Rotation-Invariant Turning-Angle Matching

Integrate this at the output of a 2D trajectory predictor, recurrent state-space model, or world model that generates \(\hat{\mathbf x}_{1:T}\) from context \(\mathbf c\). First compute observed increments and generated increments, discard turns whose adjacent increment norms are below a threshold \(\varepsilon\), and calculate signed angles with atan2(cross(v[t],v[t+1]), dot(v[t],v[t+1])). Use \(K=2\) or \(K=4\) circular Fourier moments and add lambda_turn * L_turn to the ordinary pointwise likelihood or displacement loss. To make the loss differentiable, compute moments directly from predicted coordinates; for stochastic models average moments over several generated rollouts, or use a minibatch mean. Pseudocode: v=diff(x); valid=(norm(v[:-1])>eps)&(norm(v[1:])>eps); theta=atan2(cross(v[:-1],v[1:]),dot(v[:-1],v[1:])); m[k]=mean(exp(1j*k*theta)[valid]); loss=base_loss+lambda_turn*sum(w[k]*abs(m_data[k]-m_model[k])**2). The paper supplies the FBM covariance, \(\alpha=2H\), and anisotropy parameterization; the model does not need to know \(H\) or \(r\), which should be estimated separately by simulation or maximum likelihood for controlled tests. Begin with synthetic 2D FBM sequences at \(H\in\{0.3,0.5,0.7\}\) and \(r\in\{1,0.5,0.2\}\), randomly rotating every sequence. Compare an RNN or transformer predictor trained with and without the loss. The quantitative prediction is that turning-moment error will increase monotonically as \(r\) departs from one, remain unchanged under random global rotations, and the regularized model will recover this dependence: across rotations, the relative variation of each \(|m_k|\) should be below 5%, while an isotropic-trained model should show a measurable mismatch for \(r\leq0.5\). A useful falsification is no improvement in held-out turning-moment error or loss of rotation invariance.