Spherical-Design Prototype Initialization

Use the construction in a cosine classifier, vector-quantized embedding layer, random-feature MLP, or attention head with N directional prototypes in dimension d. Obtain a spherical t-design X offline, normalize its columns, and initialize the prototype weight matrix W so that row i is x_i. For a sample embedding z, normalize it to z divided by its Euclidean norm and compute directional logits l_i = tau z^T W_i, where tau is a learnable or fixed temperature. During the first warmup period freeze W or permit only tangent-space updates: after each optimizer step replace each row w_i by w_i divided by its norm, preserving the spherical representation. If prototype adaptation is needed, add the moment regularizer L_mom using m = 1 and m = 2 first; for unit vectors the target is M_1 = 0 and M_2 = I divided by d, so the cheapest implementation is L_mom = beta_1 norm(mean_i z_i)^2 + beta_2 norm((1/N) sum_i z_i z_i^T - I/d)_F^2. The mathematical quantities are the exact design moment identities; the choice of t, beta values, temperature, and freezing schedule is empirical. To test it cheaply, train a CIFAR-10 classifier with a 16-dimensional normalized embedding and N = 64 prototypes, comparing iid Gaussian initialization, orthogonal initialization, Fibonacci-sphere initialization, and a spherical 3-design. Use identical optimizers, learning rates, augmentations, and parameter counts. Record training loss at fixed optimizer steps, final accuracy, expected calibration error, sensitivity to random seed, and empirical first- and second-moment errors. Success is lower seed variance or faster early convergence at equal compute, or equal accuracy with fewer prototypes than random initialization.