Interactive Visualizations

Explore how the ⵟ-product differs from traditional similarity measures

Similarity Measure Comparison

Drag the anchor point (⭐) to see how each measure responds. Notice how the ⵟ-product creates a localized potential well.

Dot Product
w · x
Euclidean Distance²
‖w − x‖²
ⵟ-Product
(w·x)² / (‖w−x‖² + ε)
Cosine Similarity
w·x / (‖w‖‖x‖)

Gradient Vector Fields

The gradient fields reveal optimization dynamics. The ⵟ-product creates a vortex-like attractor around the weight vector.

Dot Product ∇
Euclidean ∇
ⵟ-Product ∇
Cosine ∇

XOR Problem: Single Neuron Solution

The classic XOR problem cannot be solved by a single linear neuron. The ⵟ-product's intrinsic non-linearity enables a single unit to solve it.

Linear Neuron

Cannot separate XOR

ⵟ-Product Neuron

Solves XOR with w = [1, -1]

Input xXOR Outputw·xⵟ(w,x)
(0,0)000
(0,1)1-1>0
(1,0)11>0
(1,1)000

Decision Boundary Comparison

Linear neurons create hyperplane boundaries. ⵟ-product neurons create vortex-like territorial fields around learned prototypes.

Linear Model Boundaries

Unbounded half-space partitions

ⵟ-Product Boundaries

Localized vortex territories

Loss Landscape Comparison

The optimization landscape for the XOR problem. Dot product has a spurious minimum; ⵟ-product creates exploitable valleys.

Dot Product + Sigmoid

ⵟ-Product