is O(1) time complexity with respect to graph size: deep-reasoning accuracy at the speed of a standard embedding lookup.

Multi-Faceted Geometric Scoring (MGS)

Not all relationships look the same. PICASO scores every candidate fact through five geometric lenses simultaneously, and learns how much to trust each one.

Plain English: \(s_{geo}\) is variance-normalised distance, \(s_{trans}\) is uncertainty-weighted translation, \(s_{KL}\) measures distribution overlap, \(s_{bil}\) captures multiplicative interactions, and \(s_{cpx}\) scores in the complex plane. Hierarchical facts ("frigate is a type of naval vessel") lean on containment; symmetric facts ("UK is ally of France") lean on equality; asymmetric facts ("General X commanded Unit Y") lean on direction. The weights \(w_i\) are learned per relation type — all still in closed form.

Calibrated Uncertainty, Not Bolted On

A dedicated calibration loss ensures that stated confidence aligns with actual accuracy — no post-hoc Platt scaling, no patching after the fact. Uncertainty falls directly out of the Gaussian geometry itself. (Full evidence for this claim is in the Results section below.)

PROCS surpasses the computationally heavy NBFNet on FB15k-237 in both MRR (0.417 vs. 0.415) and strict top-1 retrieval (0.339 vs. 0.321) — without the overhead of explicit subgraph traversal, and while additionally providing calibrated uncertainty no baseline above offers natively.

Given (NATO, instance of, Military Alliance) — can the model tell if it's true or false?

Operational meaning: nearly 90% accuracy in flagging logically inconsistent facts. This acts as an automated triage filter — instantly flagging potentially poisoned intelligence for human review.

Biggest impact — MGS: removing the five geometric scoring functions costs 14.9% MRR; this is the core engine. Rotation matters: phase-based rotation captures directional relationships, −9.2% without it. An honest exception: dropping semantic-type constraints slightly improves top-1/top-3 ranking, but Hits@10 drops 3.6% — its one clear, if modest, contribution.

Uncertainty-Stratified Accuracy — FB15k-237 (Hits@1)

Lower-uncertainty bins consistently trend toward higher accuracy. On WN18RR, confident predictions achieve 59% accuracy while uncertain ones drop to 15% — a 4x difference, with no post-hoc calibration applied.

Selective Prediction: Act on What's Reliable

3,000 test triples, split into three tiers by model uncertainty:

The 18.8-point gap between high- and low-confidence tiers means a decision-maker can act on confident predictions and route uncertain ones to a human analyst — a capability no deterministic baseline offers natively. (The medium tier edging out the high tier is a real, honestly-reported quirk of this sparse, 355-relation-type domain, not a cherry-picked result.)

Not a Competitor to LLMs — a Foundation Underneath Them

Where Language Models Get Their Confidence From.

Large language models are extraordinary at fluent generation and at absorbing enormous amounts of world knowledge implicitly. What they are not built to do is give you a calibrated probability that one specific fact is true — and asking one to check is expensive: billions of opaque parameters, prone to confidently hallucinating an answer rather than admitting uncertainty. PICASO isn't trying to replace that generative fluency. It's a fast, calibrated, geometrically verifiable reasoning layer that can sit underneath it.

Already Demonstrated: Text-Enhanced Entity Initialisation

PICASO already uses an LLM to seed its own geometry. Llama 3.2 (3B, run entirely locally) generated encyclopedic descriptions for 20,607 entities lacking Wikipedia text — combined with 20,808 entities that already had it, for 41,411 entities enriched with text in total. Each description is passed through a sentence-transformer to initialise that entity's Gaussian mean \(\mu_e\), replacing random initialisation with something semantically meaningful.

The result wasn't just better accuracy — it improved uncertainty calibration, with almost no hallucination, entirely on local hardware.

The Bigger Picture

An LLM-based agent could query PICASO as an O(1) fact-verification layer before asserting a claim — giving generative AI a principled, geometric way to say "I don't know" instead of hallucinating confidently. This is Pillar I of my research — probabilistic foundations — doing exactly the job it was designed for: keeping today's language models honest.