Frobenius Quasi-Projection for Signed Orthogonal Tensors: Fixed-Order Algorithms and a Gaussian ReLU Learning Application

For each fixed tensor order p ≥ 3, we give a randomized spectral real-arithmetic algorithm for constant-factor Frobenius quasi-projection onto signed symmetric orthogonally decomposable tensors. The input is arbitrary, rank and coefficients are unknown, and no minimum coefficient magnitude, eigengap, or small-noise condition is assumed. Constants and runtime exponents may depend on p. The algorithm combines a dyadic singular-value bandpass square function, sign-aware Gaussian extraction and local stationarity, observed-weight merging into one orthonormal frame, and one final signed coefficient projection. The squared bandpass errors sum to the original squared perturbation without a multiplicative scale-count loss. Both eigenvector orientations and both tensor signs are handled explicitly, including order three and negative even-order coefficients. We separate elementary aggregate matching geometry from the computational approximation theorem. A fourth-order application gives constant-factor Gaussian agnostic prediction for one balanced paired orthogonal ReLU cluster with unknown center, rank, weights, and individual angles, under only a finite second moment of labels. Its output is a positive network, not necessarily an orthogonal-frame cluster.