For each point i, a Gaussian kernel with bandwidth σᵢ defines conditional probabilities p_{j|i} that neighbour j is picked; σᵢ is calibrated per point so the induced distribution has a fixed perplexity, roughly the effective number of neighbours. These are symmetrized to p_{ij} = (p_{j|i} + p_{i|j})/(2n).
In the low-dimensional map, similarity uses a heavy-tailed Student-t kernel q_{ij} ∝ (1 + ‖yᵢ − yⱼ‖²)⁻¹, chosen specifically to let moderately distant points repel strongly and give tight, well-separated clumps room to form — this is the mechanism behind the crowding problem, and it is also the mechanism that manufactures visual clusters that need not exist.
The objective is the KL divergence KL(P‖Q) = Σ p_{ij} log(p_{ij}/q_{ij}), minimized by gradient descent. Early exaggeration multiplies every p_{ij} by a constant for the first iterations, forcing points that are even mildly similar to clump tightly before the layout relaxes — a deliberate distortion for visual legibility, not a claim about true separation. Because the embedding only tries to preserve small pairwise probabilities (local neighbourhoods), nothing in the objective constrains the distances *between* far-apart groups: those large-scale layout choices are essentially arbitrary and depend on initialization.
H(Pᵢ) = −Σⱼ p_{j|i} log₂ p_{j|i} = log₂(perplexity)p_{ij} = (p_{j|i} + p_{i|j}) / (2n)q_{ij} = (1 + ‖yᵢ − yⱼ‖²)⁻¹ / Σ_{k≠l} (1 + ‖y_k − y_l‖²)⁻¹∂KL/∂yᵢ = 4 Σⱼ (p_{ij} − q_{ij})(yᵢ − yⱼ)(1 + ‖yᵢ − yⱼ‖²)⁻¹