A Neural Sheet Discovers the Shape of Data
A self-organizing map (SOM) starts as a flat sheet of neurons with no knowledge of the data around it. For every input sample, it finds the best-matching neuron,
c = argminᵢ ‖x - wᵢ‖,
and updates that neuron together with its neighbours,
wᵢ ← wᵢ + η(t)hᶜᵢ(t)(x - wᵢ).
As the neighbourhood radius shrinks during training, the sheet gradually bends and folds onto the hidden geometry of the dataset while preserving its neighbourhood structure.
Developed by Finnish computer scientist Teuvo Kohonen, the Self-Organizing Map remains one of the most elegant examples of competitive learning, where order emerges from thousands of simple local updates.
#MachineLearning #ArtificialIntelligence #SelfOrganizingMap #NeuralNetworks #DataVisualization #Mathematics #ComputerScience #Finland #TeuvoKohonen
Tensors generalize vectors as lists and matrices as grids into higher-order arrays like 3D cubes.
Comparisons include a vector as a list, a matrix as a grid, and a 3-tensor as a cube, along with the stress tensor σij on a cube featuring arrows for components like σ11 and σ23, the product derivative rule, the Riemann curvature tensor R(u,v)w = ∇u ∇_v w − ∇_v ∇_u w − ∇[u,v]w, quantum superposition 1/√2(|00⟩ + |11⟩), and algebraic operations like the tensor product.
It is used to analyze internal forces and deformations in engineering materials and to describe the geometry of spacetime under gravity in general relativity.
Excited to share CrystalReasoner, a reasoning model for crystal structure generation with LLMs and property-conditioned generation through RL:
Website: https://t.co/249N2224on
Paper: https://t.co/W3n8wJN25P
Code: https://t.co/gIQj75p13p
The Helmholtz decomposition is one of the fundamental results of vector calculus.
It says any well-behaved vector field can be split into two parts, one capturing sources and sinks through divergence, and one capturing rotation through curl.